Abstract

The interpretation of Fizeau interferograms of optical flats is not straightforward because they are composed of more than two reflections. This results in a confusing fringe pattern. There are three main contributions to the interferogram given by the reflections from the reference surface, the front and the rear surface of the sample. We present a new to the best of our knowledge solution to the problem. We use phase shifting measurements of the wave fields, which are reflected by and transmitted through the sample. This eliminates the need for the suppression of reflections by immersion or other methods. As an illustration of this method, several examples will also be presented.

© 2007 Optical Society of America

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References

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  1. FizCam 2000, 4D Technology Corporation, 3280 E. Hemisphere Loop, Suite 146, Tucson, Ariz. 85706, http://www.4dtechnology.com.
  2. J. S. Oh and S.-W. Kim, "Femtosecond laser pulses for surface profile metrology," Opt. Lett. 30, 2650-2652 (2005).
    [CrossRef] [PubMed]
  3. L. L. Deck, "Absolute distance measurements using FTPSI with a widely tunable IR laser," in Interferometry XI, Applications, Wolfgang Osten, ed., in Proc. SPIE 4778, 218-226 (2002).
    [CrossRef]
  4. L. L. Deck, "Multiple surface phase shifting interferometry," in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE 4451, 424-431 (2001).
    [CrossRef]
  5. P. de Groot, "Measurement of transparent plates with wavelength-tuned phase-shifting interferometry," Appl. Opt. 39, 2658-2663 (2000).
    [CrossRef]
  6. ZYGO verifire, Zygo Corporation, Laurel Brook Road, Middlefield, Conn. 06455-0448, http://www.zygo.com.
  7. M. Küchel, "Spatial coherence in interferometry, Zygos's new method to reduce intrinsic noise in interferometers," http://www.zygo.com/library/papers, and patents referenced therein.
  8. M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), p. 42.
    [PubMed]
  9. K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 94-140 and references therein.
  10. ZYGO GPI-4" XP Fizeau interferometer, Zygo Corporation, Laurel Brook Road, Middlefield, Conn. 06455-0448, http://www.zygo.com.
  11. Opaline polish, Satisloh AG, Neuhofstrasse 12, CH-6340 Baar, Switzerland, http://www.loh-optic.com.
  12. K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
    [CrossRef]
  13. G.-Soo Han and S.-Woo Kim, "Numerical correction of reference phases in phase-shifting interferometry by iterative least squares fitting," Appl. Opt. 33, 7321-7325 (1994).
    [CrossRef] [PubMed]
  14. I.-B. Kong and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least squares fitting," Opt. Eng. 34, 183-188 (1995).
    [CrossRef]
  15. S.-W. Kim, M.-G. Kang, and G.-S. Han, "Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry," Opt. Eng. 36, 3101-3106 (1997).
    [CrossRef]
  16. MetroPro 7.3.2 interferometric software, Zygo Corporation, Laurel Brook Road, Middlefield, Conn. 06455-0448, http://www.zygo.com.
  17. C1: Araldit AY 105-1 (epoxy resin), Huntsman Advanced Materials (Europe)BVBA, Everslaan, Belgium.
  18. C2: Ultramoll III (adipic acid polyester) Lanxess Deutschland GmbH, Leverkusen, Germany.
  19. C3: Araldit CY 223 (epoxy resin), Huntsman Advanced Materials (Europe)BVBA, Everslaan, Belgium.
  20. C4: Aerosil 200 (hydrophilic fumed silica) Degussa AG, Aerosil & Silanes, Hanau, Germany.
  21. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms and Software (Wiley, 1998).
  22. C. Roddier and F. Roddier, "Interferogram analysis using Fourier transform techniques," Appl. Opt. 26, 1668-1673 (1987).
    [CrossRef] [PubMed]
  23. M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), pp. 464-466; 767-772.
    [PubMed]

2005 (1)

2002 (1)

L. L. Deck, "Absolute distance measurements using FTPSI with a widely tunable IR laser," in Interferometry XI, Applications, Wolfgang Osten, ed., in Proc. SPIE 4778, 218-226 (2002).
[CrossRef]

2001 (1)

L. L. Deck, "Multiple surface phase shifting interferometry," in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE 4451, 424-431 (2001).
[CrossRef]

2000 (1)

1998 (1)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms and Software (Wiley, 1998).

1997 (1)

S.-W. Kim, M.-G. Kang, and G.-S. Han, "Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry," Opt. Eng. 36, 3101-3106 (1997).
[CrossRef]

1995 (1)

I.-B. Kong and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least squares fitting," Opt. Eng. 34, 183-188 (1995).
[CrossRef]

1994 (1)

1993 (1)

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 94-140 and references therein.

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

1987 (1)

1980 (2)

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), p. 42.
[PubMed]

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), pp. 464-466; 767-772.
[PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), p. 42.
[PubMed]

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), pp. 464-466; 767-772.
[PubMed]

Creath, K.

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 94-140 and references therein.

de Groot, P.

Deck, L. L.

L. L. Deck, "Absolute distance measurements using FTPSI with a widely tunable IR laser," in Interferometry XI, Applications, Wolfgang Osten, ed., in Proc. SPIE 4778, 218-226 (2002).
[CrossRef]

L. L. Deck, "Multiple surface phase shifting interferometry," in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE 4451, 424-431 (2001).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms and Software (Wiley, 1998).

Han, G.-S.

S.-W. Kim, M.-G. Kang, and G.-S. Han, "Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry," Opt. Eng. 36, 3101-3106 (1997).
[CrossRef]

Han, G.-Soo

Kang, M.-G.

S.-W. Kim, M.-G. Kang, and G.-S. Han, "Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry," Opt. Eng. 36, 3101-3106 (1997).
[CrossRef]

Kim, S.-W.

J. S. Oh and S.-W. Kim, "Femtosecond laser pulses for surface profile metrology," Opt. Lett. 30, 2650-2652 (2005).
[CrossRef] [PubMed]

S.-W. Kim, M.-G. Kang, and G.-S. Han, "Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry," Opt. Eng. 36, 3101-3106 (1997).
[CrossRef]

I.-B. Kong and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least squares fitting," Opt. Eng. 34, 183-188 (1995).
[CrossRef]

Kim, S.-Woo

Kong, I.-B.

I.-B. Kong and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least squares fitting," Opt. Eng. 34, 183-188 (1995).
[CrossRef]

Küchel, M.

M. Küchel, "Spatial coherence in interferometry, Zygos's new method to reduce intrinsic noise in interferometers," http://www.zygo.com/library/papers, and patents referenced therein.

Oh, J. S.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms and Software (Wiley, 1998).

Roddier, C.

Roddier, F.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), p. 42.
[PubMed]

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), pp. 464-466; 767-772.
[PubMed]

Appl. Opt. (3)

Opt. Commun. (1)

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Opt. Eng. (2)

I.-B. Kong and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least squares fitting," Opt. Eng. 34, 183-188 (1995).
[CrossRef]

S.-W. Kim, M.-G. Kang, and G.-S. Han, "Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry," Opt. Eng. 36, 3101-3106 (1997).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

L. L. Deck, "Absolute distance measurements using FTPSI with a widely tunable IR laser," in Interferometry XI, Applications, Wolfgang Osten, ed., in Proc. SPIE 4778, 218-226 (2002).
[CrossRef]

L. L. Deck, "Multiple surface phase shifting interferometry," in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE 4451, 424-431 (2001).
[CrossRef]

Other (14)

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), pp. 464-466; 767-772.
[PubMed]

ZYGO verifire, Zygo Corporation, Laurel Brook Road, Middlefield, Conn. 06455-0448, http://www.zygo.com.

M. Küchel, "Spatial coherence in interferometry, Zygos's new method to reduce intrinsic noise in interferometers," http://www.zygo.com/library/papers, and patents referenced therein.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980), p. 42.
[PubMed]

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 94-140 and references therein.

ZYGO GPI-4" XP Fizeau interferometer, Zygo Corporation, Laurel Brook Road, Middlefield, Conn. 06455-0448, http://www.zygo.com.

Opaline polish, Satisloh AG, Neuhofstrasse 12, CH-6340 Baar, Switzerland, http://www.loh-optic.com.

MetroPro 7.3.2 interferometric software, Zygo Corporation, Laurel Brook Road, Middlefield, Conn. 06455-0448, http://www.zygo.com.

C1: Araldit AY 105-1 (epoxy resin), Huntsman Advanced Materials (Europe)BVBA, Everslaan, Belgium.

C2: Ultramoll III (adipic acid polyester) Lanxess Deutschland GmbH, Leverkusen, Germany.

C3: Araldit CY 223 (epoxy resin), Huntsman Advanced Materials (Europe)BVBA, Everslaan, Belgium.

C4: Aerosil 200 (hydrophilic fumed silica) Degussa AG, Aerosil & Silanes, Hanau, Germany.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms and Software (Wiley, 1998).

FizCam 2000, 4D Technology Corporation, 3280 E. Hemisphere Loop, Suite 146, Tucson, Ariz. 85706, http://www.4dtechnology.com.

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Figures (15)

Fig. 1
Fig. 1

(a) and (b) Two reflection Fizeau interferograms of an aberrated optical flat for two different tilts of the reference wavefront. Two types of fringe pattern can be observed: first, an almost linear pattern that changes its orientation and spacing with the tilt of the reference wavefront; second, an annular fringe pattern that does not vary when the reference wavefront tilt is changed. A characteristic feature is the phase step of π in the linear fringe pattern where the linear fringes cross the annular fringes.

Fig. 2
Fig. 2

Distributions of the intensity of various reflections and transmissions at a plane-parallel flat with refractive index n s in air ignoring absorption.

Fig. 3
Fig. 3

Scheme of a Fizeau interferometer setup for testing plane-parallel flats. (i) illumination axis, (v) view axis for interferogram observation, and (a) align axis for reference surface and test piece adjustment. (1) laser, (2) focusing lens, (3) pinhole, (4) beam splitter, (5) collimating lens, (6) wedged reference flat, the reference surface is outlined by the heavy line, (7) flat under test, and (8) additional wedged flat. A stop (9) with an adjustable lens system (10), and the collimating lens (5) form an afocal system, which images the test piece onto a rotating diffuser (11). A zoom lens (12) and a lens with fixed focal length (13) image the interferogram onto a CCD chip (14). A mirror (15) steers the focused object and reference waves onto a screen with a cross hair (16). An additional lens system (17) images them onto a CCD (14); when properly aligned, the focal spots coincide with the cross hair. (R) reflection mode, (T) transmission mode, and (C) cavity mode, see text for details. The tilt of the auxiliary flat (8) in the reflection mode and the sample (7) in the transmission mode are exaggerated for clarity.

Fig. 4
Fig. 4

Set of interferometric images generated with the setup shown in Fig. 3. The samples are disks with a diameter of d = 15   mm and a thickness of t 0 = 1   mm ; (a) flat with minor radially symmetric aberrations; (b) is like (a), but a shallow dimple polished in the front surface; (c) is like (a), but a plus sign is polished in the front surface, and a minus sign is polished in the back surface of the sample. (R) reflection mode, (T) transmission mode, (C) cavity mode, see text for details.

Fig. 5
Fig. 5

(a) Amplitude | A | and (b) wrapped phase ϕ distributions for the wave fields in Fig. 4(a). The phase is linearly gray coded from 0 to 2π from black to white. (R) reflection mode, (T) transmission mode, and (C) cavity mode. The wave fields are measured using a five-bucket phase shifting method and the distributions presented here are computed using an iterative phase analysis scheme [12, 13, 14, 15] to minimize the influence of improper phase stepping. The reflection wave field exposes regions with vanishing amplitude associated with phase steps of π that result from destructive interference of front and back surface reflections.

Fig. 6
Fig. 6

(a) Real part ( A t , 0 ) , (b) imaginary part ( A t , 0 ) , (c) wrapped phase, and (d) unwrapped phase of the transmission wave for the example in Figs. 4(a) and 5 after the subtraction of the cavity wave phase according to Eqs. (13).

Fig. 7
Fig. 7

(a) Amplitude | A r rec | and (b) wrapped phase ϕ r rec of a simulated interferogram computed from the sample thickness values, obtained from the phase distribution in Fig. 6(d), according to Eqs. (16)–(18). (c) and (d) are like (a) and (b) with an iteratively adjusted thickness offset t o f f .

Fig. 8
Fig. 8

(a) Real part ( A err, n ) and (b) imaginary part ( A err, n ) of the error wavefront, which results when dividing the measured reflection wave field shown in Figs. 7(c) and 7(d) by the simulated reflection wave field shown in Figs. 5(a) and 5(b) according to Eqs. (19). Regions with low amplitude values have been masked out in order to avoid division-by-zero errors. (c) and (d) are the same as (a) and (b) after the application of an iterative Fourier algorithm to interpolate the error wave field into the masked regions; (e) amplitude and (f) wrapped phase of the extrapolated error wave.

Fig. 9
Fig. 9

(a) Amplitude | A r rec | and (b) wrapped phase ϕ r rec of the reconstructed reflection wave field resulting from adding the phase of the error wave A err shown in Figs. 8 (c) and 8(d) to the intermediate result shown in Figs. 7(c) and 7(d). This result may be compared with the measured reflection wave field (R) shown in Fig. 5.

Fig. 10
Fig. 10

Perspective representation of (a) front and (b) back surface topographies z f rec and z b rec resulting from the reconstruction of the reflection wave field A r rec . The contour lines are spaced in λ / 20 intervals. (c) front and (d) back surface topographies z f and z b measured with a commercial interferometer [10] with the reflection of the corresponding opposite sample surface suppressed by immersion (see Subsection 3.D for details).

Fig. 11
Fig. 11

Profiles of the topographic data shown in Figs. 10(a) and 10(b) along the curve AB on the front surface and the corresponding curve A*B* on the back surface of the sample. The curves i and i* show corresponding profiles of the measurement on the sample with immersed back surface as presented in Fig. 10(c) and 10(d). Note that the differences in the profiles are λ / 50 .

Fig. 12
Fig. 12

Like Fig. 10 for the sample with the shallow dimple polished in its front surface.

Fig. 13
Fig. 13

Like Fig. 11 for the sample with the shallow dimple polished in its front surface; the correspondence of the profiles is in the λ / 25 range.

Fig. 14
Fig. 14

Like Fig. 10 for the sample with the plus and minus signs polished in its front and back surfaces respectively. While the plus sign is nearly perfectly associated with the front side, the steep structures of the minus sign cause massive crosstalk on the opposite sample surface because the interpolation routine described in Subsection 5.F fails to reproduce the surfaces' high spatial frequency content.

Fig. 15
Fig. 15

Like Fig. 11 for the sample with the plus and minus signs polished in its surfaces. Although the high frequency cross talk between both sample surfaces is clearly visible (especially at the location of the minus sign), its peak amplitude remains below λ / 25 .

Equations (21)

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r = ( n s 1 n s + 1 ) 2 ,
z f , b ( x , y ) = z 0 ( x , y ) ± Δ z ( x , y ) / 2 .
n i n s 1 2 | A b | ( n s 1 ) | A f | ( n s + 1 ) .
A f = | A f | exp [ i ( 2 k z f + ϕ 0 ) ] ,
A b = | A b | exp { i [ 2 k ( z f + n s t s ) + ϕ 0 ] } = | A b | exp ( i { 2 k [ n s ( z b z f ) + z f + n s t 0 ] + ϕ 0 } ) ,
A r = A f + A b ,
| A f | = A ill [ ( 1 r r e f ) 2 r f ] 1 / 2 ,
| A b | = A ill [ ( 1 r ref ) 2 ( 1 r f ) 2 r b ] 1 / 2 ,
| A r | = ( A r A r ¯ ) 1 / 2 , = { | A f | 2 + | A b | 2 + 2 | A f | | A b | cos [ 2 k n s ( z b z f ) ] } 1 / 2 ,
A t = | A t | exp ( i { 2 k [ n s t s + ( t 0 t s ) ] + ϕ 0 } ) ,
A t = | A t | exp ( i { 2 k [ ( n s 1 ) ( z b z f ) + n s t 0 ] } + ϕ 0 ) .
A c = | A c | exp ( i ϕ 0 ) ,
A c , n = A c / | A c | , A t ,0 = A t / A c , n .
Δ z rec = ϕ t / [ 2 k ( n s 1 ) ] .
z f , b rec = ± Δ z rec / 2 ,
A f rec = | A f | exp ( i 2 k z f rec ) ,
A b rec = | A b | exp ( i { 2 k [ n s ( z b rec z f rec ) + z f rec ] } ) ,
A r rec = A f rec + A b rec .
A err = A r / A r rec , A err, n = A err / | A err | ,
z 0 rec = ϕ err / ( 2 k ) ;
z f , b rec = z 0 rec ± Δ z rec / 2 .

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