Optical Components and Methods

Abstract

In this chapter, after presenting Maxwell equations inside matter and describing reflection and refraction processes, the interaction of optical waves with simple optical components such as mirrors, prisms, and lenses is examined. The different methods for measuring the power spectrum of an optical beam are described and compared in Sect. 2.3. In Sect. 2.4, after treating wave propagation in anisotropic materials, the methods used for fixing or modifying the polarization state of an optical beam are presented. Finally, Sect. 2.5 introduces the important subject of optical waveguides.

Problems

2.1

Given that the wavelength of a light signal in vacuum is 600 nm, what will it be in a glass block having a refractive index $n = 1.50$?

2.2

A glass block having an index of 1.55 is coated with a layer of magnesium fluoride of index 1.32. For light traveling in the glass, what is the total reflection angle at the interface?

2.3

Yellow light from a sodium lamp ($\lambda = 589~\mathrm{nm}$) traverses a tank of benzene (of index 1.50), which is 30-m long, in a time $\tau _1$. If it takes time $\tau _2$ for the light to pass through the same tank when filled with carbon disulfide (of index 1.63), determine the value of $\tau _2 - \tau _1$.

2.4

Light having a vacuum wavelength of 600 nm, traveling in a glass ($n = 1.48$) block, is incident at $45^{\circ }$ on a glass-air interface. It is then totally internally reflected. Determine the distance into the air at which the amplitude of the evanescent wave has dropped to a value of 1/e of its maximum value at the interface.

2.5

A beam of light in air strikes the surface of a transparent material having an index of refraction of 1.5 at an angle with the normal of $40^{\circ }$. The incident light has component E-field amplitudes parallel and perpendicular to the plane-of-incidence of 10 and 20 V/m, respectively. Determine the corresponding reflected field amplitudes.

2.6

The complex index of refraction of silver at $\lambda =532$ nm is $n'+in''=0.14-3.05i$. Calculate: (i) the reflectance of an air-silver surface at normal incidence; (ii) the transmittance of a $100\,{\upmu }$m silver plate.

2.7

A Gaussian beam at wavelength $\lambda = 0.63~{\upmu }$m having a spot size at the beam waist of $w_o = 0.5$ mm is to be focused to a spot size of $30~{\upmu }$m by a lens positioned at a distance of 0.5 m from the beam waist. What focal length should the lens have?

2.8

A Gaussian beam with $\lambda = 800$ nm and $w_o = 0.1$ mm at $z =0$ is propagating along the z axis. A plano-convex lens, made of a spherical glass cap having a radius of curvature of 4 cm, is placed at $z_1=20$ cm. The refractive index of the glass is $n = 1.8$. Determine: (i) at which coordinate $z_2$ the beam is focused; (ii) the beam spot size at $z_2$; (iii) the lens diameter to be chosen in order to collect $99\,\%$ of the beam power.

2.9

A blazed reflection grating with pitch $d = 1200$ nm and blazing angle $\theta _g=20^{\circ }$ is used in the Littrow configuration. Determine which incident wavelength is diffracted at second order in the backward direction.

2.10

A Fabry-Perot interferometer made by two identical mirrors at distance $d=1$ mm is illuminated by a light beam containing two wavelengths $\lambda _1=1064$ nm and $\lambda _2=\lambda _1+\varDelta \lambda$, where $\varDelta \lambda =0.05$ nm. Determine which is the minimum reflectance of the two mirrors required to obtain a resolving power $P_r \ge \lambda _1/\varDelta \lambda$.

2.11

Consider a light beam at $\lambda =589$ nm traveling along the z-axis and linearly polarized along the x-axis. The light beam crosses, in succession, a quarter-wave plate and a polarizer. The optical axis of the quarter-wave plate is in the x-y plane, forming an angle $\pi /4$ with the x-axis. The optical axis of the polarizer is parallel to the y-axis. Write the Jones matrix of the quarter-wave plate in the x-y reference frame, and calculate the fraction of the incident power that is transmitted by the polarizer.

2.12

Consider the following sequence of components: a polarizer $P_1$, a Faraday rotator that rotates counterclockwise the linear polarization direction of the incoming beam by $45^{\circ }$, a calcite half-wave plate, and a second polarizer $P_2$. Assume that the incoming beam travels along y and is linearly polarized along x, that the two polarizers have optical axes parallel to x, and that the optical axis of the calcite plate lies in the x-z plane. (i) Calculate which angle the optical axis of the plate should form with the x axis in order to ensure a unitary transmission. (ii) Write the Jones matrix of the half-wave plate in the x-z reference frame.