1. Introduction

A method was presented to use an optical fiber as a cylindrical wavefront reference [1–4]. Since then, no further study has been published on the accuracy of this method. One issue in particular is how this optical element presents misalignment errors in the resulting interferograms, especially given the unusual 1-D filtering inherent in using the fiber reference. A major challenge in optical testing is to avoid or recognize and remove aberrations due to misalignment. These aberrations can reduce the accuracy of the measurement and, more importantly, indicate errors which could be erroneously attributed to the test optic. Misalignments always occur at some level in real optical systems. Therefore, it is important to determine fiber misalignment induced errors so they can be correctly numerically removed [5–8], or the part realigned. This paper presents a study of fiber reference misalignment induced errors in testing cylindrical wavefronts.

The paper is organized as follows. Section 2 briefly describes the fiber reference technique. The coordinate definition and experimental plan for measuring the fiber reference misalignment-induced errors is provided in section 3. Section 4 presents the decomposition method we employ: 2-D Chebyshev polynomials. Section 5 presents the experimental results. Finally, section 6 presents an analysis of the experimental data, section 7 provides a qualitative and quantitative analysis of the experiment using geometrical optics, and section 8 shows that the misaligned fiber reference tests can be numerically realigned.

2. Principle of fiber optics reference method

The optical fiber reference method employs a reflective metal coated fiber cladding as the reference surface for a cylindrical-wave test [1–4]. To null test a cylindrical wave generating optic, the fiber is made coincident with the line focus of the optic under test, such as the cylindrical lens shown in Fig. 1. The wavefront returning to the test optic from the fiber acts as a cylindrical wavefront reference. This wave goes back through the optic, picks up its aberrations, and then returns to the interferometer. There it combines with the reference wave and the resulting interferogram is recorded. However, the fiber reference, because of its extremely small radius of curvature, filters the incident aberrations in the powered direction, thus, errors measured are single pass in this direction, but double pass in the plane containing the fiber axis [2]. Because of this unusual behavior, aberrations due to fiber reference misalignment is, for the first time, being tested to see how this 1-D filtering affects it.

 

Fig. 1 Test configuration to analyze fiber reference misalignment.

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3. Fiber reference misalignments in testing cylindrical surface

To determine the misalignment induced errors, one need only track how the aberrations change in the interferogram of a fixed optic given specific misalignments. The setup to perform this test is shown in Fig. 1. A collimated output beam from the interferometer, a Zygo GPI XP, is focused by a cylindrical null (CN) made from assembling a COTS cylindrical lens with an aberration corrector attached on the back [9]. This focusing cylindrical wave is incident on the fiber reference. The reflected beam serves as the cylindrical wavefront reference which propagates back to the CN. In preliminary experiments, retrace errors for this interferometer were found to be negligible. Since, in this experiment, only aberrations introduced by rigid body misalignment are of interest, changes in the measurements induced by misalignments yield direct results of the misalignment sensitivity. Note that each interferogram also contains information indicating CN misalignment to the interferometer. However, since misalignment aberrations from the fiber reference are of interest in this paper, not the performance of the CN, only aberration changes are used.

The rigid body fiber reference misalignments introduced are position and angular misalignment. For cylindrical wavefronts and perfect fibers, translation along and rotation about the cylinder axis $\left(y\right)$ yield no changes [7]. Therefore, only four degrees of freedom are sensitive to the fiber alignment. Using a right hand coordinate system, Fig. 2 illustrates the errors investigated. The misalignments are: decenter, $t_x$, in the $x$- direction; defocus, $t_z$, in the $z$-direction; tilt, ${\varphi }_x$, around the $x$-axis; clocking, ${\varphi }_z$, around the $z$-axis.

 

Fig. 2 Coordinates for fiber misalignments.

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4. Analytical basis for cylindrical interferograms

Using 2-D Chebyshev polynomials as a fitting and analysis basis of cylindrical interferograms has been reported [10,11]. Chebyshev polynomials form a complete set of functions or modes that are orthogonal over a rectangular aperture which makes them suitable for accurately describing aberrations as well as for data fitting. Wavefront data can be expressed by a series of orthonormal polynomials $C_i(x,y)$ with coefficient $a_i$

Table 1. First 15 terms of 2D Chebyshev polynomial

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5. Fiber misalignment experiment setup

The testing was performed using the ZYGO GPI XP interferometer with a 4 inch aperture and a $\lambda /20$ transmission flat (TF). A cylindrical null (CN) with a 44mm by 41mm $\left[x\text{by}\text{}y\right]$ aperture was positioned in front of the TF [9]. The fiber reference (FR) with a 0.125mm diameter is mounted in a two axis adjustment stage, enabling crude tilt $\left({\varphi }_x\right)$and clocking$\left({\varphi }_z\right)$ adjustment as shown in Fig. 3(a). This fiber mount is attached to a three-axis $\left(x,y,z\right)$translation stage. The setup is shown in Fig. 3(b). Since we were only interested in changes with misalignments, and previous experiments have shown that this interferometer is insensitive to retrace error, the individual measurement results are not particularly important, though we note that the P-V errors are on the order of a few waves at best null position (BNP), which is a somewhat qualitative position selected by the metrologist. To set up the measurement system to BNP, the axis of the CN focus and the fiber reference should be coincident. Experiments were then conducted by introducing different misalignment errors to the fiber reference as follows.

 

Fig. 3 (a) Front view of the fiber reference stage. (b) Experimental set up.

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5.1 Fiber decenter

In this test, the fiber was decentered about BNP in the $+t_x$ and $-t_x$ direction. Figure 4 shows the captured interferograms at different decenter positions where the total range of decenter was ~0.08mm. An average of three measurements was taken to reduce random error.

 

Fig. 4 Recorded interferograms at different fiber decenter positions about BNP.

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5.2 Fiber defocus

In this test, the fiber was defocused about BNP to $+t_z$ and $-t_z$ positions. To get each result, Fig. 5, an average of three measurements were taken at each position to reduce random error. The total range of defocus of the fiber reference was ~1.8mm.

 

Fig. 5 Recorded interferograms at different defocus positions from BNP.

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5.3 Fiber clocking

In this test, the fiber was clocked about the $z$-axis only in the $+{\varphi }_z$ direction from BNP due to setup limitation. To determine the clocking angles, a small mirror was attached to the fiber reference stage. A HeNe laser beam was projected onto the mirror which reflected it onto a screen $L=5.18$ meters away where the spot motion between tilts, $D_m$, was measured and the angle calculated, as shown in Fig. 6 and Eq. (2). The recorded interferograms at different ${\varphi }_z$ angles are illustrated in Fig. 7.

 

Fig. 6 Geometry to calculate the fiber clocking angle.

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Fig. 7 Recorded interferograms at clocking positions relative to BNP.

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The total range of clocking was 2.12 mrad and the angle from BNP for each measurement is calculated as follows

5.4 Fiber tilt

The fiber was tilted about BNP in both the $+{\varphi }_x$ and $-{\varphi }_x$ directions. The total range of tilting was ~0.64 mrad. The tilting angle was measured using the same procedure for the clocking tests. The interferograms for each tilt angle are shown in Fig. 8.

 

Fig. 8 Recorded interferograms at different tilted angles relative to BNP.

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6. Analysis of results

Calculating the misalignment sensitivity of the fiber reference was done as follows. First, the surface data from every interferogram, including BNP, were decomposed into Chebyshev polynomials. This creates data pairs for every misalignment, $\delta$, made of the misalignment position and the Chebyshev polynomial coefficient,$(\delta ,C_n)$. These data pairs were then fit to a line, and the slope is the misalignment sensitivity of a specific Chebyshev form to a specific misalignment,$d{C}_n/d\delta$. In this section, the measured sensitivities are presented for each misalignment experiment.

6.1 Misalignment aberrations introduced by fiber decenter

One would expect that the decenter would primarily induce $x$-tilt $C_1$, but also, depending on the optic being tested, $x$-coma $C_6$and $x$-astigmatism, $C_3$. Figure 9(b) shows the dominant coefficient modes from the decenter data, which is $C_1$ as expected, and all the other terms are negligible.

 

Fig. 9 (a) Misalignment aberration due to 1mm decenter. (b) First 15 Chebyshev coefficients of misalignment aberrations

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6.2 Misalignment aberrations introduced by defocus

When the fiber reference is defocused, the power along the$x$-axis is changed leaving the plane axis the same. Therefore, aberrations expected to arise from fiber defocus are $x$- astigmatism,$C_3$, and possibly some higher order $x$- spherical aberration such as$C_{10}$. The change in the surface error map due to $t_z$and their fitting coefficients are plotted in Figs. 10(a) and 10(b), respectively. The dominant measured misalignment aberration is clearly$C_3$.

 

Fig. 10 (a) Misalignment aberration due to 1mm defocus. (b) First 15 Chebyshev coefficients of misalignment aberrations.

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The measured change of $C_{10}$ is insignificant, as shown in Fig. 10(b). Additionally, we note that there should be no contributions of tilt in $x$ or $y$, so we attribute the values for $C_1$ and$C_2$ to instabilities in the translation stage assembly coupled with the high sensitivity of these terms to $t_x$ and ${\varphi }_x$. From sections 6.1 and 6.4, $t_x\sim 0.004mm$ and ${\varphi }_x\sim 20\mu rad$ yield comparable$C_1$ and $C_2$ values, respectively. All the other terms are negligible.

6.3 Misalignment aberrations introduced by the fiber clocking

Clocking,${\varphi }_z$, appears as $+x$decenter at the top of the fiber, and $-x$ decenter at the bottom. This is $x$-tilt that depends on$y$, or, $W(x,y)=Axy\propto a_4{C}_4$, 45 degree astigmatism. This was measured, Fig. 11. Coefficient $C_4$ is the most affected by clocking misalignment. The $C_2$ is again likely due to small inadvertent${\varphi }_x$. The other terms can be ignored as there is no physical reason they should be present, and the $R^2$ of the fits were less than 0.2.

 

Fig. 11 (a) Misalignment aberrations due to 1 mrad clocking. (b) First 15 Chebyshev coefficients of misalignment aberrations

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6.4 Misalignment aberrations introduced by the fiber tilting

Fiber tilt, ${\varphi }_x$, moves the fiber closer to the CN for $+y$, and equally further away for $-y$. As was shown for defocus, moving the fiber in $z$ creates horizontal astigmatism. Thus, one expects that, in addition to$C_2$, this misalignment will create horizontal astigmatism that depends on $y$, therefore,$W(x,y)=A{x}^{2}y\propto a_7{C}_7$. In these tests, the tilt and the numerical aperture of the system were both small enough that the amount of $C_7$ was negligible as shown in Fig. 12(b), as were all the other terms.

 

Fig. 12 (a) Misalignment aberrations due to 1 mrad tilting. (b) First 15 Chebyshev coefficients of misalignment aberrations

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7. Quantitative geometrical optics analysis

An intuitive but quantitative interpretation of the misalignment behavior was pursued to corroborate and understand the magnitudes of the measured misalignment errors. To accomplish this, the behavior of the lens and the fiber were individually analyzed. Thus, the errors could be assigned to the fiber reference or the cylindrical wave generating optic. The techniques used in this analysis are geometrical optics derivations and modeling using the optical design software CODE V.

7.1 Geometrical optics model for the cylindrical null system

First, a sequential lens model of the interferometric test was created using CODE V without the fiber reference, Fig. 13. Collimated light from the interferometer is incident from the left on the cylindrical null (CN) [9], creating the converging cylinder wave. Instead of encountering the fiber reference at its line focus, the rays in this model still go back to the CN, in reversed surface order, but without any reflection. Thus, the model is an unfolded version of the system without the fiber reference. The reversed CN re-collimates the diverging cylindrical wave which would then reenter the interferometer. Now, the misalignment aberrations induced by the cylindrical optic alone can be evaluated by misaligning the second CN relative to the line focus. In this way, the fiber reference is assumed to still be generating a perfect, but misaligned cylindrical wave. To achieve this, tilts and decenters are applied at the line focus. For example, an $x$-decenter will shift the second reversed CN in$x$, relative to the centerline of the cylindrical wave. Similarly, a ${\varphi }_x$ rotation, will rotate the second CN lens about the $x$-axis at the line focus, both shifting in $y$ and tilting the lens. Therefore, this tilted and shifted CN will see a perfect but shifted cylindrical wave directed at some angle other than 0 in the $y-z$plane, as is shown in Fig. 13.

 

Fig. 13 CODE V model showing a very large 5 degree tilt about the x-axis which both tilts and shifts the identical, reversed cylinder lens.

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This model reveals both the fundamental aberrations from the misalignments and whether the lens would be contributing any higher order misalignment aberrations. Since the lens operates at ~f/4.5 in the powered $x$-direction, misalignment in $z$(defocus) contributes 9.65 waves (P-V)/mm of horizontal astigmatism and no significant 1-D spherical aberration. An $x$-decenter of ~2.02mm is required before 1 wave P-V of coma is generated. From basic Seidel aberration theory, the element has field curvature and astigmatism which forms a tangential focal surface of radius ~55mm. Thus, it requires ~3.3mm of fiber decenter to induce 1 wave P-V of a 1-D equivalent of fourth order astigmatism. Therefore, accounting only for the aberrations from the lens, the CODE V model indicates that the significant misalignment coefficient for $t_x$ in this system should be $x$-tilt with a sensitivity of - 348.65 waves (P-V)/mm. Clocking, ${\varphi }_z$, will induce a $y$- dependent $x$-decenter, and tilting, ${\varphi }_x$, induces $y$-tilt and $y$-dependent defocus. Given the 42mm long fiber, one can estimate tilt and clocking sensitivities which were verified in another CODE V model to be 63.860 waves (P-V)/mrad of tilt, $C_1$, for ${\varphi }_x$, and - 6.765 waves (P-V)/mrad of the 45 degree astigmatism error resembling $C_4$ for clocking, ${\varphi }_z$.

7.2 Geometrical optics model for fiber misalignments

The fiber is a convex cylindrical mirror (negative powered) with an extremely short radius of curvature$\left(R_F=0.0625\text{}\text{mm}\right)$. So for the fiber, which has an $EFL=-0.03125\text{mm}$, the misalignments are comparably far larger than for the lens, which has an $EFL=200mm$, ~6000X larger. This different relative scale allows a geometrical optics-based analysis, mostly employing ray-based approaches.

There is a cat’s eye and a null position for the fiber reference, however they are only separated by 0.0625mm. Using the simple equation relating the object and image distance from a thin powered element, one can derive the position of the image line of the incident focusing wave – this image becomes the source line of the cylindrical wave going back to the lens. Since the fiber reference radius is so short, the image position rapidly but asymptotically appears to be displaced from the incident line focus by $t_z-R_F/2$, essentially following the fiber motion. Spherical aberration is zero at the null and cat’s eye position, and is otherwise inherently small due to the small radius of the fiber coupled to the relatively slow beam captured by the cylindrical lens, f/4.5. This results in essentially no additional misalignment induced aberration contribution from defocusing the fiber.

Evaluating $x$-decenter is a bit more complicated. At the null position, displacing the optical fiber in $x$ will yield a displaced image of the incident focusing wave. This image is at unit magnification, therefore it will move twice the $x$-motion of the fiber. At cat’s eye, paraxially, the image does not move since it is in the plane of the vertex of the powered surface. Extending the paraxial optics approximation, the fiber is actually curved, so the approximate image distance can be calculated given that the object distance equals the sag. This virtual image distance again rapidly but asymptotically approaches a location inside the fiber a distance $R_F/2$ positioned along the fiber surface normal containing the line focus and its image, Fig. 14, thus the reflected line translates less than the motion of the fiber.

 

Fig. 14 Incident light, red rays, focusing at the solid red star. The image of that focus from the fiber mirror displaced by $t_x$is the yellow star. The incident focus and the image are on a radial line $\sim R_F/2$ from the center of the fiber as $t_x$ increases in magnitude.

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The fiber analyses up to now employed purely geometrical ray optics, but the focused wavefront is, as discussed by Geary, a planar wave [2]. With this assumption, the image line will be virtual and positioned between the fiber axis of symmetry and the reflecting surface, $\left(R_F/2\right)$ behind the reflecting surface. From this perspective, the image line will translate in $x$ with the fiber. We note here that this is the effect we have experienced in the lab, a 1 to 1 correspondence of the apparent return wave decenter and the fiber decenter. This is as expected, as the motions are generally perturbations about the line focus. A subsequent paper is being developed which provides a more detailed theoretical and experimental study of these misalignments in the focus region.

Finally, the effects of tilt and clocking can again be understood from the previous analyses as generating a reference wave that emanates from a line source that tilts or clocks with the fiber reference. So misalignments of the fiber effectively move the focus of the unaberrated returning cylinder wave with the fiber.

7.3 Combining the fiber and lens model results

Combining the effects of the fiber and lens models is achieved by applying the misalignment sensitivities of the lens to the misalignment sensitivities of the fiber. The fiber misalignment sensitivities are equivalent to the line source that projects back to the cylindrical null moving with the fiber, thus the aberrations are due only to the lens, scaled by the motion of the fiber. We can convert the P-V parameters from the model into the equivalent measured units of our experiment, Table 2, which shows a very good match.

Table 2. Measured and modeled misalignment sensitivities of the fiber reference and a 105mm radius cylinder

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Also in this table, we have included the sensitivities calculated for the 105mm radius of curvature cylinder using CODE V, this time with a reflective 105mm cylinder as the reference surface [9]. Note that the fiber reference has approximately half the misalignment sensitivities. Thus, to apply the misalignment removal process presented in [9] to correcting fiber reference misalignment would yield erroneous misalignment calculations; the fiber reference requires these new, modified misalignment sensitivities.

8. Evaluation of numerically realigned measurements

This section provides a brief analysis of numerically realigned data from these test results showing that the misalignments can be removed from the measurements. To numerically realign the measured interferograms, a previously published approach [9] is employed. Since position-dependent fiber-surface aberration is expected, the evaluation in this section is made only within each misalignment test set. The misalignment position data from two widely separated misalignments in each test was realigned and then compared. The following describes the difference in RMS surface error and the average RMS surface error for each test. For decenter, the realigned data from the tests for $t_x=+0.04\text{mm}$ and $t_x=-0.04\text{mm}$ matched to within 0.005 waves, with an averaged RMS value of 0.097 waves. For defocus, the realigned data from the tests for $t_z=+0.00\text{mm}$ and $t_z=+0.46\text{mm}$ matched to within 0.001 waves, with an averaged RMS value of 0.091 waves. For clocking, the realigned data from tests ${\varphi }_z=0.00\text{mrad}$ and ${\varphi }_z=2.12\text{mrad}$ differed by 0.001 waves about an average value of 0.198 waves. Finally, for tilt, the realigned data from the tests ${\varphi }_x=+0.21\text{mrad}$and ${\varphi }_x=-0.27\text{mrad}$ differed by only 0.004 waves, with the average RMS value of 0.090 waves. Thus, the misalignments sensitivities appear to be appropriate and consistent, and the realignment method can be employed with the fiber reference.

9. Conclusion

An experimental study was performed to investigate fiber reference misalignment behavior in cylindrical optic testing. The fiber misalignment sensitivity was measured for decenter, defocus, clocking and tilt and the results were fit to 2-D Chebyshev polynomials. It was determined that each misalignment induced an error with a different polynomial form. The experimental results matched the intuitive geometrical and extended geometrical optics analyses, requiring the assumption of a planar wavefront to properly account for the behavior of $x$-decenters of the fiber. With these alignment sensitivities, misalignment induced aberration removal techniques can be correctly applied to measurements. This was shown by numerically realigning and comparing measurements from the four misalignment tests. Perhaps more importantly, the fiber can be accurately realigned based on these newly measured sensitivities.

Finally, we emphasize the differences regarding the behavior of a fiber reference. First, the test optic wavefront aberrations seen in an interferometric setup using the fiber reference are double the test optic’s aberrations in the un-powered direction, but they are not doubled in the powered direction; this the well-known 1-D filtering property of the fiber. However, what was shown in this paper is that the 1-D filtering is not evident in the misalignment sensitivities. Second, the misalignment sensitivities of a fiber reference are uniformly half those of the far larger 105mm radius cylinder. Thus, although the previously published misalignment algorithms would remove the correct forms of aberration. However, if they were used to determine the amount of misalignment in order to realign the fiber, the results would indicate incorrect realignment values by a factor of 2. In a subsequent paper, we will show the limiting reference radius at which this reduced misalignment sensitivity will arise.