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

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

[equation] Wave propagation in free space or charge free media for time varying fields:
- Maxwell's equations
[equation], [equation], [equation], [equation]
[equation] -- magnetic field intensity (A/m), [equation] -- electric field intensity (V/m)
- Application of Maxwell's equations:
[equation] by [equation]
[equation] by [equation]

- Wave equation in free space: [equation]
where [equation] is the speed of light in vacuum.

- EM Wave in air: [equation] where position vector [equation].
Description -- y polarization, A(.) & B(.) envelops, g(.) propagation factor of the carrier, [equation] wave number, [equation] wavelength, [equation] speed of light
Propagation direction of envelop A -- [equation]
Propagation direction of envelop B -- [equation]

[equation] Effects of material (polarization [equation], magnetization [equation]):
[equation], [equation] where [equation] and [equation].

[equation] In charge free medium, replace [equation] and [equation]

[equation] Medium with charges: [equation], [equation] and [equation].

[equation] Power carried by wave: instantaneous Poynting Vector [equation] which measures intensity.

[equation] Boundary conditions: [equation], [equation], [equation], [equation]
[equation] Medium description: [equation] (Note: we concentrate on nonmagnetic media.)
Linear, nondispersive, homogeneous, isotropic and [equation] [equation] [equation] is constant and [equation] where [equation] is refractive index.
Inhomogeneous [equation] [equation] is a function of space.
Anisotropic [equation] [equation] is a matrix and [equation] depends on orientation of [equation]
Dispersive [equation] [equation] does not respond instantaneously and depends on previous values of [equation], i.e. the system has memory [equation] require convolution to model [equation].
In freq domain, [equation] is freq dependent.
Nonlinear [equation] [equation] is a nonlinear function of [equation]

[equation] Monochromatic wave (time harmonic or phasor):
Notation -- upper case regular letters to denote phasors
Relation with time harmonic: [equation]
[equation], [equation], [equation], [equation].

[equation] Complex Poynting vector: [equation]
average Poynting vector [equation]

- 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 = [equation]; Average rate of angular momentum = [equation]

[equation] Wave equation: [equation] (vector Helmholtz Eqn.) which is composed of 3 scalar Helmholtz eqns where [equation], [equation] and [equation]
Wave solution: [equation]
Dispersive medium (freq dependent of [equation], i.e. [equation]).

[equation] Various forms of wave:
- Plane wave (far field) -- [equation]
[equation] and [equation] (transverse electromagnetic (TEM) wave) where [equation] is a complex constant, [equation] or [equation] is wave impedance and [equation] is the free space impedance. [equation]
e.g. [equation] in free space; polarization, prop. direction, freq, wave number.
Phasor form -- [equation]
Off coordinate axis prop. direction [equation]
[equation] where wave vector [equation] or [equation], position vector [equation]

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

[equation] Absorption and dispersion in terms of susceptibility (Sect. 5.5):
Electric property measured by permittivity [equation] and [equation]; real part relates to phase (dispersion) and imaginary part relates to amplitude (absorption) since propagation factor [equation] has complex [equation] where [equation] is the absorption (attenuation) coef. and is positive by convention.

[equation] Weakly absorbing media: [equation],
[equation], [equation] and [equation]
Further assuming, [equation],
[equation] and [equation]
Note: [equation] and [equation] are functions of freq. [equation] for absorption.

[equation] Laser medium: nonresonant host lattice and resonant laser atoms, i.e. [equation] or [equation] where [equation] is the permittivity of the host.
There may be charges. [equation] where complex [equation]
For [equation] and [equation], [equation], [equation] where [equation] is the refractive index of the host.

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

[equation] 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 [equation] and charge [equation] can be modeled as a spring mass system where [equation] is charge of an electron.
[equation] where [equation], [equation] is the resonant angular frequency, [equation] is damping coef. amd [equation] is mass of an electron.
We construct volume density of electric dipole [equation] and obtain
[equation] where [equation] is the number of electrons per volume.
At DC steady state, [equation]. From phasor calculations, we obtain
[equation]; [equation] is width of the resonance.

[equation] Near resonance ([equation]), [equation]

[equation] Measure of absorption and dispersion:
Attenuation coef in dB/km or [equation]
Describing freq dependent [equation] -- group velocity [equation], group index [equation] where [equation] is wave number.
Material dispersion coef [equation] (ps / km-nm).
pulse widening or delay [equation] (ps), where [equation] is linewidth in nm, z is length of fiber in km.
[equation] is called anomalous dispersion where long wavelengths (low frequencies) have longer delay, i.e. long wavelengths are behind short wavelengths.
[equation] 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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