# Chapter 8 Summary

# Concepts

- The relations describing the reflection and transmission behavior of a plane EM wave at the boundary between two different media are the consequence of satisfying the conditions of continuity of the tangential components of E and H across the boundary.

- Snell’s laws state that θ i=θ r and

  sin θ t=(n 1/n 2) sin θ i.

  For media such that n 2<n 1, the incident wave is reflected totally by the boundary when θ i≥θ c, where θ c is the critical angle given by θ c=sin−1(n 2/n 1).

- By successive multiple reflections, light can be guided through optical fibers. The maximum data rate of digital pulses that can be transmitted along optical fibers is dictated by modal dispersion.

- At the Brewster angle for a given polarization, the incident wave is transmitted totally across the boundary. For nonmagnetic materials, the Brewster angle exists for parallel polarization only.

- Any plane wave incident on a plane boundary can be synthesized as the sum of a perpendicularly polarized wave and a parallel polarized wave.

- Transmission-line equivalent models can be used to characterize wave propagation, reflection by, and transmission through boundaries between different media.

- Waves can travel through a metal waveguide in the form of transverse electric (TE) and transverse magnetic (TM) modes. For each mode, the waveguide has a cutoff frequency below which a wave cannot propagate.

- A cavity resonator can support standing waves at specific resonant frequencies.

# Mathematical and Physical Models

Normal Incidence

Γ=E 0 r E 0 i=η 2−η 1 η 2+η 1 τ=E 0 t E 0 i=2 η 2 η 2+η 1 τ=1+Γ Γ=ε r 1−ε r 2 ε r 1+ε r 2(if μ 1=μ 2)

Snell’s Laws

θ i=θ r sin θ t sin θ i=u p 2 u p 1=μ 1 ε 1 μ 2 ε 2

Oblique Incidence

Perpendicular Polarization

Γ⊥=E⊥0 r E⊥0 i=η 2 cos θ i−η 1 cos θ t η 2 cos θ i+η 1 cos θ t τ⊥=E⊥0 t E⊥0 i=2 η 2 cos θ i η 2 cos θ i+η 1 cos θ t

Parallel Polarization

Γ∥=E∥0 r E∥0 i=η 2 cos θ t−η 1 cos θ i η 2 cos θ t+η 1 cos θ i τ∥=E∥0 t E∥0 i=2 η 2 cos θ i η 2 cos θ t+η 1 cos θ i

Brewster Angle

θ B∥=sin−1 1 1+(ε 1/ε 2)=tan−1 ε 2 ε 1

Waveguides

β=ω 2 μ ε−m π a 2−n π b 2 f m n=u p 0 2 m a 2+n b 2 u p=ω β=u p 0 1−(f m n/f)2 u p u g=u p 0 2 Z TE=η 1−(f m n/f)2 Z TM=η 1−f m n f 2

Resonant Cavity

f m n p=u p 0 2 m a 2+n b 2+p d 2 Q≈f m n p Δ f

# Important Terms

Provide definitions or explain the meaning of the following terms:

- ω-β diagram

- acceptance angle θ a

- angles of incidence, reflection, and transmission

- Brewster angle θ B

- cladding

- critical angle θ c

- cutoff frequency f m n

- cutoff wavenumber k c

- dominant mode

- evanescent wave

- fiber core

- grazing incidence

- group velocity u g

- index of refraction n

- modal dispersion

- modes

- optical fibers

- parallel polarization

- perpendicular polarization

- phase-matching condition

- plane of incidence

- polarizing angle

- quality factor Q

- reflection coefficient Γ

- reflectivity (reflectance) R

- refraction angle

- resonant cavity

- resonant frequency

- Snell’s laws

- standing-wave ratio S

- surface wave

- total internal reflection

- transmission coefficient τ

- transmissivity (transmittance) T

- transverse electric (TE) polarization

- transverse magnetic (TM) polarization

- unbounded-medium wavenumber

- unpolarized

- wavefront