Optical Activity

Whereas a birefringent material has different refractive indices for different linear optical polarizations, an optically active material has different indices for the two circular polarizations. In mathematical notation, we can denote the two indices by n+ and n₋ for RCP and LCP light, respectively.

Optical activity is characteristic of media with helical molecules. The helicity of the medium breaks the symmetry between the two polarizations. Examples of such media include quartz and sugar in the solid phase, or sugar solution and turpentine in the liquid phase. Note that given a material with a particular helicity, the same material with the opposite helicity exists in principle as well—it would just be the mirror image of the orginal material. For example, quartz can have n+> n₋ or n+< n₋, depending on the crystal structure. Glucose and fructose are two different sugars that give rise to opposite senses of optical activity.

One other important manifestation of optical activity is the Faraday effect. Some crystals, such as YIG (yttrium-indium-garnet), exhibit optical activity when they are placed in a uniform magnetic field.

Optical activity causes optical rotation of input linear polarization. To see this, note that linearly polarized light (say, linear-x) is a linear combination of RCP and LCP:

E⁽⁺⁾_0,in= After passing through an optically active medium of thickness d, the Jones vector becomes

E⁽⁺⁾_0,out=1 more useful to specify a rotation per unit length,

dφ

dz = π(n₋− n+) λ0

. (8.53)

(polarization rotation per unit length)

146 Chapter 8. Polarization

For the Faraday effect, it is conventional to specify the rotation per unit length as dφ

dz = V B, (8.54)

(Faraday effect) where the constant V is the Verdet constant. The rotation is typically proportional to the magnetic field in regimes of interest for laboratory polarization rotators.

8.8 Exercises 147

8.8 Exercises

Problem 8.1

Write down the Jones matrix for an ideal polarizer that transmits x-polarized light. A realistic polarizer along the x-direction blocks some fraction α (of the electric field) of the x polarization, and transmits some fraction β of the y polarization. (Both α and β should be small for a decent polarizer.) Write down the Jones matrix for this realistic polarizer.

Problem 8.2

(a) Consider a linear polarizer and a wave linearly polarized at an angle θ with respect to the polarizer’s transmission axis. Show that the intensity of the wave is reduced by cos²θ after passing through the polarizer (this is called the Law of Malus).

(b) Consider a system of N cascaded polarizers. The polarizers have their transmission axes at angles π/2N, 2π/2N, 3π/2N, . . . , π/2 from the x-axis, in the order that an input wave sees them. That is, the last polarizer is oriented along the y-direction. Suppose that an input wave is polarized in the x-direction. Compute the intensity transmission coefficient for the system. Show that the transmission coefficient approaches unity as N −→ ∞. This is a simple realization of the quantum Zeno effect, where each polarizer acts as a “measurement” of the polarization state—the polarization is “dragged”

by the measurements as long as they are sufficiently frequent.

Note that it isn’t sufficient to merely argue that cos(π/2N) −→ ∞ as N −→ ∞, because the interplay of the cosine with the exponent is nontrivial. In particular, it is critical that the first-order term in 1/N vanishes for the cosine, while the exponent scales as N . For example, if the exponent scales more quickly with N,

then we can have convergence to other values.

Problem 8.3

One important optical polarization system is the optical isolator, which prevents laser light from being reflected back into the laser (which could cause instability or even damage). An optical isolator typically consists of a polarizer (say, along x), followed by a 45^◦ polarization rotator, followed by a polarizer at 45^◦. Note that if T is the Jones matrix for an optic, the Jones matrix for the optic rotated by an angle θ is Tθ= R(−θ) T R(θ), where R(θ) is the rotation matrix.

(a) Derive the Jones matrices for forward and backward propagation through the isolator. Assume ideal polarizers and note that the rotator produces the same rotation independent of the light’s direc-tion. Show that x-polarized light passes forward through the system unattenuated, but that any light traveling backwards through the isolator will be completely extinguished. (Realistic isolators attenuate the reverse beam by about 40 dB.)

(b) A cheaper isolator is a polarizer (say, along x) followed by a quarter-wave retarder oriented at 45^◦. Show that if a laser passes through this isolator, the isolator will block any light returning from a direct mirror reflection, but will not block arbitrary return polarizations.

Problem 8.4

A beam of light can be described by its position and direction as a geometrical ray as well as by its polarization Jones vector. Suppose the position/direction vector and polarization vector transform respectively according to

148 Chapter 8. Polarization

Write down an expression for the matrix Ω that models both aspects of the optical system; that is, the matrix that gives the transformation







 y2

θ2

Ax,2

Ay,2









= Ω







 y1

θ1

Ax,1

Ay,1









. (8.57)

Problem 8.5

One method of Q-switching a laser is to include a polarizer and a switchable retarder (Pockels cell) oriented at 45^◦in the cavity. If we ignore losses other than due to the polarization rotation, the cavity finesse is unchanged. To “Q-spoil” the cavity, we can switch on some phase retardation to introduce extra loss. Calculate the cavity finesse for the ring cavity of Problem 1 without the gain medium, for retardations of ∆φ = 0, ∆φ = π/4, and ∆φ = π/2. (Obviously you should not make any low-loss approximations here.)

Problem 8.6

Suppose you have a laser beam parallel to the optical table, and the light polarization is perpendicular to the table. You need light that is polarized parallel to the table. How can you do this with only two mirrors? (And without turning the laser on its side!)

Problem 8.7

Consider a planar cavity. Suppose that an ideal (linear) polarizer and a polarization rotator (which rotates the polarization by ∆θ) are placed inside the cavity. Assuming the mirrors are perfect reflectors, what is the finesse of the cavity?

Chapter 9