Electromagnetic Optics(Ch. 5) Sept. 11, 2017

Notation — lower case (`e`) or script or italic (`ccE`) letters to denote time varying fields,
upper case (`E`) to denote phasors,
`vec{{:( ),( ):}}`(`vec E`) or boldface (`bbE`) as vector,
`hat{{:( ),( ):}}`(`hat x`) as unit vector
and double boldface (`bbbM`) or underline (`ulM`) as matrix.

• Wave propagation in free space or charge free media for time varying fields:
– Maxwell's equations
`grad \times vecccH = epsilon_o {partial vecccE}/{partial t}`, `grad times vecccE = - mu_o {partial vecccH}/{partial t}`,
`grad cdot vecccE = 0`, `grad cdot vecccH =0`
`vecccH` — magnetic field intensity (A/m),
`vecccE` — electric field intensity (V/m)
– Application of Maxwell's equations:
`vecccE -> vecccH` by `grad times vecccE = - mu_o {partial vecccH}/{partial t}`
`vecccH -> vecccE` by `grad times vecccH = epsilon_o {partial vecccE}/{partial t}`

– Wave equation in free space: `grad^2 vecccE - 1/c_o^2 {partial^2 vecccE}/{partial t^2}=0`
where `c_o = 3 times 10^8` m/s is the speed of light in vacuum.

– EM Wave in air: `vecccE (vecr , t) = haty[ ccA( vecr, t)g(k(c_o t - vecr cdot hat k))+ccB(vecr, t) g(k(c_o t + vecr cdot hat k))]` where position vector `vecr = x hatx + y haty + z hatz`.
Description — `haty` is polarization direction, `ccA(.)` & `ccB(.)` are envelops,
`g(.)` propagation factor of the carrier, `k = omega / c_o = {2 pi nu} /c_o = {2 pi} / lambda` is the wave number, `lambda` is wavelength, `c_o = 1 / {sqrt{epsilon_o mu_o}}` is the speed of light in free space or vacuum.
Propagation direction of envelop `ccA` — `+hatk`
Propagation direction of envelop `ccB` — `-hatk`

• Effects of material (polarization `vecccP`, magnetization `vecccM`):
`vecccD = epsilon_o vecccE + vecccP`, `vecccB = mu_o vecccH + mu_o vecccM` where `vecccP =epsilon_o chi vecccE`, and `vecccM = chi_m vecccH`.

• In charge free medium, replace `epsilon ->epsilon_o epsilon_r` and `mu_o -> mu_o mu_r` where `epsilon_r` is the relative permittivity and `mu_r` is the relative permeability.

• Medium with charges: `grad cdot vecccD = rho_v`, `grad times vecccH = {partial vecccD}/{partial t} + vec J` and `vec J = sigma vecccE`.

• Power carried by wave: instantaneous Poynting Vector `vecccS = vecccE times vecccH` (W/`tt{m^2}`) which measures intensity.

• Boundary conditions: `ccB_{1n} = ccB_{2n}`, `ccD_{1n} = ccD_{2n}`, `ccE_{1t} = ccE_{2t}`, `ccH_{1t} = ccH_{2t}` where `n` means normal to the boundary and `t` means tangential to the boundary.

• Medium description: `vecccP = epsilon_o chi vecccE` (Note: we concentrate on nonmagnetic media.)
Linear, nondispersive, homogeneous, isotropic and `mu = mu_o` `-> chi` is constant and `c = c_o/n` where `n = sqrt{epsilon /epsilon_o}=sqrt{1+chi}` is refractive index.
Inhomogeneous `-> chi (r)` is a function of space.
Anisotropic `-> chi` is a matrix and `vecccP` depends on orientation of `vecccE` which may not parallel to `vecccP`.
Dispersive `->vecccP` does not respond instantaneously and depends on previous values of `vecccE`, i.e. the system has memory and require convolution to model `vecccP`.
In freq domain, `chi (omega)` is freq dependent.
Nonlinear `-> vecccP` is a nonlinear function of 'vecccE`.

• Monochromatic wave (time harmonic or phasor):
Notation — upper case regular letters to denote phasors
Relation with time harmonic: `vecccE(vecr , t) = Re { vecE(vecr)e^{jomegat}}`
`grad times vecccH = j omega vecccD`, `grad times vecccE = - j omega vecccB`, `grad cdot vecccD =0`, `grad cdot vecccB = 0`.

• Complex Poynting vector: `vec S = {vecE times vec H^** }/ 2`
average Poynting vector `(: vecccS :) = Re {vec S}`.

– Application of average Poynting vector :- an EM wave carries (linear and angular) momentum that can put radiation pressure on objects, e.g. small particles.

– Average rate of momentum over a cross section area = `(: vecS :)/c`; Average rate of angular momentum = `vecr times (: vecS :)/c`

• Wave equation: `grad^2 vecE+k^2 vecE =0` (vector Helmholtz Eqn.) which is composed of 3 scalar Helmholtz eqns where wave number `k = omega /c = omega sqrt{mu epsilon}`, `omega = 2 pi nu` and `c = c_o/n`.
Wave solution: `vec E = hate_+ E_+ e^{-jveck cdot vecr} + hate_{-} E_{-} e^{jveck cdot vecr}` where `E_+` and `E_-` are complex function of space.
Dispersive medium (freq dependent of `chi`, i.e. `vecP = epsilon_o chi (nu) vecE`).

• Various forms of wave:
– Plane wave (far field) — `vecU(vecr)= hatU U_o e^{-jkvec cdot vecr}`
`|vecE| = |vecH| eta` and `hatk = hatE times hatH` (transverse electromagnetic (TEM) wave) where `U_o` is a complex constant, `eta = sqrt{mu/epsilon}` or `eta = eta_o/n` is wave impedance and `eta_o = 120 pi` is the free space impedance.
`(: vec S :) = hatk {|E_o|^2 }/ {2eta}`
e.g. `vecE = hatx sin (6 times 10^9 pi t +- k_oz )` in free space; polarization, prop. direction, freq, wave number.
Phasor form — `vecE = hatx E_o e^{+- jkz}`.
Off coordinate axis prop. direction `-> cos ( omega t +- veck cdot vecr )`
`-> e^{+- j veck cdot vecr}` where wave vector `veck = k_o hatk` or `veck = k_x hatx+k_y haty+k_z hatz`. position vector `vecr = x hatx + y haty + z hatz`.

More example on TEM wave and Poynting vector

– Spherical wave (near field) (see Sect. 2.2) — important for distance on the order of wavelength.

– Paraboloidal wave or Gaussian beam (Fresnel approximation) (see Sect. 3.1) — It is good approximation to spherical wave near the propagation axis (paraxial wave). We will use this in this course for beam optics.

Material descriptions Sept. 11, 2017

• Absorption and dispersion in terms of susceptibility (Sect. 5.5):
Electric property measured by permittivity `epsilon = epsilon_o (1+chi)` and `chi = chi ' + j chi''` where real part relates to phase (dispersion) and imaginary part relates to amplitude (absorption) since propagation factor `e^{-jkz}` has complex `k = omega sqrt{ epsilon mu_o} = beta - j alpha /2` where `alpha` is the absorption (attenuation) coef. and is positive by convention.

• Weakly absorbing media: `|chi ''| \<\< |1+chi'|`,
`beta ~~ k_o sqrt(1+chi')`, `n ~~ sqrt(1+chi')` and `alpha ~~ -k_o {chi''}/n`.
Further assuming, `|chi'| \<\<1`,
`n ~~ 1+{chi'}/2` and `alpha ~~ -k_o chi''`.
Note: `alpha` and `n` are functions of freq. `chi'' < 0` for absorption.

• Laser medium: nonresonant host lattice and resonant laser atoms, i.e. `vecD = epsilon_o vecE + vecP_("lattice") + vecP = epsilon_h vec E + vecP_(a\t\oms)` or `vecD = epsilon_h (1+chi_{a\t\oms})vecE` where `epsilon_h` is the permittivity of the host.
There may be charges. `grad times vecH = j omega epsilon_c vecE` where complex `epsilon_c = epsilon_h ( 1+chi_{a\t\oms}+sigma/{j omega epsilon_h})`.
For `|chi'' - sigma/{omega epsilon_h} \<\< 1` and `|chi'| \<\<1`,
`beta ~~ k_o n_h ( 1+ (chi'_{a\t\oms})/2)`, `alpha ~~ -k_o n_h((chi'')_{a\t\oms} -j sigma /(omega epsilon_h))` where `n_h` is the refractive index of the host.

• Kramers-Kronig relations:
Absorption and refractive index are connected by these relations; result of causality. (See Appendix B.1)

• Harmonic oscillator model (Lorentz model) for media:
Susceptibility is result of a sea of electric dipole driven by an external electric field. The electric dipole with separation `x` and charge `-e` can be modeled as a spring mass system where `e = q = 1.6 times 10^{-19}` (C) is charge of an electron.
`{d^2 x}/{dt^2} + sigma {dx}/{dt}+ omega_o^2 x= F/m` where `F = -eE`, `omega_o` is the resonant angular frequency, `sigma` is damping coef. and `m` is mass of an electron.
We construct volume density of electric dipole `P = -ttNex` and obtain
`{d^2 P}/{dt^2} + sigma {dP}/{dt}+ omega_o^2 P= {e^2ttNE}/m` where `ttN` is the number of electrons per volume.
At DC steady state, `chi_o = {e^2 ttN}/{epsilon_o m omega_o^2}`.
From phasor calculations, we obtain
`chi = chi_o {nu_o^2}/{v_o^2-v^2+jnu Delta nu}`; `Delta nu = sigma /{2 pi}` is width of the resonance peak.

• Near resonance (`nu~~nu_o`),
`chi ''(nu) = -chi_o {nu_o Delta nu}/4 1 / {(nu_o - nu)^2+((Delta nu) /2)^2}`
`chi'(nu) = 2 {nu-nu_o}/{Delta nu} chi''(nu)`.

• Measure of absorption and dispersion:
Attenuation coef in dB/km or `tt{m^(-1)}`
Describing freq dependent `n` —
group velocity `v_(g) = {domega}/{dbeta} = 1/ {beta'}=c_o/N`,
group index `N = n - lambda_o {dn}/{dlambda_o}` where `beta` is the wave number and `lambda_o` is the free space wavelength.
Material dispersion coef `D_lambda = - {2 pi c_o beta''}/{lambda_o^2} = -{lambda_o}/{c_o} {d^2n} /{dlambda_o^2}` (ps / km-nm).
pulse widening or delay `= |D_lambda|Delta lambda z` (ps), where `Delta lambda` is linewidth in nm, `z` is length of fiber in km.
`D_lambda >0` is called anomalous dispersion where long wavelengths (low frequencies) have longer delay, i.e. long wavelengths are behind short wavelengths.
`D_lambda < 0` is called normal dispersion where short wavelengths (high frequencies) have longer delay, i.e. short wavelengths are behind short wavelengths

Chapter 5 of textbook 2nd Ed.

Chapter 5 of textbook 1st Ed.

Last Modified: Sept 11, 2017
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