Abstract

As an alternative to the conventional optical frequency comb technique, a spatial frequency comb technique is proposed for dispersionfree optical coherence depth sensing. Instead of generating an optical frequency comb over a wide range of time spectrum, we generate a spatial frequency comb by modulating the incident angle of a monochromatic plane wave with a spatial light modulator (SLM). The use of monochromatic light combined with the SLM enables dispersion-free depth sensing that is free from mechanical moving components.

© 2006 Optical Society of America

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  1. K. Hotate and T. Okugawa, "Selective extraction of a two-dimensional optical image by synthesis of the coherence function," Opt. Lett. 17, 1529-1531 (1992).
    [CrossRef] [PubMed]
  2. K. Hotate and O. Kamatani, "Optical coherence domain reflectometry by synthesis of coherence function," J. Lightwave Technol. 11, 1701-1709 (1993).
    [CrossRef]
  3. S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
    [CrossRef]
  4. K. Hotate and T. Okugawa, "Optical information processing by synthesis of the coherence function," J. Lightwave Technol. 12, 1247-1255 (1994).
    [CrossRef]
  5. Z. He and K. Hotate, "Synthesized optical coherence tomography for imaging of scattering objects by use of a stepwise frequency-modulated tunable laser diode," Opt. Lett. 24, 1502-1504 (1999).
    [CrossRef]
  6. S. Choi, M. Yamamoto, D. Moteki, T. Shioda,Y. Tanaka, and T. Kurokawa, "Frequency-comb-based interferometer for profilometry and tomography," Opt. Lett. 31, 1976-1978 (2006).
    [CrossRef] [PubMed]
  7. P. A. Flournoy, R. W. McClure, and G. Wyntjes, "White-light interferometric thickness gauge," Appl. Opt.,  11, 1907-1915 (1972).
    [CrossRef] [PubMed]
  8. N. Tanaka, M. Takeda, and K. Matsumoto, "Interferometrically measuring the physical properties of test object," United States Patent 4,072,422 (filed October 20 1976, 1978).
  9. M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233-247 (1987).
  10. B. S. Lee and T. C. Strand, "Profilometry with a coherence scanning microscope," Appl. Opt. 29, 3784-3788 (1990).
    [CrossRef] [PubMed]
  11. T. Dresel, G. Hausler, and H. Venzke, "Three-dimensional sensing of rough surfaces by coherence radar," Appl. Opt. 31, 919-925 (1992).
    [CrossRef] [PubMed]
  12. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
    [CrossRef] [PubMed]
  13. J. Schwider and L. Zhou, "Dispersive interferometric profilometer," Opt. Lett. 19, 995-997 (1994).
    [CrossRef] [PubMed]
  14. M. Takeda, and H. Yamamoto, "Fourier-transform speckle profilometry: Three-dimensional shape measurements of diffuse objects with large height steps and /or spatially isolated surfaces," Appl. Opt. 33, 7829-7837 (1994).
    [CrossRef] [PubMed]
  15. S. Kuwamura and I. Yamaguchi, "Wavelength scanning profilometry for real-time surface shape measurement," Appl. Opt. 36, 4473-4482 (1997).
    [CrossRef] [PubMed]
  16. H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997).
    [CrossRef]
  17. M. Kinoshita, M. Takeda, H. Yago, Y. Watanabe, and T. Kurokawa, "Optical frequency-domain microprofilometry with a frequency-tunable liquid-crystal Fabry-Perot etalon device," Appl. Opt. 38, 7063-7068 (1999).
    [CrossRef]
  18. D. S. Mehta, M. Sugai, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, "Simultaneous three-dimensional step-height measurement and high-resolution tomographic imaging with a spectral interferometric microscope," Appl. Opt. 41, 3874-3885 (2002).
    [CrossRef] [PubMed]
  19. M. Takeda, "The philosophy of fringes: Analogies and dualities in fringe generation and analysis," in Fringe ’97 Automatic Processing of Fringe Patterns, W. Jueptner and W. Osten, eds., Akademie Verlag Series in Optical Metrology, (Akademie Verlag, Berlin, 1997), pp.17-26.
  20. M. Takeda, "Space-time analogy in absolute optical profilometry using frequency-scan techniques: A tutorial review," in Perspectives in Engineering Optics, K. Singh and P. K. Rastogi, eds, (Anita Publications, New Delhi, 2003), pp.192-198.
  21. J. Ch. Vienot, R. Ferriere, J. P. Goedgebur, "Conjugation of space and time variables in optics," in Optics in Four Dimensions, M. A. Machado and L. M. Narducci, eds., No. 65 of AIP Conference Proceedings, (American Institute of Physics, New York, 1981), pp.49-62.
  22. Z. Duan, Y. Miyamoto, and M. Takeda, "Dispersion-free absolute interferometry based on angular spectrum scanning," Opt. Express 14, 655-663 (2006).
    [CrossRef] [PubMed]
  23. P. D. Ruiz and J. M. Huntley, "Depth-resolved displacement measurement using tilt scanning speckle interferometry," in Fringe 2005 Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, Berlin, 2005), pp. 238-241.
  24. C. W. McCutchen, "Generalized source and the van Cittert-Zernike theorem: A study of spatial coherence required for interferometry," J. Opt. Soc. Am. 56, 727-733 (1966).
    [CrossRef]
  25. M. Kuechel, "Apparatus and method(s) for reducing the effects of coherent artifacts in an interferometry," US Patent 6804011 B2, (2004) or US Patent Application 2003/0030819 A1, (2003).
  26. J. Rosen and M. Takeda, "Longitudinal spatial coherence applied for surface profilometry," Appl. Opt. 37, 4107-4111 (2000).
    [CrossRef]
  27. M. Gokhler and J. Rosen, "Synthesis of a multiple-peak spatial degree of coherence for imaging through absorbing media," Appl. Opt. 44, 2921-2927 (2005).
    [CrossRef] [PubMed]

2006

2005

2002

2001

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
[CrossRef]

2000

J. Rosen and M. Takeda, "Longitudinal spatial coherence applied for surface profilometry," Appl. Opt. 37, 4107-4111 (2000).
[CrossRef]

1999

1997

S. Kuwamura and I. Yamaguchi, "Wavelength scanning profilometry for real-time surface shape measurement," Appl. Opt. 36, 4473-4482 (1997).
[CrossRef] [PubMed]

H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997).
[CrossRef]

1994

1993

K. Hotate and O. Kamatani, "Optical coherence domain reflectometry by synthesis of coherence function," J. Lightwave Technol. 11, 1701-1709 (1993).
[CrossRef]

1992

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1990

1972

1966

Ando, M.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Choi, S.

Dresel, T.

Duan, Z.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Flournoy, P. A.

Franze, B.

H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997).
[CrossRef]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Gokhler, M.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Haible, P.

H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997).
[CrossRef]

Hausler, G.

He, Z.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hinosugi, H.

Hotate, K.

Z. He and K. Hotate, "Synthesized optical coherence tomography for imaging of scattering objects by use of a stepwise frequency-modulated tunable laser diode," Opt. Lett. 24, 1502-1504 (1999).
[CrossRef]

K. Hotate and T. Okugawa, "Optical information processing by synthesis of the coherence function," J. Lightwave Technol. 12, 1247-1255 (1994).
[CrossRef]

K. Hotate and O. Kamatani, "Optical coherence domain reflectometry by synthesis of coherence function," J. Lightwave Technol. 11, 1701-1709 (1993).
[CrossRef]

K. Hotate and T. Okugawa, "Selective extraction of a two-dimensional optical image by synthesis of the coherence function," Opt. Lett. 17, 1529-1531 (1992).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Kamatani, O.

K. Hotate and O. Kamatani, "Optical coherence domain reflectometry by synthesis of coherence function," J. Lightwave Technol. 11, 1701-1709 (1993).
[CrossRef]

Kinoshita, M.

Kourogi, M.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
[CrossRef]

Kurokawa, T.

Kuwamura, S.

Lee, B. S.

Lee, S.-J.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
[CrossRef]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

McClure, R. W.

McCutchen, C. W.

Mehta, D. S.

Miyamoto, Y.

Moteki, D.

Ohtsu, M.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
[CrossRef]

Okugawa, T.

K. Hotate and T. Okugawa, "Optical information processing by synthesis of the coherence function," J. Lightwave Technol. 12, 1247-1255 (1994).
[CrossRef]

K. Hotate and T. Okugawa, "Selective extraction of a two-dimensional optical image by synthesis of the coherence function," Opt. Lett. 17, 1529-1531 (1992).
[CrossRef] [PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Rosen, J.

M. Gokhler and J. Rosen, "Synthesis of a multiple-peak spatial degree of coherence for imaging through absorbing media," Appl. Opt. 44, 2921-2927 (2005).
[CrossRef] [PubMed]

J. Rosen and M. Takeda, "Longitudinal spatial coherence applied for surface profilometry," Appl. Opt. 37, 4107-4111 (2000).
[CrossRef]

Saito, S.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Schwider, J.

Shioda, T.

Shishido, M.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Strand, T. C.

Sugai, M.

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Takahashi, H.

Takeda, M.

Tanaka, Y.

Tiziani, H. J.

H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997).
[CrossRef]

Venzke, H.

Watanabe, Y.

Widiyatmoko, B.

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
[CrossRef]

Wyntjes, G.

Yago, H.

Yamaguchi, I.

Yamamoto, H.

Yamamoto, M.

Yoshizawa, T.

Zhou, L.

Appl. Opt.

P. A. Flournoy, R. W. McClure, and G. Wyntjes, "White-light interferometric thickness gauge," Appl. Opt.,  11, 1907-1915 (1972).
[CrossRef] [PubMed]

B. S. Lee and T. C. Strand, "Profilometry with a coherence scanning microscope," Appl. Opt. 29, 3784-3788 (1990).
[CrossRef] [PubMed]

T. Dresel, G. Hausler, and H. Venzke, "Three-dimensional sensing of rough surfaces by coherence radar," Appl. Opt. 31, 919-925 (1992).
[CrossRef] [PubMed]

M. Takeda, and H. Yamamoto, "Fourier-transform speckle profilometry: Three-dimensional shape measurements of diffuse objects with large height steps and /or spatially isolated surfaces," Appl. Opt. 33, 7829-7837 (1994).
[CrossRef] [PubMed]

S. Kuwamura and I. Yamaguchi, "Wavelength scanning profilometry for real-time surface shape measurement," Appl. Opt. 36, 4473-4482 (1997).
[CrossRef] [PubMed]

M. Kinoshita, M. Takeda, H. Yago, Y. Watanabe, and T. Kurokawa, "Optical frequency-domain microprofilometry with a frequency-tunable liquid-crystal Fabry-Perot etalon device," Appl. Opt. 38, 7063-7068 (1999).
[CrossRef]

D. S. Mehta, M. Sugai, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, "Simultaneous three-dimensional step-height measurement and high-resolution tomographic imaging with a spectral interferometric microscope," Appl. Opt. 41, 3874-3885 (2002).
[CrossRef] [PubMed]

J. Rosen and M. Takeda, "Longitudinal spatial coherence applied for surface profilometry," Appl. Opt. 37, 4107-4111 (2000).
[CrossRef]

M. Gokhler and J. Rosen, "Synthesis of a multiple-peak spatial degree of coherence for imaging through absorbing media," Appl. Opt. 44, 2921-2927 (2005).
[CrossRef] [PubMed]

J. Lightwave Technol.

K. Hotate and O. Kamatani, "Optical coherence domain reflectometry by synthesis of coherence function," J. Lightwave Technol. 11, 1701-1709 (1993).
[CrossRef]

K. Hotate and T. Okugawa, "Optical information processing by synthesis of the coherence function," J. Lightwave Technol. 12, 1247-1255 (1994).
[CrossRef]

J. Mod. Opt.

H. J. Tiziani, B. Franze, and P. Haible, "Wavelength-shift speckle interferometry for absolute profilometry using mode-hope free external cavity diode laser," J. Mod. Opt. 44, 1485-1496 (1997).
[CrossRef]

J. Opt. Soc. Am.

Jpn. J. Appl. Phys

S.-J. Lee, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, "Ultrahigh scanning speed optical coherence tomography using optical frequency comb generators," Jpn. J. Appl. Phys. Part II,  40, L878-L880 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Other

N. Tanaka, M. Takeda, and K. Matsumoto, "Interferometrically measuring the physical properties of test object," United States Patent 4,072,422 (filed October 20 1976, 1978).

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, "An application of interference microscopy to integrated circuit inspection and metrology," in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE 775, 233-247 (1987).

M. Takeda, "The philosophy of fringes: Analogies and dualities in fringe generation and analysis," in Fringe ’97 Automatic Processing of Fringe Patterns, W. Jueptner and W. Osten, eds., Akademie Verlag Series in Optical Metrology, (Akademie Verlag, Berlin, 1997), pp.17-26.

M. Takeda, "Space-time analogy in absolute optical profilometry using frequency-scan techniques: A tutorial review," in Perspectives in Engineering Optics, K. Singh and P. K. Rastogi, eds, (Anita Publications, New Delhi, 2003), pp.192-198.

J. Ch. Vienot, R. Ferriere, J. P. Goedgebur, "Conjugation of space and time variables in optics," in Optics in Four Dimensions, M. A. Machado and L. M. Narducci, eds., No. 65 of AIP Conference Proceedings, (American Institute of Physics, New York, 1981), pp.49-62.

P. D. Ruiz and J. M. Huntley, "Depth-resolved displacement measurement using tilt scanning speckle interferometry," in Fringe 2005 Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, Berlin, 2005), pp. 238-241.

M. Kuechel, "Apparatus and method(s) for reducing the effects of coherent artifacts in an interferometry," US Patent 6804011 B2, (2004) or US Patent Application 2003/0030819 A1, (2003).

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Figures (10)

Fig. 1.
Fig. 1.

Two-beam interferometry with oblique illumination.

Fig. 2.
Fig. 2.

Ewald sphere representation of optical frequency comb (a) and spatial frequency comb (b).

Fig. 3.
Fig. 3.

Optical geometry with tilt misalignment in reference mirror.

Fig. 4.
Fig. 4.

Effect of the tilt of reference mirror and the depth sensing in h ′ direction. (a) Misaligned geometry in which the tilt causes the projected concentric ring sources to function as a continuous broadband source. (b) Tilt-compensated geometry in which decentered ring sources can generate a spatial frequency comb.

Fig. 5.
Fig. 5.

Experimental setup.

Fig. 6:
Fig. 6:

(a) Source structure for a spatial frequency comb of firework (SFCF). (b) Comb spectrum in the source plane. (c) Comb spectrum in spatial frequency domain.

Fig. 7.
Fig. 7.

Variation of fringe contrast with the object height relative to the reference mirror. Two different parameters N=8 and N=16 were adopted for the SFCF.

Fig. 8.
Fig. 8.

Examples of fringe patterns selected from the total 32 steps of SFCF scanning.

Fig. 9.
Fig. 9.

(a). Fringe pattern for the illumination with a SFCF (N=8) tuned to left surface. (b) Fringe pattern for the illumination with the first ring source of the SFCF (N=8). (c) Fringe pattern for the illumination with the third ring source of the SFCF (N=8). Note the phase shift of the fringes on the right surface as compared to (b). (d) The fringe intensity on the SFCF-tuned left surface does not oscillate with the shift of the ring source. (f) The fringe intensity on the SFCFdetuned right surface varies with the shift of the ring source. (g) The sinusoidal variation of the fringe intensity on the left surface with the shift of the ring source for the SFCF (N=32) tuned to 6.4mm height. The red circles indicate the intensity at the ring position for the SFCF (N=8) tuned to the left surface.

Fig. 10.
Fig. 10.

Comparison between SFCF and SSFC.

Equations (17)

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

Δ φ = k · 2 h = 2 k h h = 2 h k cos θ ,
u A = A ( k ) exp ( i k · r A ) d k
u A = A ( k ) exp ( i k · r A ) d k ,
J AA ( h ) = A ( k ) A * ( k ) exp [ i ( k · r A k · r A ) ] d k d k
= A ( k ) 2 exp [ i k · ( r A - r A ) ] d k = 2 π k A ( k h ) 2 exp ( i 2 k h h ) d k h ,
μ AA ( h ) = J AA ( h ) J AA ( 0 ) J A A ( 0 ) = A ( k h ) 2 exp ( i 2 k h h ) d k h A ( k h ) 2 d k h
A ( k h ) 2 = n = 0 N 1 δ [ k h ( k h max n Δ k h ) ] ,
μ AA ( h ) = sin ( N Δ k h h ) N sin ( Δ k h h ) exp { i [ 2 k ( N 1 ) Δ k h ] h } . .
k h = k cos θ = k f r 2 + f 2 k ( 1 r 2 2 f 2 ) ,
r n = f 2 Δ k h k × n .
A ( k h ) 2 = 1 + cos ( 2 π k h Δ k h ) ( k K h k h k ) .
μ AA ( h ) = { sinc [ K h h ] + 1 2 sinc [ K h ( h + π Δ k h ) ] exp { i [ 2 k K h ] ( π Δ k h ) }
+ 1 2 sinc [ K h ( h π Δ k h ) ] exp { i [ 2 k K h ] ( π Δ k h ) } } exp [ i ( 2 k K h ) h ]
A ( k ) 2 = 1 + cos { 2 π [ k ( k 2 2 k ) ] Δ k h } .
Δ k h = k ( f 3 f 2 ) 2 ( M Δ x ) 2 8 f 4 2 N = 16.19 N [ radians/ mm ] ,
Δ ν = c Δ k h 2 π = k c ( f 3 f 2 ) 2 ( M Δ x ) 2 16 π f 4 2 N = 0.773 N [ THz ] ,
Δ h = π Δ k h = c 2 Δ ν = 4 λ ( f 2 f 4 f 3 M Δ x ) 2 N = 0.194 N 0.2 N [ mm ] .

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