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.