Cavity Resonator

Abstract

After a basic description of a laser in the first parts of the book, we now are dealing with the question how we can operate a laser. For this purpose, we will first discuss laser resonators. In this chapter, we treat the cavity resonator, which is a closed resonator. In the next chapter, we will study the open resonator.

Problems

10.1

Modes of a cubic cavity. Determine the frequencies of the four modes of lowest frequencies of a cubic cavity resonator (side length $a= 1$ cm).

10.2

Degeneracy of modes of a rectangular cavity resonator.

• (a) Determine the degree of degeneracy of the 011, 110, and 101 modes if $a_1= a_2= L$.

• (b) Determine the degree of degeneracy of the 011, 110, and 101 modes if $a_1= a_2 \ne L$.

• (c) Determine the degree of degeneracy of the 011, 110, and 101 modes if $a_1 \ne a_2 \ne L$.

• (d) Determine the degree of degeneracy of the 111 mode.

10.3

Density of modes of a cavity resonator. Determine the density of modes of a cubic cavity resonator of 1 cm side length at a frequency corresponding to a vacuum wavelength $\lambda =700$ nm for the following cases.

• (a) If the cube (with metallic walls) is empty,

• (b) If the cube contains an Al$_2$O$_3$ crystal (refractive index $\mathrm{n}= 1.8$) and fills the cavity completely,

• (c) If the cube contains a GaAs crystal ($\mathrm{n}= 3.65$) that completely fills out the cube.

10.4

Number of modes. Determine the number of modes of a cubic cavity (side length 1 cm) in the frequency interval 1 $\times$ 10$^{14}$ Hz, 1.1 $\times$ 10$^{14}$ Hz.

10.5

Mode density on different scales. Determine the relations between the mode density on the frequency scale and on different other scales:

• (a) scale of photon energy $h \nu$; (b) $\omega$ scale; (c) scale of vacuum wavelength $\lambda$.

10.6

Variation of the resonance frequency of a mode . By changing the length L of a resonator, the resonance frequencies change. Determine the dependence of the frequency $\nu$ of the 101 mode on the change $\delta L$ of the length L of a long resonator ($L \gg a_1$).

10.7

Density of modes in free space. Determine the density of modes of electromagnetic waves in free space. [Hint: make use of periodic boundary conditions.]

10.8

Energy of a field in a cavity resonator.

• (a) Determine the energy of a field in the 101 mode of a rectangular cavity resonator (width $a_1$, height $a_2$, length L; field of amplitude A).

• (b) Determine the energy content in the case that $a_1=1$ cm, $a_2=0.5$ cm, $L= 2$ cm, $A=1$ V cm$^{-1}$.

10.9

Magnetic field in a rectangular cavity resonator.

• (a) Derive the wave equations describing the H field.

• (b) Solve the wave equations. [Hint: the normal component $H_\mathrm{n}$ of the magnetic field is zero everywhere on the walls, $H_\mathrm{n}$ $(\mathrm{wall})= 0$.]

10.10

TE $_\mathrm{mnl}$ modes of a rectangular cavity resonator.

• (a) Determine the fields of a TE$_\mathrm{mnl}$ mode.

• (b) Express the amplitudes of the field components by the amplitude of the z component of the H field. [Hint: take into account that $\varvec{k} \times \varvec{E}= 0$ and $\varvec{k} \times \varvec{H}= 0$.]

10.11

TM mode of a rectangular cavity resonator.

• (a) Determine the fields of a TM$_\mathrm{mnl}$ mode.

• (b) Express the amplitudes of the field components by the amplitude of the z component of the E field.

10.12

Field components of different modes of a rectangular cavity resonator.

• (a) Determine the magnetic field components of the 101 mode. [Solutions are given in (10.54) and (10.55).]

• (b) Determine the electric and magnetic field components of the 011 mode.

• (c) Determine the electric and magnetic field components of the TE$_{111}$ mode and the TM$_{111}$ mode.

• (d) Show that the 101, 011 and 110 modes exist only as TE modes.

• (e) Show that the TE$_\mathrm{mnl}$ and TM$_\mathrm{mnl}$ modes are degenerate if none of the three numbers is zero.

10.13

Rectangular waveguide. If we omit in a rectangular resonator the two walls perpendicular to the z axis, we obtain a rectangular waveguide.

• (a) Characterize the TE mode of lowest order.

• (b) Characterize the TM mode of lowest order.

10.14

Show that the number of (short-wavelength) cavity modes in a frequency interval $\hbox {d}\nu$ for a rectangular cavity is given by $(8\pi /\lambda ^{3})\hbox { }V_c \hbox { d}\nu \hbox {/}\nu ,$ where $V$ $_{c }$ is the cavity volume and $\lambda$ the free-space wavelength. Show that the number of modes in a spherical cavity is given by the same expression.