Abstract

We propose a practical theoretical model of an interference microscope that includes the imaging properties of optical systems with partially coherent illumination. We show that the effects on measured topography of a spatially extended, monochromatic light source at low numerical apertures can be approximated in a simplified model that assumes spatially coherent light and a linear, locally shift-invariant transfer function that accounts for optical aberrations and the attenuation of diffracted plane wave amplitudes with increasing spatial frequencies. Simulation of instrument response using this model agrees with methods using numerical pupil-plane integration and with an experimental measurement of surface topography.

© 2020 Optical Society of America

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References

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  7. A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
    [Crossref]
  8. J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148
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    [Crossref]
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  11. L. L. Deck and C. Evans, “High performance Fizeau and scanning white-light interferometers for mid-spatial frequency optical testing of free-form optics,” Proc. SPIE 5921, 59210A (2005).
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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]
  44. D. Maystre, O. M. Mendez, and A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
    [Crossref]
  45. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), pp. 46–47.
  46. G. Schulz, “Über Interferenzen gleicher Dicke und Längenmessung mit Lichtwellen,” Ann. Phys. 449, 177–187 (1954).
    [Crossref]
  47. C. J. R. Sheppard and K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
    [Crossref]
  48. P. de Groot, “Error compensation in phase shifting interferometry,” U.S. patent7,948,637 (May24, 2011).
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    [Crossref]
  50. G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001).
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    [Crossref]
  52. Y. Ichioka and T. Suzuki, “Image of a periodic complex object in an optical system under partially coherent illumination,” J. Opt. Soc. Am. 66, 921–932 (1976).
    [Crossref]
  53. I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
    [Crossref]
  54. E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
    [Crossref]
  55. B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the height transfer function (HTF),” Proc. SPIE 5878, 587806 (2005).
    [Crossref]
  56. E. Church, C. Dainty, D. Gale, and P. Takacs, “Comparison of optical and mechanical measurements of surface finish,” Proc. SPIE 1531, 234–250 (1992).
    [Crossref]
  57. A. Fujii, H. Suzuki, and K. Yanagi, “Development of measurement standards for verifying functional performance of surface texture measuring instruments,” J. Phys. Conf. Ser. 311, 012009 (2011).
    [Crossref]
  58. V. G. Badami, J. Liesener, C. J. Evans, and P. de Groot, “Evaluation of the measurement performance of a coherence scanning microscope using roughness specimens,” in Proc. ASPE (2011), pp. 23–26.

2020 (3)

C. Gomez, R. Su, P. de Groot, and R. K. Leach, “Noise reduction in coherence scanning interferometry for surface topography measurement,” Nanomanuf. Metrol. 3, 68–76 (2020).
[Crossref]

M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
[Crossref]

R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
[Crossref]

2019 (3)

P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, “Spectral composition of low-coherence interferograms at high numerical apertures,” J. Eur. Opt. Soc. 15, 5 (2019).
[Crossref]

P. de Groot, X. Colonna de Lega, R. Su, and R. Leach, “Does interferometry work? A critical look at the foundations of interferometric surface topography measurement,” Proc. SPIE 11102, 111020G (2019).
[Crossref]

X. Feng, N. Senin, R. Su, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

2018 (1)

A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
[Crossref]

2017 (3)

C. Gomez, R. Su, A. Thompson, J. DiSciacca, S. Lawes, and R. K. Leach, “Optimization of surface measurement for metal additive manufacturing using coherence scanning interferometry,” Opt. Eng. 56, 111714 (2017).
[Crossref]

M. F. Fay and T. Dresel, “Applications of model-based transparent surface films analysis using coherence-scanning interferometry,” Opt. Eng. 56, 111709 (2017).
[Crossref]

S. Wang, “Coherent phase transfer function degradation due to wave aberrations of a laser Fizeau interferometer,” Opt. Eng. 56, 111711 (2017).
[Crossref]

2015 (1)

R. K. Leach, C. Giusca, H. Haitjema, C. Evans, and X. Jiang, “Calibration and verification of areal surface texture measuring instruments,” CIRP Ann. 64, 797–813 (2015).
[Crossref]

2012 (1)

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
[Crossref]

2011 (1)

A. Fujii, H. Suzuki, and K. Yanagi, “Development of measurement standards for verifying functional performance of surface texture measuring instruments,” J. Phys. Conf. Ser. 311, 012009 (2011).
[Crossref]

2008 (3)

P. de Groot, X. Colonna de Lega, J. Liesener, and M. Darwin, “Metrology of optically-unresolved features using interferometric surface profiling and RCWA modeling,” Opt. Express 16, 3970–3975 (2008).
[Crossref]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
[Crossref]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[Crossref]

2005 (2)

L. L. Deck and C. Evans, “High performance Fizeau and scanning white-light interferometers for mid-spatial frequency optical testing of free-form optics,” Proc. SPIE 5921, 59210A (2005).
[Crossref]

B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the height transfer function (HTF),” Proc. SPIE 5878, 587806 (2005).
[Crossref]

2004 (1)

2002 (1)

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack–Hartmann wavefront sensor,” J. Microsc. 205, 61–75 (2002).
[Crossref]

2001 (1)

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik 112, 399–406 (2001).
[Crossref]

2000 (1)

1999 (1)

L. L. Deck and J. A. Soobitsky, “Phase-shifting via wavelength tuning in very large aperture interferometers,” Proc. SPIE 3782, 432–442 (1999).
[Crossref]

1997 (1)

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[Crossref]

1995 (2)

C. J. R. Sheppard and K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
[Crossref]

D. M. Aikens, A. Roussel, and M. Bray, “Derivation of preliminary specifications for transmitted wavefront and surface roughness for large optics used in inertial confinement fusion,” Proc. SPIE 2633, 350–360 (1995).
[Crossref]

1992 (1)

E. Church, C. Dainty, D. Gale, and P. Takacs, “Comparison of optical and mechanical measurements of surface finish,” Proc. SPIE 1531, 234–250 (1992).
[Crossref]

1987 (1)

M. Davidson, K. Kaufman, and I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

1983 (1)

D. Maystre, O. M. Mendez, and A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[Crossref]

1976 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1967 (2)

1965 (1)

M. De and P. K. Mondal, “Phase and amplitude contrast microscopy in partially coherent light,” J. Res. Natl. Bur. Stand., Sect. C 69C, 199 (1965).
[Crossref]

1957 (1)

1954 (1)

G. Schulz, “Über Interferenzen gleicher Dicke und Längenmessung mit Lichtwellen,” Ann. Phys. 449, 177–187 (1954).
[Crossref]

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikroskopische Anatomie 9, 413–418 (1873).
[Crossref]

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikroskopische Anatomie 9, 413–418 (1873).
[Crossref]

Abdulhalim, I.

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
[Crossref]

Ai, C.

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[Crossref]

Aikens, D. M.

D. M. Aikens, A. Roussel, and M. Bray, “Derivation of preliminary specifications for transmitted wavefront and surface roughness for large optics used in inertial confinement fusion,” Proc. SPIE 2633, 350–360 (1995).
[Crossref]

Allendorf, B.

P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, “Spectral composition of low-coherence interferograms at high numerical apertures,” J. Eur. Opt. Soc. 15, 5 (2019).
[Crossref]

Badami, V. G.

V. G. Badami, J. Liesener, C. J. Evans, and P. de Groot, “Evaluation of the measurement performance of a coherence scanning microscope using roughness specimens,” in Proc. ASPE (2011), pp. 23–26.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Bellouard, Y.

R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
[Crossref]

Beverage, J. L.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack–Hartmann wavefront sensor,” J. Microsc. 205, 61–75 (2002).
[Crossref]

Bisterov, I.

A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
[Crossref]

Boccara, A.-C.

Boreman, G. D.

G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), pp. 46–47.

Bray, M.

D. M. Aikens, A. Roussel, and M. Bray, “Derivation of preliminary specifications for transmitted wavefront and surface roughness for large optics used in inertial confinement fusion,” Proc. SPIE 2633, 350–360 (1995).
[Crossref]

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2006), pp. 547–666.

Church, E.

E. Church, C. Dainty, D. Gale, and P. Takacs, “Comparison of optical and mechanical measurements of surface finish,” Proc. SPIE 1531, 234–250 (1992).
[Crossref]

Clare, A. T.

A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
[Crossref]

Clark, M.

A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
[Crossref]

Colonna de Lega, X.

P. de Groot, X. Colonna de Lega, R. Su, and R. Leach, “Does interferometry work? A critical look at the foundations of interferometric surface topography measurement,” Proc. SPIE 11102, 111020G (2019).
[Crossref]

P. de Groot, X. Colonna de Lega, J. Liesener, and M. Darwin, “Metrology of optically-unresolved features using interferometric surface profiling and RCWA modeling,” Opt. Express 16, 3970–3975 (2008).
[Crossref]

P. de Groot and X. Colonna de Lega, “Signal modeling for low-coherence height-scanning interference microscopy,” Appl. Opt. 43, 4821–4830 (2004).
[Crossref]

P. de Groot and X. Colonna de Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Proceedings of the 5th International Workshop on Advanced Optical Metrology (Springer, 2006), pp. 30–37.

J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148

Colonna de Lega, X. M.

X. M. Colonna de Lega, M. F. Fay, and P. de Groot, “Optical evaluation of lenses and lens molds,” U.S. patent9,658,129 (May23, 2017).

Coupland, J.

M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
[Crossref]

R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
[Crossref]

Coupland, J. M.

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
[Crossref]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[Crossref]

Dainty, C.

E. Church, C. Dainty, D. Gale, and P. Takacs, “Comparison of optical and mechanical measurements of surface finish,” Proc. SPIE 1531, 234–250 (1992).
[Crossref]

Darwin, M.

Davidson, M.

M. Davidson, K. Kaufman, and I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

De, M.

M. De and P. K. Mondal, “Phase and amplitude contrast microscopy in partially coherent light,” J. Res. Natl. Bur. Stand., Sect. C 69C, 199 (1965).
[Crossref]

de Groot, P.

C. Gomez, R. Su, P. de Groot, and R. K. Leach, “Noise reduction in coherence scanning interferometry for surface topography measurement,” Nanomanuf. Metrol. 3, 68–76 (2020).
[Crossref]

P. de Groot, X. Colonna de Lega, R. Su, and R. Leach, “Does interferometry work? A critical look at the foundations of interferometric surface topography measurement,” Proc. SPIE 11102, 111020G (2019).
[Crossref]

P. de Groot, X. Colonna de Lega, J. Liesener, and M. Darwin, “Metrology of optically-unresolved features using interferometric surface profiling and RCWA modeling,” Opt. Express 16, 3970–3975 (2008).
[Crossref]

P. de Groot and X. Colonna de Lega, “Signal modeling for low-coherence height-scanning interference microscopy,” Appl. Opt. 43, 4821–4830 (2004).
[Crossref]

P. de Groot, “Error compensation in phase shifting interferometry,” U.S. patent7,948,637 (May24, 2011).

P. de Groot and X. Colonna de Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Proceedings of the 5th International Workshop on Advanced Optical Metrology (Springer, 2006), pp. 30–37.

X. M. Colonna de Lega, M. F. Fay, and P. de Groot, “Optical evaluation of lenses and lens molds,” U.S. patent9,658,129 (May23, 2017).

J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148

V. G. Badami, J. Liesener, C. J. Evans, and P. de Groot, “Evaluation of the measurement performance of a coherence scanning microscope using roughness specimens,” in Proc. ASPE (2011), pp. 23–26.

de Groot, P. J.

P. J. de Groot, “Interference microscopy for surface structure analysis,” in Handbook of Optical Metrology, T. Yoshizawa, ed. (CRC Press, 2015), pp. 791–828.

Deck, L. L.

L. L. Deck and C. Evans, “High performance Fizeau and scanning white-light interferometers for mid-spatial frequency optical testing of free-form optics,” Proc. SPIE 5921, 59210A (2005).
[Crossref]

L. L. Deck and J. A. Soobitsky, “Phase-shifting via wavelength tuning in very large aperture interferometers,” Proc. SPIE 3782, 432–442 (1999).
[Crossref]

L. L. Deck, “Method and apparatus for optimizing the optical performance of interferometers,” U.S. patent10,267,617 (April23, 2019).

Descour, M. R.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack–Hartmann wavefront sensor,” J. Microsc. 205, 61–75 (2002).
[Crossref]

DeVelis, J. B.

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R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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C. Gomez, R. Su, P. de Groot, and R. K. Leach, “Noise reduction in coherence scanning interferometry for surface topography measurement,” Nanomanuf. Metrol. 3, 68–76 (2020).
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P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, “Spectral composition of low-coherence interferograms at high numerical apertures,” J. Eur. Opt. Soc. 15, 5 (2019).
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R. K. Leach, C. Giusca, H. Haitjema, C. Evans, and X. Jiang, “Calibration and verification of areal surface texture measuring instruments,” CIRP Ann. 64, 797–813 (2015).
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B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the height transfer function (HTF),” Proc. SPIE 5878, 587806 (2005).
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Hüser, L.

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J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148

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M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
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R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
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X. Feng, N. Senin, R. Su, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
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P. de Groot, X. Colonna de Lega, R. Su, and R. Leach, “Does interferometry work? A critical look at the foundations of interferometric surface topography measurement,” Proc. SPIE 11102, 111020G (2019).
[Crossref]

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C. Gomez, R. Su, P. de Groot, and R. K. Leach, “Noise reduction in coherence scanning interferometry for surface topography measurement,” Nanomanuf. Metrol. 3, 68–76 (2020).
[Crossref]

C. Gomez, R. Su, A. Thompson, J. DiSciacca, S. Lawes, and R. K. Leach, “Optimization of surface measurement for metal additive manufacturing using coherence scanning interferometry,” Opt. Eng. 56, 111714 (2017).
[Crossref]

R. K. Leach, C. Giusca, H. Haitjema, C. Evans, and X. Jiang, “Calibration and verification of areal surface texture measuring instruments,” CIRP Ann. 64, 797–813 (2015).
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F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
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J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148

Lehmann, P.

P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, “Spectral composition of low-coherence interferograms at high numerical apertures,” J. Eur. Opt. Soc. 15, 5 (2019).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
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D. Maystre, O. M. Mendez, and A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
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M. Davidson, K. Kaufman, and I. Mazor, “The coherence probe microscope,” Solid State Technol. 30, 57–59 (1987).

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D. Maystre, O. M. Mendez, and A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
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G. O. Reynolds, J. B. DeVelis, J. George, B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).

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D. Maystre, O. M. Mendez, and A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
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D. M. Aikens, A. Roussel, and M. Bray, “Derivation of preliminary specifications for transmitted wavefront and surface roughness for large optics used in inertial confinement fusion,” Proc. SPIE 2633, 350–360 (1995).
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X. Feng, N. Senin, R. Su, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
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J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack–Hartmann wavefront sensor,” J. Microsc. 205, 61–75 (2002).
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L. L. Deck and J. A. Soobitsky, “Phase-shifting via wavelength tuning in very large aperture interferometers,” Proc. SPIE 3782, 432–442 (1999).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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Su, R.

C. Gomez, R. Su, P. de Groot, and R. K. Leach, “Noise reduction in coherence scanning interferometry for surface topography measurement,” Nanomanuf. Metrol. 3, 68–76 (2020).
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M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
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R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
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P. de Groot, X. Colonna de Lega, R. Su, and R. Leach, “Does interferometry work? A critical look at the foundations of interferometric surface topography measurement,” Proc. SPIE 11102, 111020G (2019).
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X. Feng, N. Senin, R. Su, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
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A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
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C. Gomez, R. Su, A. Thompson, J. DiSciacca, S. Lawes, and R. K. Leach, “Optimization of surface measurement for metal additive manufacturing using coherence scanning interferometry,” Opt. Eng. 56, 111714 (2017).
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A. Fujii, H. Suzuki, and K. Yanagi, “Development of measurement standards for verifying functional performance of surface texture measuring instruments,” J. Phys. Conf. Ser. 311, 012009 (2011).
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E. Church, C. Dainty, D. Gale, and P. Takacs, “Comparison of optical and mechanical measurements of surface finish,” Proc. SPIE 1531, 234–250 (1992).
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P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, “Spectral composition of low-coherence interferograms at high numerical apertures,” J. Eur. Opt. Soc. 15, 5 (2019).
[Crossref]

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R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
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M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
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C. Gomez, R. Su, A. Thompson, J. DiSciacca, S. Lawes, and R. K. Leach, “Optimization of surface measurement for metal additive manufacturing using coherence scanning interferometry,” Opt. Eng. 56, 111714 (2017).
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J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148

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S. Wang, “Coherent phase transfer function degradation due to wave aberrations of a laser Fizeau interferometer,” Opt. Eng. 56, 111711 (2017).
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E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
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E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
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A. Fujii, H. Suzuki, and K. Yanagi, “Development of measurement standards for verifying functional performance of surface texture measuring instruments,” J. Phys. Conf. Ser. 311, 012009 (2011).
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Acta Mater. (1)

A. Speidel, R. Su, J. Mitchell-Smith, P. Dryburgh, I. Bisterov, D. Pieris, W. Li, R. Patel, M. Clark, and A. T. Clare, “Crystallographic texture can be rapidly determined by electrochemical surface analytics,” Acta Mater. 159, 89–101 (2018).
[Crossref]

Ann. Phys. (2)

G. Schulz, “Über Interferenzen gleicher Dicke und Längenmessung mit Lichtwellen,” Ann. Phys. 449, 177–187 (1954).
[Crossref]

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
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Appl. Opt. (3)

Arch. Mikroskopische Anatomie (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikroskopische Anatomie 9, 413–418 (1873).
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CIRP Ann. (1)

R. K. Leach, C. Giusca, H. Haitjema, C. Evans, and X. Jiang, “Calibration and verification of areal surface texture measuring instruments,” CIRP Ann. 64, 797–813 (2015).
[Crossref]

J. Eur. Opt. Soc. (1)

P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, “Spectral composition of low-coherence interferograms at high numerical apertures,” J. Eur. Opt. Soc. 15, 5 (2019).
[Crossref]

J. Microsc. (1)

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack–Hartmann wavefront sensor,” J. Microsc. 205, 61–75 (2002).
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J. Opt. Soc. Am. (4)

J. Phys. Conf. Ser. (1)

A. Fujii, H. Suzuki, and K. Yanagi, “Development of measurement standards for verifying functional performance of surface texture measuring instruments,” J. Phys. Conf. Ser. 311, 012009 (2011).
[Crossref]

J. Res. Natl. Bur. Stand., Sect. C (1)

M. De and P. K. Mondal, “Phase and amplitude contrast microscopy in partially coherent light,” J. Res. Natl. Bur. Stand., Sect. C 69C, 199 (1965).
[Crossref]

Meas. Sci. Technol. (2)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[Crossref]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
[Crossref]

Nanomanuf. Metrol. (1)

C. Gomez, R. Su, P. de Groot, and R. K. Leach, “Noise reduction in coherence scanning interferometry for surface topography measurement,” Nanomanuf. Metrol. 3, 68–76 (2020).
[Crossref]

Opt. Acta (1)

D. Maystre, O. M. Mendez, and A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[Crossref]

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Opt. Eng. (4)

S. Wang, “Coherent phase transfer function degradation due to wave aberrations of a laser Fizeau interferometer,” Opt. Eng. 56, 111711 (2017).
[Crossref]

M. Thomas, R. Su, N. Nikolaev, J. Coupland, and R. Leach, “Modeling of interference microscopy beyond the linear regime,” Opt. Eng. 59, 034110 (2020).
[Crossref]

C. Gomez, R. Su, A. Thompson, J. DiSciacca, S. Lawes, and R. K. Leach, “Optimization of surface measurement for metal additive manufacturing using coherence scanning interferometry,” Opt. Eng. 56, 111714 (2017).
[Crossref]

M. F. Fay and T. Dresel, “Applications of model-based transparent surface films analysis using coherence-scanning interferometry,” Opt. Eng. 56, 111709 (2017).
[Crossref]

Opt. Express (1)

Opt. Laser Eng. (2)

X. Feng, N. Senin, R. Su, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. Leach, “Lens aberration compensation in interference microscopy,” Opt. Laser Eng. 128, 106015 (2020).
[Crossref]

Optik (1)

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik 112, 399–406 (2001).
[Crossref]

Proc. SPIE (7)

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[Crossref]

B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the height transfer function (HTF),” Proc. SPIE 5878, 587806 (2005).
[Crossref]

E. Church, C. Dainty, D. Gale, and P. Takacs, “Comparison of optical and mechanical measurements of surface finish,” Proc. SPIE 1531, 234–250 (1992).
[Crossref]

P. de Groot, X. Colonna de Lega, R. Su, and R. Leach, “Does interferometry work? A critical look at the foundations of interferometric surface topography measurement,” Proc. SPIE 11102, 111020G (2019).
[Crossref]

L. L. Deck and J. A. Soobitsky, “Phase-shifting via wavelength tuning in very large aperture interferometers,” Proc. SPIE 3782, 432–442 (1999).
[Crossref]

D. M. Aikens, A. Roussel, and M. Bray, “Derivation of preliminary specifications for transmitted wavefront and surface roughness for large optics used in inertial confinement fusion,” Proc. SPIE 2633, 350–360 (1995).
[Crossref]

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[Crossref]

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J. DiSciacca, C. Gomez, A. Thompson, S. D. A. Lawes, R. K. Leach, X. Colonna de Lega, and P. de Groot, “True-color 3D surface metrology for additive manufacturing using interference microscopy,” in Proceedings of EUSPEN (2017), pp. 145–148

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Figures (7)

Fig. 1.
Fig. 1. Conceptual diagram for an interference microscope.
Fig. 2.
Fig. 2. Coherent plane wave illumination at normal incidence to the object surface.
Fig. 3.
Fig. 3. Diffracted plane wave component of the object light field distribution.
Fig. 4.
Fig. 4. Geometry for a single off-axis incident plane wave and a diffracted plane wave.
Fig. 5.
Fig. 5. Amplitude of sinusoidal surface topography as a function of frequency for a fixed maximum slope angle equal to 50% of the acceptance angle for an NA of 0.15.
Fig. 6.
Fig. 6. Simulation of the instrument response for the sinusoidal amplitudes given by Fig. 5.
Fig. 7.
Fig. 7. Interference microscope measurement of surface topography for a sinusoidal specimen, illustrating instrument response at a spatial frequency of ${0.125}\;{{\unicode{x00B5}\text{m}}^{- 1}}$.

Tables (1)

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Table 1. Approximate EFO Model for Partially Coherent Light

Equations (71)

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β = sin 1 ( A N ) .
D field = ± λ / A N 2 .
k = 2 π / λ .
θ o ( x ) = 2 k h o ( x ) .
U o ( x ) = exp [ i θ o ( x ) ] ,
ρ ( x , z ) = exp [ i ( k x x + k z z ) ] ,
k x = 2 π f x .
cos ( α x ) = k x / k = λ f x .
U o ~ ( f x ) = U o ( x ) exp ( i 2 π f x x ) d x ,
U o ~ ( f x ) = F { U o ( x ) } .
( f x , z ) = exp [ i k z 1 ( λ f x ) 2 ]
U o ~ ( f x , z ) = ( f x , z ) U o ~ ( f x ) .
U o ( x , z ) = F 1 { U o ~ ( f x , z ) } = U o ~ ( f x , z ) exp ( i 2 π f x x ) d f x .
U I ~ ( f x , z ) = H ~ ( f x ) U o ~ ( f x , z ) ,
f N = A N / λ .
U I ( x , z ) = F 1 { U I ~ ( f x , z ) } .
θ I ( x ) = arg { U I ( x ) } .
h I ( x ) = θ I ( x ) / 2 k = λ θ I ( x ) / 4 π .
ρ o ( x , z ) = p ~ ( f x o ) exp [ i ( k x o x + k z o z ) ] ,
f x o = k x o / 2 π .
cos ( α x o ) = λ f x o .
θ o ( x , f x o ) = 2 k h o ( x ) sin [ α x o ( f x o ) ] .
u o ( x , f x o ) = u o ( x , f x o ) exp ( i 2 π f x o x ) ,
u o ( x , f x o ) = exp [ i θ o ( x , f x o ) ] ,
u o ~ ( f x , f x o ) = u o ~ ( f x f x o , f x o ) ,
u o ~ ( f x , f x o ) = F { u o ( x , f x o ) } .
λ ( f x f x o ) = cos ( α x ) cos ( α x o ) .
u I ~ ( f x , f x o ) = H ~ ( f x + f x o ) u o ~ ( f x , f x o ) .
u I ( x , f x o ) = F 1 { u I ~ ( f x f x o , f x o ) } ,
u I ( x , f x o ) = u I ( x , f x o ) exp ( i 2 π f x o x ) ,
u I ( x , f x o ) = F 1 { u I ~ ( f x , f x o ) } .
u Ro ( x , f x o ) = exp ( i θ R ) exp ( i 2 π f x o x ) .
u R ( x , f x o ) = H ~ ( f x o ) exp ( i θ R ) exp ( i 2 π f x o x ) .
ς ( x , f x o ) = | u I ( x , f x o ) + u R ( x , f x o ) | 2 .
ς ( x , f x o ) = | u R | 2 + | u I | 2 + u I u R + u I u R .
| u R | 2 = | H ~ ( f x o ) | 2 .
| u I | 2 = | u I ( x , f x o ) | 2 .
u I u R = exp ( i θ R ) H ~ ( f x o ) u I ( x , f x o ) .
Υ = Υ R + Υ I + Υ IR + Υ IR .
Υ R = P ~ ( f x o ) | H ~ ( f x o ) | 2 d f x o
Υ I = P ~ ( f x o ) | u I ( x , f x o ) | 2 d f x o ,
P ~ ( f x o ) = [ p ~ ( f x o ) ] 2
Υ IR = U R U I ,
U R = exp ( i θ R )
U I ( x ) = P ~ ( f x o ) H ~ ( f x o ) u I ( x , f x o ) d f x o .
Re { Υ IR } = | Υ IR | cos ( θ I θ R )
θ I ( x ) = arg { U I ( x ) } .
U I ( x ) = P ~ ( f x o ) H ~ ( f x o ) [ H ~ ( f x + f x o ) u o ~ ( f x , f x o ) × exp ( i 2 π f x x ) d f x ] d f x o .
U I ( x ) [ P ~ ( f x o ) H ~ ( f x o ) H ~ ( f x + f x o ) d f x o ] u o ~ ( f x ) × exp ( i 2 π f x x ) d f x .
U I ( x ) = F 1 { O ~ ( f x ) U ~ o ( f x ) } ,
O ~ ( f x ) = [ P ~ ( f x ) H ~ ( f x ) ] H ~ ( f x ) .
U o ~ ( f x ) = F { U o ( x ) } .
U o ( x ) = exp [ i θ eq ( x ) ]
θ eq ( x ) = 2 k eq h o ( x )
Ω = k / k eq .
h I ( x ) = θ I ( x ) / 2 k eq .
Ω = 2 / [ 1 cos ( β ) ] ,
O ~ ( f x ) = H ~ ( f x ) H ~ ( f x ) .
O ( x ) = | H ( x ) | 2 .
O ~ ( f x ) = 2 π [ cos 1 ( | f x | f max ) | f x | f max 1 ( | f x | f max ) 2 ] × rect { f x 2 f max } ,
f max = 2 f N = 2 A N / λ .
Υ I = P ~ ( f x o ) × | H ~ ( f x + f x o ) u o ~ ( f x ) exp ( i 2 π f x x ) d f x | 2 d f x o .
T ( ν ) = PSD I ( ν ) / PSD o ( ν ) ,
h o ( ν , x ) = b o sin ( 2 π ν x )
b I ( ν ) = h I ( ν , x ) exp ( i 2 π ν x ) d x ,
T ( ν ) = | b I ( ν ) / b o | .
U o ( x ) 1 + 2 i k eq h ( x ) ,
b I ( ν ) O ~ ( ν ) b o ,
T ( ν ) | O ~ ( ν ) | .
s ( ν ) = tan 1 ( 2 π ν b o ) .
b o ( ν ) = tan ( β / 2 ) / 2 π ν .

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