Creating Adjustable Waveplates using Cellophane

We demonstrate that retarders of any value and for any incident wavelength can be created by rotating a thin sample of cellophane about one of its axes.

Cellophane is an anisotropic polymer that has one axis of symmetry, along which all of the long celluose molecules are aligned. It is therefore a birefringent material, as it has two indices of refraction.

The axis of symmetry is defined as the optical (or "c") axis. The two indices of refraction are the extraordinary (n_e), parallel to the c-axis, and the ordinary (n_o), perpendicular to the c-axis. The birefringence of a material is defined by Δn = n_e - n_o. This can be either positive or negative, depending on the relative magnitudes of the two indices (the greater index is the slow axis). Cellophane's slow and extraordinary axies coincide. The retardance of a sample of birefringent material is Γ = 2π Δn t / λ, where t is the thickness of the material and λ is the wavelength.

If the incident light is normal to the cellophane sample, and polarized at 45°, its components will be retarded by some amount Γ₀ = 2π Δn₀ t₀ / λ. Define z to be the axis of propagation of light. The two axes of the cellophane are then parallel to the x and y coordinates, n_x and n_y, respectively.

Consider rotation of the cellophane about its ordinary (fast) axis, oriented parallel to the x-axis. The effective ordinary index after this rotation, n_o', has only x-components, so n_o' = n_x = n_o. However, n_e' has y and z components, so n_e' < n_e. Now the retardance Δn' is less than Δn. The phase shift Γ also depends on the thickness of the sample (which increases by rotation).

Now consider rotation of the cellophane about its extraordinary (slow) axis. n_e' has only x-components, so n_e' = n_x = n_e. The effective index of the ordinary axis after rotation does not change in this case: n_o' = n_o. The anisotropy of the cellophane is oriented in the x-direction, so n_y is the same as n_z. Here Δn' = Δn, so the change in phase shift Γ depends entirely on the change in thickness of the material, so it will increase with rotation angle.

EXPERIMENT

The following plot illustrates how the retardance of a typical sample of cellophane can be "tuned" over a wide range by rotating it about its fast or slow axis. The blue plot is for rotation about the ordinary axis, and the red plot is for rotation about the extraordinary axis.

A polarized HeNe laser, lambda = 632.8 nm, was used for the experiment. Its polarization axis was set accurately horizontal by observing the sharp minimum of reflected light intensity from a vertical dielectric surface (a microscope slide) when the angle of incidence is adjusted to be Brewster's angle, θ_i ≅ 57 degrees. This horizontal reference could then be used to calibrate the angle scale on rotators to which pieces of sheet polarizer were attached, by observing "extinction" of the transmitted light. Light polarized at 45 degrees to the horizontal was created by placing a calibrated polarizer to that angle or by rotating the laser to that angle. Retardance values are determined by measuring the light transmitted through an analyzer polaroid when it is set either parallel or perpendicular to the light incident on the cellophane, according to the formula .... How the cellophane is mounted. The achievement of high quaility circularly-polarized light (retardance 90 degrees, at retarder angle 42 degrees) was verified by measuring an analyzer curve in 5 degree steps, and by observing the nearly complete extinction of light reflected back through the cellophane and initial linear polarizer from a mirror.

To create perfect circularly polarized light, there must be a π/2 phase shift between two electric field components of equal amplitudes. A half wave plate requires a phase shift of π.

As the cellophane is rotated, some of the incident light will be reflected. S-polarized and P-polarized light reflects differently, so the intensities of the x and y components entering the cellophane may not be equal. This will affect the quality of the waveplate. To correct this, the incident light should be polarized slightly less off from 45°.

One can determine if you have found the fast or slow axis by rotating a sample in an interferometer. The fringes will move in opposite directions depending on which axis the linearly polarized incident light is parallel to. ???

REFERENCES

1. Eugene Hecht, Optics.
2. Paper that gives method to measure retardance.