Fundamentals of Modern Optics", FSU Jena, Prof

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Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 1

Fundamentals of Modern Optics Winter Term 2017/2018

Prof. Thomas Pertsch Abbe School of Photonics, Friedrich-Schiller-Universität Jena

Table of content 0. Introduction ...... 4 1. Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling) ...... 15 1.1 Introduction ...... 15 1.2 Postulates ...... 15 1.3 Simple rules for propagation of light ...... 16 1.4 Simple optical components ...... 16 1.5 Ray tracing in inhomogeneous media (graded-index - GRIN optics) ...... 20 1.5.1 Ray equation ...... 20 1.5.2 The eikonal equation ...... 22 1.6 Matrix optics ...... 23 1.6.1 The ray-transfer-matrix ...... 23 1.6.2 Matrices of optical elements ...... 23 1.6.3 Cascaded elements...... 24 2. Optical fields in dispersive and isotropic media ...... 25 2.1 Maxwell’s equations ...... 25 2.1.1 Adaption to optics ...... 25 2.1.2 Temporal dependence of the fields ...... 29 2.1.3 Maxwell’s equations in Fourier domain ...... 30 2.1.4 From Maxwell’s equations to the wave equation ...... 30 2.1.5 Decoupling of the vectorial wave equation ...... 32 2.2 Optical properties of matter ...... 33 2.2.1 Basics ...... 33 2.2.2 Dielectric polarization and susceptibility ...... 36 2.2.3 Conductive current and conductivity...... 38 2.2.4 The generalized complex dielectric function ...... 39 2.2.5 Material models in time domain ...... 43 2.3 The Poynting vector and energy balance ...... 45 2.3.1 Time averaged Poynting vector ...... 45 2.3.2 Time averaged energy balance ...... 46 2.4 Normal modes in homogeneous isotropic media ...... 49 2.4.1 Transverse waves ...... 50 2.4.2 Longitudinal waves ...... 51 2.4.3 Plane wave solutions in different frequency regimes ...... 52 2.4.4 Time averaged Poynting vector of plane waves ...... 58 2.5 The Kramers-Kronig relation ...... 59 2.6 Beams and pulses - analogy of diffraction and dispersion ...... 62 2.7 Diffraction of monochromatic beams in homogeneous isotropic media ...... 64 2.7.1 Arbitrarily narrow beams (general case) ...... 65 2.7.2 Fresnel- (paraxial) approximation ...... 72 2.7.3 The paraxial wave equation ...... 76 2.8 Propagation of Gaussian beams ...... 78 2.8.1 Propagation in paraxial approximation ...... 79 2.8.2 Propagation of Gauss beams with q-parameter formalism ...... 84 2.8.3 Gaussian optics ...... 84 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 2

2.8.4 Gaussian modes in a resonator ...... 87 2.9 Dispersion of pulses in homogeneous isotropic media ...... 93 2.9.1 Pulses with finite transverse width (pulsed beams) ...... 93 2.9.2 Infinite transverse extension - pulse propagation ...... 100 2.9.3 Example 1: Gaussian pulse without chirp ...... 101 2.9.4 Example 2: Chirped Gaussian pulse ...... 104 3. Diffraction theory ...... 108 3.1 Interaction with plane masks ...... 108 3.2 Propagation using different approximations ...... 109 3.2.1 The general case - small aperture ...... 109 3.2.2 Fresnel approximation (paraxial approximation) ...... 109 3.2.3 Paraxial Fraunhofer approximation (far field approximation) ...... 110 3.2.4 Non-paraxial Fraunhofer approximation ...... 112 3.3 Fraunhofer diffraction at plane masks (paraxial) ...... 112 3.4 Remarks on Fresnel diffraction ...... 117 4. Fourier optics - optical filtering ...... 119 4.1 Imaging of arbitrary optical field with thin lens ...... 119 4.1.1 Transfer function of a thin lens ...... 119 4.1.2 Optical imaging using the 2f-setup ...... 120 4.2 Optical filtering and image processing ...... 122 4.2.1 The 4f-setup ...... 122 4.2.2 Examples of aperture functions ...... 125 4.2.3 Optical resolution ...... 126 5. The polarization of electromagnetic waves ...... 129 5.1 Introduction ...... 129 5.2 Polarization of normal modes in isotropic media ...... 129 5.3 Polarization states ...... 130 6. Principles of optics in crystals ...... 132 6.1 Susceptibility and dielectric tensor ...... 132 6.2 The optical classification of crystals ...... 134 6.3 The index ellipsoid ...... 135 6.4 Normal modes in anisotropic media ...... 136 6.4.1 Normal modes propagating in principal directions ...... 137 6.4.2 Normal modes for arbitrary propagation direction ...... 138 6.4.3 Normal surfaces of normal modes ...... 143 6.4.4 Special case: uniaxial crystals ...... 145 7. Optical fields in isotropic, dispersive and piecewise homogeneous media ...... 148 7.1 Basics ...... 148 7.1.1 Definition of the problem ...... 148 7.1.2 Decoupling of the vectorial wave equation ...... 149 7.1.3 Interfaces and symmetries ...... 150 7.1.4 Transition conditions ...... 151 7.2 Fields in a layer system  matrix method ...... 151 7.2.1 Fields in one homogeneous layer ...... 151 7.2.2 The fields in a system of layers ...... 153 7.3 Reflection – transmission problem for layer systems ...... 155 7.3.1 General layer systems ...... 155 7.3.2 Single interface ...... 162 7.3.3 Periodic multi-layer systems - Bragg-mirrors - 1D photonic crystals ...... 169 7.3.4 Fabry-Perot-resonators ...... 176 7.4 Guided waves in layer systems ...... 182 7.4.1 Field structure of guided waves ...... 182 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 3

7.4.2 Dispersion relation for guided waves ...... 184 7.4.3 Guided waves at interface - surface polariton ...... 186 7.4.4 Guided waves in a layer – film waveguide ...... 188 7.4.5 How to excite guided waves ...... 192

This script originates from the lecture series “Theoretische Optik” given by Falk Lederer at the FSU Jena for many years between 1990 and 2012. Later the script was adapted by Stefan Skupin and Thomas Pertsch for the international education program of the Abbe School of Photonics. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 4

0. Introduction • 'optique' (Greek)  lore of light  'what is light'? • Is light a wave or a particle (photon)?

D.J. Lovell, Optical Anecdotes

• Light is one of the requirements for life  photosynthesis • 90% of information we get is visual A) What is light? • electromagnetic wave propagating with the speed of c=3 × 108 ms / • wave = evolution of − amplitude and phase  complex description − polarization  vectorial field description − coherence  statistical description

Spectrum of Electromagnetic Radiation Region Wavelength Wavelength Frequency Energy [nm] [m] (nm=10-9m) [Hz] (THz=1012Hz) [eV] Radio > 108 > 10-1 < 3 x 109 < 10-5 Microwave 108 - 105 10-1 – 10-4 3 x 109 - 3 x 1012 10-5 - 0.01 Infrared 105 - 700 10-4 - 7 x 10-7 3 x 1012 - 4.3 x 1014 0.01 - 2 Visible 700 - 400 7 x 10-7 - 4 x 10-7 4.3 x 1014 - 7.5 x 1014 2 - 3 Ultraviolet 400 - 1 4 x 10-7 - 10-9 7.5 x 1014 - 3 x 1017 3 - 103 X-Rays 1 - 0.01 10-9 - 10-11 3 x 1017 - 3 x 1019 103 - 105 Gamma Rays < 0.01 < 10-11 > 3 x 1019 > 105

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 5

B) Origin of light • atomic system  determines original properties of light (e.g. statistics, frequency, line width) • optical system  modifies properties of light (e.g. intensity, duration, …) • invention of laser in 1958  very important development

Schawlow and Townes, Phys. Rev. (1958). Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 6

• laser  artificial light source with new and unmatched properties (e.g. coherent, directed, focused, monochromatic) • laser  control of the emission properties of matter at the origin of emission • applications of laser: fiber-communication, DVD, surgery, microscopy, material processing, ...

Fiber laser: Limpert, Tünnermann, IAP Jena, ~10kW CW (world record)

C) Propagation of light through matter • light-matter interaction (G: Licht-Materie-Wechselwirkung)

effect dispersion diffraction absorption scattering ↓ ↓ ↓ ↓ governed by frequency spatial center of wavelength spectrum frequency frequency spectrum

• matter is the medium of propagation  the properties of the medium (natural or artificial) determine the propagation of light • light is the means to study the matter (spectroscopy)  measurement methods (interferometer)

• design media with desired properties: glasses, polymers, semiconductors, compounded media (effective media, photonic crystals, meta-materials) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 7

Two-dimensional photonic crystal membrane.

D) Light can modify matter • light induces physical, chemical and biological processes • used for lithography, material processing, or modification of biological objects (bio-photonics)

Hole “drilled” with a fs laser at Institute of Applied Physics, FSU Jena. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 8

E) Optical telecommunication • transmitting data (Terabit/s in one fiber) over transatlantic distances

1000 m telecommunication fiber is installed every second. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 9

F) Optics in medicine and life sciences

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 10

G) Light sensors and light sources • new light sources to reduce energy consumption

• new projection techniques

Deutscher Zukunftspreis 2008 - IOF Jena + OSRAM Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 11

H) Micro- and nano-optics • ultra small camera

Insect inspired camera system develop at Fraunhofer Institute IOF Jena Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 12

I) Relativistic optics

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 13

J) Schematic of optics

quantum optics

electromagnetic optics

wave optics

geometrical optics

• geometrical optics ⋅ λ << size of objects  daily experience ⋅ optical instruments, optical imaging ⋅ intensity, direction, coherence, phase, polarization, photons (G: Intensität, Richtung, Kohärenz, Phase, Polarisation, Photon)

• wave optics ⋅ λ ≈ size of objects  interference, diffraction, dispersion, coherence ⋅ laser, holography, resolution, pulse propagation ⋅ intensity, direction, coherence, phase, polarization, photons

• electromagnetic optics ⋅ reflection, transmission, guided waves, resonators ⋅ laser, integrated optics, photonic crystals, Bragg mirrors ... ⋅ intensity, direction, coherence, phase, polarization, photons

• quantum optics ⋅ small number of photons, fluctuations, light-matter interaction ⋅ quantum cryptography quantum computation, teleportation … ⋅ intensity, direction, coherence, phase, polarization, photons

• in this lecture Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 14

⋅ electromagnetic optics and wave optics ⋅ no quantum optics  subject of advanced lectures K) Literature • Fundamental 1. Saleh, Teich, 'Fundamenals of Photonics', Wiley (1992) in German: "Grundlagen der Photonik" Wiley (2008) 2. Hecht, 'Optic', Addison-Wesley (2001) in German: "Optik", Oldenbourg (2005) 3. Mansuripur, 'Classical Optics and its Applications', Cambridge (2002) 4. Menzel, 'Photonics', Springer (2000) 5. Lipson, Lipson, Tannhäuser, 'Optik'; Springer (1997) 6. Born, Wolf, 'Principles of Optics', Pergamon 7. Sommerfeld, 'Optik'

• Advanced 1. W. Silvast, 'Laser Fundamentals', 2. Agrawal, 'Fiber-Optic Communication Systems', Wiley 3. Band, 'Light and Matter', Wiley, 2006 4. Karthe, Müller, 'Integrierte Optik', Teubner 5. Diels, Rudolph, 'Ultrashort Laser Pulse Phenomena', Academic 6. Yariv, 'Optical Electronics in modern Communications', Oxford 7. Snyder, Love, 'Optical Waveguide Theory', Chapman&Hall 8. Römer, 'Theoretical Optics', Wiley,2005.

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 15

1. Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling) The topic of “Ray optics – geometrical optics” is not covered in the course “Fundamentals of modern optics”. This topic will be covered rather by the course “Introduction to optical modeling”. The following part of the script which is devoted to this topic is just included in the script for consistency. 1.1 Introduction − Ray optics or geometrical optics is the simplest theory for doing optics. − In this theory, propagation of light in various optical media can be described by simple geometrical rules. − Ray optics is based on a very rough approximation (λ→0 , no wave phenomena), but we can explain almost all daily life experiences involving light (shadows, mirrors, etc.). − In particular, we can describe optical imaging with ray optics approach. − In isotropic media, the direction of rays corresponds to the direction of energy flow. What is covered in this chapter? − It gives fundamental postulates of the theory. − It derives simple rules for propagation of light (rays). − It introduces simple optical components. − It introduces light propagation in inhomogeneous media (graded-index (GRIN) optics). − It introduces paraxial matrix optics. 1.2 Postulates A) Light propagates as rays. Those rays are emitted by light-sources and are observable by optical detectors. B) The optical medium is characterized by a function n(r), the so-called refractive index (n(r) ≥ 1 - meta-materials n(r) <0) c n = cn – speed of light in the medium cn C) optical path length ∼ delay i) homogeneous media nl ii) inhomogeneous media

B ∫ n()r ds A Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 16

D) Fermat’s principle

B d=∫ n()r ds 0 A Rays of light choose the optical path with the shortest delay. 1.3 Simple rules for propagation of light A) Homogeneous media − n = const.  minimum delay = minimum distance − Rays of light propagate on straight lines. B) Reflection by a mirror (metal, dielectric coating) − The reflected ray lies in the plane of incidence. − The angle of reflection equals the angle of incidence. C) Reflection and refraction by an interface − Incident ray  reflected ray plus refracted ray − The reflected ray obeys b). − The refracted ray lies in the plane of incidence.

− The angle of refraction θ2 depends on the angle of incidence θ1 and is given by Snell’s law:

nn1sinθ= 12 sin θ 2 − no information about amplitude ratio. 1.4 Simple optical components A) Mirror i) Planar mirror

− Rays originating from P1 are reflected and seem to originate from P2. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 17

ii) Parabolic mirror − Parallel rays converge in the focal point (focal length f). − Applications: Telescope, collimator

iii) Elliptic mirror

− Rays originating from focal point P1 converge in the second focal point P2

iv) Spherical mirror − Neither imaging like elliptical mirror nor focusing like parabolic mirror − parallel rays cross the optical axis at different points − connecting line of intersections of rays  caustic Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 18

− parallel, paraxial rays converge to the focal point f = (-R)/2 − convention: R < 0 - concave mirror; R > 0 - convex mirror. − for paraxial rays the spherical mirror acts as a focusing as well as an imaging optical element. paraxial rays emitted in point P1 are reflected and converge in point P2

11 2 +≈ (imaging formula) zz12()− R paraxial imaging: imaging formula and magnification m = -z2 /z1 (proof given in exercises) B) Planar interface

Snell’s law: nn1sinθ= 12 sin θ 2 θ= θ for paraxial rays: nn11 2 2

− external reflection (nn12< ): ray refracted away from the interface

− internal reflection (nn12> ): ray refracted towards the interface − total internal reflection (TIR) for:

π n2 θ=2 → sinθ=1 sin θ TIR = 2 n1 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 19

C) Spherical interface (paraxial) − paraxial imaging

n1 n 21− ny θ21 ≈ θ− (*) n22 nR

nn nn− 1+≈ 2 21 (imaging formula) zz12 R nz m = − 12 (magnification) nz21 (Proof: exercise) − if paraxiality is violated  aberration − rays coming from one point of the object do not intersect in one point of the image (caustic) D) Spherical thin lens (paraxial) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 20

− two spherical interfaces (R1, R2, ∆) apply (*) two times and assume y=const (∆ small)

y 1 11 θ21 ≈θ − with focal length: =−−(n 1) f f RR12 111 z +≈ (imaging formula) m = − 2 (magnification) zz12 f z1 (compare to spherical mirror) 1.5 Ray tracing in inhomogeneous media (graded-index - GRIN optics) − n()r - continuous function, fabricated by, e.g., doping − curved trajectories  graded-index layer can act as, e.g., a lens 1.5.1 Ray equation Starting point: we minimize the optical path or the delay (Fermat)

B d=∫ n()r ds 0 A computation: B L= ∫ nr( s) ds A Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 21

variation of the path: rr()ss+δ ()

BB d=dL∫∫ nds + n d ds AA

dnn =grad ⋅dr

22 dds =( drr +d d) −( d r)

2 22 =(dr) +2 dd rr ⋅d+( d d r) − ( d r) ddrrd ≈ds 12 +⋅−ds ds ds ddrrd ≈ds1 +⋅ −ds ds ds

ddrrd =ds ⋅ ds ds B ddrrd dL =grad n ⋅dr + n ⋅ ds ∫ds ds A integration by parts and A,B fix B ddr =∫grad n − n ⋅dr ds A ds ds δL =0 for arbitrary variation ddr grad nn=  ray equation ds ds Possible solutions: A) trajectory

22 x(z) , y(z) and ds=++ dz1 ( dx dz) ( dy dz) − solve for x(z) , y(z) − paraxial rays  (ds ≈ dz ) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 22

d dx dn nxyz( ,,) ≈ dz dz dx

d dy dn nxyz( ,,) ≈ dz dz dy B) homogeneous media − straight lines C) graded-index layer n(y) - paraxial, SELFOC dy paraxial  1 and dz≈ ds dz

2 222 1 2 2 n() y= n0 ( 1 −α y) ⇒ ny () ≈n0  1 − α y for a 1 2 d dy d dy d22 y d y1 dn () y ny( ) ≈ ny( ) ≈ ny( ) ⇒= ds ds dz dz dz22 dz n( y) dy

θ0 yz( )=y0 cos α+ z sin α z 2 α dy 2 for n(y)-n0<<1: = −α y  dz2 dy θ()z = =− yz αsin α +θ cos α z dz 00

1.5.2 The eikonal equation − bridge between geometrical optics and wave − eikonal S(r) = constant  planes perpendicular to rays − from S(r) we can determine direction of rays ∼ grad S(r) (like potential)

2 2 gradSn( r) = ( r) Remark: it is possible to derive Fermat’s principle from eikonal equation − geometrical optics: Fermat’s or eikonal equation BB SS(rr) −=( ) grad S( r) ds = n( r) ds BA∫∫AA eikonal  optical path length ∼ phase of the wave Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 23

1.6 Matrix optics − technique for paraxial ray tracing through optical systems − propagation in a single plane only − rays are characterized by the distance to the optical axis (y) and their inclination (θ)  two algebraic equation  2 x 2 matrix Advantage: we can trace a ray through an optical system of many elements by multiplication of matrices. 1.6.1 The ray-transfer-matrix

in paraxial approximation:

y211= Ay +θ B yy21AB  AB  =  →=M  θθ21CD  CD θ=21Cy + D θ 1

A=0: same θ1  same y2  focusing

D=0: same y1  same θ2  collimation

1.6.2 Matrices of optical elements A) free space

1 d M =  01  B) refraction on planar interface Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 24

10 M =  0 nn 12 C) refraction on spherical interface

10 M =  −−(n n) nR n n 2 1 2 12 D) thin lens

10 M =  −11f  E) reflection on planar mirror

10 M =  01  F) reflection on spherical mirror (compare to lens)

10 M =  21R  1.6.3 Cascaded elements

yN +1 ABy1 AB = →=M  M=MN….M2M θN +1 CDθ1 CD Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 25

2. Optical fields in dispersive and isotropic media 2.1 Maxwell’s equations Our general starting point is the set of Maxwell’s equations. They are the basis of the electromagnetic approach to optics, which is developed in this lecture. 2.1.1 Adaption to optics The notation of Maxwell’s equations is different for different disciplines of science and engineering, which rely on these equations to describe electromagnetic phenomena at different frequency ranges. Even though Maxwell's equations are valid for all frequencies, the physics of light matter interaction is different for different frequencies. Since light matter interaction must be included in the Maxwell's equations to solve them consistently, different ways have been established how to write down Maxwell's equations for different frequency ranges. Here we follow a notation, which was established for a convenient notation at frequencies close to visible light.

Maxwell’s equations (macroscopic) In a rigorous way the electromagnetic theory is developed starting from the properties of electromagnetic fields in vacuum. In vacuum one could write down Maxwell's equations in their so-called pure microscopic form, which includes the interaction with any kind of matter based on the consideration of point charges. Obviously this is inadequate for the description of light in condensed matter, since the number of point charges, which would need to be taken into account to describe a macroscopic object, would exceed all imaginable computational resources. To solve this problem one uses an averaging procedure, which summarizes the influence of many point charges on the electromagnetic field in a homogeneously distributed response of the solid state on the excitation by light. In turn, also the electromagnetic fields are averaged over some adequate volume. For optics this procedure is justified, since any kind of available experimental detector could not resolve the very fine spatial details of the fields in between the point charges, e.g. ions or electrons, which are lost by this averaging. These averaged electromagnetic equations have been rigorously derived in a number of fundamental text books on electrodynamic theory. Here we will not redo this derivation. We will rather start directly from the averaged Maxwell's equations. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 26

∂Br(,)t rot(,)Ert=−=div(,)Drttr (,) r ∂t ext

∂Dr(,)t rotHr (,)tt=+= j (,) r div(,)Br t 0 makr ∂t − electric field (G: elektrisches Feld) Er(,)t [V/m] − magnetic flux density Br(,)t [Vs/m2] or [tesla] or magnetic induction (G: magnetische Flussdichte oder magnetische Induktion) − electric flux density Dr(,)t [As/m2] or electric displacement field (G: elektrische Flussdichte oder dielektrische Verschiebung) − magnetic field (G: magnetisches Feld) Hr(,)t [A/m] 3 − external charge density ρext (,)ρ t [As/m ] 2 − macroscopic current density jrmakr (,)t [A/m ]

Auxiliary fields The "cost" of the introduction of macroscopic Maxwell's equations is the occurrence of two additional fields, the dielectric flux density Dr(,)t and the magnetic field Hr(,)t . These two fields are related to the electric field Er(,)t and magnetic flux density Br(,)t by two other new fields.

Dr(,)t=ε+0 Er (,) tt Pr (,) 1 Hr(,)t=[ Br (,) tt − Mr (,)] µ0 − dielectric polarization (G: dielektrische Polarisation) Pr(,)t [As/m2], − magnetic polarization Mr(,)t [Vs/m2] or magnetization (G: Magnetisierung) −12 − electric constant ε≈0 8.854 × 10 As/Vm or vacuum permittivity (G: Vakuumpermittivität) −7 − magnetic constant µ0 =4 π× 10 Vs/Am or vacuum permeability (G: Vakuumpermeabilität) electric constant and magnetic constant are connected by the speed of light in vacuum c as 1 ε=0 2 µ0c Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 27

Light matter interaction In order to solve this set of equations, i.e. Maxwell's equations and auxiliary field equations, one needs to connect the dielectric flux density Dr(,)t and the magnetic field Hr(,)t to the electric field Er(,)t and the magnetic flux density Br(,)t . This is achieved by modeling the material properties by introducing the material equations. • The effect of the medium gives rise to polarization Pr(,)tf= [ E] and magnetization Mr(,)tf= [ B].  In order to solve Maxwell’s equations we need material models which describe these quantities. • In optics at visible wavelength, we generally deal with non-magnetizable media. Hence we can assume Mr(,)t = 0. Exceptions to this general property are metamaterials which might possess some artificial effective magnetization properties resulting in Mr(,)t ≠ 0. Furthermore we need to introduce sources of the fields into our model. This is achieved by the so-called source terms, which are inhomogeneities and hence they define unique solutions of Maxwell's equations. − free charge density (G: Dichte freier Ladungsträger) ρ (,)ρ t [As/m3] ext − macroscopic current density (G: makroskopische Stromdichte) consisting of two contributions jr(,)ttt= jr (,) + jr (,) [A/m2] makr cond conv ⋅ conductive current density (G: Konduktionsstromdichte) jr(,)tf= [ E] cond ⋅ convective current density (G: Konvektionsstromdichte) j (,)rt= ρ (,) r ttv (,)ρ conv ext − In optics, we generally have no free charges, which change at speeds comparable to the frequency of light: ∂ρ (,)ρ t ext ≈→0j (,)ρt = 0 ∂t conv • With the above simplifications, we can formulate Maxwell’s equations in the context of optics: ∂Hr(,)t rotEr ( ,tt ) = −µ εdivEr ( , ) = −divP(,r t) 00∂t

∂Pr(,)t ∂Er(,)t rot Hr(,)tt=jr(,t) + +ε divH (,)r = 0 ∂t 0 ∂t − In optics, the medium (or more precisely the mathematical material model) determines the dependence of the induced polarization on the Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 28

electric field PE() and the dependence of the induced (conductive) current density on the electric field jE(). − Once we have specified these relations, we can solve Maxwell’s equations consistently. Example: − In vacuum, both polarization P and current density j are zero (most simple material model). Hence we can solve Maxwell’s equations directly. Remarks: − Even though the Maxwell's equations, in the way they have been written above, are derived to describe light, i.e. electromagnetic fields at optical frequencies, they are simultaneously valid for other frequency ranges as well. Furthermore, since Maxwell's equations are linear equations as long as the material does not introduce any nonlinearity the principle of superposition holds. Hence we can decompose the comprehensive electromagnetic fields into components of different frequency ranges. In turn this means that we do not have to take care of any slow electromagnetic phenomena, e.g. electrostatics or radio wave, in our formulation of Maxwell's equations since they can be split off from our optical problem and can be treated separately. − We can define a bound charge density (G: Dichte gebundener Ladungsträger) as the source of spatially changing polarization P .

ρb (,)ρ tt=−divP (,)ρ − Analogously we can define a bound current density (G: Stromdichte gebundener Ladungsträger) as the source of the temporal variation of the polarization P . ∂Pr(,)t jr(,)t = b ∂t − This essentially means that we can describe the same physics in two different ways since currents are in principle moving charges. Thus we

can use either jrb (,)t or ∂∂Pr( ,tt )/ (see generalized complex dielectric function below).

Complex field formalism (G: komplexer Feld-Formalismus) − Maxwell’s equations are also valid for complex fields and are easier to solve this way. − This fact can be exploited to simplify calculations, because it is easier to deal with complex exponential functions (exp(ix )) than with trigono- metric functions cos(x ) and sin(x ) . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 29

− Hence we use the following convention in this lecture to distinguish between the two types of fields.

real physical field: Err (,)t complex mathematical representation: Er(,)t − In this lecture we define their relation as E(,) rt=1  Er (,) tt += E∗ (,) r Re[ Er (,) t] r 2  However, this relation can be defined differently in different textbooks. − This means in general: For calculations we use the complex fields [Er ( ,t )] and for physical results we go back to real fields by simply omitting the imaginary part. This works because Maxwell’s equations are linear and no multiplications of fields occur. − Therefore, be careful when multiplications of fields are required in the description of the material response or the field dynamics. In this case you would have to go back to real quantities before you compute these multiplication. This becomes relevant for, e.g., calculation of the Poyn- ting vector, as can be seen in a chapter below. 2.1.2 Temporal dependence of the fields When it comes to time dependence of the electromagnetic field, we can distinguish two different types of light:

A) monochromatic light (stationary fields) − harmonic dependence on temporal coordinate − − exp(−ωi t )  phase is fixed  coherent, infinite wave train e.g.: Er( ,tt )= Er ( )exp( −ωi ) − Monochromatic light approximates very well the typical output of a continuous wave (CW) laser. Once we know the frequency we have to compute the spatial dependence of the (stationary) fields only.

B) polychromatic light (non-stationary fields) − finite wave train − With the help of Fourier transformation we can decompose the fields into infinite wave trains and use all the results from case A) (see next section)

∞ Er( ,t )=∫ Er ( , ω )exp( −ωi td ) ω −∞

1 ∞ Er( ,ω= )∫ Er ( ,t )exp(iω t ) dt 2p −∞ Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 30

Remark: The position of the sign in the exponent and the factor 1/2π can be defined differently in different textbooks. 2.1.3 Maxwell’s equations in Fourier domain In order to solve Maxwell's equations more easily we would like to introduce a Fourier decompositions of the fields in the Maxwell's equations. For this purpose, we need to find out how a time derivative of a dynamic variable can be calculated in Fourier space. A simple rule for the transformation a time derivative into Fourier space can be obtained using integration by parts: 11+∞ ∂ +∞ ∫∫dt Er( ,t) exp(iω t) =−ω i dtEr( , t) exp( iω t) =−ωi Er(,) ω 22p −∞ ∂t p −∞ Thus, a time derivative in real space transforms into a simple product with −ωi in Fourier space. ∂  rule: FT → −i ω ∂t Now we can write Maxwell’s equations in Fourier domain: rot(,Erω= )iω µ Hr (, ω ) εdiv(,Er ω=− ) div(, Pr ω ) 00 rot(,Hrω= ) jr (, ω )− iiωω Pr (, ω )− ε0 Er (, ω )div(, Hrω ) = 0

2.1.4 From Maxwell’s equations to the wave equation Maxwell's equations provide the basis to derive all possible mathematical solutions of electromagnetic problems. However very often we are interested just in the radiation fields which can be described more easily by an adapted equation, which is the so-called wave equation. From Maxwell’s equations it is straight forward to derive the wave equation by using the two curl equations. A) Time domain derivation We start from applying the curl operator (rot ) a second time on rotEr ( ,t ) = and substitute rot H with the other Maxwell equation. ∂∂Hr(,)t∂∂Pr(,)ttE (,)r rotrot E(,)r t = −µ rot = −µ j(,) r t + +ε 00∂∂tt∂∂tt0 We find the wave equation for the electric field as 1 ∂2 ∂∂jr(,)tt2 P (,)r rotrot Er(,)tt+=E (,)r −µ − µ ct2 ∂ 2 0 ∂∂tt0 2 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 31

The blue terms require knowledge of the material model. Additionally, we have to make sure that all other Maxwell’s equations are fulfilled. This holds in particular for the divergence of the electric field:

div[ε+0 E(,) rtt P (,) r ] = 0 Once we have solved the wave equation, we know the electric field. From that we can easily compute the magnetic field: ∂Hr(,)t 1 = − rot E(,) r t ∂µt 0 Remarks: − An analog procedure is possible also for the magnetic field H , i.e., we can derive a wave equation for the magnetic field as well. − Normally, the wave equation for the electric field E is more convenient, because the material model defines PE(). − However, for inhomogeneous media the wave equation for H can sometimes be the better choice for the numerical solution of the partial differential equation since it forms a hermitian operator. − analog procedure possible for H instead of E − generally, wave equation for E is more convenient, because PE() given − for inhomogeneous media H can sometimes be better choice B) Frequency domain derivation We can do the same procedure to derive the wave equation also directly in the Fourier domain and find ω2 rotrotEr(,ω− ) Er (, ω= )ω µ jr (, ω+µ )ω2 Pr (, ω ) c2 i 00 and ε ω+ ω = div0 E(, r ) P (, r ) 0 − magnetic field:

H(, rω=− ) i rot E(, r ω ) ωµ0 − transferring the results from the Fourier domain to the time domain ⋅ for stationary fields: take solution and multiply by e-itω . ⋅ for non-stationary fields and linear media  inverse Fourier transformation Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 32

∞ Er( ,t )=∫ Er ( , ω )exp( −ωi td ) ω −∞

2.1.5 Decoupling of the vectorial wave equation So far we have seen that for the general problem of electromagnetic waves all 3 vectorial field components of the electric or the magnetic field are coupled. Hence we have to solve a vectorial wave equation for the general problem. However, it would be desirable to express problems also by a scalar equation since they are much easier to solve. For problems with translational invariance in at least one direction, as e.g. for homogeneous infinite media, layers or interfaces, this can be achieved since the vectorial components of the fields can be decoupled. Let’s assume invariance in the y-direction and propagation only in the x-z- plane. Then all spatial derivatives along the y-direction disappear (∂∂=/0y ) and the operators in the wave equation simplify.

∂ ∂ E x ∂ E z (2) ∂∂xx( + ∂ z) ∆ E x  (2) rot rot E= grad div E−∆ E =0 − ∆ E y  ∂ ∂ E x ∂ E (2) + z ∆ E z ∂∂zx( ∂ z)  The decoupling becomes visible when the three components of the general vectorial field are decomposed in the following way. • decomposition of electric field

0 E x   EE=⊥ + E  EE⊥ = E y ,0 =    0 E z ∂∂x ∂∂22 ∇=(2)  ∆=(2) + with Nabla operator 0 , and Laplace 22  ∂∂xz ∂∂z

Hence we obtain two wave equations for the E⊥ and E fields. • gives two decoupled wave equations

2 (2) ω 2 ∆ Er⊥⊥(,ω+ ) Er (, ω=−ωµ ) jr (, ω−µω ) Pr⊥ (, ω ) c2 i 00⊥ 2 (2) ω (2) (2) 2 ∆ E(, rω+ ) E (, r ω− ) grad div E=−ωµ j(, r ω−µω ) P (, r ω ) c2 i 00 These two wave equations are independent as long as the material response, which is expressed by j and P , does not couple the respective field components by some anisotropic response. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 33

Properties

− propagation of perpendicularly polarized fields E⊥ and E can be treated separately

− propagation of E⊥ is described by scalar equation − similarly the other field components can be described by a scalar equation for H⊥ − alternative notations: ⊥  s  TE (transversal electric)   p  TM (transversal magnetic) 2.2 Optical properties of matter In this chapter we will derive a simple material model for the polarization and the current density. The basic idea is to write down an equation of motion for a single exemplary charged particle and assume that all other particles of the same type behave similarly. More precisely, we will use a driven harmonic oscillator model to describe the motion of bound charges giving rise to a polarization of the medium. For free charges we will use the same model but without restoring force, leading eventually to a current density. In the literature, this simple approach is often called the Drude-Lorentz model (named after Paul Drude and Hendrik Antoon Lorentz). 2.2.1 Basics We are looking for PE() and jE(). In general, this leads to a many body problem in solid state theory which is rather complex. However, in many cases phenomenological models are sufficient to describe the necessary phenomena. As already pointed out above, we use the simplest approach, the so-called Drude-Lorentz model for free or bound charge carriers (electrons). − assume an ensemble of non-coupling, driven, and damped harmonic oscillators − free charge carriers: metals and excited semiconductors (intraband) − bound charge carriers: dielectric media and semiconductors (interband) − The Drude-Lorentz model creates a link between cause (electric field) and effect (induced polarization or current). Because the resulting relations PE() and jE() are linear (no E2 etc.), we can use linear response theory.

For the polarization PE() (for jE() very similar): − description in both time and frequency domain possible Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 34

− In time domain: we introduce the response function (G: Responsfunktion) Er(,)t  medium (response function)  Pr(,)t

t =ε−′ ′′ Pi (,)r t0 ∑ ∫ Rij (, rr t t ) Ej (, t ) dt j −∞ with Rˆ in the convolution integral being a 2nd rank tensor i= xyz,, and summing over j= xyz,, − In frequency domain: we introduce the susceptibility (G: Suszeptibilität) Er(,ω )  medium (susceptibility)  Pr(,ω )

PEi (,rωω )=ε0 ∑ χij (, rr )j (,ω ) j − response function and susceptibility are linked via Fourier transform (convolution theorem) 1 ∞ = χ ω −ω ω Rij ( t )∫ ij ( )exp( i td) 2p −∞ − Obviously, things look friendlier in frequency domain. Using the wave equation from before and assuming that there are no currents (j = 0) we find ω2 rotrotEr(,ω ) − Er (, ω ) =µω2 Pr (, ω ) c2 0 or ω2 ∆E(, r ω+ ) E (, r ω− ) graddivE (, r ω=−µω )2 P (, r ω ) c2 0 − and for auxiliary fields

Dr(,ω=ε )0 Er (, ω+ ) Pr (, ω ) The general response function and the respective susceptibility given above simplifies for certain properties of the medium:

Simplification of the wave equation for different types of media A) linear, homogenous, isotropic, non-dispersive media (most simple but very unphysical case)

− homogenous  χij (,r ω=χω )ij ( )

− isotropic  χij (,rr ω ) =χ (, ω )δij

− non-dispersive  χij (,rr ω=χ )ij ()  instantaneous: Rtij (,)rr=χij ( )δ () t (Attention: This is unphysical!) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 35

 χωij (,r )  χis a scalar constant frequency domain time domain description

Pr(,ω=ε )00χχ Er (, ω ) ↔ Pr(,)tt= ε Er (,) (unphysical!)

Dr(,ω ) =εε00 Er (, ω ) ↔Dr(,)tt =εε Er (,) with ε=1 +χ Maxwell: div D = 0 → div E(, r ω= ) 0 for εω() ≠ 0 ω2 ε∂2 ∆ Er(,ω+ ) ε Er (, ω= ) 0  ∆ Er(,)tt−= Er (,) 0 c2 ct22∂ − approximation is valid only for a certain frequency range, because all media are dispersive − based on an unphysical material model B) linear, homogeneous, isotropic, dispersive media  χ()ω

Pr(,ω ) =εχω0 ( )Er (, ω ) Dr(,ω ) =εεω ( )Er (, ω ) 0

div D(, rω= ) 0≠ div E (, r ω= ) 0for εω≠ ( ) 0 ω2  ∆ Er(,ω+ ) εω( ) Er (, ω= ) 0 Helmholtz equation c2 − This description is sufficient for many materials. C) linear, inhomogeneous, isotropic, dispersive media  χω(,r ) Pr(,ω ) =εχ (, r ω ) Er (, ω ), 0 Dr(,ω ) =εε0 (, r ω ) Er (, ω ). divDr ( ,ω= ) 0

div(,D rω=εε )00 (, r ω ) div E (, r ω+ε ) E (, r ω ) grad ε (, r ω= ) 0,

gradεω(, r ) →div(,Er ω=− ) Er(,ω ). εω(,r )

ω2 gradεω(, r ) ∆ Er(,ω+ ) ε( r , ω) Er (, ω=− ) gradEr(,ω ) c2 εω(,r ) − All field components couple.

D) linear, homogeneous, anisotropic, dispersive media →χij () ω Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 36

PEi (,rrω=ε )0 ∑ χij ( ω ) j (, ω ) j  see chapter on crystal optics DEi (,rrω=ε )0 ∑ εij ( ω ) j (, ω ). j − This is the worst case for a medium with linear response.

Before we start writing down the actual material model equations, let us summarize what we want to do:

What kind of light-matter interaction do we want to consider?

I) Interaction of light with bound electrons and the lattice The contributions of bound electrons and lattice vibrations in dielectrics and semiconductors give rise to the polarization P . The lattice vibrations (phonons) are the ionic part of the material model. Because of the large mass of the ions (103 ×mass of electron) the resulting oscillation frequencies will be small. Generally speaking, phonons are responsible for thermal properties of the medium. However, some phonon modes may contribute to optical properties, but they have small dispersion (weak dependence on frequency ω). Fully understanding the electronic transitions of bound electrons requires quantum theoretical treatment, which allows an accurate computation of the transition frequencies. However, a (phenomenological) classical treatment of the oscillation of bound electrons is possible and useful.

II) Interaction of light with free electrons The contribution of free electrons in metals and excited semiconductors gives rise to a current density j. We assume a so-called (interaction-)free electron gas, where the electron charges are neutralized by the background ions. Only collisions with ions and related damping of the electron motion will be considered.

We will look at the contributions from bound electrons / lattice vibrations and free electrons separately. Later we will join the results in a generalized material model which holds for many common optical materials.. 2.2.2 Dielectric polarization and susceptibility Let us first focus on bound charges (ions, electrons). In the so-called Drude model, the electric field Er(,)t gives rise to a displacement sr(,)t of charged particles from their equilibrium positions. In the easiest approach this can be modeled by a driven harmonic oscillator: ∂∂2 q sr(,)tg+= sr (,) t+ω2 sr (,) t Er (,) t ∂∂tt2 0 m Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 37

− resonance frequency (electronic transition)  ω0 − damping  g − charge  q − mass  m The induced electric dipole moment due to the displacement of charged particles is given by pr(,)tq= sr (,), t We further assume that all bound charges of the same type behave identical, i.e., we treat an ensemble of non-coupled, driven, and damped harmonic oscillators. Then, the dipole density (polarization) is given by = = Pr(,)t Nt pr (,)Nq sr (,)t Hence, the governing equation for the polarization Pr(,)t reads as ∂∂2 qN2 Pr(,)tg+= Pr (,) t+ω2 Prt (,) Er (,)t=ε f Er (,) t ∂∂tt2 00m 1 eN2 with oscillator strength f = , for q=-e (electrons) e0 m This equation is easy to solve in Fourier domain: −ω22 ω− ω ω+ω ω=ε ω Pr(, )igf Pr (, )00 Pr (, ) Er (, ) ε f → Pr(,ω= ) 0 Er(,ω ) ω−22ω − ω ( 0 ) ig f with P(, rω ) =εχω ( )E (, r ω )  χω() = 0 ω22 −ω − ω ( 0 ) ig In general we have several different types of oscillators in a medium, i.e., several different resonance frequencies. Nevertheless, since in a good approximation they do not influence each other, all these different oscillators contribute individually to the polarization. Hence the model can be constructed by simply summing up all contributions. − several resonance frequencies

f Pr(,ω=ε ) j Er(,ω=εχω )( ) Er (, ω ) 00∑ ω22 −ω − ω j ( 0 jj) ig

f χω=( ) j ∑ ω22 −ω − ω j ( 0 jj) ig − χω( ) is the complex, frequency dependent susceptibility Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 38

Dr(,ω ) =ε Er (, ω ) +εχω Er (, ω ) =εεω Er (, ω ) 00( ) 0( ) − εω( ) is the complex frequency dependent dielectric function 2 Example: (plotted for eta and kappa with εω=( )  ηω+κω( ) i ( ) )

2.2.3 Conductive current and conductivity Let us now describe the response of a free electron gas with positively charged background (no interaction). Again we use the model of a driven harmonic oscillator, but this time with resonance frequency ω=0 0 . This corresponds to the case of zero restoring force. ∂∂2 e sr(,)tg+=− sr (,) t Er (,), t ∂∂t2 tm The resulting induced current density is given by ∂ jr(,)tt= −Ne sr (,) ∂t and the governing dynamic equation reads as ∂ eN2 jr(,)tgt+= jr (,) Er (,) t=eω2 Er (,) t ∂t m 0 p 1 eN2 with plasma frequency ω=2 f = p e0 m Again we solve this equation in Fourier domain: − ωjr(, ω ) +g jr (, ω ) =εω2 Er (, ω ) i 0 p 2 εω0  jr(,ω= )p Er (, ω=σω )( ) Er (, ω ). g −ωi Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 39

Here we introduced the complex frequency dependent conductivity εω22 εω ω σω=00pp =− . ( ) i 2 g −ωi −ωω− ig Remarks on plasma frequency We consider a cloud of electrons and positive ions described by the total charge density ρ in their self-consistent field E . Then we find according to Maxwell: ε=ρ 0divE(,)rtt (,) r For cold electrons, and because the total charge is zero, we can use our damped oscillator model from before to describe the current density (only electrons move): ∂ j+=gt jεω2 Er(,) ∂t 0 p Now we apply divergence operator and plug in from above (red terms): ∂ div j+=g divj εω22divE(,)rt =ωρ (,) r t ∂t 0 pp With the continuity equation for the charge density (from Maxwell's equations) ∂ ρ+divj= 0, ∂t We can substitute the divergence of the current density and find:

2 ∂∂2 − 2 ρ−g ρ=ω ρ ∂∂ttp ∂∂2 ρ+g ρ+ω2 =0  harmonic oscillator equation ∂∂tt2 p Hence, the plasma frequency ω is the eigen-frequency of such a charge p density. 2.2.4 The generalized complex dielectric function In the sections above we have derived expressions for both polarization (bound charges) and conductive current density (free charges). Let us now plug our jr(,ω ) and Pr(,ω ) into the wave equation (in Fourier domain) ω2 rotrotEr(,ω− ) Er (, ω=µω )2 Pr (, ω+ ) ωµ jr (, ω ) c2 00i

= µεω2cω + ωµσω ω 00 () i 0 ( ) Er(, ) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 40

Hence we can collect all terms proportional to Er(,ω ) and write 2  ω i rotrot E(, r ωω )= 2 1+c ()ω + σ( ω) E(, r ) c ωε0 ω2 rotrot E(,) r ω=εω( )E( r ,) ω c2 Here, we introduced the generalized complex dielectric function

i ′ ′′ εω() = 1 +χ(ω) +σ(ω) =εω()+ωiε () ωε0 So, in general we have

f ω2 εω() = 1+ j + p , ∑ ω2 −ω2 − ω −ω2 − gω j ( 0 jj) ig i because (from before)

 2 f j ε ωω χω= σω=− 0 p ( ) ∑22 , ( ) 2 . ω −ω − ω i −ω −g ω j ( 0 jj) ig i εω()contains contributions from vacuum, phonons (lattice vibrations), bound and free electrons.

Some special cases for materials in the infrared and visible spectral range:

A) Dielectrics (insulators) in the infrared (IR) spectral range near phonon resonance If we are interested in dielectrics (insulators) near phonon resonance in the infrared spectral range we can simplify the dielectric function as follows:  f j f εω() =1+ + with ω 0 ω and ωω ∑ ω22 −ω − ω ω22−ω − ω 0 0 j 0 j ( 0 jj) ig  ( 0 ) ig f  εω() =ε + ∞ ω22 −ω − ω ( 0 ) ig The contribution from electronic transitions shows almost no frequency dependence (dispersion) in this frequency range far away from the electronic resonances. hence it can be expressed together with the vacuum contribution as a constant ε∞ . Let us study the real and the imaginary part of the resulting εω() separately:

ε∞  vacuum and electronic transitions Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 41

 ′ ′′ εω() =ℜεω+ ()ii ℑεω=εω+ () () ε () ω 22 f (ω0 −ω ) εω=ε+′() ∞ , 2 22 22 (ω0 −ω) +g ω gf ω ε′′() ω= .  Lorentz curve 2 22 22 (ω0 −ω) +g ω

properties:

− resonance frequency: ω0 − width of resonance peak: g f − ω→ ε=ε+0 static dielectric constant in the limit 0 : ∞ 2 ω0 − so called longitudinal frequency ω : ε′( ω=ω )0 = L L − ε′′() ω≠ 0: absorption and dispersion appear always together − near resonance we find εω<′() 0 (damping, i.e. decay of field, without absorption if ε≈'' 0 ) − frequency range with normal dispersion: ∂ε′( ω )/ ∂ω > 0 − frequency range with anomalous dispersion: ∂ε′( ω )/ ∂ω < 0 Simplified example: sharp resonance for undamped oscillator g → 0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 42

− relation between resonance frequency ω0 and longitudinal frequency ω (Lyddane-Sachs-Teller relation) L f εω′ =ε+ = = ε02 −ε ω ()L ∞ 220, f ( ∞ ) 0 (from above) (ω0 −ωL )

ε0 → ω=ω . L 0 ε∞

B) Dielectrics in the visible (VIS) spectral range Dielectric media in visible (VIS) spectral range can be described by a so- called double resonance model, where a phonon resonance exists in the infrared (IR) and an electronic transition exists in the ultraviolet (UV). f f p ε εω() =ε∞ + + , with ω00 ωω ω22 −ω −gg ω ω22 −ω − ω 00pe ( 00p) ii p( e) e

ε∞  contribution of vacuum and other (far away) resonances

The generalization of this approach in the transparent spectral range leads to the so-called Sellmeier formula. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 43

f ω2 εω−′ = j 0 j () 1 ∑ 2 2 , j (ω0 j −ω ) − with j being the number of resonances taken into account − describes many media very well (dispersion of absorption is neglected) − oscillator strengths and resonance frequencies are often fit parameters to match experimental data

C) Metals in the visible spectral range If we want to describe metals in the visible spectral range we find ω2 εω() = 1 − p with ω >> ω 2 p ω+ig ω ωω22g εω=′() 1 − pp ,ε′′ () ω= . ω+22g ωω+( 22g ) Metals show a large negative real part of the dielectric function εω′() which gives rise to decay of the fields. Eventually this results in reflection of light at metallic surfaces.

2.2.5 Material models in time domain Let us now transform our results of the material models back to time domain. In Fourier domain we found for homogeneous and isotropic media: Dr(,ω=ε )εω() Er (, ω ) 0 Pr(,ω=ε )0χ()ω Er (, ω ). The response function (or Green's function) Rt() in the time domain is then given by 1 ∞ ∞ Rt( )= χω ( )exp( − ωt) d ω χω( ) = R ( t )exp( ω t) dt 2p ∫ −∞ i ∫ −∞ i Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 44

To prove this, we can use the convolution theorem ∞∞ Pr(,)t=ωωω= Pr (, )exp( td) eχω()Er(,ωω )exp( td) ω ∫∫−∞ −i 0 −∞ −i ∞∞1 =e χω() Er( ,t′ )exp( ω t ′′) dt exp( ωω t) d 0 ∫∫−∞ −∞ i −i 2p Now we switch the order of integration, and identify the response function R (red terms): ∞ 1 ∞ Pr(,)t = e χω( )exp( ω()tt −′ ) d ωEr(,t′′) dt 0 ∫ −∞ ∫ −∞ −i ((((((((((2p Rt()− t′ ∞ = e Rt()t − ′ Er(,td′′) t 0 ∫ −∞ For a “delta” excitation in the electric field we find the response or Greens function as the polarization:

Er(,t′′ )= eδ ( tt ′′ −0 )  Pr(,)t=ε−00 Rt ( t )ε  Green's function

Examples A) instantaneous media (unphysical simplification) − For instantaneous (or non-dispersive) media, which cannot not really exist in nature, we would find: Rt()= χδ () t →Pr,,t = ε χ Er t (unphysical!) ( ) 0 ( ) B) dielectrics 11∞∞f R() t= χω( )exp( −ωtd) ω= exp(−ωtd) ω , P p∫∫−∞ iip −∞ ω−ω−22 ω 220 ig

− Using the residual theorem we find:

 fg 2  exp−t sin Ω≥ tt 0 2 g Rt()= Ω 2 with Ω= ω0 −  4 00t < fgt  Pr(,)t=∫ exp − (tt −′ )sin[ Ω− ( tt ′ )]Er (, tdt ′′ ) Ω −∞ 2 C) metals 2 11∞∞eω R() t= σω( )exp( −ωtd) ω= 0 p exp(−ωtd) ω , j ∫∫−∞ ii−∞ 22p p g −ωi

− Using again the residual theorem we find: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 45

ω2 p exp(−≥gt) t 0 Rt()= g  00t <

t =eω2 − − ′ ′′ jr(,)t 0 p ∫ exp[ g ( t t )]Er (, t ) dt −∞

2.3 The Poynting vector and energy balance 2.3.1 Time averaged Poynting vector The energy flux of the electromagnetic field is given by the Poynting vector S. In practice, we always measure the energy flux through a surface (detector), Sn⋅ , where n is the normal vector of surface. To be more precise, the Poynting vector Sr(,)ttt= E (,) r × H (,) r gives the momentary energy flux. rr Note that we have to use the real electric and magnetic fields, because a product of fields occurs. In optics we have to consider the following time scales: −14 − optical cycle: T00=2 πω≤ / 10 s − pulse duration: T in general TT>> p p 0 − duration of measurement: T in general TT>> m m 0 Hence, in general the detector does not recognize the fast oscillations of the − ω optical field − e it0 (optical cycles) and only delivers a time averaged value. For the situation described above, the electro-magnetic fields factorize in slowly varying envelopes and fast carrier oscillations:

1 Er (,)expt(−ω t) + cc .. =E (,) r t i 0 r 2 For such pulses, the momentary Poynting vector reads: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 46

Sr(,)ttt= E (,) r × H (,) r rr 1 ∗ ∗ =ErHrErHr(,)t × (,) t +× (,) tt (,) 4  

1 ∗ ∗ +ErHr(,)tt × (,)exp( −ω+ 2 t) ErHr (,) t × (,)ex tp(2 ω t) 4 ii0 0

1 ∗ 1 = ℜ× + ℜ×  ω Er(,)tt H(,) r  Er(,)tt Hr (,) cos(2 0t) (((2 ((( (((((((2 ((( sloω fast

1 ∗∗ + ℑ× ω Er(,)tt Hr (,)sin( 2 0 t). ((((((((((((2 fast

We find that the momentary Poynting vector has some slow contributions which change over time scales of the pulse envelope Tp, and some fast contributions cos( 2ωω00tt) , sin( 2 ) changing over time scales of the optical cycle T0. Now, a measurement of the Poynting vector over a time interval Tm leads to a time average of Sr(,)t . 1 tT+ /2 Sr(,)t = m Sr (,')'t dt ∫ tT− /2 T m m

The fast oscillating terms ~ cos2ω0t and ~ sin 2ω0t cancel by the integration since the pulse envelope does not change much over one optical cycle. Hence we get only a contribution from the slow term.

+ 1 1 tT/2 ∗ Sr(,td) = m ℜEr( ,t ')× H(, rt ') t ' ∫ tT− /2   2 T m m Let us now have a look at the special (but important) case of stationary (monochromatic) fields. Then, the pulse envelope does not depend on time at all (infinitely long pulses). Er(,')tt= Er (), Hr (,')= Hr ()

1 ∗ Sr(,)t =ℜ× Er () H (). r 2  This is the definition for the optical intensity It= Sr(,) . We see that an intensity measurement destroys information on the phase. It= Sr(,)  measurement destroys phase information

2.3.2 Time averaged energy balance Let us motivate a little bit further the concept of the Poynting vector. Some interesting insight on the energy flow of light and hence also on the transport of information can be obtained from the Poynting theorem, which is the Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 47 equation for the energy balance of the electromagnetic field. The Poynting theorem can be derived directly from Maxwell’s equations. We multiply the two curl equations by Hr resp. Er (note that we use real fields): ∂ H ⋅+µ⋅=rotE H H 0 rrrr0 ∂t

∂∂ E ⋅rotH −εE ⋅E =E ⋅()j + P r r0 rr∂∂tt r rr Next, we subtract the two equations and get ∂∂ ∂ HrotEErotHrrr⋅−⋅+ε⋅+µ⋅=−⋅+ rrr00 E E H rrrr H Ej( P r ). ∂∂tt ∂ t This equation can be simplified by using the following vector identity:

Hr⋅ rotE rr −⋅ E rotH r = divE( rr × H )

1 2 Finally, with the substitution EErr⋅∂ ∂tt =2 ∂ E r ∂ we find Poynting's theorem

11∂∂22 ∂ ε00Er +µ Hr + div( E r × H r) =−⋅ E rr j + P r (*) 22∂∂tt ∂t This equation has the general form of a balance equation. Here it represents the energy balance. Apart from the appearance of the divergence of the Poynting vector (energy flux), we can identify the vacuum energy density 1122 u =ε2200EHrr +µ . The right-hand-side of the Poynting's theorem contains the so-called source terms. 11 where u =εEH22 +µ  vacuum energy density 2200rr In the case of stationary fields and isotropic media (simple but important) 1  E( r ,t )= Er ( )exp( −ω0t) + cc . . r 2 i

1  H( r ,t )= Hr ( )exp( −ω0t) + cc . . r 2 i Time averaging of the left hand side of Poynting’s theorem (*) yields:

11∂∂ 1 ∗ ε22 +µ + × = ℜ × 00E(,) rt H (,) r t div[ E (,) r tt H (,) r] div{  E()() r H r } 22∂∂ttr r rr 2 = div S(,). r t

Note that the time derivative removes stationary terms in Er2 (,)t and Hr2 (,)t . r r Time averaging of the right hand side of Poynting’s theorem yields (source terms): Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 48

∂ −+jr(,)t Pr(,)t Er(,)t r ∂t r r

1 − ω − ω −ω =−+σ()ω E(r )e it0 − ωec()ω)(Ere i 0tcc..E(r) eit0 c..c  4 0 i 00 0  

Now we use our generalized dielectric function:

1 σω( ) =− −ωe cω + 0 −ω + −ω + i00( 0 ) i E(r)exp( ii0t) cc . .E(r) exp( 0t) cc . . 4 ω00e

1 ∗  = ω00e e(ω0 ) −1E(r)E(r) +cc.. 4i

±ω Again, all fast oscillating terms  exp( 2i 0t) cancel due to the time average.

Finally, splitting εω( 0 ) into real and imaginary part yields

11∗∗ ′ ′′ ′′ = ωε00 ε( ω 0) −1 + ε( ω0) E(r)E(r) +cc.. =− ωεε00( ω 0)E() r E (). r 42ii Hence, the divergence of the time averaged Poynting vector is related to the imaginary part of the generalized dielectric function:

1 ∗  div S = − ω ε ε′′ (ω )E() r E (). r 2 00 0 This shows that a nonzero imaginary part of epsilon (ε′′ ( ω≠) 0) causes a drain of energy flux. In particular, we always have ε′′ ( ω>) 0 , otherwise there would be gain of energy. In particular near resonances we have ε′′ ( ω≠) 0 and therefore absorption. Further insight into the meaning of div S gives the so-called divergence theorem. If the energy of the electro-magnetic field is flowing through some volume, and we wish to know how much energy flows out of a certain region within that volume, then we need to add up the sources inside the region and subtract the sinks. The energy flux is represented by the (time averaged) Poynting vector, and the Poynting vector's divergence at a given point describes the strength of the source or sink there. So, integrating the Poynting vector's divergence over the interior of the region equals the integral of the Poynting vector over the region's boundary. ∫∫div SdV= S ⋅ n dA VA Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 49

2.4 Normal modes in homogeneous isotropic media Using the linear material models which we discussed in the previous chapters we can now look for self-consistent solutions to the wave equation including the material response. It is convenient to use the generalized complex dielectric function to derive the solution of the wave equation

i ′ ′′ εω=() 1 +χω+ () σω=εω+( ) ()i ε () ω ωε0 We will do our analysis in Fourier domain. In particular, we will focus on the most simple solution to the wave equation in Fourier domain, the so-called normal modes. These normal modes are the stationary solutions of the wave equation. Hence they usually correspond to waves which are infinitely extended in space and time. However as we will see later, by taking into account the principle of superposition we can construct the general solutions from these normal modes, by superimposing multiple normal modes to construct also the transient waves. We start from the wave equation in Fourier domain, which reads as ω2 rotrot E(, r ω ) = εω ( )E (, r ω ) c2 According to Maxwell the solutions have to fulfill additionally the divergence equation:

ε0 [1 +χ ( ω )]div E (, r ω ) = 0 In general, this additional condition implies that the electric field is free of divergence: for 1+χ () ω ≠ 0  div E(, r ω= ) 0 (normal case) Let us for a moment assume that we already know that we can find plane wave solutions of the following form in the frequency domain:

E( r ,ω= ) E ( ω )exp(i kr) , k = unknown complex wave-vector Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 50

The corresponding stationary field in time domain is given by:

E( r ,tt )= E exp i( kr − ω )  monochromatic plane wave  normal mode This is a monochromatic plane wave, the simplest solution we can expect, a so-called normal mode. Then, the divergence condition implies that these waves are transverse kE⊥ ()ω  transverse wave Here transverse means that the electric field and the wave-vector are oriented perpendicular to each other. If we split the complex wave vector into real and imaginary part k= k'+ i k'', we can define: ′ o planes of constant phase kr= const ′′ o planes of constant amplitude kr= const In the following we will call the solutions A) homogeneous waves  if these planes are identical B) evanescent waves  if these planes are perpendicular C) inhomogeneous waves  otherwise We will see that in dielectrics (σω=( ) 0) we can find a second, exotic type of wave solutions: At ω=ω →ε( ω ) = 0, so-called longitudinal waves kE ()ω LL appear. 2.4.1 Transverse waves Let us have a closer look at the transverse nature of the fields first. As pointed out above, for ω≠ω the electric field becomes free of divergence: L

εεω0 ( )divEr ( , ω ) = 0  divEr ( ,ω= ) 0 Then, the wave equation reduces to the Helmholtz equation: ω2 ∆Er(,ω+ ) εω ( )Er (, ω ) = 0. c2 Hence, we have three scalar equations for Er(,ω ) (from Helmholtz), and together with the divergence condition we are left with two independent field components. We will now construct solutions using the plane wave ansatz:

E( r ,ω= ) E ( ω )exp(i kr) Immediately we see that the wave is transversal:

0=divE ( ω= )i k ⋅ E (, r ω )→ kE⊥ (ω ). Hence, we have to solve Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 51

ω2 2 ⋅ ω= −kE +2 εω() () ω = 0 and kE( ) 0. c which leads to the following dispersion relation ω2 k 22222=kkkk = + + = εω() xyzc2 ω We see that the so-called wave-number k()ω =c εω () is a function of the frequency. We can conclude that transverse plane waves are solutions to Maxwell's equations in homogeneous, isotropic media, only if the dispersion relation k()ω is fulfilled. ′ ′′ ′ ′′ In general, k=k+ i k is complex. Alternatively if kk , it is sometimes useful to introduce the complex refractive index nˆ : ω ωω k()ω= εω= () nnˆ () ω=[ () ω+ik ()ω] c cc However, instead of assuming that nˆ()ω and εω() are just the same, one should clearly distinguish between the two. While εω() is a property of the medium, nˆ()ω is a property of a particular type of the electromagnetic field in the medium, i.e. a property of the infinitely extended monochromatic plane wave.

E( r ,ω= ) E ( ω )exp(i kr) Hence, the coincidence of the complex refractive index nˆ()ω with εω() holds only for homogeneous media since nˆ()ω cannot not be a local property while εω() could. With the knowledge of the electric field we can compute the magnetic field if desired: 1 H(, rω=− ) i rot E(, rω= ) k× E ( ω )exp( kr) ωµ ωµ i 00 1 → H(,) r ω= H ()exp ω (i kr) ,ωith H ()ω= k× E () ω ωµ0

2.4.2 Longitudinal waves Let us now have a look at the rather exotic case of longitudinal waves. These waves can only exist for εω() = 0 in dielectrics at the longitudinal frequency ω=ω . In this case, we cannot conclude that div E(, r ω= ) 0, and hence the L wave equation reads (the l.h.s. vanishes because εω() = 0):  rotrot E(, r ω= ) 0 L As for the transversal waves we try the plane wave ansatz and assume k to be real. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 52

E( r ,ω= ) E ( ω )exp(i kr)

With rot E(ω )exp(ii kr) = k ×ω E( )exp( i kr) we get from the wave equation: k×× k Er(,ω= ) 0 L Now we decompose the electric field into transversal and longitudinal components with respect to the wave vector: ω= ω = ω + ω E(,) r E ()exp(i kr) E⊥ ()exp( ii kr) E ()exp( kr)

with Ek⊥ ()ω ⊥ and Ek ()ω  This decomposed field is inserted into the wave equation: × ×+ = k k( E⊥ E ) exp(i kr) 0   k×× k E⊥ exp( kr) += k ×kE×  exp( kr) 0 ii (  ( =0

Since the cross product of k with the longitudinal field E ()ω is trivially zero the remaining wave equation is:

2 k E⊥ = 0

Hence the transversal field E⊥ must vanish and the only remaining field component is the longitudinal field E ()ω :  E( r ,ω= ) E ( ω )exp( kr) LL i 2.4.3 Plane wave solutions in different frequency regimes The dispersion relation k 22222=kkkk = + + = ω 22 c εω() for plane wave xyz( ) solutions dictates the (complex) wavenumber k only. Thus, different solutions ′ ′′ for the complex wave vector k=k+ i k are possible. In addition, the generalized dielectric function εω() is complex. In this chapter we will discuss possible scenarios and resulting plane wave solutions.

A) Positive real valued epsilon εω=εω>( ) '0( ) This is the favorable regime for optics. We have transparency, and the frequency of light is far from resonances of the medium. The dispersion relation gives ωω22 kn22=k' − k'' 2 +2 k ' ⋅=εω= k'' ′ ()2 () ω ⇒k ' ⋅= k'' 0 i cc22 There are two possibilities to fulfill this condition, either k'' = 0 or k'⊥ k'' .

A.1) Real valued wave-vector k'' = 0 • In this case the wave vector is real and we find the dispersion relation Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 53

ω ωπ2 kn()ω= () ω= = n () ω ccλ n • Because k'' = 0 these waves are homogeneous, i.e. planes of constant phase are parallel to the planes of constant amplitude. This is trivial, because the amplitude is constant. Example 1: single resonance in dielectric material − for lattice vibrations (phonons)

− Now the imaginary part of εω() is neglected, which mathematically corresponds to an undamped resonance

′ f εω=εω=() () ∞ + 22 ω0 −ω ω − We can invert the dispersion relation k()ω = εω ()  ω()k : c

Example 2: free electrons − for plasma and metal − Again the imaginary part of εω() is neglected Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 54

ω2 ′ p εω=εω=() () 1 − 2 ω ω − We again invert the dispersion relation k()ω = εω () → ω()k : c

A.2) Complex valued wave-vector k'⊥ k'' • The second possibility to fulfill the dispersion relation leads to a complex wave-vector and so-called evanescent waves. We find ω2 k 22=k' − k'' 2 = εω() and therefore k''2= k' 22 − k c2 • This means that  k''2 ≠ 0 and k'22> k • We will discuss the importance of evanescent waves in the next chapter, where we will study the propagation of arbitrary initial field distributions. What is interesting to note here is that evanescent waves can have arbitrary large k'22> k , whereas the homogeneous waves of case A.1) ( k'' = 0) obey k'22= k . If we plug our findings into the plane wave ansatz we get: for the evanescent waves:

Er( ,ω= ) E ( ω)p ex {ik'( ω)r}exp(−k''(ω)r) • The planes defined by the equation k''(ω )r = const. are the so-called planes of constant amplitude, those defined by k'(ω )r = const. are the planes of constant phase. Because of k'⊥ k'' these planes are perpendicular to each other. • The factor exp(−ωk''( )r) leads to exponential growth of evanescent waves in homogeneous space. Therefore, evanescent waves can't be physically justified normal modes of homogeneous space and can only exist in inhomogeneous space, where the exponential growth is suppressed, e.g. at interfaces. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 55

B) Negative real valued epsilon εω() =εω′ () < 0 This situation (negative but real εω() can occur near resonances in dielectrics (ω <ω<ω ) or below the plasma frequency (ω<ω ) in metals. Then the 0 L p dispersion relation gives ω2 k 22=k' − k'' 2 +2 k' ⋅ k'' = εω<′() 0 i c2

As in the previous case A), the imaginary term has to vanish and k'0⋅= k'' . Again this can be achieved by two possibilities.

B.1) k' = 0 ω2  k''2 = εω′()  E( r ,ω− )− exp( k''r)  strong damping c2

B.2) k'⋅= k'' 0  k'⊥ k''  evanescent waves ω2 22 2 ′ k =k' − k'' =−2 εω() c ω2 22′ k'' =2 εω+()k' . c As above, these evanescent waves exist only at interfaces (like for εω=εω>()′ () 0). The interesting point is that here we find evanescent waves for all values of k'2. In particular, case B.1) (k' = 0 ) is included. Hence, we can conclude that for εω() =εω′ () < 0 we find only evanescent waves!

C) Complex valued epsilon εω() This is the general case, which is relevant particularly near resonances. From our (optical) point of view only weak absorption is interesting. Therefore, in the following we will always assume ε′′() ω << ε ′ () ω . As we can see in the following sketch, we can have εω>′( ) 0, ε ′′ ( ω> ) 0, or εω<′() 0,() ε ′′ ω> 0. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 56

Let us further consider only the important special case of quasi-homogeneous plane waves, i.e., k' and k'' are almost parallel. Then, it is convenient to use the complex refractive index

22 2 22ωω ω [kk′+ik ′′ ] =22() ω= εω= ()nˆ () ω=[ n () ω+ k ()ω] cc22 c 2i Since k' and k'' are almost parallel: ωω  κ' =n(), ωκ'' = κω() cc The dispersion relation in terms of the complex refractive index gives

22 ωω 2 k22=kn = εω() = () ω+ k ()ω 22[ i ] cc Here we have εω=εω+′ ε ′′ ω=22 ω−κω+ ωκω () ()ii ()nn () () 2 ()(), ε′() ω =n22 () ω −κ () ω and therefore  ε′′() ω = 2()()n ωκω

2 ε′ 2 n (ω= ) sgn( ε′) 1 +ε( ′′ / ε ′ ) + 1 , 2 

2 ε′ 2 κ( ω= ) sgn( ε′) 1 +ε( ′′ / ε ′ ) − 1 . 2  Two important limiting cases of quasi-homogeneous plane waves:

C.1) εε>′, ′′ 0, ε′′ << ε′, (dielectric media) 1εω′′ () n()ω≈ εω′ (), κω≈ () 2 εω′() Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 57

In this regime propagation dominates ( n()ω κω ()), and we have weak absorption: ωω22 k'22− k'' = εω′( ), 2k' ⋅ k'' = εω′′( ). cc22 ω ω ω1 ωε′′ () ω κ' =n() ω≈ εω′ (), κ'' = κω≈() cc c2 cεω′() − k'⋅≈ k'' k' k'' , − k' and k'' almost parallel  homogeneous waves  in homogeneous, isotropic media, next to resonances, we find damped, homogeneous plane waves, k' k′′ ek with ek being the unit vector along k ω   ω  ω= ω = ω ω − κω Er(,) E ()exp( κr) E ()expn( )(erκκ) exp ( )(er) . iicc  

C.2) ε<′0, ε ′′ > 0, ε′′ << ε′ , (metals and dielectric media in so-called Reststrahl domain) 1εω′′ () n()ω≈ ,κω≈ () εω′ (), 2 εω′()

In this regime damping dominates (n()ω κω ()) and we find a very small refractive index. Interestingly, propagation (nonzero n) is only possible due to absorption (see time averaged Poynting vector below).

Summary of normal modes Here we summarize our previous findings about the properties of normal modes for different parameters of the material which they exist in. We do this at the example of a dielectric media for frequencies close to one resonance. Here we can identify different frequency intervals in which we find the typical behavior of the material's reponse. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 58

There are the following frequency ranges, which give rise to the typical properties of normal modes: a) Frequency far below or far above resonance where εω>′() 0, ε′′() ω≈ 0 Typical normal modes: − undamped homogeneous waves − evanescent waves b) Frequency above resonance where εω<′() 0, ε′′() ω≈ 0 Typical normal modes: − evanescent waves c) Frequency close to and below resonance where εω>′() 0, ε′′() ω> 0 Typical normal modes: − weakly damped quasi-homogeneous waves d) Frequency close to and above resonance where εω>′() 0, ε′′() ω> 0 Typical normal modes: − strongly damped quasi-homogeneous waves Optical systems work mainly in regime a) since here you find light propagating undamped over long distances through space. Furthermore one sometimes exploits also regime b) when one would like to have reflection of light at surfaces (e.g. at a metallic mirror). 2.4.4 Time averaged Poynting vector of plane waves

+ 11 tT/2 ∗ Sr(,)t =m ℜ×Er(,t′ ) H (, r t ′′ ) dt , ∫ tT− /2  2 T m m For plane waves we find: ′ ′′ ErEkrEkrkr( ,tt )= exp(ii −ω) =exp( i − −ω i t) 1 Hr(,)tt= k × Er (,) ωµ0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 59 assuming a stationary case EE(t )= ( ω )exp( −ω i t)

11κ′ 22eω  Sr( ,t )= exp[ −= 2κ′′r] E n 0 eexp − 2 κ( erE) κ' κ" 22ωµ00µ c

with ek' being the unit vector along k′ and ek'' being the unit vector along k′′ . 2.5 The Kramers-Kronig relation In the previous sections we have assumed a very simple model for the description of the material's response to the excitation by the electromagnetic field. This model was based on quite strong assumptions, like a single charge which is attached to a rigid lattice etc. Hence, one could imagine that more complex matter could give rise to arbitrarily complex response functions if adequate models would be used for its description. However we can show from basic laws of physics, that several properties are common to all possible response functions, as long as a linear response to the excitation is assumed. These fundamental properties of the response function are formulated mathematically by the Kramers-Kronig relation. It is a general relation between εω′() (dispersion) and εω′′() (absorption). This means in practice that we can compute εω′() from εω′′() and vice versa. For example, if we have access to the absorption spectrum of a medium, we can calculate the dispersion. The Kramers-Kronig relation follows from reality and causality of the response function R of a linear system. That the response function is real valued is a direct consequence from Maxwell's equations which are real valued as well. Causality is also a very fundamental property, since the polarization must not depend on some future electric field. As we have seen in the previous sections, in time-domain the polarization and the electric field are related as: t ∞ Pr(,)t=ε R ( t − t′ )Er (, t ′′ ) dt↔ Pr (,) t =ε R () t Er (, t −t ) d t r00∫∫−∞ rr0 r Reality of the response function implies: ∞∞ 11 ∗ Rd(τ) = ωχ( ω)ee-*iiωτ = d ωχ( ω) ωτ → χ( ω) = χ( −ω) ππ∫∫ 22-∞ -∞ Causality of the response function implies: 1 forτ> 0  1 Ry(τ) =θ(t) ( t) with θ(τ) = forτ= 0  Heaviside distribution 2 0 forτ< 0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 60

In the following, we will make use of the Fourier transform of Heaviside distribution: ∞ ω i 2πθ( ω) =dt θ( t)ePit = + πd( ω)  defined as integral only ∫ ω −∞ In Fourier space, the Heaviside distribution consists of the Dirac delta distribution ∞ ωd ω − ω ω = ω  ∫ d()00 ff( ) ( ) Dirac delta distribution −∞ ω and the expression P(i/ ) involving a Cauchy principal value: ∞ i−α ii∞ Pdfωω=ωω+ωω () lim df () df () ∫ωα→0 ∫∫ ωω −∞ −∞ α  Cauchy principle value As we have seen above, causality implies that the response function has to contain a multiplicative Heaviside function. Hence, in Fourier space (susceptibility) we expect a convolution: ∞∞ χω=( ) ∫∫ddτR(τ)eeiiωτ = τθ(t) y( t) ωτ −∞ −∞ 11i  θω=( ) P + δ(ω) ∞ 22πω = ∫ dyωωθ( ω−ω) ( ) −∞ 1 ∞ i yy(ωω) ( ) χω=( ) P∫ d ω +() ∗ 22π−∞ ω−ω In order to derive the Kramers-Kronig relation we can use a small trick (this trick saves us using complex integration in the derivation). Because of the Heaviside function, we can choose the function y(τ) for t < 0 arbitrarily without altering the susceptibility! In particular, we can choose: a) yy(−τ) = (τ) even function b) yy(−τ) = − (τ) odd function a) yy(−τ) = (τ) In this case yy(−τ) = (τ) is a real valued and even function. We can exploit this property and show that ∞∞ 11  yy(ω) = dyττ( )eeiiωτ = dττy( ) − ωτ = ∗ (ω) is real as well ππ∫∫ 22−∞ −∞ Hence, we can conclude from equation (*) above that Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 61

∞ ∗ 1 iy(ωω) y( )  χ( ω=−) P ∫ d ω + 22π−∞ ω−ω Here P∫ is a so called principal value integral (G: Hauptwertintegral). Now we have expressions for χω( ), χ* ( ω) and can compute real and imaginary part of the susceptibility: 11∞∞iy(ωω) y( ) iy( ωω) y( ) χ( ω) +χ* ( ω) = PP∫∫ddω+−ω+=y ( ω) 22π−∞ ω−ω22 π−∞ ω−ω ∞ ∗ 1 i y (ω) χω( ) − χω( ) = =P ∫ d ω π−∞ ω−ω Plugging the last two equations together we find the first Kramers-Kronig relation: 1 ∞ ℜχ( ω)  ℑχ( ω) =− P ∫ d ω 1. K-K relation π−∞ ωω− Knowledge of the real part of the susceptibility (dispersion) allows us to compute the imaginary part (absorption). b) yy(−τ) = − (τ) The second K-K relation can be found by a similar procedure when we assume that yy(−τ) = − (τ) is a real odd function. We can show that in this case ∞∞ 11  y (ω) = dyττ( )eeiiωτ =− dττy( ) − ωτ = −y∗ (ω) is purely imaginary π ∫∫π 22−∞ −∞ With equation (*) we then find that ∞ ∗ 1 iy(ωω) y( )  χ( ω=) P ∫ d ω − (see (*)) and 22π−∞ ω−ω Again we can then compute real and imaginary part of the susceptibility ∞∞ ∗ 11iy(ωω) y( ) iy( ωω) y( ) χ( ω) −χ( ω) =PP∫∫d ω+−ω+=d y (ω) 22πω−ωπωω−∞ 22−∞ − 1 ∞ iy(ω) χω( ) +ωχ* ( ) = =P ∫ d ω πω−∞ −ω and finally obtain 1 ∞ ℑ χω( ) → ℜχ( ω) =P ∫ d ω 2. K-K relation π−∞ ωω− Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 62

The second Kramers-Kronig relation allows us to compute the real part of the susceptibility (dispersion) when we know its imaginary part (absorption).

The Kramers-Kronig relation can also be rewritten in terms of the dielectric function, where one applies also the symmetry relation for ω : K-K relation for ε : − χ( ω) = χ∗ ( −ω)→ χ′′() ω = χ ( −ω ) χ′′() ω = −χ ′′ ( −ω ) and ′ ′′ χω=εω−() ()1 =εω−[ ()1] +i ε () ω 2∞ ωε′′ () ω εω−′() 1 = P dω, π∫ 0 ω22 −ω

2 ∞ [εω−′() 1] ε′′() ω=− ω P dω. π∫ 0 ω22 −ω − dispersion and absorption are linked, e.g., we can measure absorption and compute dispersion Example: ω ′′  ′ 0  ε() ω − δ (ω−ω0 ) εω−() 1− 22 Drude-Lorentz model ω0 −ω 2.6 Beams and pulses - analogy of diffraction and dispersion In this chapter we will analyze the propagation of light. In particular, we will answer the question how an arbitrary beam (spatial) or pulse (temporal) will change during propagation in isotropic, homogeneous, dispersive media. Relevant (linear) physical effects are diffraction and dispersion. Both phenomena can be understood very easily in the Fourier domain. Temporal effects, i.e. the dispersion of pulses, will be treated in temporal Fourier domain (temporal frequency domain). Spatial effects, i.e. the diffraction of beams, will be treated in the spatial Fourier domain (spatial frequency domain). We will see that: • Pulses with finite spatial width (i.e. pulsed beams) can be described as superpositions of normal modes in the frequency- and spatial frequency domain. • Spatio-temporally localized optical excitations delocalize during propagation because of the different phase evolution of the excited normal modes for different frequencies and spatial frequencies (different propagation directions of normal modes). Let us have a look at the different possibilities (beam, pulse, pulsed beam) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 63

A) beam  finite transverse width  diffraction

plane wave (normal mode) beam

A beam is a continuous superposition of stationary plane waves (normal modes) with different wave vectors (propagation directions).

k1 k2 k3 k4 k5

…

∞ 3 E(rE ,t )= (kk )exp ( r −ωt) d k ∫ −∞ i B) pulse  finite duration  dispersion ω

stationary wave (normal mode) pulse

A pulse is a continuous superposition of stationary plane waves (normal modes) with different frequencies. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 64

ω1 ω 2 ω 3

...

∞ Er( ,tt )= E (ωω )exp ( k⋅ r− ω ) d . ∫ −∞ i

C) pulsed beams  finite transverse width and finite duration  diffraction and dispersion A pulsed beam is a continuous superposition of stationary plane waves (normal modes) with different frequency and different propagation direction ∞ Er( ,t )= E (kk,ωω )exp ( ⋅r− ωt) dk3 d ∫ −∞ i 2.7 Diffraction of monochromatic beams in homogeneous isotropic media Let us have a look at the propagation of monochromatic beams first. In this situation, we have to deal with diffraction only. We will see later that pulses and their dispersion can be treated in a very similar way. Treating diffraction in the framework of wave-optical theory (or even Maxwell's equations) allows us to treat rigorously many important optical systems and effects, i.e., optical imaging and resolution, filtering, microscopy, gratings, ... In this chapter, we assume stationary fields and therefore ω=const . For technical convenience and because it is sufficient for many important problems, we will make the following assumptions and approximations: • εω() =εω′ () > 0,  optical transparent regime  normal modes are stationary homogeneous and evanescent plane waves • scalar approximation

Er(,ω→ )Ey (, r ω ) e yy → Eu (, r ω→ ) (, r ω ). − exact for one-dimensional beams and linear polarization − approximation in two-dimensional case In homogeneous isotropic media we have to solve the Helmholtz equation ω2 ∆ Er(,ω+ ) εω( ) Er (, ω= ) 0. c2 In scalar approximation and for fixed frequency ω it reads Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 65

ω2 ∆uu(,rrω+ )εω( ) (, ω= ) 0, c2 scalar Helmholtz equation ∆u(,rrω+ )k 2 (ω)u (, ω )= 0. In the last step we inserted the dispersion relation (wave number k()ω ). In the following we often even omit the argument in our notation since the frequency ω is fixed anyway. 2.7.1 Arbitrarily narrow beams (general case) Let us consider the following fundamental problem. We want to compute from a given field distribution uxy( , ,0) in the plane z = 0 the complete field uxyz(, ,) in the half-space z > 0 , where z is our “propagation direction”.

The governing equation is the scalar Helmholtz equation ∆u(,rrω+ ) ku2 ( ω) (, ω= ) 0 To solve this equation and to calculate the dynamics of the fields, we can switch again to the Fourier domain. We take the Fourier transform ∞ u(,)rω= U (,)exp k ω[ kr () ω ]dk3 ∫ −∞ i which can be interpreted as a superposition of normal modes with different propagation directions and wavenumbers k()ω (here the absolute value of the wave-vector k ). Naively, we could expect that we just constructed a general solution to our problem, but the solution is not correct because of the dispersion relation: ω2 k22222=kkkk = + + = εω( ) xyzc2

→ only two components of k are independent, e.g., kk,. xy Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 66

Our naming convention is in the following: kkk=α=β=γ, ,. x yz Then, the dispersion relation reads: k 2ω =α 222 +β +γ ( ) Thus, to solve our problem we need only a two-dimensional Fourier transform, with respect to the transverse coordinates x and y , when z is the direction of propagation: ∞ u(r )= U ( α , β; z )exp ( αx +β y) dd α β. ∫∫−∞ i

In analogy to the frequency ω we call α, β spatial frequencies. Now we plug this expression into the scalar Helmholtz equation ∆uku()rr+ω2 ( ) () = 0 This way we can transfer the Helmholtz equation in two spatial dimensions into Fourier space

2 d 2 22 2 +k −α −β Uz( α , β ; ) = 0, dz

2 d 2 2 +γUz( αβ , ; ) = 0. dz

This equation is easily solved and yields the general solution αβ = αβ γαβ + αβ γαβ U(,;) zU12 (,)exp[i (,)zU] (,)exp[-i (,)z] , depending on γ(,) α β =k 2 () ω −α 22 −β . We can identify two types of solutions:

A) Homogeneous waves γ≥2 0,  α222 +β ≤k , i.e., k real  homogeneous waves Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 67

B) Evanescent waves γ<2 0,  α22 +β >k 2 , i.e., k complex, because γ=k imaginary. Then, z ′+ ′′ ′ α +β ′′ γ we have k=ki k , with k= exy e and k=ez .  k'⊥ k''  evanescent waves

We see immediately that in the half-space z > 0 the solution ∼ exp(−γi z) grows exponentially. Because this does not make sense, this component of the solution must vanish U2 (,)αβ = 0. In fact, we will see later that U2 (,)αβ corresponds to backward running waves, i.e., light propagating in the opposite direction. We therefore find the solution: αβ = αβ γαβ U(,;) zU1 (,)exp[i (,)z] Furthermore the following boundary condition holds

UU10(,)αβ (,;0) αβ = U (,) αβ which determines uniquely the entire solution for z ≠ 0 as U(,;)αβ zU = (,;0)exp αβ[ γαβ (,)z] i αβ γαβ  Uz0 (,)exp[i (,)] In spatial space, we can find the optical field for z > 0 by inverse Fourier transform: ∞ u(r )= U ( α , β; z )exp ( αx +β y) dd α β. ∫∫−∞ i ∞ u(r )= U0 ( α , β )exp γ( αβ, ) z e.xp (αx +β y)  dd α β ∫∫−∞ i i Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 68

For homogeneous waves (real γ ) the red term above causes a certain phase shift for the respective plane wave during propagation. Hence, we can formulate the following result: Diffraction is due to different phase shifts of the different normal modes when these modes are traveling in propagation direction. The individual phase shifts are determined according to different spatial frequencies αβ, of the normal modes.

The initial spatial frequency spectrum or angular spectrum at z = 0 forms the initial condition of the initial value problem and follows from u0 (, xy )= uxy (, ,0) by Fourier transform:

2 1 ∞ U00(,)α β = u( x , y )exp −( α x +β y) dxdy, 2p ∫∫−∞ i

As mentioned above the wave-vector components αβ, are the so-called spatial frequencies. Another common terminology is “direction cosine” for the quantities α /,k β / k , because of the direct link to the angle of the respective pane wave. For example α=/k cos θx gives the angle of the plane wave's propagation direction with the x -axis.

Scheme for calculation of beam diffraction We can formulate a general scheme to describe the diffraction of beams:

1. initial field: u0 (, xy )

2. initial spectrum: U0 (,)αβ by Fourier transform 3. propagation: by multiplication with expiγ( αβ , ) z αβ = αβ γ αβ 4. new spectrum: U(,;) zU0 (,)expi ( ,) z 5. new field distribution: uxyz(, ,) by Fourier back transform

This scheme allows for two interpretations: 1) The resulting field distribution is the Fourier transform of the propagated spectrum ∞ u()r = Uz(,;)α β exp (αx +β y) dd α β. ∫∫−∞ i  2) The resulting field distribution is a superposition, i.e. interference, of homogeneous and evanescent plane waves ('plane-wave spectrum') which obey the dispersion relation Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 69

∞ ud(r) = αβd Uz(αβ, ) epx γ( αβ , ) exp [αxy +β ] ∫∫−∞ 0 {i } {i } ))))) ))))) ))))))))) )))))))))) interference of αmplitude of phαse fαctor which is shαpe of eiγenstαtes eiγenstαtes to form the excited αccumulαted βy the (plαn wαves) the field pαttern αfter eiγenstαtes eiγenstαtes durinγ propαγαtion propαγαtion

To understand the diffraction of beams let us now discuss the complex transfer function Hz(αβ , ; ) = exp[i γαβ ( , ) z ], which describes the beam propagation in Fourier space. For z = const. (finite propagation distance) it looks like:

amplitude phase

Obviously, Hz(αβ , ; ) = expi γ( αβ , ) z acts differently on homogeneous and evanescent waves:

A) Homogeneous waves for α222 +β ≤k  expiiγ( α ,β) zz = 1,argexp( γ( α ,β) ) ≠ 0  Upon propagation the homogeneous waves are multiplied by the phase factor

exp kz2−α 22 −β i  Each homogeneous wave keeps its amplitude.

B) Evanescent waves for α222 +β >k

 expγ α ,βz = exp  − a222+β −kz, arg expγ α ,βz = 0 ii( ) ( ( ) )

 Upon propagation the evanescent waves are multiplied by an amplitude factor <1

exp − α222 +β −kz <1  Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 70

This means that their contribution gets damped with increasing propagation distance z .  Each evanescent wave keeps its phase.

Now the question is: When do we get evanescent waves? Obviously, the answer lies in the boundary condition: Whenever u0 (, xy ) yields an angular 22 2 spectrum U0 (,)αβ ≠ 0 for α +β >k we get evanescent waves. Example: Slit Let us consider the following one-dimensional initial condition which corresponds to an aperture of a slit:

u0(x)  a 1 for x ≤ ux0 ()=  2.  0 otherwise x -a/2 a/2 a sina 2 a U00(a )= FT[ ux ( )]  =sinca a 2 a 2

U0(α)

− All spatial frequencies (-∞ → ∞) are excited. − Important spectral information is contained in the interval a=2/ π a .  Largest important spectral frequency for a structure with width a is α=2 π /2. − Evanescent waves appear for α>k . − To represent the relevant information by homogeneous waves the 22ππ λ following condition must be fulfilled: <=kn → a > a λ n Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 71

General result We have seen in the example above that evanescent waves appear for structures < wavelength in the initial condition. Information about these small structures gets lost for z >> λ .

Conclusion In homogeneous media, only information about structural details having length scales of ∆xy,/ ∆ >λ n are transmitted over macroscopic distances.  Homogeneous media act like a low-pass filter for light.

Summary of beam propagation scheme

−1 FT FT u00(,) xy→ U (,) αβ → U (,;) αβ z = H (,;) αβ zU0 (,) αβ → uxyz (,,)

with the transfer function Hz(αβ , ; ) = expi γ( αβ , ) z

Remark: diffraction free beams With our understanding of diffraction it is straight forward to construct so- called diffraction free beams, i.e., beams that do not change their amplitude distribution during propagation. Translated to Fourier space this means that all spatial frequency components have to get the same phase shift during the propagation αβ = αβ γ αβ ≡ αβ U(,;) z U00 (,)expii( ,) z U (,)exp[ Cz]  = uxyz(, ,) exp[i Czu] 0 (, xy )

Since in general γαβ( , ) ≠ const the excitation u0 (, xy ) must have a shape such that its Fourier transform has only components where the transfer function is of equivalent value

2 22 U0 (,)αβ ≠ 0 only for γ( α, β) =kC −α −β = It is straightforward to see that the excited spatial frequencies must lie on a circular ring in the (αβ, ) plane.

222 α +β =ρ0 For constant spectral amplitude on this ring the Fourier back-transform yields (see exercises):

u00(, xy )= J ( r r ) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 72

Bessel-beam (profile) Bessel-beam

2.7.2 Fresnel- (paraxial) approximation The beam propagation formalism developed in the previous chapter can be simplified for the important special case of a narrowband angular spectrum

22 2 Uk0 (,)α β ≠ 0 for α +β 0 In this situation the beam consists of plane waves having only small inclination with respect to the optical z -axis (paraxial (Fresnel) approximation). Then, we can simplify the expression for γαβ(,) by a Taylor expansion to:

22 22 2 22 α +β α +β γ(,) α β =kk −α −β ≈1 −2 = k − 22kk The resulting expression for the transfer function in Fresnel approximation reads: α22 +β = γαβ ≈ − = αβ H exp(i (,)z) exp( ii kz) expz HF (,;) z 2k

Amplitude Phase Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 73

We can see that this HzF (,;)αβ is always real valued. Hence it does not account for the physics of evanescent waves. However, we must remember that the derivation of HzF (,;)αβ as an approximation of Hz(,;)αβ required the assumption that the spatial frequency spectrum is narrow (paraxial waves). Thus, already at the beginning we had excluded the excitation of evanescent waves to justify the paraxial approximation. The assumption of a narrow frequency spectrum corresponds to the requirement that all structural details ∆∆xy, of the field distribution in the excitation plane (at z = 0 ) must be much larger than the wavelength: ∆xy, ∆> 10 λ / n λ / n This requirement applies also to the phase of the excitation. Hence it is not sufficient that only the structural details of the intensity have a large scale. The underlying phase of the excitation field must fulfill this condition as well. This includes the condition that the phase of the beam should not have a strong inclination to the principal propagation direction.

The propagation of the spectrum in Fresnel approximation works in complete analogy to the general case. We just use the modified transfer function to describe the propagation: UzHzU(,;)αβ = (,;) αβ (,) αβ FF0 Summary of Fresnel approximation

For a coarse initial field distribution u0 (, xyz ,) the angular spectrum U0 (,)αβ is nonzero for α22 +β  k 2 only. Then, only paraxial plane waves are relevant for transmitting information and the transfer function of homogeneous space can be approximated by HzF (,;)αβ .

Description in real space It is also possible to formulate beam propagation in Fresnel (paraxial) approximation in position space: ∞ u(,,) xyz= U (,;)exp α β z( α x +β y) d α d β FF∫∫−∞ i ∞ =H(,;) αβ zU (,)exp αβ( αx +β y) d α d β ∫∫−∞ F 0 i ∞ =h( x −− x′ , y y ′ ;) z u ( x ′′ , y ) dx ′′ dy ∫∫−∞ F 0

The spatial response function hF (,;) xyz follows from the convolution theorem and is the Fourier transform of HzF (,;)αβ : Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 74

2 1 ∞ h(,;) xyz = H(,;)expα β z( α x +β y) d α d β FF2p ∫∫−∞ i 2 1 ∞ α22 +β = exp( kz) exp − zexp (α x +β y) d α d β.  ii∫∫−∞  i 22p k This Fourier integral can be solved and we find:

22 kk k( xy+ ) h( x , y ; z )= exp( kz) −iiexp x22 +=− y expkz 1+ , F ii( ) i2 22ppzz2 zz2  The response function corresponds to a spherical wave in paraxial approximation. Similar to Huygens principle, where from each point in the object plane a spherical wave is emitted towards the image plane, here paraxial approximations of spherical waves are emitted.

To sum up, in position space paraxial beam propagation is given by:

kk∞ 22 u(,,) x y z =−i exp( kz) u (,)exp x′′ y ( x− x′′′′) +−( y y) dx dy . F ii∫∫−∞ 0  22pzz Of course, the two descriptions in position space and in the spatial Fourier domain are completely equivalent.

The correspondence between real and frequency space

Relation between transfer and response function: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 75

1 ∞ = α β  α +β α β hxy(;, z) 2 Hz( , ; )exp( x y) dd (2p) ∫∫−∞ i Transfer functions for homogeneous space

H(,;)expαβ z = γ αβ ,z = exp  ik2 −α 22 −β z exact solution i ( )  α22 +β αβ = HF ( , ; z ) exp[i kz] exp  i z Fresnel approximation 2k ω with k= k() ω= kn () ω= n () ω 0 c

Remark on the validity of the scalar approximation In the previous description of the propagation of arbitrary beams we have used the scalar approximation of the vectorial fields. It is interesting to see, to what extend this approximation stays in correspondence to the conditions which where necessary to derive the Fresnel approximation. ixyz(α +β +γ ) Er(,ω ) =∫∫ Eˆ ( αβω ,,) e dd α β ˆˆˆ divE(, r ω ) = 0 →αEEEx +β yz +γ =0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 76

A) One-dimensional beams − translational invariance in y-direction: β=0 ˆ ˆ − and linear polarization in y-direction: EUy = , Ex = 0  scalar approximation is exact since divergence condition is strictly fulfilled B) Two-dimensional beams − Finite beam which is localized in the x,y-plane: αβ≠,0 ˆ ˆ − and linear polarization, w.l.o.g. in y-direction: Ex = 0, EUy = ˆˆ  divergence condition: βEEyz +γ =0 ˆˆββˆ EEzy(αβω,, ) =−( αβω,, ) =− Ey(αβω,,) ≈ 0 γ k 2−α 22 −β In paraxial approximation (α22 +β  k 2) the scalar approximation is automatically justified. 2.7.3 The paraxial wave equation In paraxial approximation the propagated spectrum is given by UzHzU(,;)αβ = (,;) αβ (,) αβ FF0 α22 +β = exp( kz)exp− z U0 (,)αβ i i 2k  Let us introduce the slowly varying spectrum Vz(,;)αβ : α22 +β  U(α ,β ; z )= exp( kz)Vz(,;)α β  V(,;)αβ z = exp − zV(,).αβ F i i 0 2k Differentiation of V with respect to z gives:

∂ 1 22 i Vz(,;)αβ =( α +β) Vz(,;) αβ ∂zk2 Fourier transformation back to position space leads to the so-called paraxial wave equation: ∂ ∞ Vz(α , β ; )exp  ( α x +βy) d αdβ i ∂z ∫∫−∞ i 1 ∞ = αβ22+ V(α ,β ; z )exp ( α xy +β) d αd β ∫∫−∞ ( ) i 2k ∂ 1 ∂∂22∞  v(,xyz, ) =−+ Vz(α , β ; )exp( αx +β y) dd α β ii22∫∫−∞  ∂z 2k ∂∂xy Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 77

∂ 1  vxyz(, ,)+∆(2) vxyz (, ,) = 0 paraxial wave equation i ∂zk2 The slowly varying envelope vxyz(, ,) (Fourier transform of the slowly varying spectrum) relates to the scalar field as u(, xyz ,)= vxyz (, ,)exp kz . F (i ) Extension of the wave equation to weakly inhomogeneous media (slowly varying envelope approximation - SVEA) There is an alternative, more general way to derive the paraxial wave equation, the so-called slowly varying envelope approximation. This approximation even allows us to treat wave propagation in inhomogeneous media. We will include inhomogeneous media in this derivation even though the current chapter of this lecture is devoted to inhomogeneous media. We start from the scalar Helmholtz equation. However, we should mention that the extrapolation of our previous discussion towards inhomogeneous media requires another approximation. This approximation assumes small spatial fluctuations of εω(,r ). This is equivalent to having a weak index contrast between different spatial positions. ω2 ∆uxyz(, ,) +ω k2 (,r )(, uxyz ,) = 0 with k 2 (,r ω= ) ε (,r ω ) c2 We make the ansatz   uxyz(, ,)= vxyz (, ,)exp(ikz) with kk= where k is the spatially averaged wavenumber. This is the mean wave number in the volume, in which wave propagation is considered. Hence it is the average of the spatially varying material properties of the medium. With the SVEA condition kvz ∂∂v / we can simplify the scalar Helmholtz equation as follows: ∂∂2 vxyz(, ,)+ 2 k vxyz (, ,)+∆(2) vxyz (, ,) + k2 (,r ω ) − k2 vxyz (, ,)= 0, 2 i  ∂∂zz ((  ( ( ≈0

22 ∂ 1 (2) kk(,r ω− )   i vxyz(, ,)+∆ vxyz (, ,) +vxyz(, ,)= 0 ∂z 22kk This is the paraxial wave equation for inhomogeneous media having a weak index contrast. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 78

2.8 Propagation of Gaussian beams The propagation of Gaussian beams is an important special case. Optical beams of Gaussian shape are import because many beams in reality have a shape which is at least close to the shape of a Gauss function. In some cases they even have exactly the Gaussian shape as e.g. the transversal fundamental modes of many lasers. The second reason why Gaussian beams are important, is the fact that in paraxial approximation it is possible to compute the evolution of Gaussian beams analytically.

Fundamental Gaussian beam in focus

The general form of a Gaussian beam at one plane ( z = const ) is elliptic, with curved phase

xy22 u(, xy )= v (, xy ) = A exp −+exp[ ϕ (,xy )]. 00 0ww22 i xy Here, we will restrict ourselves to rotational symmetry ( www222= = ) and xy0 (initially) 'flat' phase ϕ=(,xy ) 0. Later we will see that this corresponds to the focus of a Gaussian beam. The Gaussian beam in such a focal plane with flat phase is characterized by amplitude A and width w , which is defined as 0 ux(22+ y = w 2 ) = A exp( −= 1) A / e. 00 00 In practice, the so-called 'full width at half maximum' (FWHM) of the intensity is often used instead of w . The FWHM of the intensity is connected to w of 0 0 the field by

2 æö2 22 ç wFWHM ÷ 1 ux0 ( + y) =-expç ÷ ç 2w2 ÷ 2 èø0 w2 -FWHM =-  2=» 22 2 ln 2 wFWHM 2ln 2w0 1.386w0 2w0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 79

2.8.1 Propagation in paraxial approximation Let us now compute the propagation of a Gaussian beam starting from the focus in paraxial approximation: 1) Field at z = 0 : xy22+ u00(, xy )= v (, xy ) = A 0 exp − . w2 0 2) Angular spectrum at z = 0 :

1 ∞ xy22+ UV(,)(,)αβ = αβ = A exp− exp[− ( αx +β y )] dxdy 002 0∫∫−∞ 2 i (2p) w 0 22 22 AA2 α +β 2 α +β =0 wwexp −=0 exp  − , 44pp004/ww22 0s  We see that the angular spectrum has a Gaussian profile as well and that the width in position space and Fourier space are linked by ww⋅=2. s0

Angular spectrum in the focal plane

Check if paraxial approximation is fulfilled: We can say that U (,)αβ ≈ 0 for α22 +β ≥16/ w 2, because 0 ( ) 0 exp(−≈ 4) 0.02 For paraxial approximation we need k 22 α 22 +β ( )  kw22216/ 0 λλ22  2 16 2   w 2 2 =  ≈ , 0 2π πnn   n λ λ  paraxial approximation works for w 10= 10 λ 0nn 3) Propagation of the angular spectrum:

V(,;)αβ zU =0 (,)H(,;z) αβF αβ Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 80

4) Fourier back-transformation to position space, when still using the slowly varying envelope

With the Rayleigh length which determines the propagation of a Gaussian beam:

Note that we use the slowly varying envelope ! Conclusion: − Gaussian beam keeps its shape, but amplitude, width, and phase change upon propagation − Two important parameters: propagation length and Rayleigh length

Some books use the “diffraction length” , a measure for the “focus depth” of the Gaussian beam. E.g.: 

From our computation above we know that the Gaussian beam evolves like:

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 81

For practical use, we can write this expression in terms of z-dependent amplitude, width, and shape of the phase front:

Here we used that . The (x,y)-independent phase is given by , the so-called Gouy phase shift. In conclusion, we see that the propagation of a Gaussian beam is given by a z-dependent amplitude, width, phase curvature and phase shift:

The normalized beam intensity as a function of the radial distance at different axial distances: (a) ; (b) ; (c) .

Discussion The amplitude evolves along z like: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 82

Hence, we get for the Intensity profile :

The normalized beam intensity at points on the beam axis ( ) as a function of the propagation distance .

The beam width evolves along like:

The beam radius has its minimum value at the waist ( ), reaches at , and increases linearly with for large .

The radius of the phase curvature is given by

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 83

The radius of curvature of the wavefronts of a Gaussian beam. The dashed line is the radius of curvature of a spherical wave.

The flat phase in the focus ( ) corresponds to an infinite radius of curvature. The strongest curvature (minimum radius) appears at the Rayleigh distance from the focus. The (x,y)-independent Gouy phase is given by

The so-called Gouy phase is the retardation of the phase of a Gaus- sian beam relative to a uniform plane wave at the points of the beam axis.

The Gouy phase is not important for many applications because it is ‘flat’. However, in resonators and in the context of nonlinear optics it can play an important role (i.e., harmonic generation in focused geometries). The wave fronts (planes of constant phase) of a Gaussian beam are given by

Wavefronts of a Gaussian beam. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 84

2.8.2 Propagation of Gauss beams with q-parameter formalism In the previous chapter we gave the expressions for Gaussian beam propagation, i.e., we know how amplitude, width, and phase change with the propagation variable z . However, the complex beam parameter = − qz() zi z0 q-parameter allows an even simpler computation of the evolution of a Gaussian beam. In fact, if we take the inverse of the “q-parameter”, 11 zz1 1 ==+=+0 22 22 2 2 −+ii + z0 z qzzz() 00 zz zz 0zz11++22 i ( zz) 0 ( ) 0 we can observe that real and imaginary part are directly linked to radius of phase curvature and beam width: 11 λ kw2 π = + n because zw=0 = 2 qz() Rz ()i π w2 () z 002 λ n

Thus, the q-parameter describes beam propagation for all z ! Example: propagation in homogeneous space by zd= 11 λ A) initial conditions: = + n qR(0) (0)i π w2 (0) B) propagation (by definition of q parameter) qd( )= q (0) + d C) q-parameter at zd= determines new width and radius of curvature 111 λ =  + n qd() q (0)+π d Rd ()i w2 () d

2.8.3 Gaussian optics We have seen in the previous chapter that the complex q-parameter formalism makes a simple description of beam propagation possible. The question is whether it is possible to treat optical elements (lens, mirror, etc.) as well.

Aim: for given Rw00, (i.e. q0 )  Rwnn, (i.e. qn ) after passing through n optical elements

We will evaluate the q-parameter at certain propagation distances, i.e., we will have values at discrete positions: qz()i → qi . Surprising property: We can use ABCD-Matrices from ray optics! This is remarkable because here we are doing wave-optics (but with Gaussian beams).

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 85

How did it work in geometrical optics? A) propagation through one optical element: AB Mˆ = . CD B) propagation through multiple elements:

ˆ ˆˆ ˆ AB M= MM−11.. M= . NN CD C) matrix connects distances to the optical axis y and inclination angles Θ before and after the element

yy21  = Mˆ . ΘΘ21 

Link to Gaussian beams Let us consider the distance to the intersection of the ray with the optical axis, as it was defined in chapter 1.6.1 on "The ray-transfer-matrix". Using the Θ ≈Θ = small angle approximation ( tan(1 ) 1yz 11) we can define this distance as: y AB1 + y1 y2 Ay 11+Θ B Θ1 Az1 + B z1 =  z2 = = = = Θ ΘCy +Θ D y1 Cz+ D 1 2 11CD+ 1 Θ1

The distances zz12, are connected by matrix elements, but not by normal matrix vector multiplication. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 86

It turns out that we can pass to Gaussian optics by replacing z by the complex beam parameter q . The propagation of q -parameters through an optical element is given by:

Aq10+ B 1 q1 = Cq10+ D 1  propagation through N elements:

Aq0 + B qn = Cq0 + D

ˆ ˆˆ ˆ AB with the matrix M= MM−11.. M= . NN CD

 This works for all ABCD matrices given in chapter 1.6 for ray optics!!! Here: we will check two important examples: i) propagation in free space by zd= :  propagation (by definition of q -parameter) qd( )= q (0) + d

ˆ 1 d M =   A11=1, B = dC , 1 = 0, D 1 = 1 01

Aq10++ B 1 q 0 d q10= = =qd + Cq10++ D 1 01 ii) thin lens with focal length f 22 2 What does a thin lens do to a Gaussian beam exp(−+ (xyw ) / 0 ) in paraxial approximation? − no change of the width 22 k ( xy+ ) − ×  but change of phase curvature Rf : exp i 2 Rf How can we see that? Trick: 2 πwf We start from the focus which is produced by the lens with zz0 =f = λn

and wf is the focal width. Hence the q-parameter is Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 87

1 λn = i 2 qw0 π f The radius of curvature evolves as: 2 z f Rz()=+≈ z 1  z for zz f z We can invert the propagation from the focal position to the lens at the

distance of the focal length f and obtain Rff = −

ˆ 10 f < 0 f > 0 Mthin lens =  −11f

double concave double convex lens lens  defocusing  focusing

Aq10+ B 1 q 0 q1 = = Cq10+−+ D 1 q 0 f 1 2 1−+qf00 11 i λn πw ==+=2 for q0 −i q10 q−π fw0 λn

Be careful: Gaussian optics describes the evolution of the beam's width and phase curvature only!  Changes of amplitude and reflection are not included! 2.8.4 Gaussian modes in a resonator In this chapter we will use our knowledge about paraxial Gaussian beam propagation to derive stability conditions for resonators. An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light (see He-Ne laser experiment in Labworks). Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 88

2.8.4.1 Transversal fundamental modes (rotational symmetry)

Wave fronts, i.e. planes of constant phase, of a Gaussian beam.

The general idea to get a stable light configuration in a resonator is that mirrors and wave fronts (planes of constant phase) coincide. Then, radiation patterns are reproduced on every round-trip of the light through the resonator. Those patterns are the so-called modes of the resonator. In paraxial approximation and Gaussian beams this condition is easily fulfilled: The radii of mirror and wave front have to be identical! In this lecture we use the following conventions (different to Labworks script, see remark below!):

− z1,2 is the position of mirror '1','2'; z=0 is the position of the focus!

− d is the distance between the two mirrors  zzd21−= z2 − because Rz()= z + 0  radius of wave front <0 for z <0 z

− from Chapter 1: beam hits concave mirror  radius Rii (= 1,2) < 0.

− beam hits convex mirror  radius Rii (= 1,2) > 0.

Examples:

A) Rz(12 ), Rz ( )> 0; RR12><0, 0; zz12>>0, 0

B) Rz(12 )<> 0, Rz ( ) 0; RR12,0< ; zz12<>0, 0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 89

According to our reasoning above, the conditions for stability are:

R11= Rz( ), R 2= − Rz() 2

22 z00z  Rz11=+,. −=+ Rz2 2 z12z

In both expressions we find the Rayleigh length z0, which we eliminate:

zR11(−=−+ z 1 )( zR 2 2 z 2 )

dR( 2 + d) with z21= zd + we find z1 = − . RR12++2 d

Now we can choose RRd12,, and compute modes in the resonator. However, we have to make sure that those modes exist. In our calculations above we have eliminated the Rayleigh length z0, a real and positive quantity. Hence, 2 we have to check that the so-called stability condition z0 > 0 is fulfilled!

22dRdR( 1+)( 2 + dR)( 12 ++ R d)  z0= Rz 11 −=− z 1 2 >0 (RR12++2 d)

2 The denominator (RR12++2 d) is always positive we need to fulfill

 −dRdR( 1 +)( 2 + dR)( 12 ++> R d) 0

If we introduce the so-called resonator parameters dd  gg12=+=+1,  1 RR 12  We can re-express the stability condition as (1− gg) RR −dd(R++ d)( R d)(RRd++) = ggRR 12 12 1 2 12 1212 d 2 =−>gg1 2(1 gg 12)( R 12R ) 0.

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 90

This inequality is fulfilled for (1) and or for (2) and which results in the following condition for the placement and radii of the mirrors for a stable cavity

which is equivalent to

This final form of the stability condition can be visualized: The range of stability of a resonator lies between coordinate axes and hyperbolas:

Resonator stability diagram. A spherical-mirror resonator is stable if the parameters and lie in the unshaded regions bounded by the lines and , and the hyperbola . is negative for a concave mirror and positive for a convex mirror. Various special configurations are indicated by letters. All symmetrical resonators lie along the line .

Examples for a stable and an unstable resonator: A) stable Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 91

B) R1, R 2< 0; R 1 < d , R 2 > d ;≠ g1 ≤ 0, 0 ≤≤ g2 1; ≠≠ gg 12 ≤ 0 unstable

Remark connection to He-Ne-Labwork script (and Wikipedia): In Labworks (he_ne_laser.pdf) a slightly different convention is used: − Direction of z-axis reversed for the two mirrors

− beam hits concave mirror  radius Rii (= 1,2)> 0.

− beam hits convex mirror  radius Rii (= 1,2)< 0.

− z1,2 is the distance of mirror '1','2' to the focus!

− d is the distance between the two mirrors  zzd21+ = Examples:

A) Rz()1 < 0, Rz()02 > RR12<>0, 0; zz12<>0, 0

B) Rz()0,()012>> Rz ; RR12,0> ; zz12>>0, 0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 92

Then the conditions for stability are:

R11= Rz( ), R 2= Rz ( 2 ) With analog calculation as above we find with for the resonator parameters dd  gg12= 1,− =  1− RR 12  the same stability condition 2 gg12(1−> gg 12)( RR 12) 0, 0<

2.8.4.2 Higher order resonator modes For the derivation of the above stability condition we needed the wave fronts only. Hence, there may exist other modes with same wave fronts but different intensity distribution. For the fundamental mode we have: w xy22++  kxy22  v( xyz , , )=A 0 exp−ϕ exp exp[ (z )] . G 2 ii  wz() w () z  2 Rz ()  higher order modes: ( xy, -dependence of phase is the same)

w0 xy ulm, (, xyz ,)= AGlm, l22 Gm × wz() wz() wz ()

kx2 + y2 exp iexp( iikz) exp[ (lm+ +ϕ 1)().z ]  2Rz () x2 GHll()x= ()exp x − 2

The functions Gl are given by the so-called Hermite polynomials:

Hl ()ξ ( HH01=1, = 2ξ and Hl+−11=2ξ H ll− 2 lH ). Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 93

Several low-order Hermite-Gaussian fuctions: (a) ; (b) ; (c) ; and (d) .

Intensity distributions of several low-order Hermite-Gaussian beams in the transverse plane. The order is indicated in each case.

2.9 Dispersion of pulses in homogeneous isotropic media 2.9.1 Pulses with finite transverse width (pulsed beams) In the previous chapters we have treated the propagation of monochromatic beams, where the frequency was fix and therefore the wavenumber was constant as well. This is the typical situation when we deal with continuous wave (cw) lasers. However, for many applications (spectroscopy, nonlinear optics, telecommunication, material processing) we need to consider the propagation of pulses. In this situation, we have a typical envelope length of . Let us compute the spectrum of the (Gaussian) pulse:

 spectral width: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 94

15 -1 − center frequency of visible light: ω0 =2 πν 4 ⋅ 10 s

 optical cycle: T =2 πω≈ / 1.6 fs s 0 Hence, we have the following order of magnitudes:

ω << ω → ω − ω = ω << ω s 00 0 In this situation it can be helpful to replace the complicated frequency dependence (dispersion relation) of the wave vector k()ω or the wave number k()ω by a Taylor expansion at the central frequency ω=ω0 .

In most cases, a parabolic (or cubic) approximation of the frequency dependence in the dispersion relation will be sufficient:

2 ∂∂kk1 2 kk()ω ≈( ω00) +( ω−ω) +2 ( ω−ω 0) +... ∂ω ω 2 ∂ω 0 ω0

The three expansion coefficients and their physical significance The following terminology for the individual expansion coefficients is commonly used in the literature. It associates the physics, which is inherited in the dispersion relation, with the three parameters of the Taylor expansion.

A) Phase velocity vPh 1 k n(ω ) kk(ω=) , → =0 = 0 00vcω Ph 0

 velocity of the phase front for the light at the central frequency ω=ω0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 95

B) Group velocity vg ∂k 1 = ∂ω ω v 0 g  group velocity is the velocity of the center of the pulse (see below) ω∂11kn ∂ kn()ω = () ω → = =n() ω00 +ω cv∂ω ωω c ∂ω g 00 c cn()ω vv= = = 0 g PH ∂n nn()ωω00 () n()ω +ω gg 00∂ω ω0 ∂n nn()ω = () ω +ω  group index g 0 00∂ω ω0 normal dispersion: ∂n/0 ∂ω > n > nv , < v g g PH anomalous dispersion: ∂n/0 ∂ω < n < nv , > v g g PH

C) Group velocity dispersion (GVD) or simply dispersion Dω ∂2k = D ∂ω2 ω ω0  GVD changes pulse shape upon propagation (see below) ∂∂2k 1 DD= = =  ω ∂ω2 ∂ωv ω0 g

∂ 11∂v D = = − g ∂ωvv2 ∂ω gg ∂v →>D 00 g < ∂ω ∂v → ∂ω Alternatively in telecommunication one often uses ∂π12 D = = − cD λω∂λv λ2 g

Let us now discuss the propagation of pulsed beams. We start with the scalar Helmholtz equation, with the full dispersion (no Taylor expansion yet): Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 96

ω2 ∆ uu(,rrωω )+ε (ω ) (, )= 0 c2 In contrast to monochromatic beam propagation, we now have for each frequency ω one Fourier component of the optical field: ω2 dispersion relation: k 2 ()ωω= ε () c2 Hence, we need to consider the propagation of the Fourier spectrum (Fourier transform in space and time): αβωω = αβ γ α β ω U(,, ;) zU0 (,, )expi ( ,, ) z with γ( α,, βω) =k 2 ()ω −α 22−β

The initial spectrum at z = 0 is U0 (,,αβω ) 1 ∞ U (,,)α β ω = u( x , y , t )exp −( α x +β y −ω t) dxdydt 003 ∫∫∫−∞  (2p) i Let us further assume that the Fresnel (paraxial) approximation is justified (k 2 ()ω >> α 22 + β ) α22 +β αβω ≈ αβ ω ω − U(,, ;) zU0 (,,)expiikz( ) exp z  2k (ω)  We see that propagation of pulsed beams in Fresnel approximation in Fourier space is described by the following propagation function (transfer function): α22 +β Hz(,,αβωω; )= expk ( ) zz exp − F ii 2k (ω) Now let us consider the Taylor expansion of k()ω from above. If the pulse is not too short, we can replace the wave number k()ω by

2 ∂∂kk1 2 kk()ω ≈( ω00) +( ω−ω) +2 ( ω−ω 0) +.. ∂ω ω 2 ∂ω 0 ω0

22 Moreover, in the second term exp[−i ( α +β )zk / {2 ( ω )}] of the transfer function (which is already small due to paraxiality) we can approximate the frequency dependence of the wave number by kk()ω≈ ( ω00 ) = k. This approximation is sufficiently accurate to describe the diffraction of pulsed beams which are not to short. But this approximation would break down for

T0 15 fs since for such short pulses the frequency spectrum would become very wide. By introducing this approximation, we obtain the so-called parabolic approximation: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 97

1 H(,,;)αβω z ≈ exp[ kz] exp (ω−ω ) z × FP ii0 v 0 g 22 D 2 α +β ×−ω−ω exp ii( 0 ) z exp z 22k0

 ω 12 11 22 =exp[ kz0 ] exp  z+ Dω −( α +β ) iivk22  g 0 

with ω=ω−ω0 Based on the last line of the above equation we can introduce a new variant of the propagation function, where the frequency argument is replaced by the frequency difference ω from the center frequency ω0 H(,,αβω ;) z ≈ exp[ kzH] (,,αβω ;) z FP i 0 FP

The new transfer function HzFP (,,αβω ;) is the propagation function for the slowly varying envelope vxyzt(, , ,): ∞ uxyzt(,,,)= exp[ kz00] U(,,αβω ) HFP (,,;) αβω z × i ∫∫∫−∞ ×exp i( αx +β y −ω t) ddd α β ω

∞ uxyzt(,,,)= exp( kz −ω t) U(,,αβω ) H (,,;) αβω z × i 000 ∫∫∫−∞ FP

× α +β −ω α β ω exp i( x y t) ddd

Illustration of the slowly varying envelope in the spectral domain uxyzt(, , ,)= vxyzt (, , ,)exp( kz−ω t) i 00 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 98

u(t) v(t)

t |u(ω)| |u(ω)|

ω ω ω 0

∞ vxyzt(,,,)= U (,, αβω ) H (,,;)exp αβω z α x +β y −ω t d α d β d ω ∫∫∫−∞ 0 FP i( )

In order to complete the formalism, we also need to define the initial spectrum of the slowly varying envelope = −ω u00(, xyt ,) v (, xyt ,)exp( i 0 t) 1 ∞  V (,,)α β ω = v( x , y , t )exp −( α x +β y −ω t) dxdydt 003 ∫∫∫−∞  (2p) i Thus, the propagation of the slowly varying envelope is given by: ∞  vxyzt(,,,)(,,)= V0 α β ω HzFP (,,α βω ;)exp (αx +βy −ω t) ddd α β ω ∫∫∫−∞ i

Co-moving reference frame The next step is to introduce a co-moving reference frame with ω H(,,αβω ;) z = exp zH (,,αβω ;) z FP i v FP g ∞ vxyzt(,,,)(,,)= V α β ω Hz (,,αβω;)exp αx +β y −ω t −z vddd α β ω ∫∫∫−∞ 0 FP i( ( g ))

Hz (,,αβω ;) The last line above involves the propagation function FP , which is the propagation function for the slowly varying envelope in the co-moving frame of the pulse: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 99

z t=t − v g This frame is called co-moving because Hz (,,αβω ;) is now purely quadratic FP in ω, i.e., the pulse does not “move” anymore. In contrast, the linear ω- dependence in Fourier space had given a shift in the time domain. Thus, the slowly varying envelope in the co-moving frame evolves as:

22 ∞ z 2 α +β vxyz(,,,)τ = V (,,)exp αβω Dω − × ∫∫∫−∞ 0 i  2 k0 ×exp{  αx +β y −ωτ } ddd α β ω i  The optical field u reads in the co-moving frame as: ω t= t −ω= t −−ωt0 uxyz(, ,,) vxyz (, ,,)expii( kz00 t) vxyz(, ,,)exp  kz0 z 0 vg

Propagation equation in real space Finally, let us derive the propagation equation for the slowly varying envelope in the co-moving frame. We start from the transfer function

22 z 2 α +β  αβω = αβ ω ω − V(,, ;) zV0 (,, )expi D 2 k0 Then we take the spatial derivative of the transfer function along the propagation direction z

22 ∂Vz(,, α β ω ;) 12 α +β =−D ω − Vz(,, αβω ;) i ∂zk2 0 As before in the case of monochromatic beams, we use Fourier back-transformation to get the differential equation in the time-position domain ∂vxyz(, ,,)τ D∂2 1 − vxyz(, ,,)τ+ ∆(2) vxyz (, ,,) t= 0 i 2  ∂zk22 ∂τ 0 This is the scalar paraxial equation for propagation of so-called pulsed beams.

Comment: Extension to inhomogeneous media By using the slowly varying envelope approximation, it is possible to generalize the scalar paraxial equation also for inhomogeneous media, when a weak index contrast is assumed. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 100

∂∂D2 1 kk22()r − vxyz(, ,,)τ− vxyz (, ,,)τ+ ∆(2) vxyz (, ,,) t+ 00vxyz(, ,,)τ= 0 i 2   ∂z22∂τ kk00 2

with kk00≈ (r) For D = 0 the equation would be reduced to simple diffraction, as in the beam propagation scheme which was derived earlier. 2.9.2 Infinite transverse extension - pulse propagation

Diffraction plays no role for sufficiently small propagation lengths zL B . For broad beams, the diffraction length LB can be rather large and we can assume α=β≈0 , corresponding to the assumption that we have a single plane wave propagating in z-direction. Later in the lecture series we will see that this case is also valid for mode propagation in waveguides, as e.g. optical telecommunication fibers.

Description in frequency domain = −ω 1) initial condition: ut00() vt ()exp( i 0 t)

2) initial spectrum: VU00()ω= () ω D 3) propagation of the spectrum: V(ω ; zV )= ( ω )exp z ω2 0 i 2 4) back-transformation to τ leads to the following evolution of the slowly varying envelope in the co-moving frame: ∞ D vz(,)τ = V ()exp ω z ω2 exp[ − ωτ]dω ∫ −∞ 0 ii2

Description in time domain A) In time domain it is possible to describe pulse propagation by means of a response function: D 2 τ2 FT-1 of Hz (ω= ; ) exp z ω2  hz (τ ; )= exp − P i P i 2 −ipDz2 Dz and the evolution is described by the convolution integral

∞ vz(,)τ = h (τ−t′ ;) zv ( t ′′ ) d t ∫ −∞ P 0 B) The evolution equation for slowly varying envelope in the co-moving frame reads ∂vz(,)τ D∂2 − vz(,)τ= 0 i ∂z 2 ∂τ2 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 101

Analogy of diffraction and dispersion DIFFRACTION DISPERSION ∂ 1 ∂ D ∂2 vz(,xy, )+ ∆(2) vz (xy, , )0=  vz(,)τ − vz (,)τ = 0 i ∂z 2k i ∂z 2 ∂τ2 (,)αβ  ω (,xy )  τ ∂ ∇  ∂τ 1  −D but D  0 can vary k0

In the following we will study two typical examples of pulse propagation. 2.9.3 Example 1: Gaussian pulse without chirp use analogies to spatial diffraction

1. Initial pulse shape pulse without chirp  corresponds to Gaussian pulse in the waist (focus) with flat phase

V0 e

t 2 τ2 ut( )= A exp − exp −ωt  vA(τ )= exp − 002 ( i 0) 002 T0 T0

2. Initial pulse spectrum 22 TT00ω VA00(ω= ) exp − 2 p 4

22  spectral width: ω=s 4/T0 Use results from propagation of Gaussian beams: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 102

2 k 2 1 T0 z0 describes Gaussian pulse zw00=   z0 = −  0 2 2 D Hence anomalious GVD is equivalent to 'normal' diffraction.

Dispersion length: LzD = 2 0

3. Evolution of the amplitude T ττ22   vz( ,τ )= A 0 exp− exp −i expϕ (z )  0 2  [i ] Tz() Tz ()  2 D Rz ()  with

2 1 z Az()= A004 2 ,T () z= T 1 +  z z 1+ ( z ) 0 0

A2 ( zT ) ( z )= const. 'Phase curvature' is not fitting to the description of pulses  introduction of new parameter Chirp Remember: The phase of a Gaussian beam Φ(x, y,z ) has the following shape: ∂∂22 k 22+ Φ=(,xyz ,) ∂∂x y Rz() For monochromatic fields the temporal dynamics of the phase is: ∂Φ()τ Φ()τ = −ωτ → − = ω ∂τ  arbitrary time dependence of phase ∂Φ()τ ∂2 Φ()τ ∂ω ()τ − =ω()τ and −=≠0  chirp ∂τ ∂τ2 ∂τ The chirp of a pulse describes the variation of the temporal frequency of the electric field in the pulse.  parabolic approximation  'chirp' constant  dimensionless chirp parameter (often just chirp) T 2 ∂Φ2 ()τ C = − 0 2 ∂τ2 integration leads to: ∂Φ()τ ττ2 − =ω()τ=ω+00 2C 22 , −Φ ()τ=ω+τ C ∂τ T00T Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 103

C >→0 up-chirp C <→0 down-chirp

leading trailing front tail

phase curvature Rz()  Chirp Cz() Complete phase: zzττ22  Φ()τ = −ω τ + −≡ −ω τ + − Cz() 00v2 DR () z v T 2 gg  0 Tz2  Cz()=00 = − 2DRz () Rz () with 22 zz+ 0 zz0 z Rz()=  Cz()=−=−22 + z2 z zz0 z 1+ 2 0 ( z ) 0

2 1 z0 T0  C(0)= 0, C( z00) =- sgn( z) , Cz (®¥ ) =- with z0 = − 2 z 2D

Attention: Chirp depends on sign of z0 and hence on D.

Complete field: T tt22 uz(,)t = A 0 exp− exp − Cz () expϕ ()expz kz−ω t 0 22i[ ii] ( 00) Tz() Tz () T0 Dynamics of a pulse is equivalent to that of a beam. T 2 important parameter  dispersion parameter z = − 0 0 2D

1) zz<< 0 : no effect

2) zz 0 : similar to beam diffraction

3) zz 0 : asymptotic dependence z 2 D T() z≈→ T T ()/ z z = T / z == const. 00zT0 0 0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 104

Gaussian pulse spreading as a function of distance . For large distances, the width increases at a rate , which is inversely proportional to the initial width .

is only important if initial pulse is chirped, since otherwise the same quadratic dependence is observed, independent from the sign of . 2.9.4 Example 2: Chirped Gaussian pulse Important because of: − short pulse lasers → chirped pulses − Chirp is introduced on purpose, for subsequent pulse compression − analogy to curved phase  focusing − chirped pulse amplification (CPA)  Petawatt lasers

1. Input pulse shape

C0 – initial chirp

2. Input pulse spectrum

 spectral width:

 spectral width of chirped pulse is larger than that of unchirped pulse only for transform limited pulses

Aim: calculation of pulse width and chirp in dependence on for given initial conditions Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 105

Gaussian beam  q -parameter  similar to Gaussian pulse Use analogy: however it is limited to homogeneous space qz( )= q (0) + z . Remember beams: 11 2 = + . q() z R () zi kw2 () z 1 1 2DC () z →−22 → → k, wz () T (), z 2 D Rz() T0 1 2DC () z 2 D  = − 22i qz() T0 T() z 12DT2 =Cz() − 0 22i (*) qz() T0  T() z

Important: T0 is the pulse width at z = 0 , which is not necessarily in the 'focus' or waist. hence at z = 0 : 12D  =C − with CC= (0) 2 [ 0 i] 0 qT(0) 0 Idea: 12D 1 a) qz( )= q (0) + z and =C − insert into  2 [ 0 i] qT(0) 0 qz() 12DT2 =Cz() − 0 . b) 22i qz() T0  T() z set a) and b) equal  Tz(), Cz() generally: 2 equations for C00, T ,, zCz (),() T z  3 values predetermined here: zd=

1) Determination of q parameter at input T 2 (C + ) = 0 0 i q(0) 2 2D (1+ C0 )

2) Evolution of q parameter T 2 (C + ) 1 q( d )= q (0) += d 0 0 i +=d 2Dd 1+ C2 + C T 22 + T 22( 0) 00i 0 2D (1++C00) 21 DC( )

3) Inversion of general equation (*) for q(d) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 106

12DT2 =Cd() − 0 22i qd() T0  T() d TT22() d qd()= 0 CdT()22 () d+ T 24+ 4i 0 2DC () dT () d T0 4) Set two equations equal

21Dd++ C22 C T +T 2 22 2 + 2 ( 0) 00 i 0 TTdCdTd0 () ( ) () T0  = i + 2  244 +  2D(1 C0 ) 2D C () dT () d T0 

++22 24 21Dd( C0) C 00 T CdTT() () d a) real part = 0 ++2 24 4 (1 C00)  C() dT () d T 1 TT22() d b) imaginary part = 0 ++2 24 4 (1C00)  C () dT () d T

If we predetermine 3 parameters (C00, T , Cd ()), we can determine the other 2 unknown parameters (dTd,()).

Important case: Where is the pulse compressed to the smallest length? given: CT00, & in the focus: Cd()= 0 unknown: dTd, () ++22 = a) real part must be zero 21Dd( C0) C 00 T 0 1 T 2CC1  =−=0 00− d 22sg(n DL ) D 22D(1++CC00) (1 )

T 2 2 = 0 b) Td() 2 (1+ C0 )

Resulting properties 1) A pulse can be compressed when the product of initial chirp and

dispersion is negative → CD0 < 0. 2) The possible compression increases with initial chirp.

Physical interpretation If e.g. C < 0 and Dv>→∂∂ω<→0 /0 'red' is faster than 'blue' 0 g Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 107

Compression of a chirped pulse in a medium with normal dispersion. The low frequency (marked R for red) occurs after the high frequency (marked B for blue) in the initial pulse, but it catches up since it travels faster. Upon further propagation, the pulse spreads again as the R component arrives earlier than the B component.

1) First the 'red tail' of the pulse catches up with the 'blue front' until (waist), i.e. the pulse is compressed. At this propagation distance the pulse has no remaining chirp. 2) Then and red is in front. Subsequently the 'red front' is faster than the 'blue tail', i.e. the pulse gets wider.

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 108

3. Diffraction theory 3.1 Interaction with plane masks In this chapter we will use our knowledge on beam propagation to analyze diffraction effects. In particular, we will treat the interaction of light with thin and plane masks/apertures. Therefore we would like to understand how a given transversally localized field distribution propagates in a half-space. There are different approximations commonly used to describe light propagation behind an amplitude mask: A) If we use geometrical optics we get a simple shadow. B) We can use scalar diffraction theory with approximated interaction, i.e., a so-called aperture is described by a complex transfer function txy(, ) with txy(, )= 0 for xy, > a (aperture)

Here we consider the description based on scalar diffraction theory. Then we can split our diffraction problem into three sequential processes: i) propagation from light source to aperture  not important, generally plane wave (no diffraction) ii) multiply field distribution of illuminating wave by transfer function u(, xyz , )= txyu (, ) (, xyz , ) +−AA iii) propagation of modified field distribution behind the aperture through homogeneous space ∞ uxyz(,,)= H( αβ ,; z − z) U (,; αβ z )exp( α x +β y) d α d β ∫∫ AA+ i −∞ or ∞ u(, x y ,) z= h( x −−− x′ , y y ′ , z z) u ( x ′′ , y , z ) dx ′′ dy ∫∫ AA+ −∞

1 −1 with hH= 2 FT [ ] (2π) In the following we will use the notation z= zz − . According to our choice of BA the propagation function H , resp. h , we can compute this propagation either exactly or in a paraxial approximation (Fresnel). In the following, we will see that a further approximation for very large z is possible, the so-called B Fraunhofer approximation. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 109

3.2 Propagation using different approximations 3.2.1 The general case - small aperture We know from before that for arbitrary fields (arbitrary wide angular spectrum) we have to use the general propagation function Hz(αβ, ;) = exp( γαβ ( , ) z) where γ2 =k 2() ω −α 22 −β . BBi Then we have no constraints with respect to the spatial frequencies αβ, . We get homogeneous and evanescent waves and can treat arbitrary small structures in the aperture by: ∞ uxyz(,,)= U (,) αβ H( αβ ,; z) exp ( α x +β y) d α d β ∫∫−∞ + B i

where U++(,)αβ =FT[ u (, xy )]

Derivation of the response function We start from the Weyl-representation of a spherical wave: ∞ 11i exp(iikr) = ∫∫ exp (αx +β y +γ z) d α d β r 2p−∞ γ Now we can compute the response function h , which we did not do in the previous chapter, where we computed only hF (Fresnel approximation). The following trick shows that ∞ ∂ 11 1−1 exp(ikr )=−∫ expi(x[ α +β y +γ z)]d α d β=−FT[H] =− 2p h ∂zr22p−∞ p and therefore 11∂  h( x,, y z) = − exp( kr) with r= xyz2 ++ 22. 2p∂zri The resulting expression in position space for the propagation of monochromatic beams is also called 'Rayleigh-formula': ∞ u(, x y ,) z= h( x −− x′ , y y ′ , z) u ( x ′′ , y , z ) dx ′′ dy ∫∫ BA+ −∞

3.2.2 Fresnel approximation (paraxial approximation) From the previous chapter we know that we can apply the Fresnel approximation if α22 + β << k 2 which is valid for a limited angular spectrum, and therefore a large size of the structures inside the aperture. Then

z 22 H(α, β ; z) = exp( kz ) exp −B ( α +β ) FBii B2k Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 110

k 22 h( x, y , z ) =−+i exp( kz ) exp ( x y ) FBλzziiB 2 BB 3.2.3 Paraxial Fraunhofer approximation (far field approximation) A further simplification of the beam propagation is possible for many diffraction problems. Let us assume a narrow angular spectrum α22 + β << k 2 and the additional condition for the so-called Fresnel number N F aa N N 0.1 with N = F F λ z B where a is the (largest) size of the aperture (like the "beam width"). Obviously, a larger aperture needs a larger distance z to fulfill N 0.1. B F N Hence the approximation, which we derive in the following, is only valid in the so-called 'far field', which means far away from the aperture.

To understand the influence of this new condition on the Fresnel number, we have a look at beam propagation in paraxial approximation: ∞ u(, x y , z )= h ( x −− x′ , y y ′ ; z ) u ( x ′′ , y ) dx ′′ dy FFBB∫∫−∞ +

∞  i k 22 =− exp( kz) u+ ( x′′ , y )exp( x− x′′′′) +−( y y) dx dy iiB ∫∫−∞  λzzBB2 In this situation it is easier to treat the beam propagation in position space, because

u+−(, xy )= txyu (, ) (, xy ), and txy(, )= 0 for xy, > a (aperture)

 u+ (, xy )= 0 for xy, > a This means that we need to integrate only over the aperture in the above integral and not over an infinite plane:

a  i k 22 u( x , y , z )=− exp( kz) u+ ( x′′ , y )exp( x− x′′′′) +−( y y) dx dy F BBii∫∫−a  λzzBB2 Now, let us have a closer look at the integral: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 111

a k 2 2 u+ ( x′′ , y )exp( x− x′′′′) +−( y y) dx dy ∫∫−a i  2zB

a k 2 22 2 = u+ ( x′′ , y )exp x−2 xx′′ + x +− y2 yy ′′ + y dx′′ dy ∫∫−a i  2zB

a k 2 2 kx ky k 2 2 =u+ ( xy′′ , )exp x ++ y exp xy′′expx′′+ y  dx′′ dy ∫∫−a i −ii   22zB zzBB  zB So far, nothing happened, we just sorted the factors differently. But here comes the trick: Because of the integration range, we have xy′′, < a and therefore k ka2  xy′′22+<=π2 N  F 2zzBB k  for N 0.1  expxy′′22+≈1 F N i  2zB This means that we can neglect the quadratic phase term in xy′′, and we get for the field far from the object, the so-called far field:  i k 22 u(,,) x y z =−+ exp( kzB ) exp (x y ) FR B λzzii2 B B ∞ kx ky ×+u+ ( x′′ , y )expx ′ y ′ dx ′′ dy ∫∫−∞ −i zzBB 2 (2p) x y k 22 =−+exp( kz) U + (k ,k )exp (x y ) iiλzzB zz i2 B B B B This is the far-field in paraxial Fraunhofer approximation. Surprisingly, the intensity distribution of the far field in position space is just given by the Fourier transform of the field distribution at the aperture

2 1 xy I(, xyz , ) 2 U+ ( k , k ; z ) FR B (λz ) zzA B BB Interpretation

For a plane in the far field at zz= B in each point xy, only one angular frequency (α=kx/; zBB β= ky / z ) with spectral amplitude U+ (/,/) kx zBB ky z contributes to the field distribution. This is in contrast to the previously considered cases, where all angular frequencies contributed to the response in a single position point. In summary, we have shown that in (paraxial) Fraunhofer approximation the propagated field, or diffraction pattern, is very simple to calculate. We just Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 112 need to Fourier transform the field at the aperture. In order to apply this approximation we have to check that: A) α22 + β << k 2  smallest features ∆xy, ∆ >> λ  narrow angular spectrum (paraxiality) a2 a2 B) N = <<1  largest feature a determines z >> λ  far field F λzB B Example: ∆∆=λxy, 10 , a = 100 λλ=µ , 1 µ →z >>104 λ≈ 1 cm B 3.2.4 Non-paraxial Fraunhofer approximation The concept that the angular components of the input spectrum separate in the far field due to diffraction works also beyond the paraxial approximation. If we have arbitrary angular frequencies in our spectrum, all α222 +β ≤k contribute to the far field distribution. Evanescent waves decay for kz>>1 → z >> λ. BB

aa1 z0 N N 0.1 with NF = = F λπ zzBBfor Gaussian beams

2 (2p) z u (,xyz , )= − B FR non−paraxiaλ B i 22 222 2 λ xyz++ xyz ++ BB kx ky 22 2 ×U+ ( , ;z )exp k xy++z 22 2 22 2A (i B ) xy++zyx ++ z B B 3.3 Fraunhofer diffraction at plane masks (paraxial) Let us now plug things together and investigate examples of diffraction patterns induced by plane masks in (paraxial) Fraunhofer approximation.

Let us consider an incident plane wave, with a wave vector perpendicular

(kkxy= = 0) to the mask t = u− ( xyz , ,A ) A exp(i kzzA) or, more general, inclined with a certain angle = ++ u− ( xyz , ,A ) A exp i( kxx ky y kz zA) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 113

For inclined incidence of the excitation the field behind the mask is given by: = = ++ u+−(, xyz ,A ) u (, xyz ,A )(,) txy A expi( kxx ky y kz zA) txy(, ) Form the previous chapter we know that the diffraction pattern in the far field in paraxial Fraunhofer approximation is given as:

2 2 1 xy Ixyz(, , ) uxyz (, , ) 2 U+ (,) k k BB(λz ) zz B BB Hence, the diffraction pattern is proportional to the spectrum of the field behind the mask at xy α=kk, β= . zz BB This spectrum is calculated by the Fourier transform of the field as: xy Uk(,) k + zz BB A∞ xy   = exp( kzA z) t ( x′′ , y )exp − k −− kx  x′ k − ky  y′′′ dx dy p 2 i∫∫ iizz (2 ) −∞ BB   xy =Aexp( kzTkkkkzA) −−x, y i zz BB Hence, the intensity distribution of the diffraction pattern is given as:

2 1 xy Ixyz(, , )− 2 Tkkkk−−xy, B (λz ) zz B BB This is the absolute square of the Fourier transform of the aperture function. In paraxial approximation an inclination of the illuminating plane wave just shifts the pattern transversely.

Examples A) Rectangular aperture illuminated by normal plane wave 1,for x ay b txy(, )=  0 elsewhere

22xy   I( x , y , z ) sinc ka  sinc kb  B zz BB   Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 114

Fraunhofer diffraction pattern from a rectangular aperture. The central lobe of the pattern has half-angular widths and .

B) Circular aperture (pinhole) illuminated by normal plane wave

 Airy disk

The Fraunhofer diffraction pattern from a circular aperture produces the Airy pattern with the radius of the central disk subtending an angle .

The first zero of the Bessel function (size of Airy disk):

with

So-called angle of aperture: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 115

C) One-dimensional periodic structure (grating) illuminated by normal plane wave For periodic arrangements of slits we can gain deeper insight in the structure of the diffraction pattern. Let us assume a periodic slit aperture with: period b and a size of each slit 2a :

Then, we can express the mask function t as: N −1 tx() for x a = − = s  t() x∑ t1 ( x nb ) with tx1()  n=0 0 elsewhere The Fourier transform of the mask is given as xxN −1 ∞   ′−−′′ T k− ∑∫ t1( x nb )expi k x  dx zzn=0 BB−∞   With the new variable x′′−= nb X we can simplify further: x N −1 a  xx   T k− t( X′′ )exp −− k Xexp k nb dX ′  ∑∫ S ii   zn=0 zz B  −a BB  

xxN −1  x  T k− T k exp − k nb S ∑ i  zzn=0 z BB  B 

We see that the Fourier transform TS of the elementary slit tS appears. The second factor has its origin in the periodic arrangement. With some math we can identify this second expression as a geometrical series and perform the summation by using the following formula for the definition of the sinc-funtion: N −1 sin(N δ ) exp −δn = 2 ∑ ( i ) δ = sin( ) n 0 2 Thus we finally write:

kx xxsin( Nb2 z ) Tk T k B S kx zzsin( 2 z b) BBB

For the particular case of a simple grating of slit apertures with txs ()= 1 we have Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 116

xx   Tk1 = sinc  k a  zzBB  

2 kx 2 x sin ( Nb2 z ) and therefore I sinc ka B 2 kx z sin ( 2 z b) B B

We find three important parameters for the diffraction pattern of a grating: − Global width of diffraction pattern  first zero of slit function T S x λz kaS = π  x = B z S 2a B

The width of the total far-field diffraction pattern xS (largest length scale in the pattern) is determined by the size a of the individual slit (smallest length scale in the mask). − Position of local maxima of diffraction pattern  maxima of grid function sin2 Nbkx ( 2 zB ) kx λz max  P bn= π  xn= B sin2 kxb 2 z p b ( 2 zB ) B These are the so-called diffraction orders, which are determined exclusively by the grating period. − Width of local maxima  zero-points of grid function Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 117



The width of a maximum in the far-field diffraction patter (smallest length scale in the pattern) is determined by which is the total size of the mask (largest length scale of the mask).

These observations are consistent with the general property of the Fourier- transform: small scales in position space give rise to a broad angular spectrum and vice versa. 3.4 Remarks on Fresnel diffraction Fresnel number diffraction length from Gaussian beams

− ( large, small, )  shadow − ( )  Fraunhofer  FT of aperture − ( )  Fresnel diffraction

Fresnel diffraction from a slit of width . (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffraction beam. (b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers and . The dashed area represents the geometrical shadow of the slit. The dashed lines at represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 118

with the edges of the geometrical shadow, the Fresnel number 2 NaF =/ λ= d 0.5 . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 119

4. Fourier optics - optical filtering From previous chapters we know how to propagate the optical field through homogeneous space, and we also know the transfer function of a thin lens. Thus, we have all tools at hand to describe optical imaging. While detailed designs of high resolution optical systems have to consider non-paraxial effects, usually the paraxial approximation is sufficient to obtain a principle understanding of optical systems. Hence we use the paraxial approximation here. Many imaging systems exploit the appearance of the Fourier transform of the original object in the so-called Fourier plane of the system in order to manipulate the angular spectrum of the object in this plane. Accordingly this field of optical science is called Fourier optics. We will see in the following that with the right setup of our imaging system we can generate the Fourier transform of the object on a much shorter distance than by far field diffraction in the Fraunhofer approximation. The general idea of Fourier optics is the following: 1) An imaging system generates the Fourier transform of the object in Fourier plane. 2) A spatial filter (e.g. an aperture) in the Fourier plane manipulates the field. 3) Another imaging system performs the Fourier back-transform and hence results in a manipulated image. Mathematical concept: • propagation in free space  calculated in Fourier domain • interaction with lens or filter  calculated in position space 4.1 Imaging of arbitrary optical field with thin lens 4.1.1 Transfer function of a thin lens A thin lens changes only the phase of the optical field, since due to its infinitesimal thickness, no diffraction occurs. By definition, it transforms a spherical wave into a plane wave. If we write down this definition in paraxial approximation we get k −iiexp( kf ) exp x22+= y t( x, y) − exp( kf ) ii( ) L i λλff2 f ((((((((((((  ( (  ( ( sphericaλ wave pλane wave

And therefore the response function for a thin lens is given as (see chapter 2.8.3): Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 120

k t( xy,) =−+ exp x22 y L i ( ) 2 f By Fourier transforming the response function we find consequently the transfer function in the Fourier domain as λff α β =− α22 +β T ( , ) 2 exp ( ) L ii(2p) 2k

4.1.2 Optical imaging using the 2f-setup Let us now consider optical imaging. We place our object in the first focus of a thin lens, with a field distribution u0 (, xy ), and follow the usual recipe for light propagation.

A) Spectrum in object plane

U00(αβ , ) = FT[ u ( xy , )]

B) Propagation from object to lens (lens positioned at distance f) Uf(,;αβ ) =H (αβ,;f )U (αβ,) − F 0

i 2 2 Uf− (,;)αββα= exp( kf ) exp −( α +β ) f U0 (,) i  2k  C) Interaction with lens (multiplication in position space or convolution in Fourier domain) u(, xyf , )= t( xyu ,) (, xyf , ) +−L Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 121

U (,;αβf ) =T (αβ, ) ∗ Uf(,; αβ ) +−L λff∞ 2 2 = − exp( kf ) exp (α−α′′) +( β−β) ⋅ 2 ∫∫−∞  i(2p) i i 2k    ⋅ −i α′′22 +β α′′ β α ′ β ′ exp( ) fU0 (,) d d  2k 

D) Propagation from lens to image plane U(,;2)αβ f =Hf(αβ, ; ) Uf(,;αβ ) F +

λff∞ 22 U(,;2)α β f =− exp(2 kf ) exp (α−α′′) +( β−β) ⋅ 2 ∫∫−∞  ii(2p)  i2k   ⋅ −i α′′22 +β −i α22 +β α′′ β α ′ β ′ exp ( ) fexp ( ) f U0 ( ,) dd 2k  2k 

 Quadratic terms −i ( α′′22 +β ) f and −i ( α22 +β ) f in the exponent 2k 2k f 22 cancel with quadratic terms from (α−α′′) +( β−β) and only the mixed i 2k  terms remain.

λff∞  U(,;2)α β f = − exp2( kf) U (,)expα′′ β −( αα′′′′ + ββ) d α d β 2 ∫∫−∞ 0  ii(2p)  ik

λf ff =− 2 exp( 2 kf) u0 − α−, β ii(2p) kk

We see that the spectrum in the image plane is given by the optical field in the object plane. E) Fourier back transform in image plane uxy(,,2)FT f=−1 [ U (,;2) αβ f ]

λf ∞ ff =− − α − β α +β α β ii2 exp( 2 kf)∫∫ u0 , exp i( x y) d d 2p −∞ kk ( )

With the coordinate transformation Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 122

we get:



The image in the second focal plane is the Fourier transform of the optical field in the object plane. This is simmilar to the far field in Fraunhofer approximation, but for . This finding allows us to perform an optical Fourier transform over shorter distances. And in the Fourier plane it is possible to manipulate the spectrum.

4.2 Optical filtering and image processing 4.2.1 The 4f-setup For image manipulation (filtering) it would be advantageous if we could perform a Fourier back-transform by means of an optical imaging setup as well. It turns out that this leads to the so-called 4f-setup: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 123

The filtering (manipulation) happens in the second focal plane (Fourier plane after 2f) by applying a transmission mask pxy(, ). In order to retrieve the filtered image we use a second lens: We know that the image in the Fourier plane is the FT of the optical field in the object plane. kk u( x , y ,2 f )= AU0  x , y ff We have to compute the imaging with the second lens after the manipulation of the spectrum in the Fourier plane. Our final goal is the transmission function Hf(αβ , ;4 ) of the complete ima- A ging system:

∞ u(− x , − y ,4 f ) = Hf(α,β;4 )U(,)α βexp ( αx +β y) dd α β ∫∫ A 0 i  −∞ Note: We will see in the following calculation that the second lens does a Fourier transform  exp i(αxy +β ) . In order to obtain a proper back transform we have to pass to mirrored coordinates x→− xy, →− y. The transmission mask pxy(, ) contains all constrains of the system (e.g. a limited aperture) and optical filtering (which we can design).

A) Field behind transmission mask kk u+ (, xy ,2) f= uxy (, ,2) f pxy (, )~ AU0  x , y pxy (, ) ff

B) Second lens  Fourier back-transform of field distribution 2 (2p) kk uxyf( , ,4 )= − exp( 2 kfU) +  x, yf ;2 iiλf ff 

Now we can make the link to the initial spectrum in the object plane U0 : Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 124

∞ k ′′ −+′′′′ u( x , y ,4 f )~∫∫ u+ ( x , y ,2 f )expi ( xx yy) dx dy −∞ f

∞    kk′ ′ ′′ k′′′′ ~U0  x , y p ( x , y )exp −+( xx yy) dx dy ∫∫ ff i f −∞    Here we do not care about the amplitudes and just write ~: To get the anticipated form we need to perform a coordinate transformation:

kk α=xy′′, β= ff Then we can write: ∞ ff uxy(,,4)~ f U0 (α , β) p ( α , β )exp  −( αx +β y) d α d β ∫∫ kk i −∞ By passing to mirrored coordinates x→− xy, →− y we get

∞ ff u(− x , − y ,4 f )~ p( α,)β U0 (α, β) exp ( αx +β y) ddαβ ∫∫ kk i −∞ Hence we can identify the transmission function of the system ff H(αβ, ;4 fp) ~ ( α , β ) A kk

Summary • Fourier amplitudes get multiplied by transmission mask • transmission mask  transfer function • coordinates of image  mirrored coordinates of object In position space we can formulate propagation through a 4f-system by using the response function h(, xy ) A ∞ u(−− x , y ,4 f ) = h ( x − x′ , y − y ′ ) u ( x ′′ , y ) dx ′′ dy ∫∫ A 0 −∞ As usual, the response function is given as the Fourier transform of the transfer function: 1 ∞ h(,) xy= H(α, β ;4 f) exp [ α x +β y] d α d β AA2 ∫∫ {i } (2p) −∞ ff From above we have H(αβ, ;4 fp) ~ ( α , β ) A kk ∞ ff h( xy , ) ~ p (αβ, )exp [αx +β y] d α d β A ∫∫ {i } −∞ kk Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 125

We introduce the coordinate transform

The response-function is proportional to the Fourier transform of the transmission mask. 4.2.2 Examples of aperture functions Example 1: The ideal image (infinite aperture) Be careful, we use paraxial approximation  limited angular spectrum

The 4-f system performs a Fourier transform followed by an inverse Fourier transform, so that the image is a perfect replica of the object (perfect only within the Fresnel approximation).

→  mirrored original

Example 2: Finite aperture

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 126

Spatial filtering. The transparencies in the object and Fourier planes have complex amplitude transmittances and . A plane wave traveling in the direction is modulated by the object transparency, Fourier transformed by the first lens, multiplied by the transmittance of the mask in the Fourier plane and inverse Fourier transformed by the second lens. As a result, the complex amplitude in the image plane is a filtered version of the original field in the object plane. The system has a transfer function .

Transmission function:

− finite aperture truncates large angular frequencies (low pass) − determines optical resolution 4.2.3 Optical resolution A finite aperture acts as a low pass filter for angular frequencies.

With we can define an upper limit for the angular frequencies which are transmitted (bandwidth of the system)



Translated to position space, the smallest transmitted structural information is given by:

A more precise definition of the optical resolution can be derived the following way: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 127

∞ kk −− ′− ′ − ′′ ′′ u( x , y ,4 f )~ ∫∫ P( x x), ( y y) u0 (,) x y dx dy −∞ ff

kD 22 ∞ J ( xx′′−+−) ( yy) 12f ~ u(, x′′ y ) dx ′ dy , ′ ∫∫ 220 −∞ kD ( xx′′−+−) ( yy) 2 f

One point of the object xy00, gives an Airy disk (pixel) in the image:

2 22 J kD (xx−+−) ( yy) 12f 0 0 −  kD 22 20f (xx−+−) ( yy 0)  We can define the limit of optical resolution: Two objects in the object plane can be independently resolved in the image plane as long as the intensity maximum of one of the objects is not closer to the other object than its first intensity minimum: kD Dr =1.22 π 2 f min 1.22λf Hence we find: Dr = min nD Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 128

Further examples for 4f filtering

Examples of object, mask, and filtered image for three spatial filters: (a) low-pass filter; (b) high-pass filter; (c) vertical pass filter. Black means the transmittance is zero and white means the transmittance is unity. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 129

5. The polarization of electromagnetic waves 5.1 Introduction We are interested in the temporal evolution of the electric field vector Er(,)t . r In the previous chapters we mostly used a scalar description, assuming linearly polarized light. However, in general one has to consider the vectorial nature, i.e. the polarization state, of the electric field vector. We know that the normal modes of homogeneous isotropic dielectric media are plane waves Er( ,tt )= E exp{i k( ω) r −ω }. If we assume propagation in z direction (k-vector points in z-direction), = divE(,) r t 0 implies that we can have two nonzero transversal field

y − EE, y components  x − and components x The orientation and shape of the area which the (real) electric field vector covers is in general an ellipse. There are two special cases: − line (linear polarization) − circle (circular polarization) 5.2 Polarization of normal modes in isotropic media 0   k = 0 propagation in z direction  k The evolution of the real electric field vector is given as Er( ,t )=ℜ E exp[ (kz −ω t )] r { i } Because the field is transversal we have two free complex field components E exp( ϕ ) xxi E =E exp ϕ with E and ϕ being real yy(i ) xy, xy,  0 Then the real electric field vector is given as Ecos(ω t − kz −ϕ ) xx Er( ,t )= E cos ω t − kz −ϕ ry( y)  0  Here, only the relative phase is interesting  δ=ϕ −ϕ yx Conclusion: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 130

Normal modes in isotropic, dispersive media are in general elliptically polarized. The field amplitudes EE, and the phase difference δ= ϕ −ϕ xy ( yx) are free parameters 5.3 Polarization states Let us have a look at different possible parameter settings: A) linear polarization  δ=n π (or E = 0 or E = 0) x y B) circular polarization  EEE= = ,δ = ±π /2 xy δ = +π /2  counterclockwise rotation

E x δ = −π /2  clockwise rotation

E x These pictures are for an observer looking contrary to the propagation direction. C) elliptic polarization  EE≠ ≠0,δ≠ n π xy 0 <δ<π  counterclockwise π<δ<2 π  clockwise Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 131

Examples

Remark A linearly polarized wave can be written as a superposition of two counter- rotating circularly polarized waves. Example: Let's observe the temporal evolution at a fixed position with .

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 132

6. Principles of optics in crystals In this chapter we will treat light propagation in anisotropic media (the worst case). Like in the isotropic case before we will seek for the normal modes, and in order to keep things simple we assume homogeneous media. 6.1 Susceptibility and dielectric tensor before: isotropy (optical properties independent of direction) now: anisotropy (optical properties depend on direction) The common reason for anisotropy in many optical media (in particular crystals) is that the polarization P depends on the direction of the electric field vector. The underlying reason is that in crystals the atoms have a periodic distribution with different symmetries in different directions. Prominent examples for anisotropic materials are: − Lithium Niobat  electro-optical material − Quartz  polarizer − liquid crystals  displays, NLO In order to keep things as simple as possible we make the following assumptions: − one frequency- (monochromatic), one angular frequency (plane wave) − no absorption From previous chapters we know that in isotropic media the normal modes are elliptically polarized, monochromatic plane waves. The question is how the normal modes of an anisotropic medium look like  ???

Before (isotropic)

Pr(,ω= )εω0χ(ω) Er (, )

Dr(,ω= )εω0ε(ω) Er (, ) In the following we will write E  E , because we assume monochromatic light and the frequency ω is just a parameter.

Now (anisotropic) 3 PEij(,rrω= ) e0 ∑ cij (ω) (),ω j=1  tensor components The linear susceptibility tensor has 3x3=9 tensor components. Direct conse- quences of this relation between polarization P and electric field E are: − PE : the polarization is not necessarily parallel to the electric field

− The tensor elements χij depend on the structure of crystal. However, we do not need to know the microscopic structure because of the different length scales involved (optics: 5⋅ 10−7m; crystal: 5⋅ 10−10 m), Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 133

but the field is influenced by the symmetries of the crystal (see next section). − In complete analogy we find for the D field: 3 ω=ε ε ω ω DEi (,rr )0 ∑ ij ( ) j (, ) j=1  Dr(,ω=εε )0 (ω ) Er (, ω ) As for the polarization we find: − DE We introduce the following notation:

− χˆ =( χij )  susceptibility tensor  − ε =( εij  )  dielectric tensor

3 − =−1 = σ  σ ω ω=ε ω σεˆˆ( ) ( ij ) inverse dielectric tensor ∑ ij( )DE j (,rr )0 i (, ) j=1 The following properties of the dielectric and inverse dielectric tensor are important:

− σεij, ij are real in the transparent region (omit ω), we have no losses (see our assumptions above) − The tensors are symmetric (hermitian), only 6 components are

independent ε=εσ=σij ji , ij ji . − It is known (see any book on linear algebra) that for such tensors a transformation to principal axes by rotation is possible (matrix is diagonalizable by orthogonal transformations).

− If we write down this for σij , it means that we are looking for directions where DE , i.e., our principal axes: 3 ε=σ λ 0Ei∑ ij DD j i j=1 This is a so-called eigenvalue problem, with eigenvalues λ. If we want to solve for the eigenvalues we get σ −λ = =d det ijII ij 0, with ij ij This leads to a third order equation in λ, hence we expect three solutions (roots) λ()α . The corresponding eigenvectors can be computed from 3 σ=λ()α () αα () ∑ ijDD j i . j=1 The eigenvectors are orthogonal:

()βα ( ) (αβ ) () DDii= 0 for λ ≠λ Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 134

The directions of the principal axes (defined by the eigenvectors) correspond to the symmetry axes of the crystal. The diagonalized dielectric and inverse dielectric tensors are linked:

The above reasoning shows that anisotropic media are characterized in general by three independent dielectric functions (in the principal coordinate system). It is easier to do all calculations in the principal coordinate system (coordinate system of the crystal) and back-transform the final results to the laboratory system. 6.2 The optical classification of crystals Let us now give a brief overview over crystal classes and their optical properties:

A) isotropic − three crystallographic equivalent orthogonal axes − cubic crystals (diamond, Si....)

 Cubic crystals behave like gas, amorphous solids, liquids, and have no anisotropy.

B) uniaxial − two crystallographic equivalent directions − trigonal (quartz, lithium niobate), tetragonal, hexagonal

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 135

C) biaxial − no crystallographic equivalent directions − orthorhombic, monoclinic, triclinic

6.3 The index ellipsoid The index ellipsoid offers a simple geometrical interpretation of the inverse dielectric tensor . The defining equation for the index ellipsoid is

which describes a surface in three dimensional space.

Remark on the physics of the index ellipsoid: The index ellipsoid defines a surface of constant electric energy density:

In the principal coordinate system the defining equation of the index ellipsoid reads:

This equation can be interpreted as the defining equation of an ellipsoid having semi-principal axes of length . From our discussion of the normal modes in isotropic media we know that corresponded to the refractive index of the normal modes. We will show in the following discussion that also for anisotropic media, there will be special cases where the determine the phase velocity of normal modes. Hence the elements of the dielectric tensor which define the semi-principal axes of the epsilon ellipsoid can be related to refractive indexes

This is the reason why the ellipsoid, which represents graphically the epsilon tensor, is called index ellipsoid. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 136

Graphical representation of the epsilon tensor of an anisotropic crystal by the so-called index ellipsoid.

Summary: • anisotropic media → tensor instead of scalar  in principal system: nii= ε • The index ellipsoid is degenerate for special cases: − isotropic crystal: sphere − uniaxial crystal: rotational symmetric with respect to z-axis and nn= 12 6.4 Normal modes in anisotropic media Let us now look for the normal modes in crystals. A normal mode is: • a solution to the wave equation, which shows only a phase dynamics during propagation while amplitude and polarization remain constant  most simple solution  exp{ikr(ωω)  t } • a solution where the spatial and temporal evolution of the phase are connected by a dispersion relation ω=ω()k or kk=() ω

Before – isotropic media In isotropic media the normal modes are monochromatic plane waves

Er( ,tt )= E exp{i k( ωω) r− } with the dispersion relation ω2 k 22(ω) =k () ω = εω () c2 with εω() > 0 and real as well as kE⋅=⋅= kD 0. The normal modes are elliptically polarized, and the polarization is conserved during propagation. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 137

Now – anisotropic media What are the normal modes in anisotropic media? 6.4.1 Normal modes propagating in principal directions Let us first calculate the normal modes for propagation in the direction of the principal axes of the index ellipsoid, which is the simple case.

We assume without loss of generality that the principal axes are in xyz,, direction and the light propagates in z direction (k → k ). Then, the fields z are arbitrary in the x,y-plane

DDxy,0≠ and

DEii=εε0 i In general we have ED , but here kD⋅=⇒⋅=00 kE , and the two polarization directions (xy= 1, = 2) are decoupled: ω2 e - ω= ϕ ω 2 = εω D11, D11 exp  ( kz tt) D1 exp[ 1 (z )]exp( ) with k112 () i i -i c ω2 e - ω= ϕ ω 2 = εω D22, D22 exp  ( kz tt) D2 exp[ 2 (z )]exp( ) with k222 () i i -i c We see that in contrast to isotropic media, normal modes can't be elliptically polarized, since the polarization direction would change during propagation. But, for linear polarization in the direction of a principal axis (x or y) only the phase changes during propagation, thus we found our normal modes: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 138

ω2 ()a = − ω →=2 2 =→ 2 D {Dt1exp(kr) } e 11 k2 n k normal mode a i a aac

ω2 ()b = − ω →=2 2 =→ 2 D {Dt2exp(kr) } e 22 k2 n k normal mode b i b bbc  For light propagation in principle direction we find two perpendicular linearly polarized normal modes with ED .

Remark on the indices in the index ellipsoid

The indices nii= ε in the index ellipsoid are connected to the indices na and nb of the normal modes propagating along the principal axis. However please be careful about the direction correspondence. For example, the two normal modes propagating along the z direction have phase velocities determined by the indices nn1 = x and nn2 = y determined by the direction of their electric field rather than by the direction of their propagation. 6.4.2 Normal modes for arbitrary propagation direction 6.4.2.1 Geometrical construction Before we will do the mathematical derivation and actually calculate normal modes and dispersion relation, let us preview the results visualized in the index ellipsoid. Actually, it is possible to construct the normal modes geometrically. We start from the normal modes which we have determined for propagation in the principal directions of the crystal and try to generalize to arbitrary propagation directions:

• For a specific crystal and a given frequency ω we take the εi in the principal axis system and construct the index ellipsoid. • We then fix the propagation direction of the normal mode which we would like to look at  ku/ k = . • We draw a plane through the origin of index ellipsoid which is perpendicular to k . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 139

• The resulting intersection is an ellipse, the so-called index ellipse. • The half-lengths of the principle axes of this ellipse equal the refractive indices of the normal modes for the propagation direction

 and

• The directions of the principal axes of the index ellipse are the polarization direction of the normal modes and . • The electric field vectors of the normal modes and follow from

• Thus, , and are not perpendicular to • This has a direct consequence on the pointing vector:

hence is not parallel to because

• If the index ellipse is a circle, the direction of this particular k-vector defines the optical axis of the crystal. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 140

6.4.2.2 Mathematical derivation of dispersion relation Let us now derive mathematically the dispersion relation for normal modes of the form

Er( ,tt )= E exp{i k(ω−) r ω }

Dr( ,tt )= D exp{i k(ω−) r ω } In the isotropic case we found the dispersion relation ω2 k 22()ω=k () ω= εω () c2 where the absolute value of the k-vector is independent of its direction. The fields of the normal modes are elliptically polarized. In the anisotropic case the normal modes are again monochromatic plane waves  exp{ikr(ωω)  t }, but the wavenumber depends on the direction u of propagation, where uk= / k . Hence kk=( ω,u) and the polarization of the normal modes is not elliptic. In the following, we start again from Maxwell’s equations and plug in the plane wave ansatz. We will use the following notation for the directional dependence of k :

ku11    = = 222++= k k22 ku  with uuu1 231   ku33 

Our aim is to derive ω=ω(,kkk123 , ) or ω=ω(,k uu123, ,u ) or k= k(, ω uuu123 , , ). We start from Maxwell's equations for the plane wave Ansatz:

kD = 0 k× E= ωµ0 H kH = 0 k× H= −ω D Now we follow the usual derivation of the wave equation: ω2 1 ω2 1 −=××  −=⋅+2 k( kE) 2 D kk( E) kE2 D c ε0 c ε0 − Here kE⋅ does not vanish as it would have in the isotropic case!

− In the principal coordinate system and with DEi=εε0 ii we find ω2 −=+ 2 ε ki∑ kE j j kE i2 ii E j c ω2 ε−2 =− 2 iikE ki ∑kEjj c j Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 141

Remark: for isotropic media the r.h.s. of this equation would vanish (k⋅ E=0). Now, we have the following problem to solve: ω2 22 2 ε−1k 2 − k 3 kk12 kk13 E 0 c 1 ω2 22  kk 2 ε−k − k kk E =0 21 c 2 1 3 23 2   ω2 22  kk kk 2 ε−k − k E3 0 31 32 c 3 1 2 The general way to solve this problem is using det[ ..] = 0, which gives the dispersion relation ω=ω()k for given kki / . However, there is an easy way to show some general properties of the dispersion relation. We start from the following trick: ω2 k ε−2 =−  = − i ik E i k i kE jj Ei 2 kEjj c2 ∑ ω ε− 2 ∑ j ( c2 i k ) j

Now we multiply this equation by ki , perform a summation over the index 'i ' and rename ij↔ on the l.h.s: k 2 kE = − i kE. ∑∑jj ω2 ε− 2 ∑jj ji( c2 i k ) j

Because divE =∑ j kEjj ≠ 0 we can divide and get the (implicit) dispersion relation: k 2 i = ∑ 2 1 i 2 ω k −εi c2  ku   u 11ω 1 =ω=ω   With kku22()  nu ()  2 we can write  c  ku33   u 3 ku22 u2 ii=→=11 ∑∑ω2  ε ii2 − i k −εi 1 2 c2 n  u2 1  i = final form of DR ∑ 2 − ε 2 i n i n

Discussion of results

For given εωi ( ) and direction (uu12, ) we can compute the refractive index 222 n(ω,, uu12) seen by the normal mode. Because uuu1++= 231, it is sufficient to fix two components (uu12, ) of u to determine the direction. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 142

A more explicit form of the dispersion relation can be obtained by multiplying with denominators:

22 2 2 22 2 2 22 2 2 un1( −ε 2)( n −ε 321) n + un( −ε)( n −ε 331) n + un( −ε)( n −ε 2) n =

22 2 (nnn−ε123)( −ε)( −ε )

The resulting equation is quadratic in ()n22 since the n6 -terms cancel. Hence, we get two (positive) solutions nn, and therefore kn=( ω /) c and ab a a kn=( ω /) c for the two orthogonally polarized normal modes D()a and D()b . bb

In particular, for the propagation in direction of the principal axis (u3 =1 and uu12= = 0 , see 6.4.1) we find:

222222  (nnnnnn−ε12)( −ε) =( −ε 123)( −ε)( −ε )

22  (nn−ε1)( −ε 23) ε =0

 nn22=ε=ε , ab12 Finally, we can derive some properties of the fields of the normal modes, i.e. the eigenfunctions. We start from the eigenvalue equation, which we had derived above ω2 ε−2 =− 2 ik E i k i∑ kE jj. c j For cases where the first factor of the l.h.s. is unequal zero (propagation directions not parallel to the principal axes) we can divide by this term k E= − i kE. i ω2 ε− 2 ∑ jj ( c2 i k ) j The sum does not depend on the index i . Hence the last term of the equation must be constant =  ∑kEjj const. j Knowing that the last term is a constant we can derive a relation of the individual field components from the first part of the equation kkk  EE:: E= 123:: 12 3ωωω22 2 ε−ε−ε−kkk222 ccc22123 2 and with DEi=εε0 ii εεkkk ε  DD:: D= 11 ::2 2 33 12 3ωωω22 2 ε−ε−ε−kkk222 ccc22123 2 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 143

Please be aware that this relation can only be applied for propagation directions not parallel to the principal axes. How are the normal modes polarized? − The ratio between the field components is real  phase difference 0 → linear polarization How do we see the orthogonality DD()a  ()b = 0? (be careful: EE()a  ()b ≠ 0 ) ε22 ()a ()b kkabi u i DD ~ ∑ 22 i 22ωω kka−ε22 ib −ε i cc   c2 kk εεuu22 = ab kk22ii − ii 2 22ab∑∑22 ω (kkba− ) ii22ωω kkai−ε2 bi −ε2 cc Since the two red terms vanish due to the dispersion relation, it follows that DD()a  ()b = 0. The vanishing of the red terms can be seen when rewriting the dispersion relation: 22 22ωω 22 kuab, − ε+ i ε i i ku cc22 ωε22u = ab, i = = + ii 11∑∑222 ∑ 2 ii22ωωc i  2 ω kkab,,−ε i ab−ε i  kab ,−ε i cc22 c 2  

6.4.3 Normal surfaces of normal modes In addition to the index ellipsoid, which is a graphical representation of the material properties of crystals from which the properties of the normal modes can be interpreted as shown above, we can derive a direct graphical representation of the dispersion relation of normal modes in crystals. This graphical representation of the dispersion relation is called normal surfaces: If we plot the refractive indices (wave number or norm of the k-vector divided by k0 ) of the normal modes in the ki -space (normal surfaces), we get a centro-symmetric, two layer surface. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 144

optical axis optical axis

biaxial uniaxial

isotrop Normal surfaces as the graphical representation of the dispersion relation of normal modes in crystals. isotrop: sphere uniaxial: 2 points with in the poles  connecting line defines the optical axis (for , the z-axis is the optical axis) biaxial: 4 points with  connecting lines define two optical axes How to read the figure: − fix propagation direction ( )  intersection with surfaces − distances from origin to intersections with surfaces correspond to refractive indices of normal modes − definition of optical axis  Summary: there are two geometrical constructions: A) index ellipsoid (visualization of dielectric tensor) − fix propagation direction index ellipse half lengths of principal axes give (refractive indices of the normal modes) − optical axis  index ellipse is a circle − for uniaxial crystals the optical axis coincides with one principal axis B) normal surfaces (visualization of dispersion relation) − fix propagation direction intersection with surfaces distances from origin give − optical axis connects points with Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 145

Conclusion In anisotropic media and for a given propagation direction we find two normal modes, which are linearly polarized monochromatic plane waves with two different phase velocities cna , cnb and two orthogonal polarization directions DD()a , ()b .

6.4.4 Special case: uniaxial crystals Let us now investigate the special and simpler case of uniaxial crystals. In biaxial crystals we do not find any other effects, just the description is more complicated. The main advantage of uniaxial crystals is that we have rotational symmetry in one plane. Therefore all three-dimensional graphs (index-ellipsoid, normal surfaces) can be reduced to two dimensions, and we can sketch them more easily. As we have seen before, uniaxial crystals have trigonal, tetragonal, or hexagonal symmetry. Let us assume (without loss of generality) that the index ellipsoid is rotationally symmetric around the z-axis, and we have ε=ε=ε, ε=ε, 12or 3ε which are the ordinary and extraordinary dielectric constants. Then, we expect two normal modes:

A) ordinary wave  na independent of propagation direction

B) extraordinary wave  nb depends on propagation direction

The z-axis is, according to definition, the optical axis with nnab= . (or)  The ordinary wave D is polarized perpendicular to the z-axis and the

k-vector and it does not interact with ee . (e)  The extraordinary wave D is polarized perpendicular to the k-vector (or) and D . Let us now derive the dispersion relation: From above we know the implicit form u2 1  i = ∑ 2 − ε 2 i n i n For uniaxial crystals this leads to u2 uu 221 1+ 23 += 222−ε  −ε  −ε  2 nnnor or ε n

22 22−ε 2 −ε 2 + 2 + 22 −ε 2 = 2 −ε 2 −ε nnε nor ( u12 u) nnor u3 nε n or A) ordinary wave: independent of direction Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 146



B) extraordinary wave (derivation is your exercise): dependent on direction

Hence for a given direction one gets the two refractive indexes . The geometrical interpretation as normal surfaces is straightforward and can be done, w.l.o.g., in the , or y, z plane ( ).

Normal surfaces for a uniaxial crystal.

The shape of the normal surfaces can be derived in the following way. We have with

A) ordinary wave

B) extraordinary wave

What about the fields? We know from before that

For the extraordinary wave all denominators are finite, and in particular implies , hence is polarized in the y-z plane. Then, implies that is polarized in x-direction. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 147

In summary, we find for the polarizations of the fields: A) ordinary: perpendicular to optical axis and ,

B) extraordinary: perpendicular to and in the plane -optical axis , because If we introduce an angle , as in the figures below, to describe the propagation direction, a simple computation of for the extraordinary wave is possible (exercise):

The following classification for uniaxial crystals is commonly used  negative uniaxial  positive uniaxial Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 148

7. Optical fields in isotropic, dispersive and piecewise homogeneous media 7.1 Basics 7.1.1 Definition of the problem Up to now, we always treated homogeneous media. However, in the context of evanescent waves we already used the concept of an interface. This was already a first step in the direction we now want to pursue. When we treated interfaces so far we never considered effects of the interface, we just fixed the incident field on an interface and described its further propagation in the half-space. In this chapter, we will go further and consider reflection and transmission properties of the following physical systems: − interface − layer (2 subsequent interfaces) − system of layers (arbitrary number of subsequent interfaces)

Aims • We will study the interaction of monochromatic plane waves with arbitrary multilayer systems  interferometers, dielectric mirrors, … by superposition of such plane waves we can then describe interaction of spatio-temporal varying fields with multilayer systems • We will see a new effect, the “trapping” of light in systems of layers  new types of normal modes in inhomogeneous space  “guided” waves (propagation of confined light beams without diffraction)

Approach • take Maxwell's transition condition for interfaces • calculate field in inhomogeneous media  matrix method • solve reflection-transmission problem for interface, layer, and system of layers, • apply the method to consider special cases like Fabry-Perot-interferometer, 1D photonic crystals, waveguide…

Background • orthogonality of normal modes of homogeneous space  no interaction of normal modes in homogeneous space • inhomogeneity breaks this orthogonality  modes interact and exchange energy Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 149

• however, locally the concept of eigenmodes is still very useful and we will see that the interaction at the inhomogeneities is limited to a small number of modes 7.1.2 Decoupling of the vectorial wave equation Before we will start treating a single interface, it is worth looking again at the wave equation in homogeneous space in frequency domain ω2 rotrotEr(,ω− ) Er (, ω= )ω µ jr (, ω+µ )ω2 Pr (, ω ) c2 i 00 In general, for isotropic media all field components are coupled due to the rot rot operator. However, for problems with translational invariance in at least one direction (homogeneous infinite media, layers or interfaces) a simplification is possible. Let us assume, e.g. translational invariance of the system in y direction and propagation in the x-z-plane  ∂∂=/0y

∂ ∂ E x ∂ E z (2) ∂∂xx( + ∂ z) ∆ E x  (2) rot rot E= grad div E−∆ E =0 − ∆ E y  ∂ ∂ E x ∂ E (2) + z ∆ E z ∂∂zx( ∂ z) 

Then, we can split the electric field as EE=⊥ + E with

0 Exx ∂∂ 22  (2) (2) ∂∂ ⊥ = = ∇= ∆= + EEE y ,  0, 0 , 22  ∂∂xz   ∂∂ 0 E z z

E⊥ is polarized perpendicular to the plane of propagation, E is polarized parallel to this plane. Common notations are: perpendicular: ⊥  s  TE (transversal electric) parallel:   p  TM (transversal magnetic) Both components are decoupled and can be treated independently:

2 (2) ω 2 ∆ Er⊥⊥(,ω+ ) Er (, ω=−ωµ ) jr (, ω−µω ) Pr⊥ (, ω ) c2 i 00⊥ 2 (2) ω (2) (2) 2 ∆ E(, rω+ ) E (, r ω− ) grad div E=−ωµ j(, r ω−µω ) P (, r ω ) c2 i 00 From

H(, rω=− ) i rot E(, r ω ) ωµ0 we can conclude that the corresponding magnetic fields are Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 150

00  H    x  EH=EE = ,0= TE y TE   00  H    z  E  00   x     EH= 0, =HH = TM  TM  y   E  00   z     7.1.3 Interfaces and symmetries Up to now we treated plane waves of the form

E( r ,tt )= E exp i( kr −ω )

− homogeneous space implies: exp(ikr) − monochromaticity leads to: exp(−ωi t) Now, we will break the homogenity in x-direction by considering an interface in y-z – plane which is infinite in y and z

medium I y interface

z medium II

W.l.o.g. we can assume k = (kk, 0, ) xz by choosing an appropriate coordinate system (plane of incidence is the xz, - plane), and then the problem does not depend on the y -coordinate. As pointed out before, we can split the fields in TE and TM polarization EE= + E and treat them separately. TE TM We still have homogeneity in z -direction, and therefore we expect solutions  exp( kz) . The wave vector component k has to be continuous at the i z z interface (follows strictly from continuity of transverse field components, see 7.1.4.). Therefore, we can write for the electric field: EE(xzt , , )= ( x) exp (kz−ω t) +E( x)exp (kz −ω t) TE iizzTM  Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 151

7.1.4 Transition conditions From Maxwell’s equations follow transition conditions for the field components. Here we will use that Et, Ht (transverse components) are continuous at an interface between two media. This implies for the:

A) Continuity of fields

TE: EE= y and H z continuous

TM: Ez and HH= y continuous

B) Continuity of wave vectors

ikz z homogeneous in z-direction  phase e  kz continuous 7.2 Fields in a layer system  matrix method We will now derive a quite powerful method to compute the electromagnetic fields in a system of layers with different dielectric properties. 7.2.1 Fields in one homogeneous layer Let us first compute the fields in one homogeneous layer of thickness d and dielectric function εωf ( ) • aim: for given fields at x = 0  calculate fields at xd= • strategy: − Do computation with transverse field components (because they are continuous). − The normal components can be calculated later. We will assume monochromatic light (one Fourier component, Exz(,;ω ) H(,; xzω )) and in the following we will often omit ω in the notation.

TE-polarization We have to solve the wave equation (no y -dependence because of translational invariance): ∂222 ∂ω + +ε = 2 22f ETE (,)xz 0 ∂∂x zc We use the ansatz from above:

EETE (x , z )= TE ( x )exp( ikz z) and HHTE (x , z )= TE ( x )exp( ikz z) ∂ω22 + ε−2 = 22fzkxETE () 0 ∂xc i with: HTE (,)xz = − rotETE (,)xz ωµ0 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 152

Now let us extract the equations for transversal fields Eyz= EH, : ∂2 ω2 +ω=2 22ω= ε ω− 2 kfx( k z ,) Ex () 0 with kkfx( z , ) 2 f( ) k z ∂x c i ∂ Hz () x= − Ex () ωµ0 ∂x This makes sense since the wave equation for the y-component of the electric field is a second order differential equation. Hence we need to specify the field and its first derivative as initial condition at x = 0 to determine a unique solution.

TM-polarization analog for transversal components Hyz= HE, : ∂2 +ω2 = 2 kfx( k z ,) Hx () 0 ∂x i ∂ Ez () x= Hx() ωε0f ε ∂x Again, we succeed describing everything in transversal components.

Now we have the following problem to solve: ∂∂ − calculate fields (E, H) and derivatives Ex(), Hx () at xd= for ∂∂xx given values at x = 0 − calculate the fields at xd=

− at the end: HTM  ETM  E = ETM + ETE Because the equations for TE and TM have identical structure, we can treat them simultaneously. We rename EH, → F generalized field 1

iωµ0 H z, − iE ωε 0z → G generalized field 2 and write down the problem to solve: ∂2 +ω=2 2 kfx( k z ,) Fx () 0 ∂x ∂ 1 Gx()= αf Fx () with α=α=fTE1, fTM ∂x εf We know the general solution of this system (harmonic oscillator equation): Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 153

F( x )= C1 exp( ik fx x) +− C 2 exp( ik fx x) ∂ G( x )=α=αf F ( x ) if k fx C 1 exp( ik fx x) −− C 2 exp( ik fx x) ∂x We have as initial conditions FG(0), (0) given: F(0) = CC + 12 G(0) =α− ikCf fx[ 1 C 2 ] from which we can compute the constants CC12, : 1 i CF1 =(0) − G (0) 2 αfk fx

1 i CF2 =(0) + G (0) 2 αfk fx The final solution of the initial value problem is therefore: 1 Fx( )= cos( kxFfx ) (0)+ sin(kxGfx ) (0) αfk fx

Gx( )= −αf k fx sin( kxF fx ) (0)+ cos( kxGfx ) (0) By resubstituting we have the electromagnetic field in the layer 0 ≤≤xd. 7.2.2 The fields in a system of layers In the previous subchapter we have seen how to compute the electromagnetic field in a single dielectric layer, dependent on the transverse field components Ey , H z (TE) and H y , Ez (TM) at x = 0. We can generalize our results to systems of dielectric layers, which are used in many optical devices: − Bragg mirrors − chirped mirrors for dispersion compensation − interferometer − multi-layer waveguides − Bragg waveguide − metallic interfaces and layers We can even go further and “discretize” an arbitrary inhomogeneous (in one dimension) refractive index distribution. This is important for so-called 'GRIN' - Graded-Index-Profiles. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 154

From above, we know the fields in one layer: 1 Fx( )= cos( kxFfx ) (0)+ sin(kxGfx ) (0) αfk fx

Gx( )= −αf k fx sin( kxF fx ) (0)+ cos( kxGfx ) (0) We can write this formally in matrix notation as Fx( )  F (0) = mˆ ()x  , Gx( )  G (0) where the 2x2-matrix mˆ describes propagation of the fields: 1 cos(kxfx ) sin(k x) ˆ = k α fx m( x) fx f −αksin( kx) cos( kx) fx f fx fx − To compute the field at the end of the layer we set xd= . − We assume no absorption in the layer  mˆ ( x) =1.

− A system of layers is characterized by εii,d If multiple layers are considered, the fields between them connect contin- uously since the field components used for the description of the fields are the continuous tangential components. Hence, we can directly write the formalism for a multilayer system since it just requires matrix multiplications:

A) Two layers FF   F = mˆ22()d = mm ˆˆ22 () dd 11 () GG G dd12+ d1 0

B) N layers F N FF  N = = ˆ ˆ =  ∏mMˆ ii()d   with Mm∏ ˆ ii()d Gdd+ ++.. d = D i=1 GG0 0 i=1 12 N Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 155

ω2 All matrices mˆ have the same form, but different αii,,dk = ε( ω−) k2 . i fifx c2 i z

Summary of matrix method

− F(0) and G(0) given ( EH, z for TE, EHz , for TM) − kd,αεi ,, given  matrix elements z fii − multiplication of matrices (in the right order)  total matrix − fields FD() and GD()

7.3 Reflection – transmission problem for layer systems 7.3.1 General layer systems 7.3.1.1 Reflection- and transmission coefficients  generalized Fresnel formulas In the previous chapter, we have learned how to link the electromagnetic field on one side of an arbitrary multilayer system with the field on the other side. We have seen that after splitting in the TE/TM-polarizations, continuous (transversal) field components are sufficient to describe the whole field. What we will do now is to link those field components with the fields, which are accessible in an experimental configuration, i.e. incident, reflected, and transmitted fields. In particular, we want to solve the reflection-transmission problem, which means that we have to compute reflected and transmitted fields for a given angle of incidence, frequency, layer system and polarization. We introduce the wave vectors of the incident (k ) , reflected ( k ) and I R transmitted (k ) fields: T k − kk   sx sx  cx kk= 0,= 0 , k= 0 IR   T  kkk    zzz    ωω22 with k= ε− kk22 =() ω− k 2 , k = ε − kk22 =() ω− k 2 , sx cc22s z s z cx c z c z where εω() and εω() are the dielectric functions of the substrate and s c cladding and kz is the tangential component of the wave vector which is continuous throughout the layer system. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 156

Multi layer system

As we have seen before, the kz component of the wave vector is conserved and ±k determines the direction of the wave (forward or backward). The x total length of the wave vector in each layer is given by the dispersion relation for dispersive, isotropic, homogeneous media. As a consequence, the kx component changes its value in each layer.

Remark on law of reflection and transmission (Snellius) It is possible to derive Snellius law just from the fact that k is a conserved z quantity: 1. kksinϕ= sin ϕ ϕ=ϕ (reflection) s Is R I R 2. kksinϕ= sin ϕ nn sin ϕ= sin ϕ (Snellius) sIcTsIcT

Let us now rewrite the fields in order to solve the reflection transmission problem: A) Field in substrate The fields in the substrate can be expressed based on the complex amplitudes of the incident F and reflected field F as: I R Fxz( , )= exp( kzF) exp( kx) +− Fexp( kx) s iiz I sx R isx

Gxz( , )=α k exp( kzF) exp( kx) −− Fexp( kx) s iis sx z I isx R isx

B) Field in layer system The fields inside the layer system can be expressed as F( xz , )= exp( kz) Fx ( ) fzi G( xz , )= exp( kzGx) ( ) fzi where the amplitudes Fx() and Gx() are given by matrix method as Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 157

FF  = Mˆ ()x  GGx 0

C) field in cladding The fields in the cladding can be expressed based on the complex amplitude of the transmitted field F as: T F( xz , )= exp( kz) F exp  k( x− D) c iiz T cx

G( xz , )=α− k exp( kz) F exp  k( x D) c iic cx z T  icx Note that in the cladding we consider a forward (transmitted) wave only. Reflection transmission problem We want to compute F and F for given F , k ( sinϕε ), ,d . We know that R T I z I ii F and G are continuous at the interfaces, in particular at x = 0 and xD= . We have: FF  = Mˆ ()D  . GGD 0 Field in cladding at xD= field in substrate at x = 0

On the other hand, we have expressions for the fields at x = 0 and xD= from our decomposition in incident, reflected and transmitted field from above. Hence, we can write:

F MD11() MD12 ()FF+ T = IR . α kF MD() MD () α−kFF( ) i c cx T 21 22 i s sx I R

We consider FI as known, and FR and FT as unknown and get: (αkM −α kM) −( M +α k α kM) FF= s sx 22 c cx 11 i 21 s sx c cx 12 RI(αkM +α kM) +( M −α k α kM) ((((((((((((((((((s sx 22 c cx 11 i 21 s sx c cx 12 N

2(α−sk sx MM11 22 MM 12 21) FFTI= (αskM sx 22 +αc kM cx 11)( + iM21 −αs k sx α c kM cx 12 ) 2α k FF= s sx TIN These are the general formulas for reflected and transmitted amplitudes. Please remember that the matrix elements depend on the polarization TE TM direction  MMij≠ ij . Let us now transform back to the physical fields, and write the solution for the results of the reflection transmission problem for TE and TM polarization: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 158

A) TE-polarization FEE==,1 α= y TE i) reflected field ETE = RETE R TE I with the reflection coefficient

TE TE TE TE (kM22 − kM11 ) −+( M21 kkM12 ) R = sx cx i sx cx TE TE TE TE TE (kM22 + kM11 ) +−( M21 kkM12 ) ((((((((((((((((sx cx i sx cx NTE ii) transmitted field ETE = TETE T TE I with the transmission coefficient 22kk T = sx = sx , TE kMTE + kMTE +− MTE kkMTE N ( sx 22 cx 11 ) i( 21 sx cx 12 ) TE  We get complex coefficients for reflection and transmission, which determine the amplitude and phase of the reflected and transmitted light.

B) TM-polarization 1 FHH= =,. α= y TM ε In the case of TM polarization we have the problem that an analog calculation to TE would lead to HH/ , i.e., relations between the magnetic field. R ,T I However, we want EETM/ TM . Therefore, we have to convert the H -field to the R ,T I ETM -field:

As can be seen in the figure, we can express the amplitude of the ETM field in terms of the Ex component: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 159

Ek xz=−sin ϕ=− EkTM

k  EETM = − , k x z

With Maxwell we can link Ex to H y : 1 1 k 1 E= − ( kH× )  E= kH  EHTM =−=− H x zy yy ωε0 ε ωε0 ε ωε0 ε cεε0

TM EH,,ε result: RT = s RT , → εε/ relevant for transmission only EHTM ε sc I sc, I Hence we find the following for TM polarization: ETM = RETM R TM I with the reflection coefficient

TM TM TM TM (εkM22 −ε kM11 ) −( ε ε M21 + kkM12 ) R = c sx s cx i s c sx cx TM TM TM TM TM (εkM22 +ε kM11 ) +( ε ε M21 − kkM12 ) ((((((((((((((((((c sx s cx i s c sx cx NTM

ETM = TETM T TM I with the transmission coefficient 22εε kkεε T = s c sx = s c sx TM εkMTM +ε kMTM + ε ε MTM − kkMTM N ( c sx 22 s cx 11 ) i( s c 21 sx cx 12 ) TM In summary, we have found different complex coefficients for reflection and transmission for TE and TM polarization. The resulting generalized Fresnel formulas for multilayer systems are

TE TE TE TE (kM22 − kM11 ) −+( M21 kkM12 ) R = sx cx i sx cx TE kMTE + kMTE +− MTE kkMTE ( sx 22 cx 11 ) i( 21 sx cx 12 ) 2k T = sx TE N TE

TM TM TM TM (εkM22 −ε kM11 ) −( ε ε M21 + kkM12 ) R = c sx s cx i s c sx cx TM εkMTM +ε kMTM + ε ε MTM − kkMTM ( c sx 22 s cx 11 ) i( s c 21 sx cx 12 ) 2 εε k T = s c sx . TM N TM Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 160

7.3.1.2 Reflectivity and transmissivity In the previous chapter we have computed the coefficients of reflection and transmission, which relate the electric fields in TE and TM polarization of incident, reflected and transmitted wave. However, in many situations it is more important to know the relation of energy fluxes, the so called reflectivity and transmissivity. In order to get information on these quantities we have to compute the energy flux perpendicular to the interface: − flux through a surface with x = const

1 ∗ Se= ℜ( E× H) e xx2

∗1 ∗∗ With H=( kE × ) ωµ0 we find

11∗ 22 Se=ℜ=ℜ ke E (k ) E . xx( ) x 22ωµ00ωµ Since in an absorption free medium the energy flux is conserved, in an absorption free layer the energy flux is also conserved. In the substrate ω2 k= ε− kk22 =() ω− k 2 sx c2 s z s z is supposed to be real-valued, because we have our incident wave coming from there. The total energy flux from the substrate to the layer system is given as

1 22 Se= kk E− E s x sx I sx R 2ωµ0 In contrast, in the cladding ω2 k= ε − kk22 =() ω− k 2 cx 2 c z c z c may be complex-valued. The energy flux from the layer system into the cladding is

1 2 Se= ℜ(k ) E . c x cx T 2ωµ0 Because we have energy conservation: Se= Se scxx

22ℜ(k ) 2 EE= + cx E IR k T sx Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 161

Now we will compute the global reflectivity ρ and transmissivity τ of a layer system. Of course, we will decompose into TE and TM polarizations and relate to the reflectivities ρTE,TM and transmissivities τTE,TM . We know:

EEE=+=+TE TM , EEETE TM RRR TTT

2 2222ℜ(k ) E =++ETE EETM cx ETE +TM I RTRTk ( ) sx

2 ℜ(k ) 2 2 2 ℜ(k ) 2 2 = RT+ cx ETE + RT+ cx ETM . TE k TE I TM k TM I  sx   sx  Here, we just substituted the reflected and transmitted field amplitudes by incident amplitudes times Fresnel coefficients. Now, we decompose the incident field as follows:

EETE =EEcosδ , TM =sinδ . II II 2 Then, we can divide by the (arbitrary) amplitude E and write I

22ℜ(kk) 2 ℜ( ) 2  1 =R + cx TT cos22δ+R + cx sin δ TE kkTE TM TM  sx sx 2 2 ℜ(k ) 22 1= RRcos22δ+ sin δ + cx TTcos22δ+ sin δ ( TE TM ) k ( TE TM ) sx The red and blue terms can be identified as 1= ρ + τ The global reflectivity and transmissivity are therefore given as ρ=rcos22 d+r sin d TE TM τ=tcos22 d+t sin d TE TM with the reflectivities

22ℜ(k ) ρ=RT,.τ= cx TE,, TM TE TM TE , TM k TE, TM sx for the two polarization states TE and TM. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 162

7.3.2 Single interface 7.3.2.1 (classical) Fresnel formulas Let us now consider the important example of the most simple layer system, namely the single interface. The relevant wave vectors are (as usual): k − kk   sx sx  cx kk= 0,= 0 , k= 0. IR   T  kkk    zzz   

The continuous component of the wave vector, expressed in terms of the angel of incidence, is ωω kn= εsin ϕ= sin ϕ zcc s I sI. Then, the discontinuous component is given as ω2 ωω 22 ω  k= ε− k2 = ε− εsin2 ϕ =nn2 − 22sin ϕ iixc2 z cc 22 isI ci sI ω ωω  k= ncos ϕ , k = nn2 − 22sin ϕ= n cos ϕ , sxc s I cx ccc s I c T As above, we can assume that k is always real, because otherwise we sx have no incident wave. k is real for nn>ϕsin , but imaginary for cx cs I nn<ϕsin (total internal reflection). cs I The matrix for a single interface is the unit matrix 10 Mmˆ =ˆ (d = 0) =  01 and it is easy to compute coefficients for reflection and transmission, and reflectivity and transmissivity. Using the formulas from above we find: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 163

A) TE-polarization

(kM22 − kM11 ) −+( M21 kkM12 ) 2k R = sx cx i sx cx , T = sx TE (kM+ kM) +−( M kkM) TE N ((((((((((((((sx 22 cx 11 i 21 sx cx 12 TE

(kk− ) ncosϕ− nn2 − 22sin ϕ nncosϕ− cos ϕ R =sx cx = s I cs I= s Ic T TE (kk+ ) ncosϕ+ nn2 − 22sin ϕ ncosϕ+ n cos ϕ sx cx s I cs I s I c T

2kn2 cosϕϕ2 n cos T =sx = sI = sI TE (kk+ ) ncosϕ+ nn2 − 22sin ϕ ncosϕ+ n cos ϕ sx cx s I cs I s I c T 2 2 kk− sx cx ρ=R = 2 TE TE kk+ sx cx ℜℜ(k) 2 4 kk( ) cx sx cx τ=T = 2 . TE k TE kk+ sx sx cx  ρ +τ =1 TE TE B) TM-Polarisation (εkM −ε kM) −( ε ε M + kkM) 2 εε k R = c sx 22 s cx 11 i s c 21 sx cx 12 T = s c sx . TM (εkM +ε kM) +( ε ε M − kkM) TM N ((((((((((((((((((c sx 22 s cx 11 i s c 21 sx cx 12 TM 10 with: Mmˆ =ˆ (d = 0) =  01

(kkεε− ) nn2cosϕ− n 22 n − n 2sin 2 ϕ nncosϕ− cos ϕ R = sxcs cx = sc I s c s I= cIsT TM (kkεε+ϕ) nn2cosϕ+ n 22 n − n 2sin 2 ϕ ncos+ϕ n cos sxcs cx sc I s c s I c I s T

2k εε 2nn2 cosϕϕ2n cos T = sx cs = sc I = s I , TM (kkε + ε ) nn2cosϕ−+ n 22 n n 2sin 2ϕ ncosϕ+ n cosϕ sxc cx s s c I sc s I c I s T 2 2 kkε− ε sx c cx s ρ=R = 2 , TM TM kkε+ ε sx c cx s ℜ(k) 2 4 kkℜ( ) εε cx sx cx s c τ=T = 2 TM k TM kkε+ ε sx sx c cx s  ρ +τ =1 TM TM Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 164

Remark It may seem that we have a problem for ϕ=0. For ϕ=0, TE and TM I I polarization should be equivalent, because the fields are always polarized parallel to the interface. However, formally we have RR= − , TE TM TT= . The “strange” behavior of the coefficient of reflection can be TE TM explained by the following figures:

7.3.2.2 Total internal reflection (TIR) for εs>εc Let us now consider the special case when all incident light is reflected from the interface. This means that the reflectivity is unity. 2 2 k − k k ε−k ε sx cx sx ccx s ρ= 2 ρ= 2 TE k + k TM k ε+k ε sx cx sx c cx s ω With k=−ϕ nn2 22sin we can compute the smallest angle of incidence cx c c s I with ρ=1: TE,TM k=0 nn = sin ϕ cx c s Itot n sinϕ=c . Itot n s For angles of incidence larger than this limit angle, ϕ >ϕ we have I Itot ωω2 kn=22sin ϕ− n 2 = µ= k2 − ε  iµaginary cx iccs I cii c z 2 c → ℜ=(k ) 0  TIR cx Obviously, we find the same angle of TIR for TE and TM polarization. The energy fluxes are given as (here TE, same result for TM): 2 k −µ 4kkℜ( ) sxi c sx cx ρ=22 =1 τ= =0. TE k+µ TE kk+ sxi c sx cx Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 165

Remark For metals in visible range (below the plasma frequency) we have ω always TIR, because: ℜε( ) <0 → kn= ε−22sin ϕ always c cx c c s I imaginary. In the case of TIR the modulus of the coefficient of reflection is one, but the coefficient itself is complex  nontrivial phase shift for reflected light: A) TE-polarization k − µ Z exp( a) sxi c i R =⋅1 exp( Θ=) =∗ = =exp( 2 a) TE iiTE kZ+ µ exp(−a) sx iic

Θ µ n22sin ϕ−n 2sin 2 ϕ− sin 2 ϕ  tana= tan TE =−c =− s I c =− I Itot . 2 k n cosϕ cosϕ sx sI I B) TM-polarization k e− µe Z exp( a) sx ci c s i R =⋅1 exp( Θ=) =∗ = =exp( 2 a) TM iiTM k e + µe Z exp(− a) sx cii c s ΘΘµ e e  tana= tan TM =− cs= s tanTE , 22k e e sx c c In conclusion, we have seen that the phase shifts of the reflected light at TIR is different for TE and TM polarization, and because εsc >ε Θ >Θ , TM TE As a consequence, incident linearly polarized light gets generally elliptically polarized after TIR  Fresnel prism

Remark The field in the cladding is evanescent − exp( kx) = exp( −µ x) . i xc c  The averaged energy flux in the cladding normal to the interface vanishes.

1122 S =ℜ=ℜ=(kE ) (k ) E 0. x x x 22ωµ00ωµ  =0

7.3.2.3 The Brewster angle There exists another special angle with particular reflection properties. For TM-polarization, for incident light at the Brewster angle ϕ we find R = 0: B TM 2 kkε− ε sx c cx s ρ= 2 =0, TM k ε+k ε sx c cx s Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 166

 kkε= ε sx c cx s ε22 ε −sin ϕε =ε 22 ε −sin ϕε c( s Bs) s( c Bs)

2 ε ε( ε −ε ) ε sin ϕ=sc s c = c B ε ε22 −ε (ε +ε ) ss( c) sc

22 εε cosϕ=− 1 sin ϕ=− 1 cs = BB(ε +ε) ( ε +ε ) sc sc With the last two lines we can write the final result for the Brewster angle: ε tanϕ= c . B ε s The Brewster angle exists only for TM polarization, but for any nn. sc There is a simple physical interpretation, why there is no reflection at the interface for the Brewster angle.

sinϕ n tan ϕ=Bc = B cosϕ n Bs π  nnsinϕ= cos ϕ= n sin −ϕ s Bc Bc2 B At the same time the angle of the transmitted light is always π nnsinϕ = sin ϕ ϕ = −ϕ , s Bc T T2 B Hence, at Brewster angle reflected and transmitted wave propagate in perpendicular directions. If we interpret the reflected light as an emission from oscillating dipoles in the cladding, no reflected wave can occur for TM polarization (no radiation in the direction of dipole oscillation). In summary, we have the following results for reflectivity and transmittivity at a single interface with εsc >ε . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 167

total internal reflection

angle of incidence

total internal reflection

angle of incidence 7.3.2.4 The Goos-Hänchen-Shift The Goos-Hänchen shift is a direct consequence of the nontrivial phase shift of the reflected light at TIR. It appears when beams undergo total internal reflection at an interface. The reflected beam appears to be shifted along the interface. As a result it seems as if the beam penetrates the cladding and reflection occurs at a plane parallel to the interface at a certain depth, the so- called penetration depth. For sake of simplicity we will treat here TE-polarization only.

Let us start with an incident plane wave in TE polarization:

Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 168 with ω ω2 α = n sinϕ ,.γ=n22 −α c sI ssc2 This gives rise to a reflected plane wave as E( xz , )= E exp ( α z −γ x) exp  Θ(α) . RIii s  The reflected plane wave gets a phase shift Θα(), which depends on the angle of incidence (here characterized by the transverse wave number α). Now we want to treat beams, which we can write as a superposition of plane waves (Fourier amplitude e (α)): I E( xz , )= d α e( α) exp  αz +γ( α) x II∫ i( s) We assume a mean angle of incidence ϕ : I0 α = ω n sinϕ mean angular frequency 0 c s I0

α=α0 +ε In the Fourier integral, we have to integrate over angular frequencies with non-zero amplitudes e (α) only (e (α +e) ≠0 for −∆≤ε≤∆ only) I I 0 ∆ E( xz , )= exp( α z) dee (α +ε)exp ( εz+γ x) Iii0 ∫ −∆ Is0 

∆ E( xz , )= exp( α z) dee (α +ε)exp ( εz−γ x) exp Θα( +e) . Ri00∫ −∆ Is ii0 Let us make the following assumptions: ω − small divergence of beam (narrow spectrum, ∆  n ) c s

− all Fourier component undergo TIR (Θ( α) >ΘItot ) Then, it is justified to expand the phase shift Θα() of the reflected wave into a Taylor series up to first order: ∂Θ Θα( 00 +ε≈Θα) ( ) + ε=Θα( 0) +Θε′ ∂α α 0 Then, the reflected beam at the interface at x = 0 is given as: ∆ Ez(0, )= exp  α z +Θ( α ) dee (α +e)exp ( z+Θ′)e R {i 00}∫ −∆ I 0 i We can identify the remaining integral as a shifted version of the incident beam profile at x = 0 E(0, z )= exp  Θ( α) −α Θ′′Ez( +Θ). RI{i00} Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 169

Thus, the reflected beam appears shifted by d = −Θ′ (Goos-Hänchen Shift).

Let us finally compute the shift d = −Θ′. We know from before that the phase shift for TIR is given as: Θ µµ tan TE = − cc= − 2 k γ sx s

22ω2 µ a−2 n Θ=−2arctanc =− 2arctan c c ω2 22 ksx 2 n −a c s

22 22α −α α(ksx +µ c ) k −µ ∂Θ 12µ sx 2kkc µ 1α ′ c sx c sx Θ= =−×22 × 2 =−2222=− ∂α µc kksx sx+ µ c µc k sx 1+ 2 ksx

Θ′ =−=− ϕ 11 aa a dx2 tan with x = = and tan ϕ= = 0 ET I0 ET µ 22ω2 I0 γ α−2 n ksx s c 0 c c  x depth of penetration ET 7.3.3 Periodic multi-layer systems - Bragg-mirrors - 1D photonic crystals In the previous chapters we have learned how to treat (finite) arbitrary multi- layer systems. Interesting effects occur when these multi-layer systems become periodic, so-called Bragg-mirrors. The reflectivity of such mirrors is almost 100% in certain frequency ranges; the more layers the closer we get to this ideal value. Bragg mirrors are important for building resonators (laser, interferometer). Furthermore periodic structures are of general importance in physics (lattices, crystals, atomic chains …). We can learn many things about the general features of such periodic systems by looking at the optical properties of periodic (dielectric) multi-layer systems In our theoretical approach, we will assume these layer systems as infinite, i.e. consisting of an infinite number of layers, and we treat them as so-called one-dimensional photonic crystals. We will discuss effects like band gaps, dispersion and diffraction in such periodic media, and gain understanding of the basics of Bragg reflection and the physics of photonic crystals. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 170

In order to keep things simple we will treat: − semi-infinite periodic multi-layer systems x>ε0,( 11 , dd) ,( ε 2 , 2) − TE-polarization only − monochromatic light At the interface between substrate and Bragg-mirror (x = 0) we have incident and reflected electric field:

∂E ′ EEE0RI= + and =E0 = ik sx ( EIR − E ) ∂x 0 Alternatively the incident and reflected field can be expressed by the field at the interface and its derivative as ′ ′ E00 iE E00 iE EI = − and ER = + 22ksx 22ksx In chapter 7.2, we developed a matrix formalism involving the generalized fields F and G . Because here we treat TE polarization only, we can use directly the electric field amplitude and its derivative with respect to x , because EF= and ∂E iωµ HG = = = E′. 0z ∂x Let us now calculate the field in the multi-layer system. From before, we know how to treat finite systems with the matrix method. Here, we want to treat an infinite periodic medium (like a one-dimensional crystal). As a particular example we will investigate an infinite system consisting of just two periodically repeated layers. The two layers should consist of homogeneous material with ε1 and ε2 having a thickness of d1 and d2 , respectively. Hence we have:

ε()xx =ε ( +Λ ) with the period Λ=dd12 + For infinite periodic media, we can make use of the Bloch-theorem to find the generalized normal modes (Bloch modes or Bloch waves). We seek for solutions like:

Exz( , ;ω= ) exp{ kk( ,ω) x+ kz}Exk () ixz z x with

Exkk(+Λ ) = Ex () xx

In other words, Exk () is a periodic function and we are looking for solutions x Exz(,;ω ) which have the same amplitude after one period of the medium, but we allow for a different phase  exp kk( ,ω) x . Here k is the (yet) {ixz } x Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 171 unknown Bloch vector. Because in this easy example we deal with a one- dimensional problem, the Bloch vector is actually a scalar. In the following, we will find a dispersion relation for the Bloch-waves ω2 kk( ,ω), in complete analogy to the DR for plane waves kk22= εω−( ) in xz xzc2 homogeneous media. In order to make the difference to the homogeneous case more obvious, we change the notation for the Bloch vector: kx  K . According to the Bloch-theorem (our ansatz) we have a relation between E and E′ when we advance by one period of the multi-layer system (from period N to period N +1): EE =exp K Λ . ''(i ) EE (N1+Λ) NΛ On the other hand, we know from our matrix method:

EEˆ  ''= M  EE  (N1+Λ) NΛ ˆ (2) (1) with Mm= ˆˆ(dd21) m( )  Mij= ∑k mm ik kj If the Bloch wave is a solution to our problem, we can set the two expressions equal: E MIˆ − exp KΛ= 0 { (i ) }' E NΛ and with µ=exp(iK Λ) we have to solve the following eigenvalue problem: E ˆ − µ= {MI()K }' 0 E NΛ This eigenvalue problem determines the Bloch vector K and will finally give our dispersion relation. As usual, we use the solvability condition det{MIˆ − µ= } 0 to compute the dispersion relation expressed in µ=exp(iK Λ). Hence we still need to compute K afterwards.

2 (MM++) ( MM) µ= Λ=11 22 ± 11 22 − ±±exp (iK ) 1. 22 Note that we used det{Mˆ } = 1, which explains why the off-diagonal elements of the matrix do not appear in the formula. Moreover, because of det{Mˆ } = 1 we have µµ+− =1. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 172

The corresponding eigenvectors (field and its derivative at xN= Λ ) can be computed from E MIˆ − exp KΛ= 0 { (i ) }' E NΛ MM−µ E  11 12 = 0 −µ ′ MM21 22 E NΛ From the first row the following condition can be derived:

(M11 −µ) E + ME12 ′ =0 Since the investigated system is linear and invariant for the phase, the absolute amplitude and phased of the E -component of the eigenvector can be chosen arbitrarily. Here we take E =1 and get for the full eigenvector E 1 = E . ' NΛ E (µ−MM11) / 12 NΛ 

If field values of the Bloch mode, i.e. the function Exkk(+Λ ) = Ex (), inside xx the layers are desired, they can be computed by using the matrix formalism and the above eigenvector (,EE′ )NΛ .

Physical properties of infinite multilayer systems We are interested in the reflection properties of an infinite Bragg mirror. Reasoning in terms of the electric field and derivative at the interface ( x = 0), E0 and E’0, we can express the reflectivity of the Bragg mirror as 2 E E iE ′ E iE ′ ρ= R with E =00 + and E =00 − from before E R 22k I 22k I sx sx 2 kE E'  ρ= sx 00+i kE E' sx 00-i With our knowledge of the eigenvector from above we can compute the reflectivity:

2 µ−M11 2 k µ−M kE E' sx +i M 11 sx 00+i 12 EE'00=  ρ= ' = M kE E µ−M11 12 sx 00−i k sx −i M12 According to this formula two scenarios are possible:

A) total internal reflection r = 1 Hence µ has to be real which results in the condition Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 173

(MM+ )  11 22 ≥1 2 with [for our example (,εε121 ,dd , 2 )] k M= cos( kd) cos( kd) − 2x sin( kd) sin( kd) 11 11x 2 x 2 k 11 xx 2 2 1x k M= cos( kd) cos( kd) − 1x sin( kd) sin( kd) . 22 11x 2 x 2 k 11 xx 2 2 2x This defines the so-called band gap, i.e. frequencies of excitation for which no propagating solutions exist.

B) propagating normal modes Hence µ must be complex which results in the condition (MM+ )  11 22 <1 2 We can compute the explicit dispersion relation:

2 (MM11++ 22 ) ( MM11 22 ) µ=exp (iK Λ) = ± −1. 22

2 µ=exp(iiKK Λ=) cos( Λ+) sin( KK Λ=) cos( Λ±) { cos( K Λ)} − 1 (MM+ )  cos{Kk( ,ωΛ) } = 11 22 z 2 In infinite periodic media, only if the Bloch vector K fulfills this DR the Bloch wave is a solution to Maxwell’s equations. This is in complete analogy to 22ω2 plane waves in homogeneous media with the DR kk=2 εω−( ) . xzc

Interpretation (MM+ ) • For the case of total internal reflection (µ real, 11 22 ≥1) the Bloch 2 vector K is complex, µ =exp(iiK Λ) = exp( ℜ( KK) Λ) exp( −ℑ( ) Λ). Hence ℜKk( , ωΛ) = n π and { z }

2 n (MM11++ 22 ) ( MM11 22 ) ℑ{Kk( , ωΛ) } =− ln( − 1) ±  −1 . z 22  The ± accounts for exponentially damped and growing solution, as we usually expect in the case of complex wave vectors and evanescent waves. Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 174

• There is an infinite number of so-called band gaps or forbidden bands, because n =1... ∞ . These band gaps are interesting for Bragg mirrors and Bragg waveguides. The band gaps correspond to "forbidden" frequency ranges, where no propagating solution exists. • The limits of the bands are given by ℜKk( , ωΛ) = n π and ℑKk( ,0 ωΛ) = { z } { z }  Kk( ,/ω=) n π Λ z • Outside the band gaps, i.e., inside the bands, we find propagating solutions which have different properties than the normal modes in homogeneous media (different dispersion relation).  We can exploit the strong curvature, i.e. frequency dependence, of DR for, e.g., dispersion compensation or diffraction free propagation

Special case: normal incidence In general there is a complex interplay between the angle of incidence and frequency of light determining the reflection properties of multilayer systems. Therefore let us have a look at the simpler case of normal incidence (k = 0). z In a graphical representation of the dispersion relation for k = 0 it is common z to use the following dimensionless quantities ω K 2π and with the scaling constant G = cG G Λ Examples for normal incidence 0.7

Re k Re k 0.4 0.6 Im k Im k 0.5 0.3

G 0.4 G c / c / ω ω 0.2 0.3

0.2 0.1 0.1

0 0 0 0.25 0.5 0.75 1 1.25 0 0.25 0.5 0.75 1 1.25 1.5 k/G k/G

n1=1.4, d1=0.5Λ n1=1.4, d1 = 17/24 Λ n2=3.4, d2=0.5Λ n2=3.4 , d2 =7/24 Λ

It is common to use the reduced band structure - Brillouin zone, where the information for all possible Bloch vectors is mapped onto the Bloch vectors in the following interval −≤0.5 (kG / ) ≤ 0.5 . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 175

1.2 Re k 0.4 Im k 1

0.3 G c

/ 0.8 ω

0.2 G c /

ω 0.6 0.1 0.4 0 0 0.25 0.5 0.75 1 1.25 k/G 0.2

Λ n1=1.4, d1=0.5 0 n2=3.4, d2=0.5Λ 0 0.25 0.5 k/G

Λ K Because of eiK we need only −π ≤K Λ ≤ π  ≤ 0.5to describe the disper- G sion relation. Inside the band gap, we find damped solutions:

1 n =1.0, n =1.4, 0.9 0 1 0.8 0.7 16 d/d 1 n o i 0.6 s s i

m 0.5 s n a r 0.4 T 0.3 0.2 0.1 0 0.3 0.4 0.5 0.6 0.7 ω/cG field in a Bragg-mirror Transmission

Let us quantify the damping. In our example (n1,n2,d1,d2) we have

(MM11+ 22 ) ω ω1 nn21 ωω =cosnd11 cos nd2 2 −+sin nd11 sin nd2 2 22c c nn12  c c In the middle of the first band gap (optimum configuration for high reflection) ωω π we have BBnd= nd = , with ω being the Bragg frequency, and cc11 2 2 2 B

(MM11+ 22 ) 1 nn therefore =−21 + <−1. If we plug this (for n =1) in our 22nn12 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 176

expression for ℑ()K and assume a small index contrast nn21− <<( nn 21 + ) we find

nn21− Λℑ()K max ≈ 2 (do derivation as an exercise) nn21+

 Damping is proportional to index contrast of the subsequent layers nn21− (MM+ ) The spectral width of the gap 11 22 ≥1 is then 2 2ω ∆ω ≈B Λℑ()K (do derivation as an exercise) gaπ π max  Spectral width is proportional to index contrast as well. 7.3.4 Fabry-Perot-resonators In this chapter we will treat a special multi-layer system, the so-called Fabry- Perot-resonator. To construct a Fabry-Perot-resonator, one can start from highly reflecting periodic multi-layer system (Bragg reflector). If one changes just a single layer somewhere in the middle of the otherwise periodic layer system, a so-called cavity is formed, and we are interested in the forward and backward propagating fields of the entire layer system. While the single layer which is distinguished from the periodic stack forms the cavity, the other layers function as mirrors, and may be periodic multilayer- systems or metal films. Fabry-Perot-resonators are very important in optics, as they appear as: • Fabry-Perot- interferometer • laser with plane mirrors  Fabry-Perot-Resonator with active medium inside the cavity • nonlinear optics  high intensities inside the cavity  nonlinear optical effects for low intensity incident light: − bistability − modulational instability − pattern formation, solitons Here, we want to compute the transmission properties of the resonator for arbitrary plane mirrors. This task could be achieved employing the matrix method which we developed in the previous sections. However we will take a different approach to achieve deeper physical insight into the cavity's behavior. For simplicity, we will restrict ourselves to TE-polarization. The figure shows our setup with two mirrors at x = 0 and xD= , characterized by coefficients of reflection and transmission R0 , T0 , RD , and TD . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 177

cladding

substrate

− EE, and E ,  amplitudes of incident, reflected and transmitted IR T external fields in substrate and cladding

− EE+−,  amplitudes of internal fields running forward and backward inside the cavity Using the known coefficients of reflection and transmission of the two mirrors, we can eliminate E+ and E− by connecting the field amplitudes: A) At the lower mirror inside the cavity: T ER+=E (0)E (0) 00I −+ B) At the upper mirror outside the cavity: E= TE() D TD+ E T And with ED( )= E (0) exp( kD) E+ (0) =exp( − kD) + + i fx T i fx D C) At the upper mirror inside the cavity: R E() D= RE () D = D E −+DTT D and with E (0) = ED( )exp( k D) − − i fx R E (0) = D EDepx ( k ) − T T i fx D

D) we substitute E+ (0) and E− (0) in A) TE+= REE(0) (0) 00I −+

RE TE +=RDT Eexp( kD) exp(− kD) 00I TTT iifx fx DD 1  E={exp( −−kD) RRexp( kD)} E . I TT iifx 0 D fx T 0 D Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 178

Thus, the coefficient of transmission for the whole FP-resonator expressed by the coefficients of the mirrors ( R0 , T0 , RD , and TD ), the cavity properties and ω2 the angle of incidence ( D , kk= ε−2 ) reads: fx c2 fz

E TT0 exp( k D) T =T = D i fx . TE E 1− R R exp( 2 kD) I 0 D i fx This is the general transmission function of a lossless Fabry-Perot resonator. In general, the mirror coefficients are complex and the fields get certain phase shifts (RT , )= ( R , T ) exp ϕRT, . Obviously, only the phase shifts 0,D 0,D (i 0D, ) induced by the coefficients of reflection RR0 , are important for the 2 D transmissivity of the FP resonator τ  T . TE TE For given RR,, TT, and ϕϕRR, , the general transmissivity of a lossless 0D 0D 0 D Fabry-Perot resonator reads:

2 2 22 TT 0D TT= = 2 2 TE 1+R R −2 R R cos(2 kD++ϕϕRR) 0 D 0D  (fx ( (0 (D δ

k 2 τ= cx T . k sx Here we introduced the phase-shift δ, which the field acquires in one round- trip in the cavity. Discussion Depending on whether the two mirrors have identical properties we distinguish between symmetric and asymmetric FP-resonators. a) asymmetric FP-resonator 2 2 k TT cx 0D τ= 2 2 k 1+−R R2 RR cosδ sx 0 D 0D Because we assume no losses we can use energy conservation at each mirror to eliminate T : 0,D

2222k k fx TT=−−sx 11 R R 0D kk( 0) ( D) fx cx k 2 2 =−−sx 11RR k ( 0D)( ) cx Note: τ and ρ for a lossless mirror are the same for both sides of the mirror. For lossy mirrors only τ is the same, ρ is then side-dependent. For discussing the effect of the phase shift δ we rewrite Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 179

dd d cosδ= cos22 − sin = 1 − 2sin 2 22 2 Plugging everything in we get

2 2 1−−RR1 ( 0D)( ) τ= 2 2 1+−R R2 RR 1− 2sin 2 δ 0 D 0D( 2 ) −1 2 (1− RR) 4 R R δ = 0D + 0 D 2 2222sin . 11− R −R 11 −− RR2 ( 0D)( ) ( 0D)( ) and with 2 2 ρ=RR,ρ= 0 0D D −1 2 (1− ρρ0 ) 4 ρρ δ τ=D + 0 D sin2 . (11−ρ00)( −ρ ) ( 11−ρ )( −ρ ) 2 DD b) symmetric FP-resonator (Airy-formula for transmissivity) 2 2 ρ=RR = =ρ=ρ, ϕ=ϕ=ϕRRR m 0D00 D D −1  22 (11−ρmm) ( −ρ ) 4ρδ2 τ= =22 + sin 2 2 δ 2 −ρ + ρ (11−ρmm) ( −ρ ) (1mm) 4 sin  (  ( 2 =1

−1 δ τ=1 +F sin 2 2 4ρ δ m = +ϕ with F = 2 and kDfx (1−ρm ) 2 The Airy-formula (see also Labworks script, where phase shifts ϕ due to the mirrors are not considered) gives the transmissivity of a symmetric, lossless Fabry-Perot-resonator. Only for this case we can get the maximum transmissivity τ=1 for δ/2=πn . Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 180

Remarks and conclusions • We can do an analog calculation for TM-polarization  RTM resp. ρTM • Resonances of the cavity with τ=1 occur for δ/2=kD +ϕ= m π with MAX fx MAX 2π k= nn2 −ϕ 22sin fx λ FSI ϕ mπ−ϕ λ (m − π )  D = = MAX k 2 nn2−ϕ 22sin fx FS I λ  cavity 2

where ϕI is the angle of incidence in the substrate • transmission properties of a given resonator depend on ϕ and λ. I 1 • minimum transmission is given as τ= MIN 1+ F • it is favorable to have large F , e.g.:

4ρm 100 =F = 2 (1−ρm )

41( −τm ) 4 ρmm=−1 τ22≈ ≈100τm= 0.2ρ m= 0.8. ττmm • pulses and beams can be treated efficiently in Fourier domain:  e.g. TE: E(,,αβω ) = TE (,, αβω ) (,, αβω ) T TE I Fourier back transformation: E( xyt , , )= FT−1 [ E ( αβω , , )] TT • beams, because they always contain a certain range of incident angles ϕ , I they produce interference rings in the farfield output (or image of lens, like in labworks). • a quantity often used to characterize a resonator is the finesse: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 181

distancε bεtwεεn rεsonancε ∆ Φ= = full width at half maximum of rεsonancε ε with ε the FWHM and ∆=π the distance in rad between two resonances. To calculate the FWHM ε we can start from: −1  2 ε 1 1+Fm sin  π±  22 For narrow resonances (small line width ε ) we can write − 2 1 2 ε 1 ε 2 1+≈F   F =1 ε= 222 F ∆π ρ F= =F =π m ε 21−ρm • The line width ε (FWHM) is inversely proportional to the finesse Φ . • The Airy -formula can be expressed in terms of the finesse

−1 2 2Φ2 δ τ=1 + sin . π 2 • A Fabry-Perot-resonator can be used as a spectroscope. Then, we can ask for its resolution (here: normal incidence). Resonances (maximum transmission) occur at: kD+ϕ= m π reduced transmission by factor 1/2 Dk ε π →kD +ϕ± D = m π± = m π± 2 22Φ 2ππ  Dkn= Dλ = λΦ2 f D

Dλ λ λ ε  = λ λ nDΦ ΦD D f example: λ=5 ⋅ 10−7m, Φ=30,nD = 4 ⋅ 10−3m  ∆λ =2 ⋅ 10−12m f • The field amplitudes (here forward field) inside the cavity are given as: −2 E= TE() D Because T −−1 / (1−ρ ) Φ these intra-cavity fields can be TD+ very high  important for nonlinear effects • Lifetime of photons in cavity: Via the "uncertainty relation": ∆ωT =const. ≈1 it is possible to define a lifetime of photons inside the c cavity: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 182

1 π Dk = n Dω = c Φ ΦD

cnπ 1 ΦD D Dω =  T = = Φ  . nDΦ c Dωc π ε Φ 7.4 Guided waves in layer systems Finally, we want to explore our layer systems as waveguides. For many applications it is interesting to have waves which propagate without diffraction. This is crucial for integrated optics, where we want to guide light in very small (micrometric or smaller) dielectric layers (film, fiber), or optical communication technology where some light encoded information is transported over long distances. Moreover, waveguides are important in nonlinear optics, due to confinement and long propagation distances nonlinear effects become important. Here, we will treat wave-guiding in one dimension only because we restrict ourselves to layer-systems, but general concepts developed can be transferred to other settings, e.g. where the waveguide is a fiber. 7.4.1 Field structure of guided waves Let us first do some general consideration about the field structure of guided waves. We want to find guided waves in a layer system. In such systems, till now, we have solved the reflection and transmission problem: For given kd,,E ε we calculated EE. zIii, RT, Inside each layer we have plane waves −E (k,ω) exp ( k x+ kz −ω t) . ααz i xz The question is, how can we trap (or guide) waves within a finite layer system? A possible hint gives the effect of total internal reflection, where the transmitted field is: EE(xz , )= exp( kz) exp(−µ x) T Tzi c Obviously, in the case of TIR we have no energy flux in the cladding medium. If TIR is the key mechanism to guide light, can we have TIR on two sides, i.e. in cladding and substrate? And what about a single interface?

We will concentrate our discussion first on a system of layers, which we will call the core of the waveguide, and which we place between semi-infinite Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 183 substrate and cladding. The single interface, where the core is absent, will be considered at a later stage. According to our discussion the guided waves have the following field structure:

• plane wave in propagation direction ~ exp(ikz z ) • evanescent waves in substrate and cladding

~ exp[−µc (xD − )] cladding

~ exp(µxs ) substrate ω2 with µ =k 2 − ε() ω> 0 sc,, zc2 sc

2 2 ω  1. condition: k >max{ εω, ( )} z c2 sc • oscillating solution (standing wave) in the core (layers, fiber core, film)

~A sin( kxfx )+ B cos( kxfx )

ω2 with kk= ε() ω−2 > 0 iixzc2

2 2 ω  2. condition: k evanescent field profiles in the core and leaves only oscillating radiation-type fields as a possible solution. In summary, the z -component of the wave vector of guided waves has to fulfill: ωω   maxnksc, << z max  ni cc i  The field structure in substrate and cladding is given as: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 184

EE(xz , )= exp( kz) exp −µ( x − D) x > D C Tzi c EE(xz , )= exp( kz) exp( µ< x) x 0 S R i zs 7.4.2 Dispersion relation for guided waves If we compare the principal shapes of the fields for the guide wave to our usual reflection transmission problem, we see that for guided waves the reflected and transmitted (evanescent) field exist for zero incident field

EI → 0 . In the following we will use this condition to derive the dispersion relation for guided waves. Thus we have EE,0≠ for E → 0 . Consequently the coefficients for TR I reflection and transmission are TE,TM TE,TM TE,TM EETE,TM  TR= TR, =  with E → 0 EEI II  and we find RT, →∞ in the case of guided waves. In this sense, guided waves are resonances of the system. Let us compare to a driven harmonic oscillator F x = 22, x = action, F = cause ω −ω0

In the case of resonance (ω=ω0 ) we get action for infinitesimal cause. Hence, we can get the dispersion relation of guided waves by looking for a vanishing denominator in the expressions for RT, This reasoning is a general principle in physics: The poles of the response function (or Greens function) are the resonances of the system. Furthermore, a resonance of a system is equivalent to an eigenmode of the system, which is somehow localized. We know the coefficients of reflection and transmission for a layer system from before, and they have the same denominator:

F (αkM22 −α kM11 ) − ( M21 +α k α kM12 ) R =R = s sx c cx i s sx c cx F (αkM +α kM) +( M −αk α kM) I s sx 22 c cx 11 i 21 s sx c cx 12 The pole is then given as: → (αkM +α kM) +( M −α k α kM)  0 s sx 22 c cx 11 i 21 s sx c cx 12 Because we have evanescent waves in substrate and cladding the imaginary valued ksx and kcx can be expressed as real numbers with ωω22 k=µ=−εω i ik22(), k =µ=−εω i ik () sx s zcc22 s cx c z c Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 185

We can write the general dispersion relation of guided modes in an arbitrary layer system

TE,TM TE,TM 1 TE,TM α µ TE,TM s s MMMM11 +µα 12 + 21 + 22  0 s s α µ α µ cccc

Here, as usual, we have α=TE 1, α=εTM 1 for the two independent polarizations. The waveguide dispersion relation gives a discrete set of solutions, so-called waveguide modes. For given εω,,d we get k (ω). ii zν We can see that the dispersion relation of guided modes depends on the material's dispersion in the individual layers as well as in substrate and cladding according to the dispersion relation of homogeneous space as ω2 k2()ω= εω= () kk22 + . ic2 iixz In addition to the material's dispersion in the layers the dispersion relation of guided modes depends on the geometry of the waveguide's layer system k (ω, ) z geometry In the case of guided waves it is easy to compute the field (mode profile) inside the layer system:

• take kz from dispersion relation • in the substrate we have: ∂ F( x )= F exp( µxs ), Gx( ) = α { Fexp( µxs )} ∂x

Hence, for given F(0) we get GF(0)= αµs (0) ( F(0) → free parameter) FF  1 =MMˆˆ()x  = () xF (0). GGαµ x 0 s

Analogy of optics to the stationary Schrödinger equation in QM Optics (e.g. TE-polarization) Quantum mechanics

2 2 2 d ω 2 d 2m 2m 2 + 2 ε(x) Ex()= k Ex () ↔ 2 − 2 V (x) ψ=()xx− 2 E ψ () dx c  z dx   guided waves ↔ discrete energy eigenvalues 2 2 ω k >εmax{ , } ↔ EV< ext z c2 sc Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 186

Tunnel effect ω2 k 2 >ε ↔ EV< z c2 Film Barriere

7.4.3 Guided waves at interface - surface polariton Let us first have a look at the most simple case where the guiding layer structure is just an interface.

Our condition for guided waves is that on both sides of the interface we have ω2 evanescent waves: k 2 >ε, because zc2 cs, ω2 µ=k 2 − ε >0 s,c zc2 cs, The general dispersion relation we derived before reads

TE,TM TE,TM 1 TE,TM αµ TE,TM MM+α µMM + + ss  0 11 ss 12 αµ21 αµ 22 cc cc 1 0 and with the matrix for a single interface: Mˆ =  0 1 αµ 10+=ss we get the dispersion relation αµ cc A) TE-polarization (α=1) µ +µ =0, → no solution because µ,0 µ> sc cs B) TM-polarization (α=1/ ε ) Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 187

µµ cs+=0 εε cs with µ,0 µ>  ε ⋅ε <0 , cs cs → on of the media has to have negative ε (dielectric near resonance or metal)

In dielectrics ω <ω<ω we can find surface-phonon-polaritons. 0T() L In metals ω<ω we can find surface-plasmon-polaritons. p

Remark Surface polaritons occur in TM polarization only, similar to the phenomenon kk of Brewster-angle (no reflection for cx−= sx 0, ) εε cs Let us now compute the explicit dispersion relation for surface polaritons.

W.o.l.g. we assumeεs () ω< 0, and because εωs () is near a resonance it will show a much stronger ω dependence than εc .

22 (µµcsε=ε) ( sc) 22 2 2ωω 2  22  εω−ε=ε−εωs ( )kkz22 c  cz s( )  cc  

ω εsc( ωε) kz (ω=) c εcs +ε( ω) There is a second condition for existence of surface polaritons: ε +ε( ω) <0 cs Conclusion: ω ε( ωε) k (ω=) sc z c ε( ω) +ε sc ε( ω<) 0 TM polarization s ε>0 ε( ω) >ε c sc Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 188

Surface polaritons may have very small effective wavelengths in z-direction: 22ππε (ωε) ε =1, ε( ω≈→) 1 k ( ω) = sc= → λ << λ cs z λλεω( ) + ε eff sceff

7.4.4 Guided waves in a layer – film waveguide The prototype of a waveguide is the film waveguide, where the waveguide ω2 consists of one guiding layer with ε( ω>) k 2 ( ω) . c2 f z Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 189

Such film waveguides are the basis of integrated optics. Typical parameters are: a few wavelengths

Fabrication of film waveguides can be achieved by coating, diffusion or ion implantation. The matrix of a single layer (film) is given as:

From this matrix we can compute the dispersion relation for guided modes: Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 190

1 αµ MM+α µ + M + ss M 0 11 ss 12 αµ21 αµ 22 cc cc

αµ αk αµ cos(kd) +−+=sssin( kd) ffxsin( kd) sscos( kd) 0 fx αk fx αµ fx αµ fx ffx cc cc

αµ 1+ ss sin(kd) αµ αk ( α µ +α µ ) fx =tan(kd) = cc = f fx s s c c cos(kd) fx αk αµ α22k −α α µ µ fx f fx− s s f fx c s c s αµ αk c c f fx k µµ fx s+ c αα α tan(kd) = fc s fx k 2 µµ fx− c s αα α2 cs f Here: TE-Polarisation (α=1) k (µ +µ ) tan(kd) = fx s c , fx k 2 −µ µ fx c s This waveguide dispersion relation is an implicit equation for k . For given z frequency ω and thickness d we get several solutions with index k zν ω Here is an example for fixed frequency ω, the effective index nkeff= z /  c versus the thickness d : Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 191

We can see in the figure that for large thickness d we have many modes. If we decrease d, more and more modes vanish at a certain cut-off thickness. Definition of cut-off: a guided mode vanishes  cut-off (here w.o.l.g. The idea of the cut-off is that a mode is not guided anymore. Guiding means evanescent fields in the substrate and cladding, so cut-off means

 no guiding

We can plug this cut-off condition in the DR:

with parameter of asymmetry a:

 we can define a cut-off frequency for when we keep d fix  we can define a cut-off thickness for when we keep fix Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 192

In a symmetric waveguide the fundamental more (ν=0 ) has cut-off = 0! If we plot the dispersion curves for each mode we get a graphical representation of the dispersion relation:

7.4.5 How to excite guided waves Finally, we want to address the question how we can excite guided waves. In principle, there are two possibilities, we can adapt the field profile or the wave vector (k ) z A) adaption of field  front face coupling

Then, inside the waveguide we have (without radiative modes): Exz( , )= aE ( x )exp( k z) ∑ νν i zν ν

≈  Ex(,0)∑ aEνν () x∫ Eµ () x ν

k ∞ 2 :P = zν E() x dx with νν∫ −∞ 2ωµ0

k ∞ a = zν Ex()Ex ()dx, ν ∫ −∞ in ν 2ωµ0Pν  mode ν couples to the incident field E (0) with amplitude a . in ν  Gauss-beam couples very good to the fundamental mode B) adaption of wave vector  coupling through the interface Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 193

We know that kz is continuous at interface. The condition for the existence of guided modes is ω k >ε z c c,s but dispersion relation for waves in bulk media dictates ω2 ω kk=ε − 2 < ε z c2 c, s s, cx c cs,  We got a problem! There are two solutions: i) coupling by prism we bring a medium with ε >ε (prism) near the waveguide pf ω ω ω2 k <ε k <ε kk= ε−2 >0 . zfc zpc px c2 p z

 light can couple to the waveguide via optical tunneling: ATR ('attenuated total reflection') ii) coupling by grating Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 194

grating on waveguide (modulated thickness of layer d): dz( ) = d +ς( z) 2π ς=( z) Asin( gz) mit g = P P− period coupling works for m’th diffraction order:

kzv= k z + mg ω =ns sin φ+ mg . c