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3.1.8 Error Analysis of Interferometric Retardance Measurements

Jack C.M. Wang

Statistical Engineering Division, ITL

Kent B. Rochford

Optoelectronics Division, EEEL

A NIST effort to develop an accurate and stable retardance SRM has necessitated the development of measurement methods for optical retardance. Retardance is a property of devices commonly known as waveplates, which are used for polarization control. Three methods have been developed. Two methods rely on polarimetric techniques. The third one is based on an interferometric technique that exhibits different error sources and complements the polarimetric measurements.

The retarder is a double-rhomb design. The largest measurement uncertainty arises from the reflectance of the rhomb faces. This is because the laser used in this measurement system has a long coherence length, multiple reflections from the rhomb faces can interfere coherently and cause variations in retardance measurements. The error in retardance due to coherent reflections is given by

\begin{displaymath}Y = Y(r,\,\delta_0,\,U)
= \tan^{-1}\left( \frac{-r\sin(U+\de...
...}\left( \frac{-r\sin(U-\delta_0)}{1-r\cos(U-\delta_0)} \right)

where r is the reflectivity, $\delta_0$ is the retardance of rhomb, and U is a random variable and is uniformly distributed over the interval $(0,\,2\pi)$. The pdf of Y is found to be

\begin{displaymath}f_Y(y) = \frac{(1-r^2)\sin(\delta_0)}
{\pi \vert\sin(\delta_...
...\delta_0-y) -
\left[\sin y - r^2\sin(2\delta_0-y)\right]^2}},

\begin{displaymath}\tan^{-1}\left( \frac{r^2\sin(2\delta_0)-2r\sin \delta_0}
...\sin \delta_0}
{1+r^2\cos(2\delta_0)+2r\cos \delta_0}\right).

It can be shown that, for a wild range of $\delta_0$, the mean of Yis 0 and the standard deviation of Y is proportional to the reflectivity r.

The double-rhomb retarder has endfaces with reflectance ra and an internal interface with reflectance rb. The total retardance error, resulting from multiple reflections between the endfaces and between the internal interface and endfaces, is given by

\begin{displaymath}Z = Y(r_b,\,\delta_0/2,\,U_1) + Y(r_b,\,\delta_0/2,\,U_2) +

where U1 and U2 are independent uniform random variables over the interval $(0,\,2\pi)$. If $\delta_0$ is close to $90^{\circ}$, it can be shown that the mean of Z is 0 and the variance of Z is well approximated by 2(r2a + r2b).

The results indicate that the noise is zero-mean and anti-reflection coatings should be applied to rhomb faces to reduce the variation. A manuscript, describing the interferometric system and the detailed error analysis, has been submitted to Applied Optics.


Figure 8: The top figure displays the sample (based on 100000 simulated values of U) and population (solid line) pdfs of Y with r=0.01 and $\delta_0=91^{\circ}$. The bottom plots the sample pdf of Z (based on 500000 simulated values of U1 and U2) with ra=0.002, rb=0.006 and $\delta_0=89^{\circ}$.

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