**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`.

– 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 :

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
where long wavelengths
(low frequencies) have longer delay, i.e. long wavelengths are
behind short wavelengths.` `

`D_lambda < 0` is called
where short
wavelengths (high frequencies) have longer delay, i.e. short
wavelengths are behind short wavelengths

Last Modified: Sept 11, 2017

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