Misalignment Study for a Dove Prism for Its Interferometric Application

Misalignment study for a Dove prism for its interferometric applications

Enoch Gutierrez, Marija Strojnik, and Gonzalo Paez Centro de Investigaciones en Optica Apdo. Postal 1-948, C.P. 37000, Leon, Guanajuato, Mexico [email protected]

Abstract

It is our interest to design, fabricate and characterize a Dove prism for interferometric use. In a previous work, we determined the prism tolerance to manufacturing errors and evaluate the change in the image quality introduced by this performance deterioration. In the present work, we determine the tolerance to misalignment of the Dove prism for its incorporation into a Rotationally-Shearing Interferometer (RSI). We show that it is possible to reduce the wave-front deviation caused by base angle errors with a specific optical misalignment. The misalignment analysis of a Dove prism with a 1.515 refraction index shows that to ensure a maximum OPD of /10 (at 633 nm) the tolerance to misalignment has to be ± 0.33 arc sec.

Keywords: Dove prism, Misalignment, Exact ray trace

1. Introduction

Dove prisms have found wide use in traditional optical systems as image erectors. Recently, the implementation of prisms in interferometers has been on the rise.1-3

In our group, we have proposed a Dove prism as the main optical element of a Rotationally-Shearing Interferometer (RSI). The RSI is built on the basis of a traditional Mach-Zehnder interferometer with the addition of a rotational system (a Dove prism) on one arm, with the purpose of shearing (rotating) the wave front.4,5 The RSI sensitivity may be increased by decreasing the shearing angle.6,7 This interferometer has been suggested for distinct applications.8,9

The manufacturing quality of the Dove prisms and a suitable alignment determine the performance of RSI.10 Formerly, other groups have analyzed alignment problems and the quality of manufacture in rotational prisms for conventional imaging systems.11,12 However, interferometric implementation of a Dove prism restricts the manufacturing errors and the misalignment that can be tolerated. For this reason, we have determined suitable parameters and tolerance to manufacturing errors.13,14

2. Misalignment analysis of a Dove prism

In this work, we determine the tolerance to misalignment of a Dove prism within a RSI. In order to measure a /5 tilt into the RSI, in accordance with Nyquist theorem, a maximum wave-front deviation of /10 (at 633 nm) is assumed as acceptable.

Our analysis considers that the Mach-Zehnder interferometer is perfectly aligned. Therefore, the obtained OPD at the observation plane is only due to manufacturing errors and misalignment of the Dove prism.

The exact ray trace program used in this analysis was previously developed by our infrared group.14 X  i Now, we introduce an Euler transformation matrix x’ x and several programming routines in order to study

X o the lack of accuracy in the alignment of the Dove y’ Y i prism. This matrix helps to model the x-axis and y-  y axis rotations of the prism. It is important to keep in mind that in this analysis we consider plane Y o surfaces within an afocal system. Fig. 1. Misalignment sketch of a Dove prism. The parameters of the analyzed Dove prism are: length = 107.56 mm, height = 25.4 mm (square aperture), base angle = 45° and refraction index = 1.515. Prism misalignment is indicated by means of the coordinate system rotation (Δα, Δ) of the prism (see Figure 1). The obtained data are taking into account an analyzed area of 60% for each surface. Along this work, we use the symbolism and the sign conventions used in Ref. 14, with exception of the base angle error to avoid a possible confusion with the coordinate system rotation in the y-axis.

We study the worst case for the pyramidal angle (1 + 2) and the base angle error () separately. This helps us to identify the consequences of a misalignment on the image affected by each

14 manufacturing error. The worst case for each error is fulfilled when: -1 = 2 and 1 = 2.

First, we discuss the OPD caused by the prism misalignment for three different magnitudes of each manufacturing error. Afterward, we determine the tolerance to misalignment of an interferometric Dove prism.14

A Dove prism with 120 arc sec base angle error (the worst case) causes a slight OPD (smaller than 0.03) when it is rotated on the x-axis in an interval of 200 arc sec. Hence, this misalignment is neglected for practical purposes in the first analysis.

Figure 2 illustrates the y-axis rotation of the real Dove prism with base angle errors of 120, 60 and 20 arc sec. This figure shows that when no rotation is introduced, an OPD also exists, due to the base angle error. However, this Fig. 2. Decrease of the OPD brings about y-axis rotation of a Dove prism. The base angle errors of the modeled prism, in problem could be eliminated with a the worst case (- =  ), are: 120, 60 and 20 arc sec. 1 2 suitable prism misalignment. OPD, through y-axis rotation, has the same gradient in the three cases. Therefore, the prism has the same tolerance to misalignment no matter the magnitude of the base angle As we can see above, the OPD error. due to any base angle error has an identical rate of change with respect to the prism rotation angle in all cases. It is important to mention that the rate of change is dependent on the refraction index of the prism. Hence, the prism always has the same tolerance to misalignment to ensure maximum OPD, regardless of the magnitude of the base angle error.

On the other hand, figure 3 depicts the y-axis rotation of the real prism with pyramidal angles of 240, 120 and 40 arc sec. In contrast to the base angle error, the OPD given by the pyramidal angle could not be eliminated by any prism rotation. Therefore, the ideal position of a prism with a pyramidal angle is when it is perfectly aligned, unlike a prism with a base angle error.

Finally, we study the misalignment of the interferometric Dove prism (tolerance to manufacturing error of ± 0.35 arc sec). In this analysis we neglect the study of the pyramidal error due to the aforementioned information. We start by rotating the prism round the x-axis. Table 1 shows Fig. 3. Increase of the OPD brings about y-axis rotation of a real Dove prism. Pyramidal angle of the modeled prisms, in the the maximum OPD obtained for this worst case ( +  ), are: 240, 120 and 40 arc sec. OPD 1 2 rotation. As we see in figure 5, the caused by pyramidal angle could not be eliminated, on the contrary it rises. Therefore, the ideal position of a prism with interferometric prism rotation round pyramidal angle is the perfect alignment. the x-axis does not modify the OPD caused by the base angle error.

Table 1. The maximum OPD due to x-axis and y-axis rotations of an interferometric Dove prism. x-axis rotation y-axis rotation α [arc sec] OPD/  [arc sec] OPD/ - 1.42 0.09 - 1.42 0.52 2.05 0.09 2.05 0.52

We also rotate the interferometric prism in the y-axis (see Figure 4). The maximum OPDs due to this misalignment are shown in table 1. The wave-front deviation is a consequence of the inversion property of a Dove prism and the base angle error.

Fig. 4. OPD due to x-axis and y-axis rotations of an interferometric Dove prism. We analyze the worst case The misalignment analysis of the in the base angle error ( = - ). To ensure a 1 2 maximum OPD of /10 due to interferometric Dove interferometric Dove prism shows that prism misalignment, the tolerance to misalignment is of to ensure a maximum OPD of /10 (at ± 0.33 arc sec. 633 nm), the tolerance to misalignment is of ± 0.33 arc sec. This tolerance to misalignment is the same for any Dove prism with a refraction index of 1.515 (BK7).

3. Summary

We analyzed the Dove prism misalignment for its incorporation into a RSI. This analysis showed that the OPD given by the pyramidal angle could not be compensated with any prism misalignment, in contrast to the base angle error. The only possibility to compensate this error is to misalign the interferometer arm that contains the prism, in the y direction, until the OPD is removed.

We saw that the Dove prism always has the same tolerance to misalignment regardless of the magnitude of the manufacturing error. The analysis of the interferometric Dove prism showed that to ensure a maximum OPD of /10 (at 633 nm), the tolerance to misalignment is of ± 0.33 arc sec. This tolerance is for any Dove prism with a refraction index of 1.515.

References

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