IVM Retarder Off-axis Effects

The IVM polarimeter uses a pair of liquid crystal variable retarders from Meadowlark Optics as its modulator. These devices rely on the asymmetry of the liquid crystal molecules to produce a uniaxial birefringent material. The birefringence can be adjusted by adjusting the amplitude of an AC voltage applied across the thickness of the material, that is, in the direction the light is propagating. The birefringence is very uniform over the surface of the device, but it does depend on the direction that the light is incident on the device. This paper describes a simple model of the IVM modulator which includes the effects due to off-axis rays. The model predicts, to a satisfactory degree, the field-angle dependence of the IVM polarimeter calibration parameters.

Last updated: Fri Sep 8 16:51:48 HST 2000
Don Mickey

Optical layout

This sketch shows part of the IVM optics. At the left is the field stop at the telescope focal plane. A square aperture 4.77 mm on a side, it selects an area on the sky that is 280 arc sec square. Behind (to the right in the figure) the field stop is a field lens which images the telescope aperture onto the collimator lens, at the right edge of this figure. Toward the center of the figure are the two liquid-crystal variable retarders (LVRs). The first one is oriented with its fast axis vertical, parallel to the sides of the field stop. The second retarder is rotated 45 degrees clockwise (as seen from the field stop, looking toward the right in this drawing). The analyzer, not shown here, is a polarizing beam splitter with its transmission axis vertical for the Data camera beam.

The LVRs are located in the f:12.8 diverging beam from the telescope focus, so rays from any field point fill a cone with a half-angle of about 0.04 radians. The telescope secondary obscuration means that the central part of the LVRs is not illuminated. In addition, the chief ray from a given field point has an angle of up to 0.003 radians from the optical axis of the system.

Liquid Crystal Retarders

Meadowlark Optics has published several useful notes on applications of retarders. One describes how variable retarders may be used in Stokes polarimetry. The IVM polarimeter is essentially the same as Figure 3 in that note. A second note called Minimizing Errors in Linear Retarders discusses angle of incidence effects in various types of retarders, mentioning that because in LVRs the optic axis is tilted there can be "large and voltage-dependent retardation changes" with angle. In this IVM note I explore, using a simple model, the effects due to the tilted optic axis in the retarders.

The LVRs consist of a layer of liquid crystal material a few microns thick, contained between two fused silica plates. The elongated molecules of the liquid crystal are aligned parallel to each other and nearly parallel to the substrate, in their quiescent condition. The asymmetry of the molecules causes the material to be birefringent. The construction of the retarders is described in the Meadowlark catalog page, which also has a cartoon cross section of the device. When a voltage is applied, the molecules in the liquid crystal are rotated about the fast axis of the retarder, reducing their apparent asymmetry as seen by the incident light and thus reducing the birefringence. The birefringence, however, depends on the angle of incidence, as shown in the following drawing. This is a view from above, looking at two rays incident on the first LVR. The ray labeled A sees a larger birefringence than does the ray labeled B.

Calculations

I calculated the transmitted intensity for several choices of input Stokes vectors and the retardance settings actually used in the IVM modulation sequence. I integrated over the pupil, for a selection of positions in the field, calculating the effective retardances at each position. In the IVM calibration, we produce fully polarized light, say with Stokes vector {1,1,0,0}, then its complement, in this example {1,-1,0,0}. A four-frame modulation sequence is obtained with each of the inputs, then they are subtracted to obtain the IVM response to the pseudo-Stokes vector {0, 2, 0, 0}. Similar sequences are done to get the response to pure U and pure V. The raw frames calculated for the Q, U and V input are shown below.

Each image is scaled to the expected value, +/- 0.57735, plus and minus a range of 0.04.

When these sets are demodulated, we get a 4 x 4 array of calibration images, as shown below. Some of the images here contain no information: the upper left one is an arbitrary scale factor, and is set to all ones. The remainder of both the first row and column contain zeros here, since we are interested in the QUV terms at this point. The QQ, UU, VV diagonal terms are shown as 0.577 +/- 0.02; the rest are shown as 0.0 +/- 0.02. There is some field position dependence in all the terms, particularly in the UV and VU terms which reach amplitudes of about +/- 0.04 at the edges. The actual observed IVM calibration matrix is very similar to this model.

A possible modification to the modulation sequence

The first two frames in the modulation sequence shown above are either nearly alike or nearly opposite, for each ot the three input vectors. The third and fourth frames are less obviously related. It occurred to me that I could change the retardances used in the fourth frame of the sequence and improve the symmetry. The new values are .875 wave for LVR A and 0.848 wave for LVR B.

Here are the individual frames in the calibration set, scaled the same as above, i.e. a range of 0.08 from black to white with a mean of 0.57735.

The derived cal matrix looks like this: The main difference is that the diagonal terms now have essentially no field variation, and the remaining field dependence is either horizontal or at 45 degrees to the horizontal, i.e. along the slow axis of one of the retarders.

Six-frame Modulation Sequence

We used to use a modulation sequence that encoded Q, U and V separately. The retarder settings were all multiples of 1/4 wave. The calibration frames and the instrument matrix are shown below.

The result is just the same as the "improved" four-frame sequence.

Conclusions

The fact that the optic axis in a liquid crystal retarder does not lie in the plane of the device means that the retardance depends quite sensitively on the angle of incidence and on the setting of the retarder. The IVM was built with the retarders at some distance from the image plane, to minimize effects of possible retarder nonuniformity and to limit the incident power density. But we see that this arrangement leads to field-dependent crosstalk with an amplitude of up to 6%. In principle the crosstalk can be calibrated and removed, but small temporal variations (e.g. due to temperature) may make this less than satisfactory.

The solution may be to arrange the retarders in a location where the pupil is collimated, so that for any field point the chief ray is normal to the retarder surface. Then even though the retardance varies over the pupil, the variation is symmetric, averages to zero, and is the same for all field points.
Last modified: Wed Jun 25 17:47:24 HST 2003