Homework 2

Problems 1 and 2: do the first two problems from HW1. (Problem 4 from HW1 will be part of HW3).

Problem 3

a. Using Jones vectors, show that R and L circular polarized waves can be combined to produce linear polarized light.

b. What constraints on the parameters of two fields are there for the result of (a) to hold?

c. How is the result of (a) affected by a relative phase shift between the two beams?

d. Calculate an expression for the E-field and the intensity at z=0 for two equal intensity beams that are propagating in the x-z plane. Beam 1 is RH polarized and is propagating at an angle +θ to the z-axis; Beam 2 is LH polarized and is propagating at an angle -θ to the z-axis. Describe how the intensity and the polarization state varies across the beam at the crossing plane.

problem 4

A circularly polarized beam ϵ = {1,i} passes through a half-wave plate. How does this affect the polarization state? What happens to the output field as the half wave plate is rotated?

problem 5

a. Calculate the precise thickness of a half-wave plate for a wavelength of 532nm where the nominal thickness is 0.5mm. Assume the optic axis is in the plane of the surface. Do the calculation for calcite and quartz.

b. For both types of material, what is the order m of the waveplate? (Remember, Δϕ=π/2+m 2π).

c. Suppose you start with vertical polarization, and use this waveplate to rotate the polarization by 90 degrees (for the design wavelength), then pass the beam through an ideal polarizer aligned to pass horizontal polarization. Plot the transmission of the beam vs wavelength for both calcite and quartz. What is the usable bandwidth of the waveplates? (measure the Δλ for T>90%).

d. A zero-order waveplate can be constructed by following the waveplate with a second one of nearly the same thickness so that its optic axis is rotated 90deg to the first. Explain how this works, and plot the transmission vs wavelength in the test setup of part (c).

problem 6

A beam is normally incident on a uniaxial crystal that has its optic axis (Z) at an angle of θ to the surface normal. The figure shows the orientations of the different vectors characterizing the wave relative the the crystal optic axis. 


Show that the walk-off angle α as shown in the figure is given by
cos α=((((Subscript[n, e])^2)(cos^2)θ+((Subscript[n, o])^2)(sin^2)θ)/(Sqrt[((Subscript[n, e])^4)(cos^2)θ+((Subscript[n, o])^4)(sin^2)θ]))
To get started, note that D ·E=(Subscript[D, 0])(Subscript[E, 0])cos α and k·D=0.