Abstract

Micro-optic components and subsystems are becoming increasingly important in optical sensors, communications, data storage, and many other diverse applications. To adequately predict the performance of the final system, it is important to understand the element's effect on the wavefront as it propagates through the system. The wavefront can be measured using interferometric means, however, random and systematic errors contribute to the measurement. Self-calibration techniques exploit symmetries of the measurement or averaging techniques to separate the systematic errors of the instrument from the errors in the test lens. If the transmitted wavefront of a ball lens is measured in a number of random orientations and the measurements are averaged, the only remaining deviations from a perfect wavefront will be spherical aberration from the ball lens and the systematic errors of the interferometer. If the radius, aperture, and focal length of the ball lens are known, the spherical aberration can be calculated and subtracted, leaving only the systematic errors of the interferometer. We develop the theory behind the technique and illustrate the approach with a description of the calibration of a microinterferometer used to measure the transmitted wavefront error of micro-optics.

© 2007 Optical Society of America

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  1. A. Davies, B. Bergner, and N. Gardner, "Improving metrology for micro-optics manufacturing," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.Suleski, ed., Proc. SPIE 5177, 67-81 (2003).
  2. D. Malacara, "Twyman-Green interferometer," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 51-94.
  3. P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measurement of aberrations of microlenses using a Shack-Hartmann wavefront sensor," in Current Developments in Lens Design and Optical Engineering III, R.Fischer, W.Smith, and B.Johnson, eds., Proc. SPIE 4767, 44-52 (2002).
  4. H. Schreiber, L. Lindlein, and J. Schwider, "A lateral shearing setup for testing microlenses in the near infrared region," in the EOS Topical Meeting (EOS, 1997), Vol. 2, pp. 65-69.
  5. N. Lindlein, H. Schreiber, J. Schwider, and H. Sickinger, "The measurement of microlens parameters using shearing interferometer," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 55-64.
  6. H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
    [CrossRef]
  7. P. H. Malyak, D. L. Kent, P. Kolodner, and J. Crawford, "Interferometric measurement of the back focal length and insertion loss of precision microlens arrays," in Gradient Index, Miniature, and Diffractive Optical Systems II, T.Suleski, ed., Proc. SPIE 4437, 161-171 (2001).
  8. M. Testorf and S. Sinzinger, "Evaluation of microlens properties in the presence of high spherical aberration," Appl. Opt. 34, 6431-6437 (1995).
    [CrossRef] [PubMed]
  9. D. Daly and M. C. Hutley, "Mircrolens measurements at NPL," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 50-54.
  10. A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).
  11. B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).
  12. C. J. Evans and R. N. Kestner, "Test optic error removal," Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  13. K. Creath and J. C. Wyant, "Absolute measurement of surface roughness," Appl. Opt. 29, 3823-3827 (1990).
    [CrossRef] [PubMed]
  14. R. E. Parks, C. J. Evans, and L. Shao, "Calibration of interferometer transmission spheres," poster presented at Optical Fabrication and Testing, Kona, Hawaii (1998).
  15. N. Gardner, T. Randolph, and A. Davies, "Self-calibration for micro-refractive lens measurements," in Optical Manufacturing and Testing V, H.P.Stahl, ed., Proc. SPIE 5180,244-252 (2003).
  16. A. C. S. van Heel, Advanced Optical Techniques (Wiley, 1967), pp. 241-254.
  17. N. Gardner, "Self-calibration for micro-refractive lens measurements," M.S. thesis (University of North Carolina at Charlotte, Charlotte, N.C., 2003).
  18. D. Malacara and S. L. DeVore, "Interferogram analysis and wavefront fitting," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 455-499.
  19. P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Scientist, 3rd ed. (McGraw-Hill, 2003).
  20. K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).
  21. E. Dietrich and A. Schulze, Statistical Procedures for Machine and Process Qualification (ASQ Quality Press, 1999).

1996

1995

1994

H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

1990

1984

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

1973

A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).

Bergner, B.

A. Davies, B. Bergner, and N. Gardner, "Improving metrology for micro-optics manufacturing," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.Suleski, ed., Proc. SPIE 5177, 67-81 (2003).

K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).

Bevington, P.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Scientist, 3rd ed. (McGraw-Hill, 2003).

Crawford, J.

P. H. Malyak, D. L. Kent, P. Kolodner, and J. Crawford, "Interferometric measurement of the back focal length and insertion loss of precision microlens arrays," in Gradient Index, Miniature, and Diffractive Optical Systems II, T.Suleski, ed., Proc. SPIE 4437, 161-171 (2001).

Creath, K.

Daly, D.

D. Daly and M. C. Hutley, "Mircrolens measurements at NPL," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 50-54.

Davies, A.

A. Davies, B. Bergner, and N. Gardner, "Improving metrology for micro-optics manufacturing," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.Suleski, ed., Proc. SPIE 5177, 67-81 (2003).

N. Gardner, T. Randolph, and A. Davies, "Self-calibration for micro-refractive lens measurements," in Optical Manufacturing and Testing V, H.P.Stahl, ed., Proc. SPIE 5180,244-252 (2003).

K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).

DeVore, S. L.

D. Malacara and S. L. DeVore, "Interferogram analysis and wavefront fitting," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 455-499.

Dietrich, E.

E. Dietrich and A. Schulze, Statistical Procedures for Machine and Process Qualification (ASQ Quality Press, 1999).

Evans, C. J.

C. J. Evans and R. N. Kestner, "Test optic error removal," Appl. Opt. 35, 1015-1021 (1996).
[CrossRef] [PubMed]

R. E. Parks, C. J. Evans, and L. Shao, "Calibration of interferometer transmission spheres," poster presented at Optical Fabrication and Testing, Kona, Hawaii (1998).

Falkenstörfer, O. R.

H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

Fritz, B. S.

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

Gardner, N.

A. Davies, B. Bergner, and N. Gardner, "Improving metrology for micro-optics manufacturing," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.Suleski, ed., Proc. SPIE 5177, 67-81 (2003).

K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).

N. Gardner, T. Randolph, and A. Davies, "Self-calibration for micro-refractive lens measurements," in Optical Manufacturing and Testing V, H.P.Stahl, ed., Proc. SPIE 5180,244-252 (2003).

N. Gardner, "Self-calibration for micro-refractive lens measurements," M.S. thesis (University of North Carolina at Charlotte, Charlotte, N.C., 2003).

Hutley, M. C.

D. Daly and M. C. Hutley, "Mircrolens measurements at NPL," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 50-54.

Jensen, A. E.

A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).

Karodkar, D.

K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).

Kent, D. L.

P. H. Malyak, D. L. Kent, P. Kolodner, and J. Crawford, "Interferometric measurement of the back focal length and insertion loss of precision microlens arrays," in Gradient Index, Miniature, and Diffractive Optical Systems II, T.Suleski, ed., Proc. SPIE 4437, 161-171 (2001).

Kestner, R. N.

Kolodner, P.

P. H. Malyak, D. L. Kent, P. Kolodner, and J. Crawford, "Interferometric measurement of the back focal length and insertion loss of precision microlens arrays," in Gradient Index, Miniature, and Diffractive Optical Systems II, T.Suleski, ed., Proc. SPIE 4437, 161-171 (2001).

Lindlein, L.

H. Schreiber, L. Lindlein, and J. Schwider, "A lateral shearing setup for testing microlenses in the near infrared region," in the EOS Topical Meeting (EOS, 1997), Vol. 2, pp. 65-69.

Lindlein, N.

H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

N. Lindlein, H. Schreiber, J. Schwider, and H. Sickinger, "The measurement of microlens parameters using shearing interferometer," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 55-64.

Malacara, D.

D. Malacara, "Twyman-Green interferometer," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 51-94.

D. Malacara and S. L. DeVore, "Interferogram analysis and wavefront fitting," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 455-499.

Malyak, P. H.

P. H. Malyak, D. L. Kent, P. Kolodner, and J. Crawford, "Interferometric measurement of the back focal length and insertion loss of precision microlens arrays," in Gradient Index, Miniature, and Diffractive Optical Systems II, T.Suleski, ed., Proc. SPIE 4437, 161-171 (2001).

Medicus, K.

K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).

Neal, D. R.

P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measurement of aberrations of microlenses using a Shack-Hartmann wavefront sensor," in Current Developments in Lens Design and Optical Engineering III, R.Fischer, W.Smith, and B.Johnson, eds., Proc. SPIE 4767, 44-52 (2002).

Parks, R. E.

R. E. Parks, C. J. Evans, and L. Shao, "Calibration of interferometer transmission spheres," poster presented at Optical Fabrication and Testing, Kona, Hawaii (1998).

Pulaski, P. D.

P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measurement of aberrations of microlenses using a Shack-Hartmann wavefront sensor," in Current Developments in Lens Design and Optical Engineering III, R.Fischer, W.Smith, and B.Johnson, eds., Proc. SPIE 4767, 44-52 (2002).

Randolph, T.

N. Gardner, T. Randolph, and A. Davies, "Self-calibration for micro-refractive lens measurements," in Optical Manufacturing and Testing V, H.P.Stahl, ed., Proc. SPIE 5180,244-252 (2003).

Ratte, K.

P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measurement of aberrations of microlenses using a Shack-Hartmann wavefront sensor," in Current Developments in Lens Design and Optical Engineering III, R.Fischer, W.Smith, and B.Johnson, eds., Proc. SPIE 4767, 44-52 (2002).

Robinson, D.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Scientist, 3rd ed. (McGraw-Hill, 2003).

Roller, J. P.

P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measurement of aberrations of microlenses using a Shack-Hartmann wavefront sensor," in Current Developments in Lens Design and Optical Engineering III, R.Fischer, W.Smith, and B.Johnson, eds., Proc. SPIE 4767, 44-52 (2002).

Schreiber, H.

H. Schreiber, L. Lindlein, and J. Schwider, "A lateral shearing setup for testing microlenses in the near infrared region," in the EOS Topical Meeting (EOS, 1997), Vol. 2, pp. 65-69.

N. Lindlein, H. Schreiber, J. Schwider, and H. Sickinger, "The measurement of microlens parameters using shearing interferometer," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 55-64.

Schulze, A.

E. Dietrich and A. Schulze, Statistical Procedures for Machine and Process Qualification (ASQ Quality Press, 1999).

Schwider, J.

H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

N. Lindlein, H. Schreiber, J. Schwider, and H. Sickinger, "The measurement of microlens parameters using shearing interferometer," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 55-64.

H. Schreiber, L. Lindlein, and J. Schwider, "A lateral shearing setup for testing microlenses in the near infrared region," in the EOS Topical Meeting (EOS, 1997), Vol. 2, pp. 65-69.

Shao, L.

R. E. Parks, C. J. Evans, and L. Shao, "Calibration of interferometer transmission spheres," poster presented at Optical Fabrication and Testing, Kona, Hawaii (1998).

Sickinger, H.

H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

N. Lindlein, H. Schreiber, J. Schwider, and H. Sickinger, "The measurement of microlens parameters using shearing interferometer," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 55-64.

Sinzinger, S.

Testorf, M.

van Heel, A. C. S.

A. C. S. van Heel, Advanced Optical Techniques (Wiley, 1967), pp. 241-254.

Wyant, J. C.

Appl. Opt.

J. Opt. Soc. Am.

A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).

Opt. Eng.

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

H. Sickinger, O. R. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

Other

P. H. Malyak, D. L. Kent, P. Kolodner, and J. Crawford, "Interferometric measurement of the back focal length and insertion loss of precision microlens arrays," in Gradient Index, Miniature, and Diffractive Optical Systems II, T.Suleski, ed., Proc. SPIE 4437, 161-171 (2001).

D. Daly and M. C. Hutley, "Mircrolens measurements at NPL," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 50-54.

A. Davies, B. Bergner, and N. Gardner, "Improving metrology for micro-optics manufacturing," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.Suleski, ed., Proc. SPIE 5177, 67-81 (2003).

D. Malacara, "Twyman-Green interferometer," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 51-94.

P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measurement of aberrations of microlenses using a Shack-Hartmann wavefront sensor," in Current Developments in Lens Design and Optical Engineering III, R.Fischer, W.Smith, and B.Johnson, eds., Proc. SPIE 4767, 44-52 (2002).

H. Schreiber, L. Lindlein, and J. Schwider, "A lateral shearing setup for testing microlenses in the near infrared region," in the EOS Topical Meeting (EOS, 1997), Vol. 2, pp. 65-69.

N. Lindlein, H. Schreiber, J. Schwider, and H. Sickinger, "The measurement of microlens parameters using shearing interferometer," in the EOS Topical Meeting (EOS, 1993), Vol. 2, pp. 55-64.

R. E. Parks, C. J. Evans, and L. Shao, "Calibration of interferometer transmission spheres," poster presented at Optical Fabrication and Testing, Kona, Hawaii (1998).

N. Gardner, T. Randolph, and A. Davies, "Self-calibration for micro-refractive lens measurements," in Optical Manufacturing and Testing V, H.P.Stahl, ed., Proc. SPIE 5180,244-252 (2003).

A. C. S. van Heel, Advanced Optical Techniques (Wiley, 1967), pp. 241-254.

N. Gardner, "Self-calibration for micro-refractive lens measurements," M.S. thesis (University of North Carolina at Charlotte, Charlotte, N.C., 2003).

D. Malacara and S. L. DeVore, "Interferogram analysis and wavefront fitting," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley, 1992), pp. 455-499.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Scientist, 3rd ed. (McGraw-Hill, 2003).

K. Medicus, D. Karodkar, B. Bergner, N. Gardner, and A. Davies, "Compact interferometer for micro-optic performance and shape characterization," in Lithographic and Micromachining Techniques for Optical Component Fabrication II, E.Kley and H.P.Herzig, eds., Proc. SPIE 5183, 85-93 (2003).

E. Dietrich and A. Schulze, Statistical Procedures for Machine and Process Qualification (ASQ Quality Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Schematic of the calibration configuration. The test lens is replaced with a ball lens and aperture in the interferometer.

Fig. 2
Fig. 2

Schematic of wavefront propagation through an imperfect ball lens.

Fig. 3
Fig. 3

(a) Wavefront map of the average of the 65 measurements with the ball lens at random orientations. (b) Wavefront map of the spherical aberration contribution from the ball lens. (c) Final wavefront map representing the interferometer bias, W INT (d) Combined uncertainty map for the calibration. The maximum uncertainty (at the edge of the map) is 0.103 waves while the minimum (at the center) is 0.008 waves.

Fig. 4
Fig. 4

Data editor and 2D layout for the ZEMAX model of the ball lens and aperture ( 0.14 NA ) .

Fig. 5
Fig. 5

Schematic of possible misalignments in the calibration procedure. (a) Angular ( δ θ ) misalignment of the aperture with respect to the optical axis of the interferometer; (b) lateral ( Δ x , Δ y ) and vertical ( Δ z ) misalignment of the ball and aperture with respect to the optical axis of the interferometer; (c) lateral misalignment ( δ x , δ y ) of the ball lens with respect to the aperture.

Tables (1)

Tables Icon

Table 1 Two Examples of Zernike Coefficients for the Calibration Result, the Uncertainty Contributions, and the Combined Uncertainty a

Equations (11)

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

W i = W INT + W i , BALL Θ ( ρ , θ ) + W BALLspherical ( ρ ) + W i , NOISE ,
1 N ( i = 1 N W i , BALLFigure ( ρ , θ ) + i = 1 N W i , NOISE ) 0 ,
W INT = 1 N i = 1 N ( W i ) W BALLspherical ( ρ ) .
W lens = W meas W INT .
W INT ( ρ , θ ) = n , m A n m Z n m = A 2 2 ρ 2 cos ( 2 θ ) + A 2 2 ρ 2 × sin ( 2 θ ) + A 3 1 ( 3 ρ 2 2 ) ρ cos ( θ ) + ,
W INT = n , m ( ( 1 N i = 1 N ( A n m ) i ) Z n m δ m 0 ( A n m ) spher Z n m ) ,
( A n m ) estimate = ( A n m ) true + ( ε n m ) NOISE + ( ε n m = 0 ) Ballspherical + ( ε n m ) misalignment + ( ε n m ) reproducibility ,
u n m = ( u n   NOISE m ) 2 + ( u n   Ballspherical m = 0 ) 2 + ( u n   misalignment m ) 2 + etc .
u c ( ρ , θ ) = n , m ( ( u n m ) 2 ( Z n m ) 2 ) = ( u 2 2 ) 2 [ ρ 2 cos ( 2 θ ) ] 2 + ( u 2 2 ) 2 [ ρ 2 sin ( 2 θ ) ] 2 + ,
u n   NOISE m = σ n m N ,
u n   misalignment m = ( u n   aper   tilt m ) 2 + ( u n   ball + aper m ) 2 + ( u n   ball m ) 2 ,

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