Abstract

Utilizing the asymptotic method of stationary phase, I derive expressions for the Fourier transform of a two-dimensional fringe pattern. The method assumes that both the amplitude and the phase of the fringe pattern are well-behaved differentiable functions. Applying the limits in two distinct ways, I show, first, that the spiral phase (or vortex) transform approaches the ideal quadrature transform asymptotically and, second, that the approximation errors increase with the relative curvature of the fringes. The results confirm the validity of the recently proposed spiral phase transform method for the direct demodulation of closed fringe patterns.

© 2001 Optical Society of America

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References

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  1. K. G. Larkin, D. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  2. Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
    [CrossRef]
  3. M. A. Stuff, J. N. Cederquist, “Coordinate transformations realizable with multiple holographic optical elements,” J. Opt. Soc. Am. A 7, 977–981 (1990).
    [CrossRef]
  4. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  5. According to Bracewell (Ref. 4, p. 83), Lerch’s theorem states that if two functions f(x, y)and g(x, y)have the same Fourier transform, then f(x, y)-g(x, y)is a null function.
  6. There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
    [CrossRef]
  7. A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
    [CrossRef]
  8. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  9. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef] [PubMed]
  10. J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
    [CrossRef]
  11. N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).
  12. We shall refrain from calling this function the 2-D analytic signal at present because there are several conflicting definitions of analyticity 13in multiple dimensions. The alternative term “monogenic” does not seem appropriate because the word now has another widespread use in molecular genetics.
  13. M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).
  14. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  15. The Gaussian curvature Cat a critical point (zero gradient) is equal to the Hessian; C=ψ2,0ψ0,2-ψ1,12(1+ψ1,02+ψ0,12)3/2=H(1+ψ1,02+ψ0,12)3/2,ψ1,0=ψ0,1=0 ⇒ C=H.
  16. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  17. A zero gradient of the orientation implies a zero Hessian. The converse is not true, as there are surfaces, such as the cone, with a zero Hessian and a nonzero orientation gradient.
  18. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).
  19. K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).
  20. G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).
  21. T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.
  22. J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1990), pp. 381–384.
  23. P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
    [CrossRef]

2001 (1)

1998 (1)

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

1997 (1)

1991 (1)

1990 (1)

1986 (1)

1982 (1)

1966 (1)

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

Bachor, H.-A.

Bara, S.

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).

Bone, D.

Bone, D. J.

Bovik, A. C.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Cederquist, J. N.

Fiddy, M. A.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

Figueroa, J. E.

Granlund, G. H.

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).

Huang, N. E.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Ina, H.

Jaroszewicz, Z.

Kaiser, J. F.

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1990), pp. 381–384.

Knutsson, H.

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).

Kobayashi, S.

Kolodziejczyk, A.

Kreis, T.

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.

Larkin, K. G.

K. G. Larkin, D. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).

Liu, H. H.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Long, S.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Maragos, P.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

Marroquin, J. L.

Mouriz, D.

Nuttall, A. H.

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

Nye, J. F.

J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

Oldfield, M. A.

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Quatieri, T. F.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

Sandeman, R. J.

Servin, M.

Shen, Z.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Shih, H. H.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Stuff, M. A.

Takeda, M.

Tung, C. C.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Wu, M. C.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Yen, N.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Zheng, Q.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Proc. IEEE (1)

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.
[CrossRef]

Other (15)

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

According to Bracewell (Ref. 4, p. 83), Lerch’s theorem states that if two functions f(x, y)and g(x, y)have the same Fourier transform, then f(x, y)-g(x, y)is a null function.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).

We shall refrain from calling this function the 2-D analytic signal at present because there are several conflicting definitions of analyticity 13in multiple dimensions. The alternative term “monogenic” does not seem appropriate because the word now has another widespread use in molecular genetics.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

The Gaussian curvature Cat a critical point (zero gradient) is equal to the Hessian; C=ψ2,0ψ0,2-ψ1,12(1+ψ1,02+ψ0,12)3/2=H(1+ψ1,02+ψ0,12)3/2,ψ1,0=ψ0,1=0 ⇒ C=H.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

A zero gradient of the orientation implies a zero Hessian. The converse is not true, as there are surfaces, such as the cone, with a zero Hessian and a nonzero orientation gradient.

J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1990), pp. 381–384.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Typical fringe pattern with smooth and differentiable amplitude and phase. (b) Underlying phase function of the fringe pattern in (a). The gray-scale representation has black for 0 rad and white for 70 rad.

Fig. 2
Fig. 2

(a) Definition of orientation angle from the phase gradient. (b) Definition of orientation angle from the fringe angle.

Fig. 3
Fig. 3

(a) Simple square root of orientation phase map (mod π). Gray-scale encoding means that black represents -π/2 and white represents +π/2. The singularity occurs at the center of curvature of the closed fringes, where the orientation is undefined. (b) Unwrapped orientation phase map (mod 2π). Gray-scale encoding means that black represents -π and white represents +π. Again, the singularity occurs at the center of curvature of the closed fringes, where the orientation is undefined. Note that the branch cut is not a real (or an imaginary) discontinuity.

Fig. 4
Fig. 4

Magnitude of the Hessian of the phase. Gray-scale encoding means that black represents zero and white represents peak value. The magnitude has been set to zero in the outer region, where the fringe amplitude is insignificant.

Fig. 5
Fig. 5

(a) Sixth root of the magnitude of the relative curvature of the phase. Gray-scale encoding means that black represents zero and white represent peak value (152, dimensionless units). The magnitude has been set to zero in the outer region, where the fringe amplitude is insignificant. The sixth root is chosen to emphasize certain features. Note that the rms value (excluding the region within one fringe of the central discontinuity) is 0.029. (b) Sixth root of the magnitude of actual error in the phase derived by using the vortex operator on the fringe pattern of Fig. 1. Gray-scale maximum (white) is 2.6 rad, and the minimum (black) is 0.0 rad. Note that the rms phase value (excluding the region within one fringe of the central discontinuity) is 0.017 rad.

Fig. 6
Fig. 6

Relative magnitude of the vortex-operator-derived magnitude. Gray-scale encoding means that black represents zero and white represents peak value. Central region values vary from 0.16 to 1.30, with most regions near the ideal value of 1.00.

Equations (77)

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

f(x, y)=b(x, y)cos[ψ(x, y)]=b(x, y)2 {exp[iψ(x, y)]+exp[-iψ(x, y)]},
fˆ(x, y)=-b(x, y)sin[ψ(x, y)]=-b(x, y)2i {exp[iψ(x, y)]-exp[-iψ(x, y)]}.
p(x, y)=f(x, y)-ifˆ(x, y)=b(x, y)exp[iψ(x, y)].
P(u, v)=-+-+p(x, y)exp[-2πi(ux+vy)]dxdy.
P(u, v)=-+-+b(x, y)exp[iΨ(u, v, x, y)]dxdy,
Pk(u, v)=-+-+b(x, y)exp[ikΨ(u, v, x, y)]dxdy.
Pk(us, vs)2πkn=1Nσ(xn, yn)|H(xn, yn)|1/2exp[ikΨ(xn, yn)]×b(xn, yn)+ik Q2(xn, yn),
Ψ(x, y)=0Ψ1,0(x, y)=Ψ0,1(x, y)=0
atx=xn,y=yn.
H(xn, yn)=Ψ2,0(xn, yn)Ψ0,2(xn, yn)-Ψ1,12(xn, yn),
Ψl,m=l+mΨxlxm.
σ=1ifH<0iifH>0,Ψ2,0>0-iifH>0,Ψ2,0<0.
Q2=α1H(xn, yn)+α2H2(xn, yn)+α3H3(xn, yn).
H(xn, yn)0.
ψ1,0(xn, yn)=2πus,ψ0,1(xn, yn)=2πvs.
ψ(xn+s, yn+t)=ψ0,0(xn, yn)+ψ1,0(xn, yn)s+ψ0,1(xn, yn)t+ψ2,0(xn, yn)s22+ψ0,2(xn, yn)t22+ ,
Ψ(xn+s, yn+t)=ψ0,0(xn, yn)+ψ2,0(xn, yn)s22+ψ0,2(xn, yn)t22+ψ1,1(xn, yn)st1+ .
[Pk(us, vs)]n2πkσ(xn, yn)|H(xn, yn)|1/2×exp[ikψ(xn, yn)]b(xn, yn).
tan[ β(x, y)]=ψ0,1(x, y)ψ1,0(x, y),0β<2π.
P˜k(u, v)=-+-+b˜(x, y)exp[ikΨ(u, v, x, y)]dxdy,
b˜(xn, yn)=b(xn, yn)exp[iβ(xn, yn)]=b(xn, yn) ψ1,0(xn, yn)+iψ0,1(xn, yn)[ψ1,02(xn, yn)+ψ0,12(xn, yn)]1/2.
b˜(xn, yn)=b(xn, yn) us+ivsus2+vs2.
u=q cos(ϕ), v=q sin(ϕ),
us=qscos(ϕs),vs=qssin(ϕs);
b˜(xn, yn)=b(xn, yn) us+ivsqs=b(xn, yn)exp(iϕs).
[P˜k(us, vs)]n2πkσ(xn, yn)|H(xn, yn)|1/2exp[ikψ(xn, yn)]×b(xn, yn)exp(iϕ).
g(x, y)=p*(x, y)=b(x, y)exp[-iψ(x, y)].
[Gk(-us, -vs)]n2πkσ(xn, yn)|H(xn, yn)|1/2×exp[-ikψ(xn, yn)]b(xn, yn).
g˜(x, y)=p*(x, y)exp[iβ(x, y)].
[G˜k(-us, -vs)]n2πkσ(xn, yn)|H(xn, yn)|1/2exp[-ikψ(xn, yn)]×b(xn, yn)us+ivsus2+vs2.
[G˜k(-us,-vs)]n-2πkσ(xn, yn)|H(xn, yn)|1/2×exp[-ikψ(xn, yn)]×b(xn, yn)exp[iϕ(-us, -vs)].
p(x, y)P(u, v)p*(x, y) P*(-u, -v).
F(u, v)=P(u, v)+P*(-u, -v)2=F*(-u, -v).
f˜(x, y)=b˜(x, y)cos[ψ(x, y)]
=b(x, y)2exp[iβ(x, y)]×{exp[iψ(x, y)]+exp[-iψ(x, y)]}.
2F˜s,n(u, v)[P˜k(us, vs)]nδ(u-us, v-vs)+[P˜k(-us, -vs)]nδ(u+us, v+vs).
2F˜s,n(u, v)2πkσ(xn, yn)|H(xn, yn)|1/2 b(xn, yn)exp[iϕ(us, vs)]×{exp[ikψ(xn, yn)]δ(u-us, v-vs)-exp[-ikψ(xn, yn)]δ(u+us, v+vs)}.
w(x, y)=b(x, y)sin[ψ(x, y)],
2iWs,n(u, v)2πkσ(xn, yn)|H(xn, yn)|1/2 b(xn, yn)×{exp[ikψ(xn, yn)]δ(u-us, v-vs)-exp[-ikψ(xn, yn)]δ(u+us, v+vs)}.
2F˜s,n(u, v)=2iWs,n(u, v)exp[iϕ(us, vs)].
F {exp(-iβ)b cos(ψ)}exp(iϕ)F {ib sin(ψ)}.
F -1{exp(iϕ)F {b sin(ψ)}}-i exp(iβ)b cos(ψ).
F -1{exp(iϕ)F {b cos(ψ)}}+i exp(iβ)b sin(ψ).
b sin(ψ)-i exp(-iβ)F -1{exp(iϕ)F{b cos(ψ)}}=V {b cos(ψ)}.
P˘k(u,v)= -+-+b(x,y)exp{k[iΨ(u,v,x,y)-iβ(x,y)]}dxdy.
[P˘k(u˘s, v˘s)]n2πkσ˘(x˘n, y˘n)|H˘(x˘n, y˘n)|1/2×exp[ikΨ˘(x˘n, y˘n)]b(x˘n, y˘n).
Ψ˘(u, v, x, y)=ψ(x, y)-2π(ux+vy)-β(x, y).
Ψ˘(u˘s, v˘s, x˘n, y˘n)=0.
ψ1,0(x˘n, y˘n)-β1,0(x˘n, y˘n)=2πu˘s=2π(us-Δus),
ψ0,1(x˘n, y˘n)-β0,1(x˘n, y˘n)=2πv˘s=2π(vs-Δvs).
2πΔus=β1,0=ψ1,0ψ1,1-ψ0,1ψ2,0ψ1,02+ψ0,12=usψ1,1-vsψ2,02πqs2,
2πΔvs=β0,1=ψ1,0ψ0,2-ψ0,1ψ1,1ψ1,02+ψ0,12=usψ0,2-vsψ1,12πqs2.
ψ2,0ψ0,2-ψ1,12=0.
tan(ϕ+Δϕ)=tanvs-Δvsus-Δus.
ΔϕusΔvs-vsΔusus2+vs2,(Δus)2us2+vs2,
(Δvs)2us2+vs2.
Δϕψ1,02ψ0,2+ψ0,12ψ2,0-2ψ1,0ψ0,1ψ1,1(ψ1,02+ψ0,12)2=ψ1,02ψ0,2+ψ0,12ψ2,0-2ψ1,0ψ0,1ψ1,1(ψ1,02+ψ0,12)3/21(ψ1,02+ψ0,12)1/2.
H˘(x˘n, y˘n)=Ψ˘0,2(x˘n, y˘n)Ψ˘2,0(x˘n, y˘n)-Ψ˘1,12(x˘n, y˘n)
=(ψ0,2-β0,2)(ψ2,0-β2,0)-(ψ1,1-β1,1)2
=H(x˘n, y˘n)+(β2,0β0,2-β1,12)-(ψ0,2-β2,0
+ψ2,0β0,2-2ψ1,1β1,1).
ΔHH
 =(β2,0β0,2-β1,12)-(ψ0,2β2,0+ψ2,0β0,2-2ψ1,1β1,1)ψ2,0ψ0,2-ψ1,12.
cos(ψ)=cos(-ψ).
f=b[cos(ψ)]-ψ[b sin(ψ)].
tan(βest)=tan(β+)=f0,1f1,0=ψ0,1 b sin(ψ)-b0,1cos(ψ)ψ1,0b sin(ψ)-b1,0cos(ψ).
tan()
=(ψ0,1b1,0-ψ1,0b0,1)cos(ψ)(ψ0,12+ψ1,02)b sin(ψ)-(ψ1,0b10+ψ0,1b0,1)cos(ψ),
b0,1b1,0=ψ0,1ψ1,0tan(β).
{exp(iψ)}=exp(iψ)ψ.
{cos(ψ)}=-sin(ψ)ψ.
β=arctan-ψ0,1sin(ψ)-ψ1,0sin(ψ).
i exp(iβ)b sin(ψ)F -1{exp(iϕ)F {b cos(ψ)}},
arg{i exp(iβ)b sin(ψ)}=arctancos(β)sin(ψ)-sin(β)sin(ψ),
[exp(iβ)]est=±exp(2iβ).
C=ψ2,0ψ0,2-ψ1,12(1+ψ1,02+ψ0,12)3/2=H(1+ψ1,02+ψ0,12)3/2,
ψ1,0=ψ0,1=0C=H.

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