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3.1.10 Statistical Analysis of Retardance Measurements

Jack C.M. Wang

Statistical Engineering Division, CAML

Paul A. Williams

Kent B. Rochford

Alan H. Rose

Optoelectronics Division, EEEL

Optical communication, data storage, ellipsometry, sensor and other optoelectronic systems often use linear retarders to control or analyze optical signals. Often these systems require retarders with specific and/or accurately known values of retardance. NIST is developing a quarterwave linear retarder designed to have a retardance stable within $0.1^{\circ}$ over a variety of operational and environmental conditions. Three methods are used to measure retardance. One of the methods uses a modified version of standard polarimetric measurements and uses rotating polarizers. Linearly polarized light, with known orientation, is incident on the retarder and the light emerges with an elliptical polarization. The intensities of the perpendicular and parallel states of the emerging light are measured. Measurements are made as the input polarizer rotating from $0^{\circ}$to $-360^{\circ}$ with increments of $-5^{\circ}$. Let $\theta_{i}, \; i=1,2,\cdots, 72$, be the ith orientation of the input polarizer, and Ri be the square root of the ratio of the measured perpendicular intensity to the measured parallel intensity at angle $\theta_i$. Define $Y_i = R_i/R_{i+18},\, i=0,1,\cdots,54$, a model relating Yi and $\theta_i$ is given by

\begin{displaymath}Y_i = \frac{ \sin^2(\phi - \theta_i)\cos^2(\delta/2) +
...ta/2) +
\sin^2(\phi + \theta_i)\sin^2(\delta/2)} + \epsilon_i

where $\epsilon_i$ are random noise, $\phi$ and $\delta$are parameters to be estimated. In particular, $\delta$ is the retardance parameter.

Many (112) experiments were run to determine the retardance of 5 rhombs. A brief summary of the findings is given below.

  • Due to experimental conditions and other interfering factors, the retardance parameter may vary from experiment to experiment. Estimates were obtained under a random-coefficients model assumption.
  • Since the model is periodic in $\theta$ with a periodicity of $180^{\circ}$, the pooled rotation error variance was used to test the lack-of-fit for each experiment.
  • The phase estimate $\hat\phi$ was found to be related to some of the experimental conditions and could be used to screen the experiments.
  • A downward trend of retardance estimates was detected for the earlier experiments. Further investigations indicated that some experiments were not properly run.
  • The goal of retardance measurement with uncertainty less than $0.1^{\circ}$ is reachable with this system.


Figure 10: The top figure displays the measurement results of intensity ratio vs. polarizer orientation for a typical experiment. The dotted line is the least-squares fit of the model. The bottom plots the rotation and residual errors for each experiment. It shows, for most cases, the residual error is smaller than the rotation error, indicating that the model is adequate for the data.

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Date created: 7/20/2001
Last updated: 7/20/2001
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