Abstract

The homogeneity test of glass plates in a Fizeau interferometer is hampered by the superposition of multiple interference signals coming from the surfaces of the glass plate as well as the empty Fizeau cavity. To evaluate interferograms resulting from such nested cavities, various approaches such as the use of broadband light sources have been applied. In this paper, we propose an adaptive frequency comb interferometer to accomplish the cavity selection. An adjustable Fabry–Perot resonator is used to generate a variable frequency comb that can be matched to the length of the desired cavity. Owing to its flexibility, the number of measurements needed for the homogeneity test can be reduced to four. Furthermore, compared to approaches using a two-beam interferometer as a filter for the broadband light source, the visibility of the fringe system is considerably higher if a Fabry–Perot filter is applied.

© 2011 Optical Society of America

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  1. G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” Prog. Opt. 13, 93–167 (1976).
    [CrossRef]
  2. J. Schwider, R. Burow, K.-E. Elssner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059–3061 (1985).
    [CrossRef] [PubMed]
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).
  4. Any interference filter, be it a two-beam or a multibeam interferometer with a finite resonator length, will produce a channelled spectrum or a frequency comb. This is also true for a pulse train produced by a femtosecond laser. For the outcome of a correlation experiment of second order like an interferometric experiment, there is no basic difference between a comb filtered out of a continuum and a comb corresponding to a pulse train on the time axis.
  5. J. R. Benoit, C. Fabry, and A. Perot, “Nouvelle determination du rapport des longueurs d’onde fondamentales avec l’unite metrique,” Trav. Mem. Bur. Int. Poids Mes. 15, 1–134 (1913).
  6. R. Patten, “Michelson interferometer as a remote gauge,” Appl. Opt. 10, 2717–2721 (1971).
    [CrossRef] [PubMed]
  7. C. Froehly, A. Lacourt, and J. Vienot, “Notions de reponse impulsionelle et de function de transfert temporelles de pupilles optiques, justifications experimentelles et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
    [CrossRef]
  8. B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low coherence vibration insensitive Fizeau interferometer,” Proc. SPIE 6292, 62920F (2006).
    [CrossRef]
  9. Ch. Fabry and H. Buisson, “Indications techniques sur les étalons interferentiels à lames argentées,”J. Phys. Theor. Appl. 9, 189–210 (1919).
    [CrossRef]
  10. C. Fabry, Les Applications des Interferences Lumineuses(Edition Rev. d’Optique, 1923).
  11. M. Cagnet, “Methodes interferometriques utilisant les franges de superposition I,” Rev. Opt. Theor. Instrum. 33, 113–125(1954).
  12. M. Cagnet, “Methodes interferometriques utilisant les franges de superposition II,” Rev. Opt. Theor. Instrum. 33, 229–241(1954).
  13. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
    [CrossRef] [PubMed]
  14. D. Malacara, Optical Shop Testing (Wiley, 2007).
    [CrossRef]
  15. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432(1983).
    [CrossRef] [PubMed]
  16. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  17. S. Schulte, B. Dörband, H. Müller, and W. Kähler, “Interferometer system and method for recording an interferogram using weighted averaging over multiple frequencies, and method for providing and manufacturing an object having a target surface,” U.S. patent 7,002,694 (21 February 2006).
  18. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [CrossRef]
  19. K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
    [CrossRef] [PubMed]
  20. M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique of fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
    [CrossRef] [PubMed]
  21. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42, 2354–2365 (2003).
    [CrossRef] [PubMed]
  22. D. Battistoni, “FT interferometry measures homogeneity,” Photonics Spectra (2004), http://www.photonics.com/Article.aspx?AID=18379.
  23. A. Fercher, W. Drexler, C. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
    [CrossRef]
  24. J. Schwider, “Interferometric homogeneity testing with compensation,” Opt. Commun. 6, 106–110 (1972).
    [CrossRef]
  25. A. G. Schott, “TIE-26: Homogeneity of optical glass” (July 2004).
  26. P. Hartmann, R. Jedamzik, S. Reichel, and B. Schreder, “Optical glass and glass ceramic historical aspects and recent developments: a Schott view,” Appl. Opt. 49, D157–D176 (2010).
    [CrossRef]
  27. J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 04.1–04.15 (2008).
  28. J. Schwider, “Multiple beam Fizeau interferometer with frequency comb illumination,” Opt. Commun. 282, 3308–3324(2009).
    [CrossRef]
  29. J. Schwider, “Informationssteigerung in der Vielstrahlinterferometrie,” Opt. Acta 15, 351–372 (1968).
    [CrossRef]
  30. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
    [CrossRef] [PubMed]
  31. G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934(1971).
    [CrossRef] [PubMed]
  32. J. Schwider, “Superposition fringes as a measuring tool in optical testing,” Appl. Opt. 18, 2364–2367 (1979).
    [CrossRef] [PubMed]
  33. J. Schwider, “Superposition fringe shear interferometer,” Appl. Opt. 19, 4233–4240 (1980).
    [CrossRef] [PubMed]
  34. J. Schwider, “White-light Fizeau interferometer,” Appl. Opt. 36, 1433–1437 (1997).
    [CrossRef] [PubMed]
  35. V. Shidlovski, Superlum Prospect (2004), http://www.superlumdiodes.com.
  36. J. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
    [CrossRef]
  37. J. Schwider and O. Falkenstorfer, “Twyman-Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975(1995).
    [CrossRef]
  38. Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
    [CrossRef]
  39. C. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. thesis (University of Arizona, 1981).
  40. S. Diddams and J.-C. Diels, “Dispersion measurements with white light interferometry,” J. Opt. Soc. Am. B 13, 1120–1129(1996).
    [CrossRef]
  41. J. Chamberlain, Principles of Interferometric Spectroscopy (Wiley, 1979).
  42. S. Debnath and M. Kothiyal, “Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer,” Appl. Opt. 46, 5103–5109 (2007).
    [CrossRef] [PubMed]
  43. J. V. Ramsay, “A rapid-scanning Fabry–Perot interferometer with automatic parallelism control,” Appl. Opt. 1, 411–413(1962).
    [CrossRef]
  44. J. H. R. Clarke, M. A. Norman, and F. L. Borsay, “A high performance Fabry–Perot interferometer for Rayleigh and Raman scattering studies,” J. Phys. E 8, 144–146 (1975).
    [CrossRef]
  45. D. J. Bradley, “Parallel movement for high finesse interferometric scanning,” J. Sci. Instrum. 39, 41–45 (1962).
    [CrossRef]

2010 (1)

2009 (1)

J. Schwider, “Multiple beam Fizeau interferometer with frequency comb illumination,” Opt. Commun. 282, 3308–3324(2009).
[CrossRef]

2008 (1)

J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 04.1–04.15 (2008).

2007 (1)

2006 (2)

J. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
[CrossRef]

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low coherence vibration insensitive Fizeau interferometer,” Proc. SPIE 6292, 62920F (2006).
[CrossRef]

2004 (1)

D. Battistoni, “FT interferometry measures homogeneity,” Photonics Spectra (2004), http://www.photonics.com/Article.aspx?AID=18379.

2003 (2)

A. Fercher, W. Drexler, C. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42, 2354–2365 (2003).
[CrossRef] [PubMed]

2000 (1)

1997 (1)

1996 (1)

1995 (2)

1991 (1)

1990 (2)

1987 (1)

1985 (1)

1983 (1)

1980 (1)

1979 (1)

1976 (1)

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” Prog. Opt. 13, 93–167 (1976).
[CrossRef]

1975 (1)

J. H. R. Clarke, M. A. Norman, and F. L. Borsay, “A high performance Fabry–Perot interferometer for Rayleigh and Raman scattering studies,” J. Phys. E 8, 144–146 (1975).
[CrossRef]

1973 (1)

C. Froehly, A. Lacourt, and J. Vienot, “Notions de reponse impulsionelle et de function de transfert temporelles de pupilles optiques, justifications experimentelles et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

1972 (1)

J. Schwider, “Interferometric homogeneity testing with compensation,” Opt. Commun. 6, 106–110 (1972).
[CrossRef]

1971 (2)

1968 (1)

J. Schwider, “Informationssteigerung in der Vielstrahlinterferometrie,” Opt. Acta 15, 351–372 (1968).
[CrossRef]

1967 (1)

1962 (2)

D. J. Bradley, “Parallel movement for high finesse interferometric scanning,” J. Sci. Instrum. 39, 41–45 (1962).
[CrossRef]

J. V. Ramsay, “A rapid-scanning Fabry–Perot interferometer with automatic parallelism control,” Appl. Opt. 1, 411–413(1962).
[CrossRef]

1954 (2)

M. Cagnet, “Methodes interferometriques utilisant les franges de superposition I,” Rev. Opt. Theor. Instrum. 33, 113–125(1954).

M. Cagnet, “Methodes interferometriques utilisant les franges de superposition II,” Rev. Opt. Theor. Instrum. 33, 229–241(1954).

1919 (1)

Ch. Fabry and H. Buisson, “Indications techniques sur les étalons interferentiels à lames argentées,”J. Phys. Theor. Appl. 9, 189–210 (1919).
[CrossRef]

1913 (1)

J. R. Benoit, C. Fabry, and A. Perot, “Nouvelle determination du rapport des longueurs d’onde fondamentales avec l’unite metrique,” Trav. Mem. Bur. Int. Poids Mes. 15, 1–134 (1913).

Battistoni, D.

D. Battistoni, “FT interferometry measures homogeneity,” Photonics Spectra (2004), http://www.photonics.com/Article.aspx?AID=18379.

Benoit, J. R.

J. R. Benoit, C. Fabry, and A. Perot, “Nouvelle determination du rapport des longueurs d’onde fondamentales avec l’unite metrique,” Trav. Mem. Bur. Int. Poids Mes. 15, 1–134 (1913).

Bor, Z.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Borsay, F. L.

J. H. R. Clarke, M. A. Norman, and F. L. Borsay, “A high performance Fabry–Perot interferometer for Rayleigh and Raman scattering studies,” J. Phys. E 8, 144–146 (1975).
[CrossRef]

Bradley, D. J.

D. J. Bradley, “Parallel movement for high finesse interferometric scanning,” J. Sci. Instrum. 39, 41–45 (1962).
[CrossRef]

Buisson, H.

Ch. Fabry and H. Buisson, “Indications techniques sur les étalons interferentiels à lames argentées,”J. Phys. Theor. Appl. 9, 189–210 (1919).
[CrossRef]

Burow, R.

Cagnet, M.

M. Cagnet, “Methodes interferometriques utilisant les franges de superposition I,” Rev. Opt. Theor. Instrum. 33, 113–125(1954).

M. Cagnet, “Methodes interferometriques utilisant les franges de superposition II,” Rev. Opt. Theor. Instrum. 33, 229–241(1954).

Chamberlain, J.

J. Chamberlain, Principles of Interferometric Spectroscopy (Wiley, 1979).

Clarke, J. H. R.

J. H. R. Clarke, M. A. Norman, and F. L. Borsay, “A high performance Fabry–Perot interferometer for Rayleigh and Raman scattering studies,” J. Phys. E 8, 144–146 (1975).
[CrossRef]

Coen, S.

J. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
[CrossRef]

Creath, K.

de Groot, P.

Debnath, S.

Deck, L.

Diddams, S.

Diels, J.-C.

Dörband, B.

S. Schulte, B. Dörband, H. Müller, and W. Kähler, “Interferometer system and method for recording an interferogram using weighted averaging over multiple frequencies, and method for providing and manufacturing an object having a target surface,” U.S. patent 7,002,694 (21 February 2006).

Drexler, W.

A. Fercher, W. Drexler, C. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

Dudley, J.

J. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
[CrossRef]

Elssner, K.-E.

Fabry, C.

J. R. Benoit, C. Fabry, and A. Perot, “Nouvelle determination du rapport des longueurs d’onde fondamentales avec l’unite metrique,” Trav. Mem. Bur. Int. Poids Mes. 15, 1–134 (1913).

C. Fabry, Les Applications des Interferences Lumineuses(Edition Rev. d’Optique, 1923).

Fabry, Ch.

Ch. Fabry and H. Buisson, “Indications techniques sur les étalons interferentiels à lames argentées,”J. Phys. Theor. Appl. 9, 189–210 (1919).
[CrossRef]

Falkenstorfer, O.

J. Schwider and O. Falkenstorfer, “Twyman-Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975(1995).
[CrossRef]

Fercher, A.

A. Fercher, W. Drexler, C. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

Froehly, C.

C. Froehly, A. Lacourt, and J. Vienot, “Notions de reponse impulsionelle et de function de transfert temporelles de pupilles optiques, justifications experimentelles et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Genty, G.

J. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
[CrossRef]

Grzanna, J.

Hariharan, P.

Hartmann, P.

Hayes, J.

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low coherence vibration insensitive Fizeau interferometer,” Proc. SPIE 6292, 62920F (2006).
[CrossRef]

Hiller, C.

Hitzenberger, C.

A. Fercher, W. Drexler, C. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

Jedamzik, R.

Kähler, W.

S. Schulte, B. Dörband, H. Müller, and W. Kähler, “Interferometer system and method for recording an interferogram using weighted averaging over multiple frequencies, and method for providing and manufacturing an object having a target surface,” U.S. patent 7,002,694 (21 February 2006).

Kicker, B.

Kimbrough, B.

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low coherence vibration insensitive Fizeau interferometer,” Proc. SPIE 6292, 62920F (2006).
[CrossRef]

Koliopoulos, C.

C. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. thesis (University of Arizona, 1981).

Kothiyal, M.

Lacourt, A.

C. Froehly, A. Lacourt, and J. Vienot, “Notions de reponse impulsionelle et de function de transfert temporelles de pupilles optiques, justifications experimentelles et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Lasser, T.

A. Fercher, W. Drexler, C. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).
[CrossRef]

Merkel, K.

Millerd, J.

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low coherence vibration insensitive Fizeau interferometer,” Proc. SPIE 6292, 62920F (2006).
[CrossRef]

Müller, H.

S. Schulte, B. Dörband, H. Müller, and W. Kähler, “Interferometer system and method for recording an interferogram using weighted averaging over multiple frequencies, and method for providing and manufacturing an object having a target surface,” U.S. patent 7,002,694 (21 February 2006).

Norman, M. A.

J. H. R. Clarke, M. A. Norman, and F. L. Borsay, “A high performance Fabry–Perot interferometer for Rayleigh and Raman scattering studies,” J. Phys. E 8, 144–146 (1975).
[CrossRef]

Okada, K.

Ose, T.

Osvay, K.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Patten, R.

Perot, A.

J. R. Benoit, C. Fabry, and A. Perot, “Nouvelle determination du rapport des longueurs d’onde fondamentales avec l’unite metrique,” Trav. Mem. Bur. Int. Poids Mes. 15, 1–134 (1913).

Racz, B.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Ramsay, J. V.

Reichel, S.

Sakuta, H.

Schmit, J.

Schreder, B.

Schulte, S.

S. Schulte, B. Dörband, H. Müller, and W. Kähler, “Interferometer system and method for recording an interferogram using weighted averaging over multiple frequencies, and method for providing and manufacturing an object having a target surface,” U.S. patent 7,002,694 (21 February 2006).

Schulz, G.

Schwider, J.

J. Schwider, “Multiple beam Fizeau interferometer with frequency comb illumination,” Opt. Commun. 282, 3308–3324(2009).
[CrossRef]

J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 04.1–04.15 (2008).

J. Schwider, “White-light Fizeau interferometer,” Appl. Opt. 36, 1433–1437 (1997).
[CrossRef] [PubMed]

J. Schwider and O. Falkenstorfer, “Twyman-Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975(1995).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059–3061 (1985).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432(1983).
[CrossRef] [PubMed]

J. Schwider, “Superposition fringe shear interferometer,” Appl. Opt. 19, 4233–4240 (1980).
[CrossRef] [PubMed]

J. Schwider, “Superposition fringes as a measuring tool in optical testing,” Appl. Opt. 18, 2364–2367 (1979).
[CrossRef] [PubMed]

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” Prog. Opt. 13, 93–167 (1976).
[CrossRef]

J. Schwider, “Interferometric homogeneity testing with compensation,” Opt. Commun. 6, 106–110 (1972).
[CrossRef]

G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934(1971).
[CrossRef] [PubMed]

J. Schwider, “Informationssteigerung in der Vielstrahlinterferometrie,” Opt. Acta 15, 351–372 (1968).
[CrossRef]

G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
[CrossRef] [PubMed]

Shidlovski, V.

V. Shidlovski, Superlum Prospect (2004), http://www.superlumdiodes.com.

Spolaczyk, R.

Suematsu, M.

Szabo, G.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Takeda, M.

Tsujiuchi, J.

Vienot, J.

C. Froehly, A. Lacourt, and J. Vienot, “Notions de reponse impulsionelle et de function de transfert temporelles de pupilles optiques, justifications experimentelles et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Wyant, J.

B. Kimbrough, J. Millerd, J. Wyant, and J. Hayes, “Low coherence vibration insensitive Fizeau interferometer,” Proc. SPIE 6292, 62920F (2006).
[CrossRef]

Appl. Opt. (17)

K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
[CrossRef] [PubMed]

M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique of fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
[CrossRef] [PubMed]

J. Schwider, “White-light Fizeau interferometer,” Appl. Opt. 36, 1433–1437 (1997).
[CrossRef] [PubMed]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
[CrossRef]

L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42, 2354–2365 (2003).
[CrossRef] [PubMed]

R. Patten, “Michelson interferometer as a remote gauge,” Appl. Opt. 10, 2717–2721 (1971).
[CrossRef] [PubMed]

G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934(1971).
[CrossRef] [PubMed]

S. Debnath and M. Kothiyal, “Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer,” Appl. Opt. 46, 5103–5109 (2007).
[CrossRef] [PubMed]

P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
[CrossRef] [PubMed]

P. Hartmann, R. Jedamzik, S. Reichel, and B. Schreder, “Optical glass and glass ceramic historical aspects and recent developments: a Schott view,” Appl. Opt. 49, D157–D176 (2010).
[CrossRef]

J. V. Ramsay, “A rapid-scanning Fabry–Perot interferometer with automatic parallelism control,” Appl. Opt. 1, 411–413(1962).
[CrossRef]

G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
[CrossRef] [PubMed]

J. Schwider, “Superposition fringes as a measuring tool in optical testing,” Appl. Opt. 18, 2364–2367 (1979).
[CrossRef] [PubMed]

J. Schwider, “Superposition fringe shear interferometer,” Appl. Opt. 19, 4233–4240 (1980).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432(1983).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K.-E. Elssner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059–3061 (1985).
[CrossRef] [PubMed]

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

J. Phys. E (1)

J. H. R. Clarke, M. A. Norman, and F. L. Borsay, “A high performance Fabry–Perot interferometer for Rayleigh and Raman scattering studies,” J. Phys. E 8, 144–146 (1975).
[CrossRef]

J. Phys. Theor. Appl. (1)

Ch. Fabry and H. Buisson, “Indications techniques sur les étalons interferentiels à lames argentées,”J. Phys. Theor. Appl. 9, 189–210 (1919).
[CrossRef]

J. Sci. Instrum. (1)

D. J. Bradley, “Parallel movement for high finesse interferometric scanning,” J. Sci. Instrum. 39, 41–45 (1962).
[CrossRef]

Nouv. Rev. Opt. (1)

C. Froehly, A. Lacourt, and J. Vienot, “Notions de reponse impulsionelle et de function de transfert temporelles de pupilles optiques, justifications experimentelles et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Any interference filter, be it a two-beam or a multibeam interferometer with a finite resonator length, will produce a channelled spectrum or a frequency comb. This is also true for a pulse train produced by a femtosecond laser. For the outcome of a correlation experiment of second order like an interferometric experiment, there is no basic difference between a comb filtered out of a continuum and a comb corresponding to a pulse train on the time axis.

V. Shidlovski, Superlum Prospect (2004), http://www.superlumdiodes.com.

C. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. thesis (University of Arizona, 1981).

J. Chamberlain, Principles of Interferometric Spectroscopy (Wiley, 1979).

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Figures (15)

Fig. 1
Fig. 1

On the left: Superposition of six interference terms in a nested Fizeau interferometer with He–Ne laser illumination. The nested resonator has four glass/air boundaries. In the rim region, the interference fringes of the empty Fizeau are visible because the test plate has a smaller diameter than the reference cavity. On the right: The last glass/air boundary has been removed, which can be inferred from the disappearance of fringes in the rim region. Nevertheless, in the measuring field of the test sample, a very involved interference pattern results, which is in fact a superposition of three interference patterns. The interference pattern on the right side would occur if the test plate is inserted in the test arm of a TWG interferometer.

Fig. 2
Fig. 2

Reflected complex amplitudes at the boundary surfaces for three-beam interference. By tilting the reference surface, either the front or back surface can be adjusted to fluffed out fringes, while the contribution from the other surface is high frequency.

Fig. 3
Fig. 3

Measurement of the deviations of a glass plate in the presence of strong spurious fringes resulting from the wedge of the glass plate. The low frequency deviation patterns show the phase distributions of the combinations u 1 / u 2 (top) and of u 1 / u 3 (bottom) in unwrapped and wrapped versions in a line. The evaluation of the He–Ne interferograms has been carried out by shifting the phase of the complex amplitude u 1 .

Fig. 4
Fig. 4

Phase error caused by parasitic fringes from competing cavities (line scan). The error is extracted from the deviation picture by subtracting a Zernike fit of the measured data. The order of magnitude is about 1 / 10 of a wave as predicted by Eqs. (9, 10).

Fig. 5
Fig. 5

Arrangement for the visualization of the eigenfrequencies of the FP illuminated with the light of an SLD. The blazed grating splits the light into its different frequencies, which appear as spatially separated lines on a CCD chip after focusing.

Fig. 6
Fig. 6

Frequency comb of an FP interferometer shown in transmitted light dispersed by a blazed grating in the second diffraction order. From the number of visible resonances of the frequency comb or wavelength comb, the bandwidth can be estimated to about 8 nm , which is in agreement with the bandwidth given by Superlum Corp.

Fig. 7
Fig. 7

Basic arrangement of a frequency comb Fizeau interferometer. The setup consists of a broadband SLD with a single-mode fiber interface and a length tunable FP filter. The filtered light is coupled into a multimode fiber that can be considered as a secondary light source having a spectral characteristic determined by the axial eigenfrequencies of the FP. The following Fizeau interferometer is illuminated with a slightly defocused image of the fiber exiting on a rotating scatterer. The light is collimated and after reflection is detected by a CCD camera connected to a personal computer. Phase shifting evaluations can be done either by fine-tuning of the FP or the Fizeau with the help of piezo transducers.

Fig. 8
Fig. 8

Method for the parallel adjustment of the mirrors of the variable FP filter using superposition fringes with an FP etalon of known cavity length. After the cavity length of the variable FP filter has been adjusted to be a multiple of the cavity length of the FP etalon, superposition fringes are obtained. The fringe pattern shown indicates a tilt of the adjustable plate (indicated by the double arrow) of the FP filter with respect to the fixed plate, which can now be corrected by adjusting the fringe pattern to fluffed out fringes.

Fig. 9
Fig. 9

On the left: Nested cavity configuration commonly used in homogeneity tests. The test sample (surfaces 2, 3) is inserted into a Fizeau cavity (surfaces 1, 4) in a somewhat asymmetrical fashion, i.e., t 1 t 3 , and the difference of the two lengths shall be greater than one coherence length of the light from the SLD (here > 40 μm ). On the right: Coherence function of the SLD used in our experiment (data taken from the SLD specification from Superlum Corp. [35]).

Fig. 10
Fig. 10

Set of interferograms corresponding to the resonator distance t of the FP filter for the resonances indicated in Table 3 in accordance with Fig. 9 (on the left). The average visibility is below 0.35 in the presence of all four boundary surfaces, which follows also from our estimates in Table 1. Only the interferogram of the empty Fizeau interferometer (third row interferogram on the left) shows a visibility similar to the configuration given in Fig. 13.

Fig. 11
Fig. 11

Phase shift evaluation of the glass plate through tuning of the FP filter. The corresponding FP filter distance was t = 17.366 mm . Upper left, wrapped phase of optical thickness; upper right, deviation picture (linear terms removed through least squares fit); below, Zernike fit (tenth degree) of the thickness variations (linear term removed).

Fig. 12
Fig. 12

Interference pattern generated with partially coherent monochromatic light from a single-mode He–Ne laser. From the pattern, a visibility V > 0.93 can be inferred, which is very close to 1 (this pattern delivers a realistic reference concerning the maximal attainable fringe contrast).

Fig. 13
Fig. 13

Interference pattern generated by a frequency comb produced by an FP filter with reflectivity of 0.75 ± 0.02 . From the pattern, a visibility V > 0.64 can be inferred, which is close to the theoretical value of V = 0.69 .

Fig. 14
Fig. 14

Interference pattern between the two boundaries of a glass plate with 11 mm thickness with an additional reflection from the reference surface being out of resonance with the filtering FP cavity. From the pattern, a visibility V 0.4 can be inferred, which is worse than the theoretical value of V = 0.49 .

Fig. 15
Fig. 15

Reestablishment of high contrast through tilt of the sample under test relative to the Fizeau cavity. A movable spatial filter can be used for the selection of interference patterns belonging either to the resonator 1 / 4 or the resonator 2 / 3 .

Tables (3)

Tables Icon

Table 1 Measured Optical Cavity Length for the Cavities of the Nested Configuration via the Resonant Cavity Lengths of the FP Filter a

Tables Icon

Table 2 Visibility Figures for Single Cavities in Presence of Incoherent Intensity Contributions from Competing Cavities a

Tables Icon

Table 3 Visibility Figures for the Case of Spatially Blocked Intensity Contributions from Parasitic Cavities, Assuming Spatially Coherent Light a

Equations (27)

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u ges = u 1 + u 2 + u 3 .
u 1 = | u o | R exp ( i φ 1 ) , u 2 = | u o | R [ 1 R ] exp ( i φ 2 ) , u 3 = | u o | R [ 1 R ] 2 exp ( i φ 3 ) .
I = { u 1 + u 2 + u 3 } { u 1 * + u 2 * + u 3 * } = | u 1 | 2 + | u 2 | 2 + | u 3 | 2 + u 1 u 2 * + u 2 u 1 * + u 3 u 1 * + u 1 u 3 * + u 2 u 3 * + u 3 u 2 * , I = | u 0 | 2 R [ 1 + ( 1 R ) 2 + ( 1 R ) 4 + 2 ( 1 R ) cos ( φ 2 φ 1 ) + 2 ( 1 R ) 2 cos ( φ 3 φ 1 ) + 2 ( 1 R ) 3 cos ( φ 3 φ 2 ) ] ,
( 1 R ) = 0.96 , ( 1 R ) 2 = 0.9216 , ( 1 R ) 3 = 0.8847 , ( 1 R ) 4 = 0.8493 .
I = | u 0 | 2 R [ 2.7709 + 1.92 cos ( φ 2 φ 1 ) + 1.8432 cos ( φ 3 φ 1 ) + 1.7694 cos ( φ 3 φ 2 ) ] , I 1 + A cos ( φ 2 φ 1 ) + B cos ( φ 3 φ 1 ) + C cos ( φ 3 φ 2 ) ,
tan Φ = I 2 I 4 I 1 I 3 = N D .
I 1 1 + A cos φ 2 + B cos φ 3 + C cos ( φ 3 φ 2 ) , I 2 1 + A sin φ 2 + B sin φ 3 + C cos ( φ 3 φ 2 ) , I 3 1 A cos φ 2 B cos φ 3 + C cos ( φ 3 φ 2 ) , I 4 1 A sin φ 2 B sin φ 3 + C cos ( φ 3 φ 2 ) .
tan φ 2 = A sin φ 2 + B sin φ 3 A cos φ 2 + B cos φ 3 .
Δ φ 2 = arctan { tan φ 2 } arctan { tan φ 2 } .
Δ φ 2 = arctan sin ( φ 3 φ 2 ) A / B + cos ( φ 3 φ 2 ) .
Δ φ 3 = arctan sin ( φ 3 φ 2 ) B / A + cos ( φ 3 φ 2 ) .
OPD FP = OPD res .
t res = t FP .
Φ λ 0 = λ 0 { 4 π λ 0 ( n air t FP n t gl ) } = 0 .
n air t FP = t gl { n λ 0 n λ 0 } = N t gl , t FP = N n air t gl ,
Δ φ 1 V 1 S / N ,
Δ OPD t u 2 .
I = ν a ν b I 0 ( ν ) { 1 + V cos 4 π ν c t Fiz } T 2 ( 1 R FP ) 2 { 1 + 2 m = 1 R FP m cos 4 π ν m c t FP } d ν
I m = 1 = I 0 { 1 + R FP V cos 4 π ν ¯ ( t Fiz t FP ) c } ,
u = u 1 + u 2 = u 0 R exp ( i φ 1 ) + u 0 R ( 1 R ) exp ( i φ 2 ) , I = I 0 R { 1 + ( 1 R ) 2 + 2 ( 1 R ) cos ( φ 2 φ 1 ) } .
V = 2 ( 1 R ) 1 + ( 1 R ) 2 = 0.999 .
I = | u 1 + u 2 | 2 + | u 3 | 2 I = | u 0 | 2 R [ 1 + ( 1 R ) 2 + ( 1 R ) 4 + 2 ( 1 R ) cos ( φ 2 φ 1 ) ] .
V = 2 ( 1 R ) 1 + ( 1 R ) 2 + ( 1 R ) 4 = 0.693 .
u 1 = | u o | R exp ( i φ 1 ) , u 2 = | u o | R [ 1 R ] exp ( i φ 2 ) , u 3 = | u o | R [ 1 R ] 2 exp ( i φ 3 ) , u 4 = | u o | R [ 1 R ] 3 exp ( i φ 4 ) .
I = | u 0 | 2 R [ 1 + ( 1 R ) 2 + ( 1 R ) 4 + ( 1 R ) 6 + 2 ( 1 R ) cos ( φ 2 φ 1 ) + 2 ( 1 R ) 2 cos ( φ 3 φ 1 ) + 2 ( 1 R ) 3 cos ( φ 4 φ 1 ) + 2 ( 1 R ) 3 cos ( φ 3 φ 2 ) + 2 ( 1 R ) 4 cos ( φ 4 φ 2 ) + 2 ( 1 R ) 5 cos ( φ 4 φ 3 ) ] .
V 12 = 2 | u 1 | | u 2 | R FP ( | u 1 | 2 + | u 2 | 2 + | u 3 | 2 + | u 4 | 2 ) 1 2 R FP ,
Δ Φ ( 5 ) = 2.1 * 10 5 rad , Δ Φ ( 6 ) = 4.4 * 10 10 rad .

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