ECE 182 Birefringence and wave plates

Objective: To study how light propagates in birefringent crystals, and how retardation wave plates can modify the polarization state of the light.

Background Some materials, such as calcite, quartz or mica are optically anisotropic. Their optical properties are not the same in all directions. If a light beam enters into anisotropic crystal, the electric displacement vector is not necessarily parallel to the electric field vector (as in isotropic crystal where = , ε is the permittivity or dielectric constant), but has tensor relation = . By 𝐷𝐷��⃗ 𝐸𝐸�⃗ rotating the coordinate system the permittivity tensor can be simplified to have only diagonal 𝐷𝐷��⃗ 𝜀𝜀𝐸𝐸�⃗ 𝐷𝐷��⃗ �𝜀𝜀𝑖𝑖𝑖𝑖�𝐸𝐸�⃗ elements εii. This coordinate system defines the principal axes of the crystal. The principal dielectric �𝜀𝜀𝑖𝑖𝑖𝑖� constants εii correspond to principal indices of refraction = . If n1 = n2 ≠ n3 the crystal is called uniaxial. Usually indices n1 = n2 = no and n3 = ne are called the ordinary and extraordinary 𝑖𝑖 𝑖𝑖𝑖𝑖 indices. If ne < no the crystal is negative uniaxial, if ne > n𝑛𝑛o it is� positive𝜀𝜀 uniaxial crystal. If unpolarized light beam entering into uniaxial crystal, the beam will be split into two parts (ordinary and extraordinary), polarized at right angles to each other. In an ordinary beam the electric field will be polarized perpendicular to the optic axis and will have a phase velocity independent of the direction. In an extraordinary beam the electric field will be polarized in the same plane as the optic axis and will have a phase velocity depending on the direction of propagation.

The index ellipsoid is shown in Fig. 1. For a uniaxial crystal, the equation of the index ellipsoid is given by:

(1) Solving for the index of refraction as a function of angle θbetween optical axis and direction of propagation gives:

(2)

If ne - no << no and ne the expression (2) can be simplified as:

(3) Fig. 1: Index ellipsoid for a uniaxial crystal.

Let’s consider a uniaxial crystal cut as a thin plate of thickness d, and with its optic axis parallel to the surfaces. The speed with which linearly polarized light will travel through the plate will depend on the orientation of electric field. A phase difference will be introduced between the ray with parallel to the optic axis and the ray with perpendicular to the optic axis. Thus, one ray is retarded relative to the other, and the path difference between two rays is given by 𝐸𝐸�⃗ 𝐸𝐸�⃗ mλo = d(ne - no) (4) Value of mλo can be considered as the retardation and expressed in fractions of a wavelength. For example, m = 1/4 for a quarter-wave plate, and m = 1/2 for a half-wave plate. In term of phase difference: one beam is delayed from other by π/2 in quarter-wave and by π in half-wave plates. The phase difference δ is δ = 2πm = 2π d(ne - no) / λo (5)

In a positive uniaxial crystal ne > no, so the extraordinary ray travels more slowly than the ordinary ray. The fast axis is defined as the direction in which the faster-moving ray vibrates. Thus in a positive uniaxial crystal, the fast axis (ordinary ray) is perpendicular to the optic axis, while the slow axis (extraordinary ray) coincides with the optic axis. For a negative uniaxial crystal, the fast axis coincides with the optic axis. If input light is linear polarized at ±45˚with respect to the fast (or slow) axis of the wave plate, the quarter-wave plate will generate circular polarized light. And opposite: circular polarized light incident on a quarter-wave plate will generate a linear output polarization at +45˚ or -45˚ depending on right or left circular polarization was used. A half-wave plate can change the linear polarization to any desired direction. Linear polarized light with at angle -θ˚ to the plate’s optic axis will exit from such a plate with the plane of

polarization rotated at angle 2θ such that is now at angle +�θ˚ to the optic axis. A diagram how the �⃗ wave plate𝐸𝐸s work is shown in Fig.2. 𝐸𝐸�⃗

Figure 2. Modification of polarization state by half-wave and quarter-wave plate.

Now let’s consider a case when the retardation plate is placed between two polarizers. What will be the light intensity after passing through this optical system? If there is no absorption in the retardation plate, amplitudes of the ordinary and extraordinary rays will not be changed, but the rays will be phase delayed by δ. After passing through the analyzer these two rays will interfere because they are polarized in the same direction. The resulting intensity will be the function of two angles α and β which polarizers are making with optic axis of the plate. If polarizers are crossed (α - β = π/2):

(6) If polarizers are parallel (α = β):

(7)

Experiments Part 1. Wave plates

Note: Place a wave plate at normal incidence to the input laser beam (λo = 633nm). A. Procedure how to find optic axis of λ/4-wave plate. Place the quarter-wave plate between horizontally and vertically oriented polarizers (polarizers are crossed). Rotate the λ/4-plate until you see a maximum intensity. This corresponds to the λ/4-plate axis making angle α =45o with respect to the polarizers. - Rotate the λ/4-plate through 360o with 10o-step and measure the intensity as a function of angle α. Plot the result and explain. - Rotate the λ/4-plate to precise α =45o with respect to the incident polarization. Verify that λ/4-wave plate changes linear polarization to circular polarization by rotating the analyzer through 180o. Plot the result and explain. - Verify that λ/4-wave plate can be used as an optical isolator, device that prevents light from reflecting back in optical system. (Hint: Use a mirror and a microscope slide to separate backward reflected beam.)

B. Procedure how to find optic axis of λ/2-wave plate. Place the half-wave plate between two polarizers with transmission axes oriented horizontally (polarizers are parallel). Rotate the λ/2-wave plate until you see a minimum intensity. This corresponds to the λ/2-wave plate axis making angle α =45o with respect to the polarizers. - Rotate the first polarizer making α = 0o between input polarization and the wave plate’s axis. Rotate analyzer to find polarization after the λ/2-wave plate. - Repeat step 2 for α = -10o, -20o and -40o. Plot the output polarization angle vs the input polarization angle α. Do your data correspond with prediction how λ/2-wave plate works?

C. Procedure how to find retardation of unknown wave plate. - Place an unknown retardation wave plate between two parallel polarizers (say polarizers axes oriented horizontally). Rotate the wave plate 360o and find the maximum and the minimum output intensities. - Unknown retardation δ can be calculated using equation cos( /2) = / .

D. Variable retardation plate. δ �𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 Retardation δ = 2π d(ne - no)/λo can be varied by tilting the wave plate from the normal incidence. The optical path d will change depending on the angle φ between the incident beam and the normal to the plate surface as d = d0 /cosφ (d0 is the thickness of the wave plate). - Place λ/2-wave plate perpendicular to the input beam (φ =0o). (Axis of this wave plate is fixed horizontally). Rotate input polarizer to make α =45o and analyzer parallel to it. Verify that no light through after the λ/2-wave plate and analyzer. Calculate angle φ needed to tilt the λ/2-wave plate to behave as a λ/4-wave plate. Rotate the wave plate at φ and verify that output polarization is circular. (You might need fine tune angle φ for the best performance.) - Can you tilt a λ/4-wave plate to behave as a λ/2-wave plate? If yes, what should be the tilt angle φ?

Part 2. Birefringent crystals. A. Wollaston prism

- Place a Wollaston prism at normal incidence in unpolarized (or circular polarized) laser beam. - Measure the angles of refraction and polarizations of the refracted beams. - If the laser beam enters the first prism at normal incidence and the second prism at 45o to the normal, calculate the angles of refraction for the output beams. At a wavelength of 633nm the ordinary index of calcite is no = 1.6557 and the extraordinary index is ne = 1.4852. You can use approximation (3) in your calculations.

B. Mica plate. - Place a mica plate into a white light and between two parallel polarizers. - While tilting the plate observe the output light. Explain what is going on.