Abstract

A unified theory for the optical transfer function (OTF) is presented. The method permits the factoring of light-scattering effects out of the overall value, which also involves diffraction and aberration. OTF’s of multicomponent systems are evaluated for all cases, including those in which the conditions for the multiplication of individual OTF’s are not met. The formalism treats afocal and imaging systems on the same level. The Wigner distribution function is used to show that in most cases the OTF may be calculated from ray traces without losing the accuracy of wave theory.

© 1988 Optical Society of America

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References

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  1. M. M. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–577 (1956).
    [CrossRef]
  2. M. M. Hopkins, “Interferometric methods for the study of diffraction images,” Opt. Acta 2, 23–29 (1955).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
  4. R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,”J. Opt. Soc. Am. 54, 52–61 (1964).
    [CrossRef]
  5. V. I. Takarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  6. D. Kelsall, “Optical ‘seeing’ through the atmosphere by an interferometric technique,”J. Opt. Soc. Am. 63, 1472–1484 (1973).
    [CrossRef]
  7. Melles Griot Optics Guide (Melles Griot, Irvine, Calif., 1982), Vol. 2, Chap. 2.
  8. Z. Karny, S. Lavi, O. Kafri, “Direct determination of the number of transverse modes of a light beam,” Opt. Lett. 8, 409–411 (1983).
    [CrossRef] [PubMed]
  9. I. Glatt, O. Kafri, “Analysis of turbulent mixing in liquids by moiré deflectometry,” Chem. Eng. Sci. 39, 1637–1638 (1984).
    [CrossRef]
  10. I. Glatt, A. Livnat, O. Kafri, “Surface finish determination by moiré deflectometry,” Exp. Mech. 24, 248–251 (1984).
    [CrossRef]
  11. I. Glatt, A. Livnat, O. Kafri, “Direct determination of modulation transfer function by moiré deflectometry,” J. Opt. Soc. Am. A 2, 107–110 (1985).
    [CrossRef]
  12. O. Kafri, Y. B. Band, T. Chin, D. F. Heller, J. C. Walling, “Real-time moiré vibration analysis of diffusive objects,” Appl. Opt. 24, 240–242 (1985).
    [CrossRef] [PubMed]
  13. O. Kafri, H. Samelson, T. Chin, D. F. Heller, “Moiré modulation transfer function of alexandrite rods and their thresholds as lasers,” Opt. Lett. 11, 201–203 (1986).
    [CrossRef] [PubMed]
  14. E. Keren, I. Glatt, O. Kafri, “Propagator for the modulation transfer function of a wide-angle scatterer,” Opt. Lett. 11, 554–556 (1986).
    [CrossRef] [PubMed]
  15. O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett. 5, 555–557 (1980).
    [CrossRef] [PubMed]
  16. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  17. O. Kafri, I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 10.
  19. J. C. Wyant, “OTF measurements with a white light source: an interferometric technique,” Appl. Opt. 14, 1613–1615 (1975).
    [CrossRef] [PubMed]
  20. E. Keren, A. Livnat, I. Glatt, “Moiré deflectometry with pure sinusoidal gratings,” Opt. Lett. 10, 167–169 (1985).
    [CrossRef] [PubMed]
  21. See Ref. 3, Sec. 3.7.
  22. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Secs. 1.3 and 2.3.
  23. E. Keren, “Immunity to shock and vibration in moiré deflectometry,” Appl. Opt. 24, 3028–3031 (1985).
    [CrossRef] [PubMed]
  24. H. Talbot, “Facts relating to optical science no. 4,” Phil. Mag. 9, 401–407 (1836).
  25. See Ref. 22, Sec. 1.4.
  26. F. Zernike, Physica 5, 785, 791 (1936).
    [CrossRef]
  27. P. M. van Cittert, Physica 1, 201 (1934).
    [CrossRef]
  28. See Ref. 22, Sec. 3.6.
  29. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  30. H. Mori, I. Oppenheim, I. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.
  31. P. M. Woodward, Probability and Information Theory with Applications to Radar (McGraw-Hill, New York, 1953).
  32. I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,”J. Math. Phys. 17, 734–743 (1976).
    [CrossRef]
  33. A. Papoulis, “Ambiguity function in Fourier optics,”J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  34. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  35. J.-P. Guigay, “The ambiguity function in diffraction and iso-planatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
    [CrossRef]
  36. K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  37. K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [CrossRef]

1986 (2)

1985 (6)

1984 (3)

I. Glatt, O. Kafri, “Analysis of turbulent mixing in liquids by moiré deflectometry,” Chem. Eng. Sci. 39, 1637–1638 (1984).
[CrossRef]

I. Glatt, A. Livnat, O. Kafri, “Surface finish determination by moiré deflectometry,” Exp. Mech. 24, 248–251 (1984).
[CrossRef]

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

1983 (2)

Z. Karny, S. Lavi, O. Kafri, “Direct determination of the number of transverse modes of a light beam,” Opt. Lett. 8, 409–411 (1983).
[CrossRef] [PubMed]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1980 (1)

1978 (2)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

J.-P. Guigay, “The ambiguity function in diffraction and iso-planatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

1976 (1)

I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,”J. Math. Phys. 17, 734–743 (1976).
[CrossRef]

1975 (1)

1974 (1)

1973 (1)

1964 (1)

1956 (1)

M. M. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–577 (1956).
[CrossRef]

1955 (1)

M. M. Hopkins, “Interferometric methods for the study of diffraction images,” Opt. Acta 2, 23–29 (1955).
[CrossRef]

1936 (1)

F. Zernike, Physica 5, 785, 791 (1936).
[CrossRef]

1934 (1)

P. M. van Cittert, Physica 1, 201 (1934).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

1836 (1)

H. Talbot, “Facts relating to optical science no. 4,” Phil. Mag. 9, 401–407 (1836).

Band, Y. B.

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Besieris, I. M.

I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,”J. Math. Phys. 17, 734–743 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 10.

Brenner, K.-H.

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Chin, T.

Glatt, I.

E. Keren, I. Glatt, O. Kafri, “Propagator for the modulation transfer function of a wide-angle scatterer,” Opt. Lett. 11, 554–556 (1986).
[CrossRef] [PubMed]

I. Glatt, A. Livnat, O. Kafri, “Direct determination of modulation transfer function by moiré deflectometry,” J. Opt. Soc. Am. A 2, 107–110 (1985).
[CrossRef]

O. Kafri, I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

E. Keren, A. Livnat, I. Glatt, “Moiré deflectometry with pure sinusoidal gratings,” Opt. Lett. 10, 167–169 (1985).
[CrossRef] [PubMed]

I. Glatt, A. Livnat, O. Kafri, “Surface finish determination by moiré deflectometry,” Exp. Mech. 24, 248–251 (1984).
[CrossRef]

I. Glatt, O. Kafri, “Analysis of turbulent mixing in liquids by moiré deflectometry,” Chem. Eng. Sci. 39, 1637–1638 (1984).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

Guigay, J.-P.

J.-P. Guigay, “The ambiguity function in diffraction and iso-planatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

Heller, D. F.

Hopkins, M. M.

M. M. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–577 (1956).
[CrossRef]

M. M. Hopkins, “Interferometric methods for the study of diffraction images,” Opt. Acta 2, 23–29 (1955).
[CrossRef]

Hufnagel, R. E.

Kafri, O.

O. Kafri, H. Samelson, T. Chin, D. F. Heller, “Moiré modulation transfer function of alexandrite rods and their thresholds as lasers,” Opt. Lett. 11, 201–203 (1986).
[CrossRef] [PubMed]

E. Keren, I. Glatt, O. Kafri, “Propagator for the modulation transfer function of a wide-angle scatterer,” Opt. Lett. 11, 554–556 (1986).
[CrossRef] [PubMed]

E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
[CrossRef]

O. Kafri, I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

I. Glatt, A. Livnat, O. Kafri, “Direct determination of modulation transfer function by moiré deflectometry,” J. Opt. Soc. Am. A 2, 107–110 (1985).
[CrossRef]

O. Kafri, Y. B. Band, T. Chin, D. F. Heller, J. C. Walling, “Real-time moiré vibration analysis of diffusive objects,” Appl. Opt. 24, 240–242 (1985).
[CrossRef] [PubMed]

I. Glatt, O. Kafri, “Analysis of turbulent mixing in liquids by moiré deflectometry,” Chem. Eng. Sci. 39, 1637–1638 (1984).
[CrossRef]

I. Glatt, A. Livnat, O. Kafri, “Surface finish determination by moiré deflectometry,” Exp. Mech. 24, 248–251 (1984).
[CrossRef]

Z. Karny, S. Lavi, O. Kafri, “Direct determination of the number of transverse modes of a light beam,” Opt. Lett. 8, 409–411 (1983).
[CrossRef] [PubMed]

O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett. 5, 555–557 (1980).
[CrossRef] [PubMed]

Karny, Z.

Kelsall, D.

Keren, E.

Lavi, S.

Livnat, A.

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Secs. 1.3 and 2.3.

Mori, H.

H. Mori, I. Oppenheim, I. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

Ojeda-Castañeda, J.

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Oppenheim, I.

H. Mori, I. Oppenheim, I. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

Papoulis, A.

Ross, I.

H. Mori, I. Oppenheim, I. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

Samelson, H.

Stanley, N. R.

Takarski, V. I.

V. I. Takarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Talbot, H.

H. Talbot, “Facts relating to optical science no. 4,” Phil. Mag. 9, 401–407 (1836).

Tappert, F. D.

I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,”J. Math. Phys. 17, 734–743 (1976).
[CrossRef]

van Cittert, P. M.

P. M. van Cittert, Physica 1, 201 (1934).
[CrossRef]

Walling, J. C.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 10.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (McGraw-Hill, New York, 1953).

Wyant, J. C.

Zernike, F.

F. Zernike, Physica 5, 785, 791 (1936).
[CrossRef]

Appl. Opt. (3)

Chem. Eng. Sci. (1)

I. Glatt, O. Kafri, “Analysis of turbulent mixing in liquids by moiré deflectometry,” Chem. Eng. Sci. 39, 1637–1638 (1984).
[CrossRef]

Exp. Mech. (1)

I. Glatt, A. Livnat, O. Kafri, “Surface finish determination by moiré deflectometry,” Exp. Mech. 24, 248–251 (1984).
[CrossRef]

J. Math. Phys. (1)

I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,”J. Math. Phys. 17, 734–743 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

M. M. Hopkins, “Interferometric methods for the study of diffraction images,” Opt. Acta 2, 23–29 (1955).
[CrossRef]

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Opt. Commun. (3)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

J.-P. Guigay, “The ambiguity function in diffraction and iso-planatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Eng. (1)

O. Kafri, I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

Opt. Lett. (5)

Phil. Mag. (1)

H. Talbot, “Facts relating to optical science no. 4,” Phil. Mag. 9, 401–407 (1836).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Physica (2)

F. Zernike, Physica 5, 785, 791 (1936).
[CrossRef]

P. M. van Cittert, Physica 1, 201 (1934).
[CrossRef]

Proc. Phys. Soc. London Sect. B (1)

M. M. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–577 (1956).
[CrossRef]

Other (10)

Melles Griot Optics Guide (Melles Griot, Irvine, Calif., 1982), Vol. 2, Chap. 2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

V. I. Takarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 10.

See Ref. 22, Sec. 3.6.

See Ref. 22, Sec. 1.4.

See Ref. 3, Sec. 3.7.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Secs. 1.3 and 2.3.

H. Mori, I. Oppenheim, I. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

P. M. Woodward, Probability and Information Theory with Applications to Radar (McGraw-Hill, New York, 1953).

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Figures (5)

Fig. 1
Fig. 1

A deflectometer arrangement for measuring the OTF of an arbitrary object. The detector measures the intensity of radiation transmitted through the object and two moiré gratings, G1 and G2.

Fig. 2
Fig. 2

The angular spread of an incoherent source of dimension ρ (in geometrical optics) is ρ divided by F, the focal length of the collimating lens.

Fig. 3
Fig. 3

Propagation of the mutual intensity function expressed in angle space as a straight-line propagation of an angular ray distribution.

Fig. 4
Fig. 4

A pencil of rays pictured as a plane wave of cross section s. The angular distribution cannot be defined to a spatial resolution better than s.

Fig. 5
Fig. 5

Total transmitted light intensity of a double-slit aperture (slit width, 2 mm; slit separation, 2 mm) as a function of the shear s. The solid-envelope line is the theoretical MTF.

Equations (89)

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

f ( x , z ) = cos ( q x / 2 ) ,
q = 2 π / p .
s = q Δ z / k ,
f ( x , z 1 ) = f 0 ( x , z 1 ) cos ( q x / 2 ) .
ψ 0 ( k x , z 1 ) = d x f 0 ( x , z 1 ) exp ( i k x x ) ,
ψ ( k x , z 1 ) = ψ 0 ( k x , z 1 ) 1 / 2 [ δ ( k x q / 2 ) + δ ( k x + q / 2 ) ] ,
k x = α k .
ψ ( k x , z 1 ) = 1 / 2 [ ψ 0 ( k x q / 2 , z 1 ) + ψ 0 ( k x + q / 2 , z 1 ) ] .
ψ ( k x , z 2 ) = 1 / 2 { [ ψ 0 ( k x q / 2 , z 1 ) + ψ 0 ( k x + q / 2 , z 1 ) ] × exp [ i Δ z ( k k x 2 / 2 k ) ] } .
f ( x , z 2 ) = 1 / 2 π d k x ψ ( k x , z 2 ) exp ( i k x x ) .
f ( x , z 2 ) = 1 / 2 exp ( i q s / 8 ) [ exp ( i q x / 2 ) f 0 ( x s / 2 , z 2 ) + exp ( i q x / 2 f 0 ( x + s / 2 , z 2 ) ] ,
I ( x , z 2 ) = 1 / 4 [ I 0 ( x s / 2 ) + I 0 ( x + s / 2 ) ] + 1 / 2 Re { exp ( i q x ) f 0 ( x + s / 2 , z 2 ) f 0 * ( x s / 2 , z 2 ) } ,
I ( x , z 2 ) = 1 / 2 [ 1 + cos ( q x ) ] ,
g 2 ( x ) 1 / 2 [ 1 + cos q ( x + s ) ] ,
I tot = c / 4 d x [ 1 + cos q ( x + s ) ] [ I 0 ( x s / 2 ) + I 0 ( x + s / 2 ) + 2 Re { exp ( i q x ) f 0 ( x + s / 2 , z 2 ) f 0 * ( x s / 2 , z 2 ) } ] ,
I tot = c / 4 [ 2 I 0 + Re { exp ( i q s ) × f 0 ( x + s / 2 , z 2 ) f 0 * ( x s / 2 , z 2 ) d x } ] ,
I 0 = d x I 0 ( x ) .
D ( s ) = 1 / I 0 d x f 0 ( x + s / 2 ) f 0 * ( x s / 2 ) ,
ν x = s / λ F ,
I tot = c 1 + c 2 Re { exp ( i q s ) D ( s ) } .
f 0 ( x ) = exp [ i k w ( x ) ] .
D ( s ) = d x exp { i k [ w ( x + s / 2 ) w ( x s / 2 ) ] } .
γ ( x , s ) 1 / p x p / 2 x + p / 2 d x exp { i k [ w ( x + s / 2 ) w ( x s / 2 ) ] } .
γ ( x , s ) exp [ i k s ϕ ( x ) ] ,
ϕ ( x ) d w / d x
I ( x ) = c 1 + c 2 Re { exp ( i q s ) γ ( x , s ) }
I ( x ) c 1 + c 2 cos [ k s ϕ ( x ) q s ] .
P ( x , a ) { 1 , | x | < a / 2 0 , | x | a / 2 .
f 0 ( x ) = P ( x , a ) exp [ i k w ( x ) ] .
I ( x ) = c 1 + c 2 P ( x + s / 2 , a ) P ( x s / 2 , a ) cos [ k s ϕ ( x ) q s ] .
I tot = c 1 + c 2 ( s / 2 ) ( a / 2 ) ( a / 2 ) ( s / 2 ) d x cos [ k s ϕ ( x ) q s ] , s a .
w ( x ) = w ( x ) + ϕ ( x ) ( x x ) + ( x x ) 2 2 F ( x ) + O ( x x ) 3 ,
f 0 ( x , z ) = exp [ i ( k x x + k z z + k w 0 ) ] ,
k z = ( k 2 k x 2 ) 1 / 2 k k x 2 / 2 k .
D ( s ) = exp ( i k x s ) ,
I tot = c 1 + c 2 cos ( k x s q s ) .
f 0 ( x , z ) = d k x [ h ( k x ) ] 1 / 2 exp { i [ k x x + k z z + k w 0 ( k x ) ] } ,
D ( s ) = c d k x h ( k x ) exp ( i k x s ) ,
Γ ( x , s ) = f 0 ( x + s / 2 ) f 0 * ( x s / 2 ) .
D ( s ) = γ ( x , s ) f 0 ( x + s / 2 ) f 0 * ( x s / 2 ) | ( f 0 ( x ) | 2 ,
f ( x , z ) = d k x [ S ( k x ) ] 1 / 2 P ( x , a ) × exp { i [ k x x + k z z + k w 1 ( k x ) + k w ( x ) ] } .
D ( s ) = γ 1 ( s ) a d x P ( x + s / 2 , a ) P ( x s / 2 , a ) × exp [ i k s ϕ ( x ) ] ,
γ 1 ( s ) = d k x S ( k x ) exp ( i k x s ) d k x S ( k x )
f ( x , z ) = d k x [ S ( k x k x ) ] 1 / 2 P ( x , a ) × exp { i [ k x x + k z z + k w 0 ( x ) ] } .
D ( s ) = c d x d k x S ( k x k x ) P ( x + s / 2 , a ) × P ( x s / 2 , a ) exp [ i k x s i k s f ( x ) ] ,
D ( s ) = γ ( s ) a d x P ( x + s / 2 , a ) P ( x s / 2 , a ) × exp [ i k s ϕ ( x ) ] ,
γ ( s ) = c d k x exp ( i k x s ) d k x h ( k x ) × S ( k x k x ) .
γ ( s ) = γ 0 ( s ) γ 1 ( s ) ,
γ 0 ( s ) = sinc ( s δ k 2 ) , δ k = k ρ / F .
γ 1 ( s ) = exp ( δ k 2 s 2 / 4 ) .
δ s δ k 1.
Γ ( x , s ) = d k x h ( x , k x ) exp ( i k x s ) ,
x 1 = x + s / 2 , x 2 = x s / 2 .
γ ( x , s ) = f 0 ( x + s / 2 ) f 0 * ( x s / 2 ) [ I 0 ( x + s / 2 ) I 0 ( x s / 2 ) ] 1 / 2 .
γ ( x , s ) = γ 0 ( x , s ) γ 1 ( x , s ) P ( x + s / 2 , a ) P ( x s / 2 , a ) × exp [ i k s ϕ ( x ) ] .
h ( x , k x , z 1 ) = h ( x α z 1 , k x , 0 ) , k x = α k .
h ( x , k x , z 1 ) = d x 1 δ ( x 1 x + α z 1 ) h ( x 1 , k x , 0 ) .
Γ ( x , s , z 1 ) = 1 / λ z 1 d x 1 d s 1 Γ ( x 1 , s 1 , 0 ) × exp [ i k ( x x 1 ) ( s s 1 ) / z 1 ] .
h ( x , k x , z 1 ) = I ( x α z 1 , 0 ) .
δ x δ k s δ k 1.
D ( s ) = d x Γ ( x , s ) d x Γ ( x , 0 ) .
ν x ν cut = s P eff , ν cut = 2 N.A. λ .
ψ 0 ( k x , 0 ) = d x f 0 ( x , 0 ) exp ( i k x x ) ,
ψ 0 ( k x , z 1 ) = ψ 0 ( k x , 0 ) exp [ i z 1 ( k 2 k x 2 ) 1 / 2 ] ψ 0 ( k x , 0 ) exp [ i z 1 ( k k x 2 / 2 k ) ] .
C ( s ) d x f 0 ( x + s / 2 , 0 ) f 0 * ( x s / 2 , 0 ) = d x f 0 ( x + s / 2 , z ) f 0 * ( x s / 2 , z )
g 1 ( x ) = n = a n exp ( i q n x ) ,
ψ ( k x , z 1 ) = d k x ψ 0 ( k x , 0 ) exp [ i z 1 ( k k x 2 / 2 k ) ] × n = a n δ ( k x k x + q n ) = exp [ i z 1 ( k k x 2 / 2 k ) ] n = a n ψ 0 ( k x + q n , 0 ) × exp [ + i z l ( k x q n / k + q 2 n 2 / 2 k ) ] .
f ( x , z 2 ) = 1 / 2 π d k x exp [ i Δ z ( k k x 2 / 2 k ) ] × exp [ i z 1 ( k k x 2 / 2 k ) ] × a n ψ 0 ( k x + q n , 0 ) × exp [ i z 1 ( k x q n / k + q 2 n 2 / 2 k ) ] exp ( i k x x ) .
f 0 ( x , z 2 ) = 1 / 2 π exp ( i k z 2 ) × d k x exp [ i ( k x 2 z 2 / 2 k k x x ) ] ψ 0 ( k x , 0 ) .
f ( x , z 2 ) = n = a n exp [ i q ( n x + n 2 s / 2 ) ] f 0 ( x + n s , z 2 ) .
s = q Δ z / k = λ Δ z / p ,
| g 2 ( x , s ) | 2 = j = b j exp [ i q j ( x + s ) ] ,
I tot = d x n = a n exp [ i q ( n x + n 2 s / 2 ) ] f 0 ( x + n s , z 2 ) × m = a m * exp [ i q ( m x + m 2 s / 2 ) ] f 0 * ( x + m s , z 2 ) × j = b j exp [ i q j ( x + s ) ] .
I tot = j = b j d j exp ( i q j s ) C ( j s ) ,
d j ( s ) = n = a n a n j * exp [ i q s j ( n j / 2 ) ] = d x g 1 ( x + j s / 2 ) g 1 * ( x j s / 2 ) exp ( i q j x ) .
Δ z = p 2 m / λ , s = m p , m = 1,2 ,
d j ( s = m p ) = exp ( i π m j 2 ) a n a n j * .
d 0 = 1 / 2 , d 2 = d 2 = 1 / 4 ,
I tot = 1 / 2 C ( 0 ) + 1 / 2 Re { exp ( i q s ) C ( s ) } .
D ( s ) = C ( s ) / C ( 0 ) = C ( s ) / I 0 ,
I tot = I 0 / 2 [ 1 + Re { exp ( i q s ) D ( s ) } ] .
I tot = C ( 0 ) 4 + 1 π 2 j = cos [ q s / 2 ( 2 j + 1 ) 2 ] ( 2 j + 1 ) 2 × exp [ i q ( 2 j + 1 ) s ] C [ ( 2 j + 1 ) s ]
I tot = C ( 0 ) 4 + 2 π 2 j = 0 cos [ q s / 2 ( 2 j + 1 ) 2 ] ( 2 j + 1 ) 2 × Re { exp [ i q ( 2 j + 1 ) s ] C [ ( 2 j + 1 ) s ] } .
I ( x ) = I 0 4 + 1 π 2 j = cos [ q s / 2 ( 2 j + 1 ) 2 ] ( 2 j + 1 ) 2 exp ( i q j s ) Γ ( x , j s ) ,
Γ ( x , j s ) = 1 / 2 [ f 0 ( x + j s ) f 0 * ( x ) + f 0 ( x ) f 0 * ( x j s ) ] .
Γ ( x , j s ) = Γ * ( x , j s ) .
I ( x ) = I 0 ( 1 4 + 2 π 2 j = 0 cos [ q s / 2 ( 2 j + 1 ) 2 ] ( 2 j + 1 ) 2 × Re { exp ( i q j s ) γ ( x , j s ) } ) ,
γ ( x , j s ) = Γ ( x , j s ) I 0 f 0 ( x + j s / 2 ) f 0 * ( x j s / 2 ) I 0 .
f 0 ( x ) = exp [ i k w ( x ) ] ,

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