Polarization is a core technology in products ranging from fiber-optic isolators to LCDs. Retarders, also called waveplates, introduce a phase difference between the exiting beams of two orthogonal incident polarization states. This phase difference is the retardance, which we express as a fraction of a wave, in degrees, or in nanometers. The fast axis of a retarder defines the polarization state that will propagate through the retarder the fastest; the polarization state orthogonal to that state will propagate the slowest. The retardance is the phase difference between these two states upon their exit from the waveplate. For most waveplates, the fast axis corresponds to a linear polarization state, but in general, the fast axis can include any polarization state, including linear, elliptical, or circular. The three most common defects found in waveplates are probably elliptical fast axes, incorrect retardance, or inaccurately marked fast-axis orientations.
Retardance spectrum of a cemented zero-order quartz waveplate shows a ripple caused by plate-to-plate misalignments (left). The plot of the fast axis of the retarder varying as a function of wavelength on the Poincaré sphere shows the desired linear fast-axis state periodically varying between linear and elliptical states as a function of wavelength (right).
The most common method for measuring retardance and fast-axis orientation involves placing two linear polarizers with horizontally aligned axes between a collimated optical source and a detector. Place the waveplate between the polarizers with its marked fast axis approximately horizontal, then rotate the plate until the detector signal reaches its maximum. Now, the true fast axis is horizontal. Record the detector signal and then rotate the retarder by 45° to find the minimum detector signal. The retardance δ is given by
Though simple, this parallel polarizer technique requires careful use to ensure accurate results. First, the method cannot differentiate between δ = λ/4, 3λ/4, 5λ/4, etc. In single-wavelength applications, the user will not typically care whether a retarder has λ/4 or 5λ/4 retardance, as both will behave identically. The retardance of a 5λ/4 waveplate will vary with wavelength more strongly than the retardance of a λ/4 waveplate, however, so a lower-order retarder is generally preferred for polychromatic applications. To ensure an accurate assessment of wavelength dependence, take measurements at multiple wavelengths. The technique also cannot distinguish between the fast and slow axes of the retarder because of an ambiguity in the test. To detect whether your vendor has mistakenly swapped the axes, tilt the waveplate along what is marked as the fast axis. For a positive uniaxial material such as quartz, the retardance will decrease when the waveplate is tilted in this way.
Finally, the technique gives an erroneous retardance if the fast axis is elliptical instead of linear. To test for an elliptical fast axis, place the retarder between two orthogonal polarizers and align either the fast or the slow axis of the retarder with a polarizer axis. If the fast axis is linear, no light should leak through. Light leaking through the system indicates that the element is an elliptical retarder. In such a case, iteratively rotate the retarder and the second polarizers to achieve the best null possible. The final angle between the polarizers provides a figure of merit related to the retarder ellipticity. Determining the actual fast axis is more involved.
Completely characterizing a retarder requires a test setup that measures the complete Mueller matrix or complete Jones matrix. Consider a retarder consisting of two individual quartz plates laminated together to create a zero-order waveplate. A slight misalignment of the two plates causes a ripple in the retardance spectrum, which would otherwise vary linearly (see figure). This behavior is also common in two-material waveplates such as achromatic waveplates constructed from quartz and magnesium fluoride plates. Even more complex behaviors are found in improperly fabricated achromatic and super-achromatic Pancharatnum polymer waveplates. In many high-precision applications, each waveplate should be tested. For accurate results, use the simpler techniques with care, and choose more sophisticated test systems for more complex designs. oe
Matthew Smith is chief technology officer of Axometrics Inc., Huntsville, AL.