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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8856–8864
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Separating misalignment from misfigure in interferograms on cylindrical optics

Fei Liu, Brian M. Robinson, Patrick J. Reardon, and Joseph M. Geary  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8856-8864 (2013)
http://dx.doi.org/10.1364/OE.21.008856


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Abstract

This paper presents an analytical method that allows for unambiguous separation of misalignment from the interferometric measurement of cylindrical optics with rectangular apertures. This method not only removes the misalignment-induced aberration from the measured wavefront data, but also yields the amount of misalignment in the test setup. We verified this method during testing of a convex cylindrical optic.

© 2013 OSA

1. Introduction

Metrologists using interferometry are constantly battling against misalignment-induced aberrations. For example, in the test of a concave mirror at its confocal position, a defocus of the test object would introduce power and spherical aberration, optical path difference (OPD) that is not part of the true surface error. If the uncorrected test data were used by the optician for figure correction, these measurement errors would end up being polished into the surface.

An experienced metrologist may be able to minimize such error by tweaking the alignment for a best visual null (BVN) fringe and removing any remnant defocus from the interferogram. However, there are higher order contributions to the interferogram from misalignment and BVN is very subjective, so it is no surprise that different people see a different BVN. This is particularly the case when the optics under test exhibits a large figure error, yielding a fringe pattern too complicated to null, such as the test optic in this paper. Therefore, a procedure that allows the metrologist to unambiguously know the “best” surface correction profiles needed in the fabrication is absolutely critical.

Several good articles can be found on this subject. Rimmer studied various perturbations and their effects on the system performance in a lens system including lens tolerances and misalignments [1

1. M. Rimmer, “Analysis of Perturbed Lens Systems,” Appl. Opt. 9(3), 533–537 (1970). [CrossRef] [PubMed]

]. Because the system under study is assumed to be rotationally symmetric, only standard surface tilt/decenter were included. Dente and Young studied misalignment-induced aberrations on the interferometric testing of an off-axis parabola (OAP) and proposed two different ways of removing them from the measurement [2

2. G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE. 429, 187–193 (1983).

,3

3. E. W. D. Young and G. C. Dente, “The effects of rigid body motion in interferometric tests of large-aperture, off-axis, aspheric optics,” Southwest Conference on Optics. Albuquerque, NM, SPIE. 540: 59–68 (1983).

]. Because the OAP has a circular aperture and it exhibits some kind of rotational symmetry, Zernike polynonmials were used as the orthogonal, meaningful aberration descriptor. On the other hand, interferometric testing of cylindrical optics poses its own unique challenges. In addition to the need for null optics, cylindrical optics often have rectangular apertures, over which Zernike polynomials are no longer orthogonal.

This paper describes an analytical method for separating the misalignment-induced aberrations from measured wavefront data for cylindrical optics. It accomplishes this by minimizing the mean-square difference between the wavefront formed by the generally misaligned optic and that of the ideal, aligned surface that one is trying to fabricate. This procedure uses 2-D Chebyshev polynomials as the meaningful aberration descriptor for rectangular cylindrical optics and sifts out the misalignment aberration from the residual misfigure of the test optic using the linear model developed by Robinson and Reardon [4

4. F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011). [CrossRef]

]. Furthermore, we can regress the rigid body misalignment of the test optic, so that the metrologist can align the optic to its best performance position. In this paper, we used this method during testing of a convex cylindrical mirror in the center-of-curvature configuration, as illustrated in Fig. 1
Fig. 1 (a) The test layout of a convex cylindrical mirror in the center-of-curvature configuration. (b). The line plot of the test layout illustrates the assigned coordinate system, where the surface under test sits at the origin of the coordinate.
. However, the results are generalized to include testing of cylindrical optics in essentially any configuration.

2. Misalignment-induced aberrations in testing cylindrical optics

For the remainder of this paper, we assume the cylindrical axis is in the X-direction, hence the power is in the y-direction, as shown in Figs. 1. The cylindrical mirror under test might suffer unwanted alignment errors corresponding to the six degrees of freedom of the optic in 3-D Cartesian space. Specifically there are three translations (εx, εyand εz) and three rotations (ϕx, ϕyand ϕz) about respective axes.

Out of the six possible misalignments described above, translation along X-axis (εx) does not introduce any aberration, as the movement is along the axis of the cylinder. The other five misalignments, however, will alter the OPD distribution across the pupil and hence result in a different interferogram. We modeled a cylinder of 105 mm in radius in CODE V® and simulated the effects on the interferogram by introducing small misalignments, as illustrated in Fig. 2
Fig. 2 Sample interferogram (column No.1) and OPD maps (column No.2) simulated in ray tracing software (CODE V®) under different amount of movements.
.

The amount of rigid body movements (misalignments) are in column three. Careful readers might notice that both rotation about X (row No.3) and Y translation (row No.1) yield the same kind of fringe pattern, a tilt along Y-axis. This means that these two misalignments are degenerate, as long as they are small, and that the same tilt can be compensated by either alignment adjustment or a mix of both. At this moment, we have yet to include the actual surface figure error. All fringes in Fig. 2 are purely caused by misalignments.

3. Analytical basis for cylindrical interferogram

To make the correspondence more visual, we created interferograms from the first 15 Chebyshev polynomials, as illustrated in Fig. 3
Fig. 3 The artificial interferogram generated from the 2-D Chebyshev surface contour map. Each of them was assigned a P-V of 4λ.
. Each of these interferograms is assigned a Peak-to-Valley (P-V) of 4λ to give readers a good idea of their fringe patterns.

4. Misalignment Subtraction procedure that minimize the mean-square wavefront errors

The misalignment-induced aberration can therefore be expressed as a linear combination of the 2-D Chebyshev polynomials. In other words, it is reproducible by a linear addition of 2-D Chebyshev polynomials whose coefficients depend on the shape of the surface. Based on the criteria chosen (minimization of the mean square difference), separating misalignment-induced aberratioins can now be restated as solving for the coefficientsAithat minimize the mean square of the wavefront OPD.

The mathematical representation of the above statement is to solve for the coefficients {Ai} that minimize the integral of Eq. (1),
S=x,y(WAiCi)2dxdy,
(1)
where Wis the measured OPD data (containing both figure error and misalignments) andCiis theith 2-D Chebyshev polynomial. Aican be easily solved by least-squares fitting [6

6. Mathworld. “Least-squarea fitting,” Retrieved Jan 3, 2012, from http://mathworld.wolfram.com/LeastSquaresFitting.html.

].

5. Linear Model of Misalignment-induced Aberrations

In order to regress the contribution of the misalignments, we will have to include the shape information of our test object. For example, the same amount of tilt fringes is caused by a smaller misalignment on a steeper cylinder than on one that has larger radius curvature.

The linear model developed by Robinson and Reardon bridges the misalignment-induced aberrations δWto the nominal shape information of the optic under test.

The model equation can be written as [7

7. B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52(18), 2625–2636 (2005). [CrossRef]

]
δW=2[tn^+ϕ(r×n^)].
(2)
where t=[εxεyεz],ϕ=[ϕxϕyϕz],n^=[nxnynz],r=[xyz].

In this notation, t is the vector of translational misalignments and ϕdenotes the vector of angular misalignments, n^ represents the surface normal vectors at an arbitrary point of the unperturbed ideal surface and r denotes the position vector of any point on the surface with respect to the coordinate origin.

This model is based on two assumptions: 1) surface under test is close enough to the ideal surface so that the surface normals are equal to within some very small error; 2) the OPD change is linear with respect to the rigid body movements, whose magnitudes are small in nature. We can be reasonably sure about these assumptions if the interferogram is not “too” dense with fringes and because the changes of surface normal are of second order in perturbations to OPD. Additionally, this avoids the confounding effects of retrace error on measurements with large OPD errors.

6. Tracking the amount of rigid body movements

We have shown previously that the misalignment-induced OPD map δW is a linear combination of 2-D Chebyshev polynomials{Ci},

δW(x,y)=AiCi(x,y)
(3)

By using the orthogonality law of the 2-D Chebyshev polynomials, individual coefficient {Ai} can be found via Eq. (4) below, where 11x21y2 is an auxiliary weighting factor. We give an integral of this form the notation of an inner product, in this case it is(Ci,δW).

Ai=pupilCi(x,y)δW(x,y)11x21y2dxdy.=(Ci,δW).
(4)

We have also shown that the same OPD induced by misalignments can also be described by the linear model in Eq. (2). We can expand it as follows,
δW(x,y)=2(δrxnx+δryny+δrznz)=2(εx+ϕyzϕzy)nx+2(εy+ϕzxϕxz)ny+2(εz+ϕxyϕyx)nz
(5)
By substituting δWin Eq. (4) with the expanded Eq. (5), the coefficients {Ai} of the misalignment-induced aberrations can now be expressed by,
Ai=(Ci,δW)=2(Ci,δrxnx)+2(Ci,δryny)+2(Ci,δrznz)=2εx(Ci,nx)+2εy(Ci,ny)+2εz(Ci,nz)+2ϕx(Ci,ynzzny)+2ϕy(Ci,znxxnz)+2ϕz(Ci,xnyynx)=2εx(Ci,nx)+2εy(Ci,ny)+2εz(Ci,nz)+2ϕx(Ci,lx)+2ϕy(Ci,ly)+2ϕz(Ci,lz)
(6)
where lx,ly,lzare the components of normal momentsr×n^. Equation (6) can be also reduced to a much more compact form by using matrix format,
A=.
(7)
where the vector A is the expansion coefficients {Ai}, the matrix Xis the misalignment influence matrix, and the vectorδdescribes the misalignment movements imparted to the mirror,
δ=[εxεyεzϕxϕyϕz]T.
(8)
Each element in X is exclusively determined by the surface normal function n^(x,y) and the spatial coordinates on the mirror, in other words, by the shape of the ideal cylindrical optics under test,
X=[(C1,nx)(C1,ny)(C1,nz)(C1,lx)(C1,ly)(C1,lz)(C2,nx)(C2,ny)(C2,nz)(C2,lx)(C2,ly)(C2,lz)(C3,nx)(C3,ny)(C3,nz)(C3,lx)(C3,ly)(C3,lz)(C4,nx)(C4,ny)(C4,nz)(C4,lx)(C4,ly)(C4,lz)(C5,nx)(C5,ny)(C5,nz)(C5,lx)(C5,ly)(C5,lz)(C6,nx)(C6,ny)(C6,nz)(C6,lx)(C6,ly)(C6,lz)].
(9)
The vector of the rigid body movementsδcan therefore be calculated via a least-squares fit by minimizing the RMS wavefront error via variation of the vector δ (Dente and Young, Robinson and Reardon). The least-squares solution is generally found by multiplying the total aberration vector Aby the pseudo-inverse of the misalignment influence matrix X

δ=(XTX)1XTA=X+A.
(10)

Generally speaking, the rigid body movement vector δ in Eq. (10) contains six elements, three translations and three rotations. However, as discussed earlier, εxdoes not introduce any OPD change, as it is along the rotational axis of the cylinder, so we can ignoreεx. And as we discussed previously, εy and ϕx are degenerate, thus we can only assume that only one of these two movements exists in the setup. Otherwise the matrix X becomes rank deficient (or nearly so) and Eq. (10) does not yield a stable solution. We choose to include the εyterm in X.

Under these assumptions, the output vector δ now does not containεxorϕx, and therefore the coefficient vector A only has four independent components.

As an example, we use Eq. (10) to solve for misalignments for the general case illustrated the row No. 6 in Fig. 4. The retrieved results in this example are,
δ'=[εyεzϕyϕz]T=[0.0011mm0.1941mm0.005°0.01°]T
Note that the retrieved numbers are, for the most part, the same as those in Fig. 4. We expect the value forεy to differ since we are now accounting for the φxeffect withεyas well. What this means in practice is that even though both φxand εymisalignments exist for a cylindrical reflective surface, we are able to restore the ideal alignment position by only adjusting lateral displacement along Y axis (by a negative amount of εy). This is similar to the lens making process where a tilted lens surface can be corrected by purposely decentering it by a small amount. The value for εz differs because the derived solution required some of the residual error to reduce the RMS surface error to a minimum. Thus, the interferogram and OPD map formed by this set of parameters matches to that in the table, as illustrated in Fig. 5
Fig. 5 Interferogram (a) and OPD maps (b) from misalignments using the retrieved results of the reduced matrix.
.

Note that we could have assumed the output vector δ only contains ϕx and doesn’t have εy, by which we are only using tilt about X axis to compensate both εyand ϕx misalignments.

7. Experimental verification

Figure 6
Fig. 6 Interferometric measurements done by four metrologists independently, before separation of the misalignment aberrations (left column) and after separation (right column).
illustrates the four independent measurements alongside the calculated residual aberrations after misalignment removal. The comparison before (left) and after (right) the separation of misalignment illustrates a much improved measurement consistency and demonstrates the validity of our misalignment sifting algorithm. We have also included in Table 1

Table 1. P-V and RMS Error Before and After Separating Misalignment Errors

table-icon
View This Table
the peak-to-valley (P-V) and RMS error in each interferogram before misalignment removal and the P-V and RMS error for each case after removal. The square deviation in the P-V aberration obtained after misalignment removal is about 1/50th wave.

The small variations among the four cases, after misalignment removal, is a result of the noise in the interferometer measurement and the small error introduced by the first order approximation (Eq. (2) that forms the basis of this linear method (and indeed the basis of traditional interferometry in general). These errors are amplified by the condition number of the misalignment influence matrix X through the inversion (Eq. (10) that yields the misalignment components. Using the first 6 Chebyshev basis functions and the four non-degenerate misalignment components, the condition number is 5749. As a general rule of thumb, a condition number of 10k will result in a loss of k digits accuracy. The typical floating point accuracy on today’s PC is below 10−12 and therefore it has a negligible effect on the final derived misalignment error.

8. Conclusion

We have described and demonstrated a method to remove misalignment induced OPD error in measured wavefront data on cylindrical optics. A convex cylindrical reflective optic was independently measured by the four authors, and the results were processed via the method presented in this paper. The post processed data showed a more consistent result. This method could also be used to analytically align the optics under test to its ideal location.

Acknowledgments

Thank you to Synopsys, Inc / Optical Research Associates, for the free student use of CodeV. This work was partially funded through the NSF Grant ECCS-0923157.

References and links

1.

M. Rimmer, “Analysis of Perturbed Lens Systems,” Appl. Opt. 9(3), 533–537 (1970). [CrossRef] [PubMed]

2.

G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE. 429, 187–193 (1983).

3.

E. W. D. Young and G. C. Dente, “The effects of rigid body motion in interferometric tests of large-aperture, off-axis, aspheric optics,” Southwest Conference on Optics. Albuquerque, NM, SPIE. 540: 59–68 (1983).

4.

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011). [CrossRef]

5.

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49(5), 053002 (2010). [CrossRef]

6.

Mathworld. “Least-squarea fitting,” Retrieved Jan 3, 2012, from http://mathworld.wolfram.com/LeastSquaresFitting.html.

7.

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52(18), 2625–2636 (2005). [CrossRef]

OCIS Codes
(120.3940) Instrumentation, measurement, and metrology : Metrology
(220.4840) Optical design and fabrication : Testing

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 19, 2012
Revised Manuscript: January 8, 2013
Manuscript Accepted: January 16, 2013
Published: April 3, 2013

Citation
Fei Liu, Brian M. Robinson, Patrick J. Reardon, and Joseph M. Geary, "Separating misalignment from misfigure in interferograms on cylindrical optics," Opt. Express 21, 8856-8864 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8856


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References

  1. M. Rimmer, “Analysis of Perturbed Lens Systems,” Appl. Opt.9(3), 533–537 (1970). [CrossRef] [PubMed]
  2. G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE.429, 187–193 (1983).
  3. E. W. D. Young and G. C. Dente, “The effects of rigid body motion in interferometric tests of large-aperture, off-axis, aspheric optics,” Southwest Conference on Optics. Albuquerque, NM, SPIE. 540: 59–68 (1983).
  4. F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011). [CrossRef]
  5. P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010). [CrossRef]
  6. Mathworld. “Least-squarea fitting,” Retrieved Jan 3, 2012, from http://mathworld.wolfram.com/LeastSquaresFitting.html .
  7. B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt.52(18), 2625–2636 (2005). [CrossRef]

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