We present a remote, differential method for measuring the prescription of aspheric mirrors using null interferometry in the center-of-curvature configuration. The method requires no equipment beyond that used in a basic interferometery setup (i.e., there are no shearing elements or absolute distance meters). We chose this configuration because of its widespread use. However, the method is generalizable to other configurations with an adjustment of the governing equation. The method involves taking a series of interferograms before and after small, known misalignments are applied to the mirror in the interferometry setup and calculating the prescription (e.g., radius of curvature and conic constant) of the mirror, based on these differential measurements, using a nonlinear regression. We apply this method successfully to the testing of a Space Optics Research Lab off-axis parabola with a known focal length of 152.4 mm, a diameter of 76.2 mm, and an off-axis angle of 12°.

An important problem in the realm of optical testing concerns the ability to remotely measure radius of curvature and conic constant of large reflective optical surfaces. Primary mirrors destined for space applications require cryogenic testing on the ground, during which their radii of curvature and conic constant can vary significantly and must be monitored. In such situations, the metrology instruments are located in a separate environment from the test mirror behind an optical window. Such remote measurements should be achievable with little or no equipment auxiliary to that normally used in interferometric figure testing. The radius of curvature and conic constant of large aperture optical surfaces in spaceborne instruments must also be monitored. In this situation as well, there is often a need to conduct an nonintrusive test of these shape parameters.

Methods currently exist for measuring the shapes of three-dimensional diffuse and specular objects in general [

In this section, building on earlier work by the authors [

A mathematical surface describing the mirror, with its explicit dependence on shape parameters, is written as the level surface

Equation (

The measurements in our test consist of arrays of OPD values (i.e., interferograms from a phase-shifting inteferometer), one for each pixel of the interferometer detector. We will now perform a decomposition of

Our regression of the shape of the mirror, embodied in the parameters

The matrix

In our test, we apply only translations, and not rotations, to misalign the mirror under test. Under these conditions, the equation which relates translations of the mirror to the OPD function, reduces to

If we let

The orientation of the mirror’s native coordinate system, with respect to the axes along which we apply misalignments during our testing, will contain small uncertainties that lead to some small measurement errors. One can attempt to account for these errors by introducing more free parameters though the addition of more free parameters generally reduces the precision to which all parameters can be calculated. We have approached this by developing two more detailed models, in addition to (

In the first model, rotations of the orthogonal system of translation axes about the

In the second model, independent rotations of each of the three translation stage axes about the

A single inteferometric difference measurement in our test (i.e., the difference between the OPD measured before a controlled misalignment is introduced and the OPD measured in the misaligned state) results in the recording of a number of coefficients with correlated errors. This multiplicity of responses from each measurement and their mutual correlations imply that a simple linear least squares regression does not deliver the most likely values for the shape parameters. It is a different quantity that must be minimized in order to find the probabilistically best answers. A detailed derivation of the multiresponse optimization criterion can be found in [

To recap, we are going to apply known translational misalignments to our mirror while it is nearly nulled in a center-of-curvature null test, and we are going to record the interferograms both before and after misalignments are introduced. Each measurement that we perform for a particular experimental setting yields several responses. These responses will comprise the several coefficients in the expansion of the interferometrically measured, misalignment-induced OPD. We are performing this Zernike expansion and only using the first few terms in order to reduce the amount of data and the complexity involved in the calculations. Each of these responses (the low-order Zernike terms) will, in general, be a function of the experimental settings and the shape parameters. To perform the regression of the mirror shape, the measured responses are arrayed in an

In this multiresponse situation, the most likely values of the parameters are thus obtained by minimizing the quantity

A heuristic explanation for this final result is that

To minimize the error measure

Since our regression will proceed iteratively, we will use the linear models developed above. We perform three different regressions on our data, each based on the three models embodied in (

Experimental settings

The quantities measured under each of the

During measurement, the matrix

To begin the regression, we form the residual, matrix

The optimization algorithm is illustrated in the flowchart of Figure

Flowchart for the multiresponse regression algorithm.

We tested our method using an off-axis parabola with known radius of curvature

For our measurements, we used a WYKO 400 phase shifting interferometer with an

The mirror was mounted so that its parent optical axis coincided with the optical axis of the null corrector and the axes of translation coincided nominally with the home axes of the parent parabola. That is, the plane defined by the

Setup for testing the off-axis parabola.

The surface altitude function from which we derived the model functions is the general conic of rotation with pupil offset

The presence of zero elements in the matrix

The circular pupil of the off-axis parabola was distorted by the null corrector in such a way that it appeared oblong at the image plane of the interferometer. This presented a problem for the measurement, since the theory on which the regression is based requires that the OPD recorded by each detector in the array be associated with its true pupil coordinate. If we had not corrected for the distortion and just applied a circular analysis mask, then the Zernike decomposition would be distorted and the model functions would be inaccurate.

To correct for pupil image distortion, a procedure was employed which the authors have elaborated in a previously published paper [

At each iteration of the multiresponse regression, the Zernike decomposition of the linearized model functions is performed numerically. In contrast, the raw data supplied by the interferometer need only to be decomposed into Zernike components once, at the beginning of the regression. This decomposition is accomplished by minimizing the variance between the data and a linear combination of sampled basis functions.

To take a measurement, we recorded an interferogram with the mirror aligned, and let its Zernike decomposition be

We took five sets of measurements, each set containing three series of data, one for

Two sample OPD maps are shown below in Figure

OPD Maps for (a) an aligned state and (b) for

(a)

We see in the above figures that the mirror symmetry has had its predicted effect, however, the contribution to the translation induced wavefront aberration from components that should be zero, by symmetry arguments, is not quite zero due to mechanical errors. It is also important to note that the maximum displacement introduced to the mirror was kept within certain bounds (calculated based on a relationship derived in earlier, published work by the authors [

After data are Zernike decomposed and the data matrix

In performing the regression, we found that the algorithm was most stable when we only considered the tilt and focus coefficients (i.e., coefficients one through three) in the Zernike expansions of the OPD. The other coefficients were small enough, that, when we retained them, the residual matrix was badly scaled. In other words, the columns of the residual matrix corresponding to the higher order Zernike polynomials were very small (close to zero) compared to the lower order columns. The more Zernike terms we included, the closer the residual matrix came to column rank deficiency and the more unstable the algorithm became. As discussed below, this simplification, in addition to being necessary, supplied reasonable, stable answers, and so it is justified on the grounds of correctness and economy. The reader can also be assured based on earlier work by Young and Dente [

We applied the three regression algorithms corresponding to three models to three different cases, for a total of nine final measurement results. The three cases to which we applied the regression corresponded to three different ways that the data were conditioned before being sent to the regression algorithm. In the first case, all three hundred data points were used; in the second case, five aberrant data points were first removed; and in the third case, the aberrant data points were excluded and, in addition, the OPD coefficients that should have been zero, by the symmetry arguments presented in Section

The results of our measurement are listed in Table

Results of measurement.

Algorithm | Data removed? | Symmetry zeroing? | Average | RMS | % Diff. |
---|---|---|---|---|---|

1 | N | N | 304.3/−1.030 | 1.0/0.028 | 0.16/3.00 |

2 | N | N | 304.5/−1.023 | 1.7/0.040 | 0.10/2.30 |

3 | N | N | 316.2/−0.743 | 1.3/0.061 | 3.74/25.7 |

1 | Y | N | 304.1/−1.031 | 1.0/0.027 | 0.23/3.10 |

2 | Y | N | 304.6/−1.026 | 1.6/0.042 | 0.07/2.60 |

3 | Y | N | 314.5/−0.762 | 2.5/0.074 | 3.18/23.8 |

1 | Y | Y | 305.3/−1.025 | 2.7/0.032 | 0.16/2.50 |

2 | Y | Y | 305.3/−1.024 | 2.7/0.040 | 0.16/2.40 |

3 | Y | Y | 317.1/−0.659 | 6.9/0.271 | 4.04/34.1 |

The results that were closest to the specifications given by the manufacturer of

To examine the effect that different starting values have on the outcome of the regression, and to allay any concern that the algorithm trivially reproduces the guess values, we started with the values 335 mm for

Results of measurement using different starting values for

Algorithm | Data removed? | Zeroing? | Avg. | RMS | %Diff. |
---|---|---|---|---|---|

1 | N | N | 304.3/−1.030 | 1.0/0.029 | 0.16/3.00 |

2 | N | N | 304.7/−1.022 | 1.7/0.039 | 0.03/2.30 |

1 | Y | N | 304.1/−1.031 | 1.0/0.027 | 0.23/3.10 |

2 | Y | N | 305.1/−1.027 | 1.3/0.041 | 0.10/2.70 |

If, as in (

The precision and accuracy in the answers supplied by algorithms two and three suffer from the inclusion of more free parameters. This imprecision is minimal for algorithm two, but algorithm three drives the errors to intolerable levels. Merely removing five bad data points out of three hundred causes a large change in the answers and in the errors produced by the third algorithm. This demonstrates the numerical instability introduced by the included additional parameters in algorithm three.

Possible sources of error are errors in magnitudes of displacement as well as unaccounted for rotations due to moments applied to the stage when the micrometer knobs are turned. The random departure of the relationship between OPD and displacement from linearity indicates errors of the former kind, while the presence of significant Zernike components that should be disallowed by mirror symmetries indicates errors of the second kind.

To analyze errors introduced by coupling between translations and rotations, we examined the calculated radius and conic values for the case in which the orientation of the translation axes was allowed to vary, yet it remains orthogonal. The results for this case included, in addition to radius and conic values, the estimated rotations of the mirror coordinate system about the ^{−1}). These results suggest that measurement of the radius of curvature with this method is most sensitive to rotations of the system of translation axes about the

Another possible source of error comes from the imaging distortion produced by the interferometer-null combination. The measured Zernikes coefficients and, therefore, the regressed radius and conic values are subject to errors in registration of the pixels of OPD data to coordinates in physical space. A thorough, quantitative study of this relationship will be left for future work for the sake of brevity but it should be noted that this distortion compensation process was fully reported in an earlier paper in this journal [

Since this method requires data being taken at a number of alignment positions, it is likely that significant OPD may be induced. In some intereferometers, and especially in combination with null optics, retrace error may confound the measurements by producing alignment sensitive errors in the measurement. This error can be ameliorated by empirical measurement of the retrace error sensitivity of the interferometric system, modeling of the null system according to previously published work by the coauthors [

Environmental and instrumental errors such as air turbulence (which was mitigated by the placement of a cardboard box over the airpath) and errors introduced by finite accuracy of the interferometer are likely overwhelmed by the mechanical contributions to error that were discussed above. The OPD measurement accuracy of the WYKO 400 interferometer is on the order of

It is important to note one source of error to which our method is highly insensitive: null errors. Since our method is differential in nature, the mild errors in the null are subtracted out during each measurement.

We have presented an elegant, flexible method for measuring the prescription of mirrors of any nominal shape, which is insensitive to null wavefront and alignment errors. Development of this method involved identifying the fundamental requirement that the testing be performed in a differential interferometric modality; realizing that the most elegant way to implement this differential measurement scheme was by introducing rigid body motions through a mechanical support that is already common in interferometric test setups and, perhaps most importantly, developing a sophisticated yet economical regression algorithm, based on the correct assumptions about the statistics of the measurement errors. We have successfully employed a method for measuring the prescription of an aspheric mirror using only the usual equipment necessary for performing interferometric figure tests. The method is suited to arbitrary mirror prescriptions and can be used in the conjugate null and center-of-curvature null test configuration. Such a method is suitable for the profile testing of reflective optics in a cryogenic testing situation, such as that encountered in James Webb Space Telescope testing, since it is a remote method and requires no more equipment than is necessary in making basic figure measurements.

With our setup, we were able to achieve vertex radius of curvature measurement accuracy of better than 0.1% and conic constant accuracy of better than 3% relative to the manufacturers measured values on an off-axis paraboloid with over

The accuracy of our measurement technique would no doubt be greatly improved by enhancing the mechanical support for the mirror and using more accurate motion control. An electronically controlled translation stage, like those used in most cryogenic tests, would reduce the amount of extraneous mechanical movements of the mirror and would provide better translation accuracy and relief from backlash, especially if the magnitudes of the translations were monitored with distance measuring interferometers. Future work toward improving this method of shape measurement should concentrate on two general areas: improvement of mechanical support and motion induction and improvement of data acquisition. More accurately controlled movement, via, for example, the introduction of distance measuring interferometers to the translation stage, would reduce the amount of data that would have to be taken in order to achieve good accuracy and would reduce measurement time through automation. For

The authors would like to thank NASA Marshall Spaceflight Center and the National Science Foundation for funding portions of the work described here.