Reflectors of Quantum Well Lasers and of Other Lasers

Abstract

We discuss different reflectors: distributed feedback reflector; Bragg reflector and photonic crystal reflector; total internal reflector leading to whispering gallery modes. The reflectors are suited as reflectors not only in quantum well lasers but also in quantum wire and quantum dot lasers (Chap. 27).

Problems

25.1

Bragg reflection. Formulate the conditions for the occurrence of Bragg reflection of Bragg reflection of electromagnetic radiation in different systems. (a) A 1D photonic crystal, (b) 2D photonic crystal, (c) 3D photonic crystal.

25.2

Bragg reflection of X-rays.

• (a) Formulate the conditions for the occurrence of Bragg reflection of X-rays.

• (b) Why are Bragg peaks at X-rays extremely sharp?

• (c) Estimate the width of an energy gap expected for X-rays. [Hint: estimate the refractive index of X-rays—it is slightly smaller than unity—and describe a crystal (e.g., with respect to the 100 direction) as a 1D photonic crystal with the electrons distributed in thin layers perpendicular to the propagation direction ofthe X-rays.]

25.3

One-dimensional photonic crystal.

• (a) Estimate the widths of forbidden frequency bands in the case that $n_2 - n_1 \ll 1$.

• (b) Estimate, for radiation of the vacuum wavelength 1 $\upmu$m, the widths of forbidden frequency bands in the case that the photonic crystal consists of a stack of GaAs/AlAs quarter-wavelength films.

25.4

One-dimensional photonic crystal consisting of freestanding plates.

• (a) Determine the effective refractive index, the Bragg frequency and the Bragg wavelength of thin freestanding silicon plates (thickness 1 $\upmu$m, refractive index $n =4$) separated by air under the assumption that the plates and the space between two plates have the same optical thickness.

• (b) Calculate the dispersion relation of radiation in such a one-dimensional photonic crystal.

25.5

How many quarter-wavelength films of GaAs and AlAs films on a GaAs substrate are necessary to obtain reflectivities $R \sim 70$, 80, 90, 95 or, 99, 99.9%?

25.6

Antireflecting coating.

• (a) Show, by use of the matrix method, that the reflectivity of the surface of an optical substrate (refractive index $n_s$) covered with a quarter-wavelength film, thickness $\lambda /(4n)$, is zero if the refractive index n of the film satisfies the condition $n_s - n^2 =0$. [Hint: assume that the substrate has infinite thickness, so that no reflection from the end surface of the substrate occurs.]

• (b) Show that the multiple beam method (introduced in Sect. 3.5) yields the same result. [Hint: add all beams reflected by the two surfaces of the film, taking multiple reflection into account.]

25.7

Determine the Airy formula (Sect. 3.5) by use of the matrix method.

25.8

Double-resonator. We consider a double-resonator (Fabry–Perot resonator) with three lossless mirrors of equal reflectivity R. The distance between mirror 1 and mirror 2 is $L_1$ and the distance between mirror 2 and mirror 3 is $L_2$. Derive, by the use of the matrix method, the transmission curve of a double resonator for (a) $L_1 =L_2$; (b) $L_2 \ll L_1$, (c) $L_1 =\lambda /2$. (d) Choose $R =0.95$ and $\lambda =1\,\upmu$m for a discussion of the results.

25.9

Boundary between two dielectric media.

• (a) Show that the boundary conditions for normal incidence are consistent with the requirement that the energy flux density is the same in medium 2 as in medium 1. [Hint: describe the energy flux density by the Poynting vector $\varvec{P} =\varvec{E} \times \varvec{H}$.]

• (b) Derive the Fresnel coefficient of reflection for normal incidence by the use of the matrix method.

25.10

Bloch theorem. . Derive the Bloch theorem for the one-dimensional photonic crystal, i.e., justify the ansatz (25.26). [Hint: make use of periodic boundary conditions.]

25.11

Propagating of radiation in a one-dimensional crystal. Discuss the dependence of group and phase velocity on the wave vector of radiation belonging to the two lowest branches of the dispersion curves shown in Fig. 25.11.

25.12

Determine, by use of the matrix method, the halfwidth of the resonance curve of a Fabry–Perot resonator (Sect. 3.6) that has a reflector of a reflectivity of unity and a partial reflector.

25.13

Derive the Airy formula for a Fabry–Perot resonator containing an active medium (Sect. 3.7), by the use of the matrix method. [Hint: assume that one of the mirrors has a reflectivity of unity.]

25.14

Reflection of radiation by a perfect conductor.

• (a) Show that the reflectivity of a perfect conductor is 1.

• (b) The radiation penetrates into the conductor. Derive an expression of the penetration depth of the electric field and of the radiation energy.

• (c) Calculate the penetration depth of radiation reflected by a perfect conductor, which contains electrons of a concentration $N =10^{28}\,\mathrm{m}^3$, for radiation of of 1 mm and of 0.5  $\upmu$m wavelength.

25.15

A perfect mirror. A thin film consisting of a perfectly conducting material can act as a partial mirror. [Hint: a perfect conductor for currents at microwave frequencies is superconducting lead at a temperature well below the superconducting transition temperature of 7 K.]

• (a) Determine the complex transmission coefficient $\tilde{t}$, the complex reflection coefficient $\tilde{r}$, the phase $\varphi$ of the reflected beam, the phase $\varphi _t$ of the transmitted beam, transmissivity T and the reflectivity R (see Sect. 3.4). [Hint: make use of the matrix method; treat the film as a free-standing film surrounded by air].

• (b) Design partial mirrors that have reflectivities $R \sim 70$, 80, 90, 95, 99, 99.9% for radiation of 1 mm wavelength, assuming that the mirror is perfectly conducting and contains electrons of a concentration $N =10^{28}\,\mathrm{m}^3$.

• (c) Calculate, for a Fabry–Perot resonator resonator formed by two (perfect) partial mirrors as reflectors, the change of phase per round trip transit of radiation of 1 mm wavelength in the case that the reflectivity of each mirror is $R =0.9$.

25.16

Methods of describing the field in a resonator. Show that the three methods of describing a field in a resonator lead to the same result:

• (a) The method of multiple reflection (Sect. 3.5).

• (b) The method directly based on the boundary conditions (this chapter).

• (c) A method directly based on the boundary conditions but that immediately introduces the complex transmission coefficient $\tilde{t} =B_1/A_1$ and the complex reflection coefficient $\tilde{r} =B_2/A_1$ of a mirror; use this method to derive the Airy formula.

25.17

Bulk metal. We study the optical properties of a metal like copper (free-electron concentration $N =10^{28}$ m$^{-3}$, relaxation time $\tau = 10^{-13}$ s).

• (a) Determine, by use of the complex optical constants, the frequency dependence of the reflectivity.

• (b) Compare the reflectivity of the metal with the reflectivity of a perfect conductor that contains electrons of the same density.

• (c) Determine the optical constants and the reflectivity of a metal for radiation of long wavelengths (i.e., for $\omega \ll \omega _\mathrm{p} =\sqrt{Ne^2/ \epsilon _0 m_0}$ $=$ plasma frequency).

25.18

Metal film. Study optical properties of a metal film (e.g., a copper film). Restrict the discussion to long wavelengths.

• (a) Determine the dependence of transmissivity T, reflectivity R and absorptivity A of a metal film on the thickness of the film by use of the matrix method.

• (b) Show that there is a film thickness where $T =R =0.25$ and $A =0.5$, and that $T \ll A$ for thicker films.