sign in sign up
<<
Home , Geometrical optics

Script Fundamentals of Modern , FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 1 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 2

Fundamentals of Modern Optics 2.6 (The Kramers-Kronig relation, covered by lecture Structure of Matter) ...... 100 3. theory ...... 104 Winter Term 2014/2015 3.1 Interaction with masks ...... 104 3.2 Propagation using different approximations ...... 105 Prof. Thomas Pertsch 3.2.1 The general case - small aperture ...... 105 Abbe School of 3.2.2 Fresnel approximation (paraxial approximation) ...... 105 3.2.3 Paraxial Fraunhofer approximation (far field approximation) ...... 106 Friedrich-Schiller-Universität Jena 3.2.4 Non-paraxial Fraunhofer approximation ...... 108 3.3 Fraunhofer diffraction at plane masks (paraxial) ...... 108 3.3.1 Fraunhofer diffraction pattern ...... 108 3.4 Remarks on Fresnel diffraction ...... 113 Table of content 4. - optical filtering ...... 114 0. Introduction ...... 4 4.1 Imaging of arbitrary optical field with thin ...... 114 4.1.1 Transfer function of a thin lens ...... 114 1. optics - geometrical optics (covered by lecture Introduction to Optical 4.1.2 Optical imaging ...... 115

Modeling) ...... 16 4.2 Optical filtering and image processing ...... 117 1.1 Introduction ...... 16 4.2.1 The 4f-setup ...... 117 1.2 Postulates ...... 16 4.2.2 Examples of aperture functions ...... 119 1.3 Simple rules for propagation of ...... 17 4.2.3 Optical resolution ...... 120 1.4 Simple optical components...... 17 5. The polarization of electromagnetic waves ...... 123 1.5 in inhomogeneous media (graded-index - GRIN optics) ...... 21 5.1 Introduction ...... 123 1.5.1 Ray equation ...... 21 5.2 Polarization of modes in isotropic media ...... 123 1.5.2 The eikonal equation ...... 23 5.3 Polarization states ...... 124 1.6 optics ...... 24 1.6.1 The ray-transfer-matrix ...... 24 6. Principles of optics in crystals ...... 126 1.6.2 Matrices of optical elements ...... 24 6.1 Susceptibility and dielectric tensor ...... 126 1.6.3 Cascaded elements ...... 25 6.2 The optical classification of crystals ...... 128 6.3 The index ellipsoid ...... 129 2. Optical fields in dispersive and isotropic media ...... 26 6.4 Normal modes in anisotropic media ...... 130 2.1 Maxwell’s equations ...... 26 6.4.1 Normal modes propagating in principal directions ...... 131 2.1.1 Adaption to optics ...... 26 6.4.2 Normal modes for arbitrary propagation direction ...... 132 2.1.2 Temporal dependence of the fields ...... 31 6.4.3 Normal surfaces of normal modes ...... 136 2.1.3 Maxwell’s equations in Fourier domain ...... 31 6.4.4 Special case: uniaxial crystals ...... 138 2.1.4 From Maxwell’s equations to the wave equation ...... 32 2.1.5 Decoupling of the vectorial wave equation ...... 33 7. Optical fields in isotropic, dispersive and piecewise homogeneous media 2.2 Optical properties of matter ...... 34 ...... 141 2.2.1 Basics ...... 34 7.1 Basics ...... 141 2.2.2 Dielectric polarization and susceptibility ...... 37 7.1.1 Definition of the problem ...... 141 2.2.3 Conductive current and conductivity ...... 39 7.1.2 Decoupling of the vectorial wave equation ...... 142 2.2.4 The generalized complex dielectric function ...... 40 7.1.3 Interfaces and symmetries ...... 143 2.2.5 Material models in time domain ...... 44 7.1.4 Transition conditions ...... 143 2.3 The Poynting vector and energy balance ...... 46 7.2 Fields in a layer system  matrix method ...... 144 2.3.1 Time averaged Poynting vector ...... 46 7.2.1 Fields in one homogeneous layer ...... 144 2.3.2 Time averaged energy balance ...... 47 7.2.2 The fields in a system of layers ...... 146 2.4 Normal modes in homogeneous isotropic media ...... 49 7.3 – transmission problem for layer systems ...... 148 2.4.1 Transversal waves ...... 50 7.3.1 General layer systems ...... 148 2.4.2 Longitudinal waves ...... 51 7.3.2 Single interface ...... 154 2.4.3 Plane wave solutions in different frequency regimes ...... 52 7.3.3 Periodic multi-layer systems - Bragg- - 1D photonic crystals ...... 161 2.4.4 Time averaged Poynting vector of plane waves ...... 58 7.3.4 Fabry-Perot-resonators ...... 168 2.5 Beams and pulses - analogy of diffraction and ...... 58 7.4 Guided waves in layer systems ...... 174 2.5.1 Diffraction of monochromatic beams in homogeneous isotropic media ...... 60 7.4.1 Field structure of guided waves ...... 174 2.5.2 Propagation of Gaussian beams ...... 71 7.4.2 for guided waves ...... 175 2.5.3 Gaussian optics ...... 78 7.4.3 Guided waves at interface - surface polariton ...... 177 2.5.4 Gaussian modes in a resonator ...... 82 7.4.4 Guided waves in a layer – film waveguide ...... 179 2.5.5 Pulse propagation ...... 87 7.4.5 how to excite guided waves...... 183 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 3 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 4

8. Statistical optics - coherence theory ...... 186 8.1 Basics ...... 186 0. Introduction 8.2 Statistical properties of light...... 188 8.3 Interference of partially coherent light ...... 190  'optique' (Greek)  lore of light  'what is light'?  Is light a wave or a particle (photon)?

D.J. Lovell, Optical Anecdotes

 Light is the origin and requirement for life  photosynthesis  90% of information we get is visual A) Origin of light  atomic system  determines properties of light (e.g. statistics, frequency, width)  optical system  other properties of light (e.g. intensity, duration, …)  invention of laser in 1958  very important development

Schawlow and Townes, Phys. Rev. (1958).

 laser  artificial light source with new and unmatched properties (e.g. coherent, directed, focused, monochromatic)  applications of laser: fiber-communication, DVD, surgery, microscopy, material processing, ... Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 5 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 6

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

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

B) Propagation of light through matter  light-matter interaction Hole “drilled” with a fs laser at Institute of Applied , FSU Jena. effect dispersion diffraction absorption ↓ ↓ ↓ ↓ governed by frequency spatial center of 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: , polymers, semiconductors, compounded media (effective media, photonic crystals, meta-materials)

Two-dimensional photonic crystal membrane. Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 7 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 8

D) Optics in our daily life E) Optics in telecommunications  transmitting data (Terabit/s in one fiber) over transatlantic distances

A small story describing the importance of light for everyday life, where all things which rely on optics are marked in red.

1000 m telecommunication fiber is installed every second. Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 9 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 10

F) Optics in medicine, life sciences G) Optical 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_2014-10-24s.docx 11 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 12

H) Micro- and nano-optics I) Relativistic 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_2014-10-24s.docx 13 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 14

K) What is light? L) Schematic of optics  electromagnetic wave propagating with speed of c= 3*108 m/s  leading to evolution of:  amplitude and phase  complex description  polarization  vectorial field description  coherence  statistical description electromagnetic optics Spectrum of Electromagnetic Radiation Region Wavelength Wavelength Frequency Energy wave optics (nanometers) (centimeters) (Hz) (eV) Radio > 108 > 10 < 3 x 109 < 10-5 8 5 9 12 -5 Microwave 10 - 10 10 - 0.01 3 x 10 - 3 x 10 10 - 0.01 geometrical optics Infrared 105 - 700 0.01 - 7 x 10-5 3 x 1012 - 4.3 x 1014 0.01 - 2 Visible 700 - 400 7 x 10-5 - 4 x 10-5 4.3 x 1014 - 7.5 x 1014 2 - 3 Ultraviolet 400 - 1 4 x 10-5 - 10-7 7.5 x 1014 - 3 x 1017 3 - 103 X-Rays 1 - 0.01 10-7 - 10-9 3 x 1017 - 3 x 1019 103 - 105  geometrical optics Gamma Rays < 0.01 < 10-9 > 3 x 1019 > 105   << size of objects  daily experiences  optical instruments, optical imaging  intensity, direction, coherence, phase, polarization, photons

 wave optics    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  intensity, direction, coherence, phase, polarization, photons

 in this lecture  electromagnetic optics and wave optics  no quantum optics  subject of advanced lecture Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 15 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 16

M) Literature  Fundamental 1. Ray optics - geometrical optics (covered by 1. Saleh, Teich, 'Fundamenals of Photonics', Wiley, 1992 lecture Introduction to Optical Modeling) 2. Mansuripur, 'Classical Optics and its Applications', Cambridge, 2002 The topic of “Ray optics – geometrical optics” is not covered in the course 3. Hecht, 'Optik', Oldenbourg, 2001 “Fundamentals of modern optics”. This topic will be covered rather by the 4. Menzel, 'Photonics', Springer, 2000 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. 5. Lipson, Lipson, Tannhäuser, 'Optik'; Springer, 1997 6. Born, Wolf, 'Principles of Optics', Pergamon 1.1 Introduction 7. Sommerfeld, 'Optik'  Ray optics or geometrical optics is the simplest theory for doing optics.

 In this theory, propagation of light in various optical media can be  Advanced described by simple geometrical rules. 1. W. Silvast, 'Laser Fundamentals',  Ray optics is based on a very rough approximation (0, no wave 2. Agrawal, 'Fiber-Optic Communication Systems', Wiley phenomena), but we can explain almost all daily life experiences 3. Band, 'Light and Matter', Wiley, 2006 involving light (shadows, mirrors, etc.). 4. Karthe, Müller, 'Integrierte Optik', Teubner  In particular, we can describe optical imaging with ray optics approach. 5. Diels, Rudolph, 'Ultrashort Laser Pulse Phenomena', Academic  In isotropic media, the direction of rays corresponds to the direction of 6. Yariv, 'Optical Electronics in modern Communications', Oxford energy flow. 7. Snyder, Love, 'Optical Waveguide Theory', Chapman&Hall What is covered in this chapter? 8. Römer, 'Theoretical Optics', Wiley,2005.  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 (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 nds()r A Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 17 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 18

D) Fermat’s principle

B nds()r 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 ii) Parabolic  Rays of light propagate on straight lines.  Parallel rays converge in the focal point (focal length f). B) Reflection by a mirror (metal, dielectric coating)  Applications: Telescope, collimator  The reflected ray lies in the plane of incidence.  The angle of reflection equals the angle of incidence. C) Reflection and by an interface  Incident ray  reflected ray plus refracted ray  The reflected ray obeys b).  The refracted ray lies in the plane of incidence.

iii) Elliptic mirror

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

 The angle of refraction 2 depends on the angle of incidence 1 and is given by Snell’s law:

nn1122sin sin   no information about amplitude ratio.

1.4 Simple optical components iv) Spherical mirror A) Mirror  Neither imaging like elliptical mirror nor focusing like parabolic mirror i) Planar mirror  parallel rays cross the optical axis at different points  Rays originating from P1 are reflected and seem to originate from P2.  connecting line of intersections of rays  caustic Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 19 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 20

 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 C) Spherical interface (paraxial) imaging optical element. paraxial rays emitted in point P1 are reflected and converge in point P2  paraxial imaging

nnny121 21  (*) nnR22

11 2  (imaging formula) zz12() R paraxial imaging: imaging formula and magnification m = -z2 /z1 (proof given in exercises) nn nn B) Planar interface 12 21 (imaging formula) zz12 R Snell’s law: nn1122sin sin  nz nn  12 for paraxial rays: 11 2 2 m  (magnification) nz21  external reflection ( nn ): ray refracted away from the interface 12 (Proof: exercise)  internal reflection (nn ): ray refracted towards the interface 12  if paraxiality is violated  aberration  total internal reflection (TIR) for:  rays coming from one point of the object do not intersect in one point  n of the image (caustic)   sin sin   2 2 1TIR D) Spherical thin lense (paraxial) 2 n1 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 21 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 22

variation of the path: rr()s   ()s  two spherical interfaces (R1, R2, ) apply (*) two times and assume BB y=const ( small) Lndsnds   AA

nngrad r

22 ds drr  d d r 222 y 111 ddddrrrr2  d  r 21 with focal length: n 1   f fRR 12 ddrr ds12 ds 111 z ds ds  (imaging formula) m  2 (magnification) zz12 f z1 ddrr ds1 ds (compare to spherical mirror) ds ds

ddrr ds 1.5 Ray tracing in inhomogeneous media (graded-index ds ds - GRIN optics) B ddrr  n()r - continuous function, fabricated by, e.g., doping Lnndsgrad r  ds ds A integration by parts and A,B fix  curved trajectories  graded-index layer can act as, e.g., a lens B ddr grad nn r ds 1.5.1 Ray equation A ds ds Starting point: we minimize the optical path or the delay (Fermat) L 0 for arbitrary variation B ddr nds()r 0 gradnn  ray equation A ds ds computation: Possible solutions: B A) trajectory Lnsds r   22 A 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_2014-10-24s.docx 23 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 24

ddxdn 1.6 Matrix optics nxyz,,  dz dz dx  technique for paraxial ray tracing through optical systems  propagation in a single plane only ddydn  rays are characterized by the distance to the optical axis (y) and their nxyz,,  dz dz dy inclination ()  two algebraic equation  2 x 2 matrix B) homogeneous media Advantage: we can trace a ray through an optical system of many elements by multiplication of matrices.  straight lines C) graded-index layer n(y) - paraxial, SELFOC 1.6.1 The ray-transfer-matrix dy paraxial  1 and dz ds dz

2222 1 2 2 ny() n0  1 y ny ()n0  1  y for a 1 2 ddyddydydydny 221() ny  ny ny ds ds dz dz dz22 dz n y dy

0 yz()y0 cos z sin z 2  dy 2 for n(y)-n0<<1:  y  dz2 dy ()zyzz 00 sin cos dz in paraxial approximation:

yAyB211 yy21AB   AB    M  21CD   CD 21Cy  D  1

A=0: same   same y2  focusing 1.5.2 The eikonal equation 1 D=0: same y1  same 2  collimation  bridge between geometrical optics and wave  eikonal S(r) = constant  planes to rays 1.6.2 Matrices of optical elements  from S(r) we can determine direction of rays  grad S(r) (like potential) A) free space

2 2 gradSn r r  Remark: it is possible to derive Fermat’s principle from eikonal equation  geometrical optics: Fermat’s or eikonal equation 1 d M   BB 01  SSrrBA grad Sdsnds r   r AA B) refraction on planar interface eikonal  optical path length  phase of the wave Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 25 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 26

2. Optical fields in dispersive and isotropic media 10 M  2.1 Maxwell’s equations 0 nn 12 Our general starting point is the set of Maxwell’s equations. They are the C) refraction on spherical interface basis of the electromagnetic approach to optics 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 10 M   electromagnetic phenomena at different frequency ranges. Even though nnnRnn 212 12 Maxwell's equations are valid for all frequencies, the physics of light matter interaction is different for different frequencies. Since light matter interaction D) thin lens must be included in the Maxwell's equations to solve them consistently, different ways have been established how to write down Maxwell's equations 10 for different frequency ranges. Here we follow a notation which was M  11f established for a convenient notation at frequencies close to visible light.  E) reflection on planar mirror 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 10 M  includes the interaction with any kind of matter based on the consideration of 01 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 F) reflection on spherical mirror (compare to lens) 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 homo- 10 geneously distributed response of the solid state on the excitation by light. In M   turn, also the electromagnetic fields are averaged over some adequate 21R volume. For optics this procedure is justified, since any kind of available 1.6.3 Cascaded elements 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

yN 1 ABy1  AB this averaging.  M  M=MN….M2M N 1 CD1  CD 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. Br(,)t rot(,)Erttt div(,) Dr  (,) r t ext

Dr(,)t rot(,)Hrtt j (,) r div(,) Br t  0 makr t  electric field Er(,)t [V/m] Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 27 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 28

 magnetic flux density Br(,)t [Vs/m2] or [tesla]  In optics at visible , we generally deal with non-magnetizable or magnetic induction media. Hence we can assume Mr(,)t  0. Exceptions to this general pro-  dielectric flux density Dr(,)t [As/m2] perty are metamaterials which might possess some artificial effective magnetization Mr(,)t  0.  Hr(,)t [A/m] 3 Furthermore we need to introduce sources of the fields into our model. This is  external charge density ext (,)r t [As/m ] achieved by the so-called source terms, which are inhomogeneities and 2 hence they define unique solutions of Maxwell's equations.  macroscopic current density jmakr (,)r t [A/m ]  free charge density Auxiliary fields 3  (,)r t [As/m ] The "cost" of the introduction of macroscopic Maxwell's equations is the ext occurrence of two additional fields, the dielectric flux density Dr(,)t and the  macroscopic current density consisting of two contributions magnetic field . These two fields are related to the electric field 2 Hr(,)t Er(,)t jmakr(,)r ttt j cond (,)r  j conv (,)r [A/m ] and magnetic flux density Br(,)t by two other new fields.  conductive current density

Dr(,)ttt 0 Er (,) Pr (,) jcond (,)rEtf   1 Hr(,)ttt Br (,) Mr (,)  convective current density 0 jconv(,)rrvrttt  ext (,)(,)  dielectric polarization Pr(,)t [As/m2], 2  In optics, we generally have no free charges which change at speeds  magnetic polarization Mr(,)t [Vs/m ] comparable to the frequency of light: or magnetization  (,)r t  electric constant (vacuum permittivity) ext  0(,)0jrt t conv 8.854  1012 As/Vm 0 Thus only conductive currents are taken into account and in the future

 magnetic constant (vacuum permeability) we might just omit all subscripts by writing j(,)r tt jmakr+cond (,)r . 4107 Vs/Am  With the above simplifications, we can formulate Maxwell’s equations in the 0 context of optics: electric constant and magnetic constant are connected by the speed of Hr(,)t light in vacuum c as rot(,)Ertt  div(,) Er divP(,r t) 00t 1  Pr(,)t Er(,)t 0  c2 rot Hr(,)ttjr(,t)  div(,) Hr  0 0 t 0 t Light matter interaction  In optics, the medium (or more precisely the mathematical material In order to solve this set of equations, i.e. Maxwell's equations and auxiliary model) determines the dependence of the induced polarization on the field equations, one needs to connect the dielectric flux density Dr(,)t and electric field PE() and the dependence of the induced (conductive) the magnetic field Hr(,)t to the electric field Er(,)t and the magnetic flux current density on the electric field j()E . density Br(,)t . This is achieved by modeling the material properties by  Once we have specified these relations, we can solve Maxwell’s introducing the material equations. equations consistently. Example:  The effect of the medium gives rise to polarization Pr(,)tf  E and  In vacuum, both polarization P and current density j are zero (most magnetization Mr(,)tf  B.  In order to solve Maxwell’s equations we simple material model). Hence we can solve Maxwell’s equations need material models which describe these quantities. directly. Remarks: Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 29 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_2014-10-24s.docx 30

 Even though the Maxwell's equations, in the way they have been  Therefore, be careful when multiplications of fields are required to written above are derived to describe light, i.e. electromagnetic fields describe relevant some material response of some field dynamics. You at optical frequencies, they are simultaneously valid for other frequen- would have to go back to real quantities before you compute these cy ranges as well. Furthermore, since Maxwell's equations are linear multiplications. This becomes relevant for, e.g., calculation of the equations as long as the material does not introduce any nonlinearity Poynting vector, as can be seen in a Chapter below. 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. 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 as the source of spatially changing polarization P .

b (,)rdivPrtt (,)  Analogously we can define a bound current density 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 (see generalized complex dielectric function below). Complex field formalism  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 ) ].  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  Here we define their relation as E(,) rttt1  Er (,) 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.