Limitation of Retardation Measurement by Berek Compensator

Berek compensator is a variable phase plate used with a polarized light microscope to measure the retardation of a sample. It has a plate of a uniaxial birefringent crystal such as calcite or magnesium fluoride. The optical axis of the crystal is perpendicular to the surface. By rotating the plate, retardation continuously increases. The retardation is a function of the rotation angle.

Fig. 1 Berek Compensator

Fig. 2 Relationship between Retardation and Rotation Angle

Figure 2 shows the relationship between the angle and the retardation. We have found three equations to express the relationship. One is the equation in the Olympus website.

The second one is the approximation formula for tilted uniaxial crystal in the textbook of polarized light microscope written by Tsuboi.

The third one is an empirical formula obtained by fitting correction table data of a Berek compensator.

These three equations give the same result practically. You can use any of these equations.

Each compensator has a correction table for it. Since the thickness of the plate differs among compensators, the correction table is only valid for the compensator. When you buy a new compensator, you get the table with it. You should keep the table carefully. It is a good idea to make a spreadsheet file to calculate retardation from the measured angles. Using the third formula above, you can easily make such a file. If you do not have a correction table for your compensator, you can make it by yourself. Prepare phase plates having known retardation values. Measure the rotation angle of the compensator for each phase plate. Using the relationship between retardation values and angles, you can decide the fitting parameters of the third equation.

Since refractive indices have dispersion, birefringence also has dispersion. The dispersions of the sample and the compensator are different unless they are made of the same material. The difference makes compensation imperfect. This imperfection does not cause a serious problem in the measurement when the difference is not large. Figure 3 shows the simulation results of compensation a quartz plate with retardation of 2000nm by a compensator made of calcite.

Fig. 3 Compensation Simulation of Quartz plate with retardation of 2000 nm by calcite

As shown in the figure, polarization color becomes darkest at 2000nm of the compensator. The brightness also becomes the darkest at this value. For such materials, birefringence measurement by a Berek compensator gives good results.

In contrast, measurements give false values when the birefringence dispersion is large so as to show anomalous polarization color.  Figure 4 shows the compensation simulation of 5CB by a calcite compensator. The reference wavelength of the retardation is 593 nm, the sodium D-line. When the retardation is 250 nm, the darkest position is the same as the retardation of the sample. The retardation of the sample increases, not only the same value but also one order higher value becomes dark. By plotting Y value of XYZ chromaticity diagram, the darkest value can be evaluated numerically. The darkest value is the same as the sample till the retardation value of 750 nm. The one order higher value becomes darkest for retardation from 1000 nm and higher.

Fig. 4 Compensation of 5CB by Calcite

Sometimes a monochromatic filter is used with the compensator to make an estimation of the dark position easier. Fig. 5 is the simulation of compensation with the Nikon GF green filter. Though the wavelength range is smaller than without the filter, the brightness of the false value becomes the darkest in higher sample retardation values. Usage of a bandpass filter doesn't help the decision of the true compensation value.

 
Fig. 5 Compensation of 5CB with green filter

Figure 6 shows the compensation simulation of MBBA by the compensator. Birefringence dispersion of MBBA is bigger than that of 5CB. Therefore, the darkest value changes to the first higher minimum values at 750 nm. And the second higher minimum becomes darkest at the sample retardation of 3000 nm.

Fig 6. Compensation of MBBA by Calcite