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

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