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

**Notation** -- lower case (e) or script or italic (*E*)
letters to denote time varying
fields, () or
boldface (**E**) as vector, () as unit vector
and double boldface (**M**) or underline ()
as matrix.` `

Wave propagation in free space or charge free
media for time varying fields:

- Maxwell's equations

,
,
,

-- magnetic field intensity (A/m),
-- electric field intensity (V/m)

- Application of Maxwell's equations:

by

by

- Wave equation in free space:

where is the speed of light in vacuum.` `

- EM Wave in air:
where position vector .` `

Description -- y polarization, A(.) & B(.) envelops, g(.)
propagation factor of the carrier, wave number,
wavelength,
speed of light

Propagation direction of envelop A --

Propagation direction of envelop B --

Effects of material (polarization , magnetization ):

,
where
and .` `

In charge free medium, replace and

Medium with charges:
,
and
.` `

Power carried by wave: instantaneous
Poynting Vector which
measures intensity.` `

Boundary conditions:
, ,
,

Medium description:
(Note: we concentrate on nonmagnetic media.)

Linear, nondispersive, homogeneous, isotropic and
is constant and
where is refractive index.` `

Inhomogeneous is a function of space.` `

Anisotropic is a matrix and depends on orientation of

Dispersive does not respond instantaneously and depends on
previous values of , i.e. the system has memory require
convolution to model .` `

In freq domain, is freq dependent.` `

Nonlinear is a nonlinear function of

Monochromatic wave (time harmonic or phasor):

**Notation -- upper case regular letters to denote phasors**

Relation with time harmonic:

,
,
,
.` `

Complex Poynting vector:

average Poynting vector

- 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 = ; Average rate of angular momentum =

Wave equation:
(vector Helmholtz Eqn.) which is composed of 3 scalar
Helmholtz eqns where
,
and

Wave solution:

Dispersive medium (freq dependent of , i.e.` `
).` `

Various forms of wave:

- Plane wave (far field) --

and
(transverse electromagnetic (TEM) wave)
where is a complex constant,
or is wave
impedance and is the free space impedance.` `

e.g. in
free space; polarization, prop. direction, freq, wave
number.` `

Phasor form --

Off coordinate axis prop. direction

where wave vector or
,
position 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
and ;
real part relates to phase (dispersion) and imaginary part relates to
amplitude (absorption) since propagation factor has
complex
where is the absorption (attenuation) coef. and is
positive by convention.` `

Weakly absorbing media: ,

,
and

Further assuming, ,

and

Note: and are functions of freq.` `
for absorption.` `

Laser medium: nonresonant host lattice and resonant laser atoms,
i.e.` `
or
where is the permittivity of the host.` `

There may be charges.` `
where
complex

For and ,
,
where 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 and charge
can be modeled as a spring mass system where
is charge of an electron.` `

where , is the
resonant angular frequency, is damping coef.` `
amd is mass of an electron.` `

We construct volume density of electric dipole
and obtain

where is the number of electrons per volume.` `

At DC steady state,
.` `
From phasor calculations, we obtain

;
is width of the resonance.` `

Near resonance (),

Measure of absorption and :

Attenuation coef in dB/km or

Describing freq dependent -- group velocity
, group index where is wave number.` `

Material dispersion coef
(ps / km-nm).` `

pulse widening or delay
(ps),
where is linewidth in nm, z is length of fiber in km.` `

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

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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