Ray Theory Transmission Theories of Optics

• Light is an electromagnetic phenomenon described by the same theoretical principles that govern all forms of electromagnetic radiation

• Electromagnetic optics provides the most complete treatment of light phenomena in the context of classical optics.

• quantum theory provides the successful explanation for light-matter interaction(emission & absorption) • Electromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric field- wave & magnetic field wave(classical optics)

• Many optical phenomena such as diffraction is described by a single scalar wave function(scalar wave optics or wave optics)

• Ray optics is the limit of wave optics when the wavelength is very short. Quantum Optics

Electromagnetic Optics

Wave Optics

Ray Optics Engineering Model

• Depending on the structure, we may use ray optics or electromagnetic optics Electromagnetic Optics

• Electromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric field wave & a magnetic field wave. Both are vector functions of position & time. • In a source-free, linear, homogeneous, isotropic & non-dispersive media, such as free space, these electric & magnetic fields satisfy the following partial differential equations, known as Maxwell’ equations:   E   H   [1] t   H   E   [2] t    E  0 [3]    H  0 [4] • In Maxwell’s equations, E is the electric field expressed in [V/m], H is the magnetic field expressed in [A/m].

 [F/m] : Electric permittivi ty  [H/m] : Magnetic permeabili ty

  : is divergence operation  : is curl operation

• The solution of Maxwell’s equations in free space, through the wave equation, can be easily obtained for monochromatic electromagnetic wave. All electric & magnetic fields are harmonic functions of time of the same frequency. Electric & magnetic fields are perpendicular to each other & both perpendicular to the direction of propagation, k, known as transverse wave (TEM). E, H & k form a set of orthogonal vectors. Electromagnetic Plane wave in Free space

Ex Direction of Propagation k x

z z y

An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and the direction of propagation, . Linearly Polarized Electromagnetic Plane wave

 E  e E cos(ωt - kz) [5]  x 0x H  e H cos(ωt  kz) y 0 y [6] where : ω  2f : Angular frequency [rad/m] [7] k : Wavenumber or wave propagation constant [1/m] 2   :Wavelength [m] k

E0x     [] : intrinsic (wave) impedance [8] H 0 y  1 v  [m/s] : velocity of wave propagation [9]  E and B have constant phase in this xy plane; a wavefront z E E k Propagation B

E x E = E sin(wt–kz) x o

, has the same A plane EM wave travelling alongz Ex (or By) at any point in a given xy plane. All electric field vectors in a givenxy plane are therefore in phase. The xy planes are of infinite extent in thex and y directions. Wavelength & free space

• Wavelength is the distance over which the phase changes by 2  . v   [10] f

• In vacuum (free space):

9 10 7  0  [F/m]  0  410 [H/m] 36 [11] 8 v  c  310 m/s 0  120 [] EM wave in Media

• Refractive index of a medium is defined as:

c velocity of light (EM wave) in vacuum  n      r r v velocity of light (EM wave) in medium 0 0 [12]  r : Relative magnetic permeability

 r : Relative electric permittivity

(r 1) • For non-magnetic media :

[13] n   r Intensity & power flow of TEM wave

1   • The poynting vector S  E  H  for TEM wave is parallel to the 2 wavevector k so that the power flows along in a direction normal to the wavefront or parallel to k. The magnitude of the poynting vector is the intensity of TEM wave as follows:

E 2 I  0 [W/m 2 ] [14] 2 Connection between em wave optics & ray optics

• The EM waves radiated by a small optical source can be represented by a train of spherical wave fronts with the source at the center

• A wave front is the locus of all points in the wave train which exhibit the same phase

• Large scale optical effects such as reflection & refraction can be analyzed by simple geometrical process called ray tracing(geometrical optics) Wave fronts (constant phase surfaces) Wave fronts Wave fronts k

  P E r rays k P  O

z A perfect plane wave A perfect spherical wave A divergent beam (a) (b) (c)

Examples of possible EM waves Modes in an optical fiber Meridional and Skew rays

• Meridional rays: The rays which always pass through the axis of fiber giving high optical intensity at the center of the core of the fiber.

• Skew Rays : The rays which never intersect the axis of the fiber, giving low optical intensity at the center and high intensity Meridional and Skew rays Number of fiber modes vs. the V number Limitations of the Ray-model

• (1) The ray model gives an impression that during total internal reflection the energy is confined to the core only. However, it is not so. In reality the optical energy spreads in cladding also. • (2) The ray model does not speak of the discrete field patterns for propagation inside a fiber. • (3) The ray model breaks down when the core size becomes comparable to the wavelength of light. The ray model therefore is not quite justified for a SM fiber. • The limitations of the Ray model are overcome in the wave model General form of linearly polarized plane waves

Any two orthogonal plane waves Can be combined into a linearly Polarized wave. Conversely, any arbitrary linearly polarized wave can be resolved into two independent Orthogonal plane waves that are in phase.

 E  e x E0x cos(ωt  kz)  e y E0 y cos(ωt  kz)  E  E  E 2  E 2 0x 0 y [15] E   tan 1 ( 0 y ) E0x Elliptically Polarized plane waves

 E  e x Ex  e y E y

 e x E0x cos(ωt  kz)  e y cos(ωt  kz  )

2 2  E   E   E  E   x    y   2 x  y cos  sin 2          E0x   E0 y   E0x  E0 y  [16]

2E0x E0 y cos tan(2)  2 2 E0x  E0 y Circularly polarized waves

 Circular polarizati on : E  E  E &    [17] 0x 0 y 0 2  : right circularly polarized, - : left circularly polarized Laws of Reflection & Refraction

Reflection law: angle of incidence=angle of reflection

Snell’s law of refraction:

n1 sin 1  n2 sin 2 [18] Total internal reflection, Critical angle

Transmitted (refracted) light k  t 2  90 2 n 2 Evanescent wave

n 1 > n2  k k 1 i r 1 c Critical angle 1  c TIR Incident Reflected light light

n2 (a) sin c  (b) (c) n1 Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence anglewrt to c , which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a) 1  c and total internal reflection (TIR). (c)    (b) 1  c 1 c n2 [19] sin c  n1 Phase shift due to TIR

• The totally reflected wave experiences a phase shift however which is given by:

 n 2 cos 2  1  n n2 cos 2  1 tan N  1 ; tan p  1 [20] 2 nsin 1 2 sin 1 n n  1 n2

• Where (p,N) refer to the electric field components parallel or normal to the plane of incidence respectively. Optical wave guiding by TIR: Dielectric Slab Waveguide

Propagation mechanism in an ideal step-index optical waveguide. Launching optical rays to slab waveguide

n2 sin min  ; minimum angle that supports TIR n1 [21]

Maximum entrance angle,  0 max is found from the Snell’s relation written at the fiber end face.

2 2 [22] nsin 0max  n1 sin c  n1  n2

Numerical aperture:

2 2 NA  nsin  0 max  n1  n2  n1 2 [23]

n1  n2   [24] n1 Optical rays transmission through dielectric slab waveguide

 n  n ;      1 2 c 2 c O

For TE-case, when electric waves are normal to the plane of incidence  must be satisfied with following relationship: 2 2 n d sin  m   n cos 2   n  tan 1     1 2  [25]  2  n sin      1  EM analysis of Slab waveguide

• For each particular angle, in which light ray can be faithfully transmitted along slab waveguide, we can obtain one possible propagating wave solution from a Maxwell’s equations or mode. • The modes with electric field perpendicular to the plane of incidence (page) are called TE (Transverse Electric) and numbered as: TE 0 ,TE 1,TE 2 ,... Electric field distribution of these modes for 2D slab waveguide can be expressed as:  Em (x, y, z,t)  e x f m (y)cos(ωt   m z) [26] m  0,1,2,3 (mode number)

wave transmission along slab waveguides, fibers & other type of optical waveguides can be fully described by time & z dependency of the mode:

j(wtm z) cos(ωt   m z) or e TE modes in slab waveguide

 Em (x, y, z,t)  e x f m (y)cos(ωt   m z) m  0,1,2,3 (mode number) Modes in slab waveguide

• The order of the mode is equal to the # of field zeros across the guide. The order of the mode is also related to the angle in which the ray congruence corresponding to this mode makes with the plane of the waveguide (or axis of the fiber). The steeper the angle, the higher the order of the mode.

• For higher order modes the fields are distributed more toward the edges of the guide and penetrate further into the cladding region. • Radiation modes in fibers are not trapped in the core & guided by the fiber but they are still solutions of the Maxwell’ eqs. with the same boundary conditions. These infinite continuum of the modes results from the optical power that is outside the fiber acceptance angle being refracted out of the core.

• In addition to bound & refracted (radiation) modes, there are leaky modes in optical fiber. They are partially confined to the core & attenuated by continuously radiating this power out of the core as they traverse along the fiber (results from Tunneling effect which is quantum mechanical phenomenon.) A mode remains guided as long as

n2k    n1k Optical Fibers: Modal Theory (Guided or Propagating modes) & Ray Optics Theory

n 1 n2

n1  n2

Step Index Fiber Modal Theory of Step Index fiber

• General expression of EM-wave in the circular fiber can be written as:

   j(ωtm z) E(r,, z,t)   Am Em (r,, z,t)  AmU m (r,)e  m  m  j(ωtm z) H(r,, z,t)   Am H m (r,, z,t)   AmVm (r,)e m m [27]   Em (r,, z,t) & Hm (r,, z,t) • Each of the characteristic solutions is called mth mode of the optical fiber. • It is often sufficient to give the E-field of the mode. j(ωtmz) U m (r,)e m 1,2,3...  • The modal field distribution, , and the mode U m (r,) propagation constant,  are obtained from solving the Maxwell’s equations subjectm to the boundary conditions given by the cross sectional dimensions and the dielectric constants of the fiber.

• Most important characteristics of the EM transmission along the fiber are determined by the mode propagation constant,  m ( ω ) , which depends on the mode & in general varies with frequency or wavelength. This quantity is always between the plane propagation constant (wave number) of the core & the cladding media .

n2k   m (ω)  n1k [28] • At each frequency or wavelength, there exists only a finite number of guided or propagating modes that can carry light energy over a long distance along the fiber. Each of these modes can propagate in the fiber only if the frequency is above the cut-off frequency, ω c , (or the source wavelength is smaller than the cut-off wavelength) obtained from cut-off condition that is:

[29]  m (ωc )  n2k

• To minimize the signal distortion, the fiber is often operated in a single mode regime. In this regime only the lowest order mode (fundamental mode) can propagate in the fiber and all higher order modes are under cut- off condition (non-propagating). • Multi-mode fibers are also extensively used for many applications. In these fibers many modes carry the optical signal collectively & simultaneously. Fundamental Mode Field Distribution

Mode field diameter Polarizations of fundamental mode Ray Optics Theory (Step-Index Fiber)

Skew rays

Each particular guided mode in a fiber can be represented by a group of rays which make the same angle with the axis of the fiber. Mode designation in circular cylindrical waveguide (Optical Fiber)

TE lm modes : The electric field vector lies in transverse plane.

TM lmmodes :The magnetic field vector lies in transverse plane.

Hybrid HE lmmodes :TE component is larger than TM component.

Hybrid EH lmmodes : TM component is larger than TE component. y l= # of variation cycles or zeros in  direction. r m= # of variation cycles or zeros in r direction.  x

Linearly Polarized (LP) modes in weakly-guided fibers ( n 1  n 2  1 )

LP0m (HE 1m ), LP1m (TE 0m  TM 0m  HE 0m ) Fundamental Mode: LP01(HE 11) Two degenerate fundamental modes in Fibers (Horizontal & Vertical Modes)

HE 11 Mode propagation constant as a function of frequency

• Mode propagation constant, , is the most important transmission lm (ω) characteristic of an optical fiber, because the field distribution can be easily written in the form of eq. [27]. • In order to find a mode propagation constant and cut-off frequencies of various modes of the optical fiber, first we have to calculate the normalized frequency, V, defined by:

2a 2 2 2a V  n  n  NA [30]  1 2  a: radius of the core, is the optical free space wavelength, n1 & n2 are the refractive indices of the core & cladding. Plots of the propagation constant as a function of normalized frequency for a few of the lowest-order modes Single mode Operation

• The cut-off wavelength or frequency for each mode is obtained from:

2n w n 2 c 2 [31] lm (ωc )  n2k   c c

• Single mode operation is possible (Single mode fiber) when: V  2.405 [32]

Only HE 11 can propagate faithfully along optical fiber Single-Mode Fibers

  0.1% to 1% ; a  6 to 12 m ; V  2.3 to 2.4 @ max frequency or min 

• Example: A fiber with a radius of 4 micrometer and n1 1.500 & n2 1.498 has a normalized frequency of V=2.38 at a wavelength 1 micrometer. The fiber is single-mode for all wavelengths greater and equal to 1 micrometer.

MFD (Mode Field Diameter): The electric field of the first fundamental mode can be written as: r 2 [33] E(r)  E0 exp( 2 ); MFD  2W0 W0 Birefringence in single-mode fibers

• Because of asymmetries the refractive indices for the two degenerate modes (vertical & horizontal polarizations) are different. This difference is referred to as birefringence, : B f

B f  n y  nx [34] Fiber Beat Length

• A linearly polarized mode is a combination of both of the degenerate modes. As the modal wave travels along the fiber, the difference in the refractive indices would change the phase difference between these two components & thereby the state of the polarization of the mode.For fiber beat length, the modal wave will produce its original state of polarization. :

2 Lp  [35] kBf Multi-Mode Operation

• Total number of modes, M, supported by a multi-mode fiber is approximately (When V is large) given by:

V 2 M  [36] 2

• Power distribution in the core & the cladding: the ratio of the mode power in the cladding P c lad , to the total optical power in the fiber, P, which at the wavelengths (or frequencies) far from the cut-off is given by:

P 4 clad  [37] P 3 M Mode field diameter

• The mode field diameter (MFD) is an expression of distribution of the irradiance, i.e., the optical power per unit area, across the end face of a single- mode fiber.

• For a Gaussian intensity (i.e., power density, W/m2) distribution in a single-mode optical fiber, the mode field diameter is that at which the electric and magnetic field strengths are reduced to 1/e of their maximum values, i.e., the diameter at which power density is reduced to 1/e2 of the maximum power density, because the power density is proportional to the square of the field strength.