Abstract

Interferometry with a null corrector can be used to test cylindrical surfaces. The requirement for accurate measurement is a null fringe pattern. When the tested cylindrical surface is not perfect or seriously misaligned, a nonzero fringe pattern might be obtained. As a result, high-order misalignment aberrations (e.g., coma and spherical aberration) are introduced into the measurement. The sources and types of high-order misalignment aberrations are analyzed by orthogonal Legendre polynomials. Based on the analysis, a mathematical model was proposed to estimate the high-order misalignment aberrations. Then a wavefront difference method was proposed to calibrate the coefficients of this model. With the calibrated coefficients, the high-order misalignment aberrations can be determined and separated from the measurement results. Several experiments were conducted to demonstrate the validity of the proposed method. Compared with the lower-order misalignment aberrations removal method, the proposed method can reduce the high-order misalignment aberrations by at least half, and highly accurate results can be achieved by the proposed method.

© 2014 Optical Society of America

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References

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2013 (2)

2012 (1)

2010 (2)

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

V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Opt. 49, 6924–6929 (2010).
[CrossRef]

2009 (1)

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

2007 (1)

2003 (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

2002 (1)

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10 (2002).
[CrossRef]

2000 (1)

1993 (1)

C. Evans and J. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals-Manufact. Technol. 42, 577–580 (1993).

1990 (1)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Brown, T. G.

Bryan, J.

C. Evans and J. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals-Manufact. Technol. 42, 577–580 (1993).

Burge, J. H.

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Carpio-Valadez, M.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Chen, M.

Dai, G.-M.

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Evans, C.

C. Evans and J. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals-Manufact. Technol. 42, 577–580 (1993).

Fleig, J.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Forbes, G.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Ge, D.

Geary, J.

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

Geary, J. M.

Lamprecht, J.

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10 (2002).
[CrossRef]

Lindlein, N.

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10 (2002).
[CrossRef]

Liu, F.

Mahajan, V. N.

Malacara-Hernandez, D.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Moore, D. T.

Murphy, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Murphy, P. E.

Peng, J.

Reardon, P.

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

Reardon, P. J.

Robinson, B. M.

Sanchez-Mondragon, J. J.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Schreiner, R.

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10 (2002).
[CrossRef]

Schwider, J.

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10 (2002).
[CrossRef]

Tian, C.

Tricard, M.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Wang, K.

Yang, Y.

Yu, Y.

Zhou, P.

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Zhuo, Y.

Appl. Opt. (4)

CIRP Annals-Manufact. Technol. (1)

C. Evans and J. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals-Manufact. Technol. 42, 577–580 (1993).

J. Opt. A (1)

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

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

Opt. Express (1)

Opt. Photon. News (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Proc. SPIE (1)

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Other (2)

http://diffraction.com/cylinder.php .

“Encoding and fabrication report: Cgh cylinder null h45f15c,” Tech. Rep. (Diffraction International, 2012).

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Figures (14)

Fig. 1.
Fig. 1.

Wavefront maps of the first 15 terms of 2D OLP in a unit square.

Fig. 2.
Fig. 2.

Diagram of the simulated system.

Fig. 3.
Fig. 3.

Recorded interferograms and residual phase maps (upper row, the recorded interferograms; bottom row, the residual maps after removing the lower-order misalignment aberrations): (a) defocus, (b) tilt, (c) tip, and (d) twist.

Fig. 4.
Fig. 4.

High-order aberrations caused by misalignment errors.

Fig. 5.
Fig. 5.

Residual maps after removing the high-order misalignment aberrations: (a) defocus, (b) tilt, (c) tip, and (d) twist.

Fig. 6.
Fig. 6.

Relationship between the misalignment errors and the high-order misalignment aberrations: (a) defocus, (b) tilt, (c) tip, and (d) twist.

Fig. 7.
Fig. 7.

Experimental setup.

Fig. 8.
Fig. 8.

Measurement result at approximate null condition. (a) Interferogram. (b) Real surface figure.

Fig. 9.
Fig. 9.

Measurement results under certain defocus: (a) interferogram, (b) without high-order misalignment aberrations compensated, and (c) with high-order misalignment aberrations compensated.

Fig. 10.
Fig. 10.

Comparison results when the tested cylindrical surface is misaligned with certain defocus. (a) Residual errors obtained by subtracting Fig. 9(b) from Fig. 8(b). (b) Residual errors obtained by subtracting Fig. 9(c) from Fig. 8(b).

Fig. 11.
Fig. 11.

Measurement results under composite misalignment errors: (a) interferogram, (b) without high-order misalignment aberrations compensated, and (c) with high-order misalignment aberrations compensated.

Fig. 12.
Fig. 12.

Comparison of residual errors when the tested cylindrical surface is misaligned with composite misalignment errors. (a) Residual errors obtained by subtracting Fig. 11(b) from Fig. 8(b). (b) Residual errors obtained by subtracting Fig. 11(c) from Fig. 8(b).

Fig. 13.
Fig. 13.

Measurement result of a rectangular pupil. (a) Interferogram with null-fringe pattern. (b) Phase map of Fig. 13(a). (c) Interferogram with misalignment errors. (d) Phase map of Fig. 13(c).

Fig. 14.
Fig. 14.

Comparison of residual errors when measuring the cylindrical lens with rectangular pupil: (a) without the high-order misalignment compensated and (b) with high-order misalignment aberrations compensated.

Tables (5)

Tables Icon

Table 1. First 15 Terms of 2D OLP

Tables Icon

Table 2. Comparison of Residual Errors without and with High-Order Misalignment Aberrations Compensation in Simulation

Tables Icon

Table 3. Calibrated Results of Scale Factors

Tables Icon

Table 4. Comparison of Residual Errors without and with High-Order Aberrations Compensation

Tables Icon

Table 5. Reproducibility Investigation by Introducing Different Misalignment Errors

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

Qj(x,y)=Ll(x)Lm(y),
W=j=1NajQj,
a15=k1a6,
Whdef=k1a6Q15,
a10=k2a3,
Whtil=k2a3Q10,
a9=k3a2,
Whtip=k3a2Q9,
a14=k4a5,
Whtwi=k4a5Q14,
Whma=k1a6Q15+k2a3Q10+k3a2Q9+k4a5Q14,
Whma=k1a4Q11+k2a2Q7+k3a3Q8+k4a5Q12.

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