Abstract

We consider an inverse source problem for partially coherent light propagating in the Fresnel regime. The data are the coherence of the field measured away from the source. The reconstruction is based on a minimum residue formulation, which uses the authors’ recent closed-form approximation formula for the coherence of the propagated field. The developed algorithms require a small data sample for convergence and yield stable inversion by exploiting information in the coherence as opposed to intensity-only measurements. Examples with both simulated and experimental data demonstrate the ability of the proposed approach to simultaneously recover complex sources in different planes transverse to the direction of propagation.

© 2018 Optical Society of America

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References

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  1. N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
    [Crossref]
  2. J. W. Goodman, Statistical Optics (Wiley, 1985).
  3. E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
    [Crossref]
  4. H. E. Kondakci, A. Beckus, A. E. Halawany, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Coherence measurements of scattered incoherent light for lensless identification of an object’s location and size,” Opt. Express 25, 13087–13100 (2017).
    [Crossref]
  5. A. El-Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” Opt. Lett. 42, 3089–3092 (2017).
    [Crossref]
  6. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957).
    [Crossref]
  7. S. Divitt, Z. J. Lapin, and L. Novotny, “Measuring coherence functions using non-parallel double slits,” Opt. Express 22, 8277–8290 (2014).
    [Crossref]
  8. H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).
    [Crossref]
  9. C. Iaconis and I. A. Walmsley, “Direct measurement of the two-point correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [Crossref]
  10. C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
    [Crossref]
  11. R. R. Naraghi, H. Gemar, M. Batarseh, A. Beckus, G. Atia, S. Sukhov, and A. Dogariu, “Wide-field interferometric measurement of a nonstationary complex coherence function,” Opt. Lett. 42, 4929–4932 (2017).
    [Crossref]
  12. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
    [Crossref]
  13. I. J. LaHaie, “Uniqueness of the inverse source problem for quasi-homogeneous, partially coherent sources,” J. Opt. Soc. Am. A 3, 1073–1079 (1986).
    [Crossref]
  14. G. Gbur, “Uniqueness of the solution to the inverse source problem for quasi-homogeneous sources,” Opt. Commun. 187, 301–309 (2001).
    [Crossref]
  15. B. E. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007).
  16. W. H. Garter and E. Wolf, “Inverse problem with quasi-homogeneous sources,” J. Opt. Soc. Am. A 2, 1994–2000 (1985).
    [Crossref]
  17. A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
    [Crossref]
  18. I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
    [Crossref]
  19. D. Kohler and L. Mandel, “Source reconstruction from the modulus of the correlation function: a practical approach to the phase problem of optical coherence theory,” J. Opt. Soc. Am. 63, 126–134 (1973).
    [Crossref]
  20. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref]
  21. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [Crossref]
  22. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
    [Crossref]
  23. T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Prob. 13, 47–61 (1997).
    [Crossref]
  24. M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
    [Crossref]
  25. A. Beckus, A. Tamasan, A. Dogariu, A. F. Abouraddy, and G. K. Atia, “Spatial coherence of fields from generalized sources in the Fresnel regime,” J. Opt. Soc. Am. A 34, 2213–2221 (2017).
    [Crossref]
  26. S. Sukhov, M. Batarseh, R. R. Naraghi, H. Gemar, A. C. Tamasan, and A. Dogariu, “Babinet’s principle for mutual intensity,” Opt. Lett. 42, 3980–3983 (2017).
    [Crossref]
  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  28. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [Crossref]
  29. F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [Crossref]
  30. E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 4th ed. (Wiley, 2013).
  31. G. Seber and C. Wild, Nonlinear Regression (Wiley, 1989).

2017 (5)

2014 (3)

2006 (1)

2003 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

2001 (1)

G. Gbur, “Uniqueness of the solution to the inverse source problem for quasi-homogeneous sources,” Opt. Commun. 187, 301–309 (2001).
[Crossref]

2000 (1)

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

1997 (2)

T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Prob. 13, 47–61 (1997).
[Crossref]

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[Crossref]

1996 (1)

1987 (1)

1986 (1)

1985 (2)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

1980 (1)

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[Crossref]

1979 (1)

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[Crossref]

1978 (1)

1973 (1)

1957 (1)

Abouraddy, A. F.

Atia, G.

Atia, G. K.

Batarseh, M.

Beckus, A.

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Cheng, C.-C.

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Chong, E. K. P.

E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 4th ed. (Wiley, 2013).

Collett, E.

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[Crossref]

Devaney, A. J.

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[Crossref]

Divitt, S.

Dogariu, A.

El-Halawany, A.

Fienup, J. R.

Friberg, A. T.

T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Prob. 13, 47–61 (1997).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Garter, W. H.

Gbur, G.

G. Gbur, “Uniqueness of the solution to the inverse source problem for quasi-homogeneous sources,” Opt. Commun. 187, 301–309 (2001).
[Crossref]

Gemar, H.

George, N.

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

Habashy, T.

T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Prob. 13, 47–61 (1997).
[Crossref]

Halawany, A. E.

Heier, H.

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Iaconis, C.

Kohler, D.

Kondakci, H. E.

LaHaie, I. J.

Lapin, Z. J.

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Mandel, L.

Mohammadian, N.

Monroe, M.

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Naraghi, R. R.

Novotny, L.

Partanen, H.

Raymer, M. G.

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Rehacek, J.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Sáchez-Soto, L. L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Saleh, B. E.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007).

Seber, G.

G. Seber and C. Wild, Nonlinear Regression (Wiley, 1989).

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Sukhov, S.

Tamasan, A.

Tamasan, A. C.

Teich, M. C.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007).

Tervo, J.

Thompson, B. J.

Turunen, J.

Unser, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Walmsley, I. A.

Wild, C.

G. Seber and C. Wild, Nonlinear Regression (Wiley, 1989).

Wolf, E.

T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Prob. 13, 47–61 (1997).
[Crossref]

W. H. Garter and E. Wolf, “Inverse problem with quasi-homogeneous sources,” J. Opt. Soc. Am. A 2, 1994–2000 (1985).
[Crossref]

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[Crossref]

B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Zak, S. H.

E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 4th ed. (Wiley, 2013).

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Inverse Prob. (1)

T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Prob. 13, 47–61 (1997).
[Crossref]

J. Math. Phys. (1)

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[Crossref]

J. Mod. Opt. (1)

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Nat. Commun. (1)

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Opt. Commun. (5)

G. Gbur, “Uniqueness of the solution to the inverse source problem for quasi-homogeneous sources,” Opt. Commun. 187, 301–309 (2001).
[Crossref]

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[Crossref]

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Other (5)

E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 4th ed. (Wiley, 2013).

G. Seber and C. Wild, Nonlinear Regression (Wiley, 1989).

J. W. Goodman, Statistical Optics (Wiley, 1985).

B. E. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Supplementary Material (1)

NameDescription
» Visualization 1       This video shows the progression of the gradient-descent algorithm for the example described in Fig. 4 of the article. The example shows convergence of the minimum residual approach to the parameters of the scene based on coherence measurements.

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Figures (10)

Fig. 1.
Fig. 1. (a) Illustration of rotated coordinates. An example of a generalized source is shown in (b) unrotated coordinates and (c) rotated coordinates. For this example, the Gauss–Schell source parameters are A = 1 , w = 1    mm , and σ = 50    μm . The transmission function is such that t ( x ) = 0 for x [ a 1 , a 2 ) , and t ( x ) = 1 otherwise, where a 1 = 0.4    mm and a 2 = 0.2    mm . Dotted white lines indicate the regions affected by the transmission function. Reprinted from [25].
Fig. 2.
Fig. 2. Single object scene with x 0 = 1.5    mm , l = 0.5    mm , d 0 = 10    cm , and d = 100    cm . The normalized magnitude of the coherence function is shown at the bottom of the diagram in three planes: in the plane of the Gaussian source, immediately after interacting with the object (i.e., at the secondary source), and at the measurement plane.
Fig. 3.
Fig. 3. Reconstruction results for one object at known distance. (a) Normalized modulus of coherence function in the measurement plane with sample points marked. (b) Modulus and phase of coherence function at measurement plane. Both measured samples and final estimate are shown. (c) Path of the gradient descent algorithm. The top two plots show the estimates of the two parameters, with horizontal dashed lines indicating the actual values of the parameters. The bottom plot shows the maximum residual value among all sample points with the threshold ε indicated by a dashed line. A vertical dotted line indicates a restart of the algorithm with a new initialization.
Fig. 4.
Fig. 4. Video showing gradient descent for single object (see Visualization 1 in the supplementary material). The plots in (a) are the same as in Fig. 3(c) and are included in the video to show the progression of the algorithm in each frame. The real and imaginary parts of the coherence functions are shown in (b). Bars extending vertically from a sample point indicate the magnitude and direction of the step contribution from that point (up indicates the parameter increases at the next step). The algorithm calculates the next step by accumulating the individual contributions from each point. (c) Contains a map of residual function F from Eq. (13) plotted with regard to the two object parameters. The location of the actual parameters is marked by a green “ x .” The estimate at the current iteration is indicated with a red circle in (a) and (c).
Fig. 5.
Fig. 5. Gradient descent algorithm estimating three object parameters: x 0 , l , and d . The configuration and sample points are the same as in Fig. 3. The plot labels are the same as those defined in Fig. 3(c), with an additional plot included for parameter d .
Fig. 6.
Fig. 6. Example showing estimation of positions of two objects in the same axial plane. (a) Diagram of scenario. (b) Normalized modulus of coherence function in the measurement plane with sample points marked. (c) Path of the gradient descent algorithm. The top three plots show the estimates of the five parameters (blue lines correspond to Object A and orange lines to Object B), with dashed lines indicating the actual values of the parameters. The bottom plot shows the maximum residual value among all sample points with the threshold ε indicated by a dashed line. The Gaussian source parameters are the same as in the one object example. The object parameters are x 0 A = 2.5    mm and l A = 500    μm for Object A, and x 0 B = 1.5    mm and l B = 750    μm for Object B. The two objects are located in the same plane, and the actual distances are d 0 = 0.1    m and d = 1    m . The final estimates are x 0 A = 2.483    mm , l A = 495.6    μm , x 0 B = 1.492    mm , l B = 741.3    μm , and d = 0.944    m .
Fig. 7.
Fig. 7. Example showing estimation of positions assuming two objects in the same axial plane when only one object is actually present. Panels (a)–(c), as well as the source parameters, are the same as in Fig. 6. The parameters for Object A are x 0 A = 2.5    mm and l A = 500    μm , and Object B is absent from the scene. The actual distances are d 0 = 0.1    m and d = 1    m . The final estimates are x 0 A = 2.494    mm , l A = 495.5    μm , x 0 B = 1.198    mm , l B = 0.45    μm , and d = 0.998    m . Note that the estimate of l B 0 , indicating no Object B, is present (thus rendering the estimate of x 0 B irrelevant).
Fig. 8.
Fig. 8. Example showing estimation of positions of two objects in different axial planes. Panels (a)–(c) are the same as in Fig. 6. The Gaussian source and object parameters are also the same as used in Fig. 6. The distances are d 0 = 0.1    m , d A = 1.2    m , and d B = 1    m . The final estimates are x 0 A = 2.516    mm , l A = 501.1    μm , x 0 B = 1.508    mm , l B = 755.7    μm , d A = 1.20    m , and d B = 0.998    m .
Fig. 9.
Fig. 9. Results of gradient descent algorithm using experimental data. (a) Diagram of setup. (b) Modulus and phase of coherence function at measurement plane. Both measured samples and final estimate are shown. (c) Path of the gradient descent algorithm. The top two plots show the estimates of the two parameters, with dashed lines indicating the actual value of the parameters. The bottom plot shows the cardinality of the “vote” set | κ | at each iteration, with the threshold p indicated by a dashed line.
Fig. 10.
Fig. 10. Comparison of intensity and coherence measurements. The modulus of the simulated coherence function is shown in (a) and (d) with intensity sample points indicated by black “x” marks and coherence sample points indicated by white “x” marks. The corresponding residual maps F ( x 0 , l ) for the two scenarios are shown in (b) and (e). For comparison purposes, the functions are normalized against 1 M k = 1 M | G d ( y 1 k , y 2 k ) | 2 , and plotted on the same scale. As can be seen in this example, the residual map for intensity measurements exhibits a larger area of minima than that of the coherence measurements. This may lead to more ambiguity in the reconstruction, although results will vary depending on physical factors such as the signal-to-noise ratio of the measurements. (c) and (e) show the residual f plot as a function of the sample point (along the horizontal) and parameter (vertical). Each plot shows variation with regard to one parameter, while the other is fixed at the correct value, and all plots use the same scale. The actual parameter values are indicated in red. The parameters are the same as used in Fig. 3.

Tables (1)

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Table 1. Experimental Resultsa

Equations (42)

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y 1 = x 1 + x 2 2 , y 2 = x 1 x 2 2 ,
G d ( y 1 , y 2 ) = k π d R 2 G ( y 1 , y 2 ) L ( y 1 , y 1 , y 2 , y 2 ) d y 1 d y 2 ,
L ( y 1 , y 1 , y 2 , y 2 ) = exp { i 2 k d ( y 1 y 1 ) ( y 2 y 2 ) } ,
G ( y 1 , y 2 ) = A exp { i y 1 y 2 / R 2 } N w ( y 1 ) N σ ( y 2 )
G ˜ d ( y 1 , y 2 ) = A ˜ exp ( i y 1 y 2 / R ˜ 2 ) N w ˜ ( y 1 ) N σ ˜ ( y 2 ) ,
A ˜ = A ( 1 + δ ) 1 + ξ 2 ,
R ˜ = R ( 1 + δ ) ( 1 + ξ 2 ) 1 + ( 1 + 1 δ ) ξ 2 ,
w ˜ = w ( 1 + δ ) 1 + ξ 2 ,
σ ˜ = σ ( 1 + δ ) 1 + ξ 2 ,
G ( y 1 , y 2 ) = G ( y 1 , y 2 ) t ( y 1 + y 2 ) t * ( y 1 y 2 ) ,
H u f ( ω ) exp ( i ω u ) p.v. 1 π exp ( i s u ) f ( s ) ω s d s ,
w > 10 2 σ > 10 3 λ ,
j = 1 N N w ( | a j | 3 σ ) < 4 ,
min j = 2 , , N ( a j a j 1 ) > 3 σ ,
G ¯ d ( y 1 , y 2 ) = G ˜ d ( y 1 , y 2 ) i 2    N η σ ˜ ( y 2 ) × j = 2 N T j , j [ ( H b j ( y 1 ) H b j 1 ( y 1 ) ) N σ ˜ / η ] ( y 2 ) ,
η = 1 + σ 2 σ ˜ 2 4 ,
b j ( y 1 ) = 1 η 2 2 ( a j y 1 ( 1 + δ ) ( 1 + ξ 2 ) ) .
f ( y 1 , y 2 ; a , d ) = G ¯ d ( y 1 , y 2 ; a , d ) G d ( y 1 , y 2 ) ,
F ( a , d ) = 1 M k = 1 M | f ( y 1 k , y 2 k ; a , d ) | 2 .
a ( n + 1 ) = a ( n ) μ a F a ,
d ( n + 1 ) = d ( n ) μ d F d ,
| f ( y 1 k , y 2 k ) | < ε , 1 k M .
F a j = 2 π A ˜ σ ˜ η 3 2 M ( T j , j T j + 1 , j + 1 ) × k = 1 M { R e [ f ( y 1 k , y 2 k ) ] R e [ Ψ j ( y 1 k , y 2 k ) ] I m [ f ( y 1 k , y 2 k ) ] I m [ Ψ j ( y 1 k , y 2 k ) ] } ,
Ψ j ( y 1 , y 2 ) = Z j ( y 1 , y 2 ) exp { i y 2 b j ( y 1 ) i y 1 y 2 / R ˜ 2 } ,
Z j ( y 1 , y 2 ) = N w ˜ ( y 1 ) N 2 η σ ( y 2 ) N η / σ ˜ ( b j ( y 1 ) ) ;
f ˜ ( y 1 , y 2 ; a , d ) = g ¯ d ( y 1 , y 2 ; a , d ) g d ( y 1 , y 2 ) ,
Ψ ˜ j ( y 1 , y 2 ) = 1 I ¯ 1 I ¯ 2 { Ψ j ( y 1 , y 2 ) g ¯ d * ( y 1 , y 2 ) × [ I ¯ 1 Z j ( y 1 y 2 , 0 ) + I ¯ 2 Z j ( y 1 + y 2 , 0 ) ] } .
Ğ d ( y 1 , y 2 ) = G ( y 1 , y 2 ) G ¯ d A ( y 1 , y 2 ) G ¯ d B ( y 1 , y 2 ) ,
κ = { k | | f ( y 1 k , y 2 k ) | < ε , 1 k M } .
| κ | p and F < γ ,
G ¯ d ( y 1 , y 2 ) = G ˜ d ( y 1 , y 2 ) i 2 N η σ ˜ ( y 2 ) j = 1 N ( T j , j T j + 1 , j + 1 ) × exp { i y 2 b j ( y 1 ) } × p.v. 1 π exp { i s b j ( y 1 ) } N σ ˜ / η ( s ) y 2 s d s .
G ¯ d ( y 1 , y 2 ) = i η 3 2 2 π σ ˜ C ( y 1 , y 2 ) j = 1 N ( T j , j T j + 1 , j + 1 ) × exp { i y 2 b j ( y 1 ) } × p.v. 1 π exp { i s b j ( y 1 ) } N σ ˜ / η ( s ) y 2 s d s = i η 3 2 2 π σ ˜ C ( y 1 , y 2 ) j = 1 N ( T j , j T j + 1 , j + 1 ) × p.v. exp { i ( s y 2 ) b j ( y 1 ) + i y 1 y 2 / R ˜ 2 } N σ ˜ / η ( s ) y 2 s d s ,
C ( y 1 , y 2 ) = A ˜ σ ˜ 2 π η 3 2 N w ˜ ( y 1 ) N 2 η / σ ( y 2 ) .
R e [ G ¯ d ] a j = ( T j , j T j + 1 , j + 1 ) 1 2 π σ ˜ C ( y 1 , y 2 ) × cos ( y 2 b j ( y 1 ) y 1 y 2 / R ˜ 2 ) × cos ( s b j ( y 1 ) ) N σ ˜ / η ( s ) d s .
D ( y 1 , y 2 ) = ( T j , j T j + 1 , j + 1 ) C ( y 1 , y 2 ) × exp ( i y 2 b j ( y 1 ) i y 1 y 2 / R ˜ 2 ) N η / σ ˜ ( b j ( y 1 ) ) ,
R e [ G ¯ d ] a j = ( T j , j T j + 1 , j + 1 ) C ( y 1 , y 2 ) × cos ( y 2 b j ( y 1 ) y 1 y 2 / R ˜ 2 ) N η / σ ˜ ( b j ( y 1 ) ) = R e [ D ( y 1 , y 2 ) ] .
I m [ G ¯ d ] a j = I m [ D ( y 1 , y 2 ) ] .
f ( y 1 , y 2 ; a , d ) = G ¯ d ( y 1 , y 2 ; a , d ) P ¯ ( y 1 , y 2 ) G d ( y 1 , y 2 ) P ( y 1 , y 2 ) .
a j | f | 2 = 2 R e [ f ] R e [ f ] a j + 2 I m [ f ] I m [ f ] a j = 2 R e [ f ] R e [ G ¯ d ] a j P ¯ P ¯ a j R e [ G ¯ d ] P ¯ 2 + 2 I m [ f ] I m [ G ¯ d ] a j P ¯ P ¯ a j I m [ G ¯ d ] P ¯ 2 = 2 P ¯ { R e [ f ] ( R e [ D ] P ¯ a j R e [ G ¯ d P ¯ ] ) I m [ f ] ( I m [ D ] P ¯ a j I m [ G ¯ d * P ¯ ] ) } .
a j | f | 2 = 2 { R e [ f ] R e [ D ] I m [ f ] I m [ D ] } = 2 π A ˜ σ ˜ η 3 2 ( T j , j T j + 1 , j + 1 ) × { R e [ f ] R e [ Ψ j ] I m [ f ] I m [ Ψ j ] } ,
I ¯ 1 I ¯ 2 a j = 1 2 ( I ¯ 1 I ¯ 2 ) 1 / 2 [ I ¯ 1 G ¯ d ( y 1 y 2 , 0 ) a j + I ¯ 2 G ¯ d ( y 1 + y 2 , 0 ) a j ] = T j , j T j + 1 , j + 1 2 I ¯ 1 I ¯ 2 { I ¯ 1 C ( y 1 y 2 , 0 ) N η / σ ˜ ( b j ( y 1 y 2 ) ) + I ¯ 2 C ( y 1 + y 2 , 0 ) N η / σ ˜ ( b j ( y 1 + y 2 ) ) } .
a j | f ˜ | 2 = 2 I ¯ 1 I ¯ 2 { R e [ f ˜ ] ( R e [ D ] I ¯ 1 I ¯ 2 a j R e [ g ¯ d ] ) I m [ f ˜ ] ( I m [ D ] I ¯ 1 I ¯ 2 a j I m [ g ¯ d * ] ) } = 2 π A ˜ σ ˜ η 3 2 ( T j , j T j + 1 , j + 1 ) × { R e [ f ˜ ] R e [ Ψ ˜ j ] I m [ f ˜ ] I m [ Ψ ˜ j ] } ,

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