Abstract

A new double-grating-based phase-contrast imaging technique is described. This technique differs from the conventional double-grating imaging method by the image acquisition strategy. The novelty of the proposed method is in lateral scanning of both gratings simultaneously while an image is collected. The collected image is not contaminated by a Moiré pattern and can be recorded even by using a high-spatial-resolution integrating detector (e.g. X-ray film), thus facilitating improved resolution and/or contrast in the image. A detailed theoretical analysis of image formation in the scanning-double-grating method is carried out within the rigorous wave-optical formalism. The transfer function for the scanning-double- grating imaging system is derived. An approximate geometrical-optics solution for the image intensity distribution is derived from the exact wave-optical formula using the stationary-phase approach. Based on the present formalism, the effects of finite source size on the preferred operating conditions and of polychromaticity on the image contrast and resolution are investigated.

© 2008 Optical Society of America

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  1. U.  Bonse and M.  Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
    [CrossRef]
  2. M.  Ando and S.  Hosoya, "An attempt at x-ray phase-contrast microscopy," in Proc. 6th Intern. Conf. On X-ray Optics and Microanalysis, G. Shinoda, K. Kohra and T. Ichinokawa Eds. (Univ. of Tokyo Press, Tokyo, 1972) pp. 63-68.
  3. A.  Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods A 352, 622-628 (1995).
    [CrossRef]
  4. K.  Goetz, M. P.  Kalashnikov, Yu. A.  Mikhailov, G. V.  Sklizkov, S. I.  Fedotov, E.  Foerster, and P.  Zaumseil, "Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method," Sov. J. Quantum Electron. 9, 607-610 (1979).
    [CrossRef]
  5. V. A.  Somenkov, A. K.  Tkalich, and S. Sh.  Shil’shtein, "Refraction contrast in x-ray introscopy," Sov. Phys. Tech. Phys. 36, 1309-1311 (1991).
  6. V. N.  Ingal and E. A.  Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D: Appl. Phys. 28, 2314-2317 (1995).
    [CrossRef]
  7. T. J.  Davis, D.  Gao, T. E.  Gureyev, A. W.  Stevenson, and S. W.  Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature 373, 595-598 (1995).
    [CrossRef]
  8. J. F.  Clauser, "Ultrahigh resolution interferometric X-ray imaging," US patent No. 5,812,629 (1998).
  9. C.  David, B.  Nöhammer, H. H.  Solak, and E.  Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
    [CrossRef]
  10. A.  Momose, S.  Kawamoto, I.  Koyama, Y.  Hamaishi, K.  Takai, and Y.  Suzuki, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
    [CrossRef]
  11. C.  David, "Apparatus and method to obtain phase contrast x-ray images," Europ. patent No. EP 1,447,046-A1; Internat. publ. No. WO 2004/071298-A1; Aust. patent No. AU 2003/275964-A1 (2004).
  12. T.  Weitkamp, B.  Nöhammer, A.  Diaz, C.  David, and E.  Ziegler, "X-ray wavefront analysis and optics characterization with a grating interferometer," Appl. Phys. Lett. 86, 054101 (2005).
    [CrossRef]
  13. T.  Weitkamp, A.  Diaz, C.  David, F.  Pfeiffer, M.  Stampanoni, P.  Cloetens, and E.  Ziegler, "X-ray phase imaging with a grating interferometer," Opt. Express 13, 6295-6304 (2005).
    [CrossRef]
  14. C.  David, T.  Weitkamp, and F.  Pfeiffer, "Interferometer for quantitative phase contrast imaging and tomography with an incoherent polychromatic x-ray source," Europ. patent No. EP 1,731,099-A1; Internat. publ. No. WO 2006/131235-A1 (2006).
  15. F.  Pfeiffer, T.  Weitkamp, O.  Bunk, and C.  David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nat. Phys. 2, 258-261 (2006).
    [CrossRef]
  16. T.  Weitkamp, C.  David, C.  Kottler, O.  Bunk, and F.  Pfeiffer, "Tomography with grating interferometers at low-brilliance sources," Proc. SPIE 6318, 63180S (2006).
    [CrossRef]
  17. A.  Momose, W.  Yashiro, Y.  Takeda, Y.  Suzuki, and T.  Hattori, "Phase Tomography by X-ray Talbot Interferometry for Biological Imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
    [CrossRef]
  18. Y.  Takeda, W.  Yashiro, Y.  Suzuki, S.  Aoki, T.  Hattori, and A.  Momose, "X-Ray Phase Imaging with Single Phase Grating," Jpn. J. Appl. Phys. 46, L89-L91 (2007).
    [CrossRef]
  19. A.  Snigirev, I.  Snigireva, V.  Kohn, S.  Kuznetsov, and I.  Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
    [CrossRef]
  20. S. W.  Wilkins, T. E.  Gureyev, D.  Gao, A.  Pogany, and A. W.  Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-338 (1996).
    [CrossRef]
  21. P.  Cloetens, R.  Barrett, J.  Baruchel, J.-P.  Guigay, and M.  Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
    [CrossRef]
  22. T. E.  Gureyev, Ya. I.  Nesterets, D. M.  Paganin, A.  Pogany, and S. W.  Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
    [CrossRef]
  23. L.  Mandel and E.  Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  24. Ya. I.  Nesterets, P.  Coan, T. E.  Gureyev, A.  Bravin, P.  Cloetens, and S. W.  Wilkins, "On qualitative and quantitative analysis in analyser-based imaging," Acta Cryst. A 62, 296-308 (2006).
    [CrossRef]
  25. M. V.  Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26, 65-115 (1971).
    [CrossRef]

2007 (1)

Y.  Takeda, W.  Yashiro, Y.  Suzuki, S.  Aoki, T.  Hattori, and A.  Momose, "X-Ray Phase Imaging with Single Phase Grating," Jpn. J. Appl. Phys. 46, L89-L91 (2007).
[CrossRef]

2006 (5)

F.  Pfeiffer, T.  Weitkamp, O.  Bunk, and C.  David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nat. Phys. 2, 258-261 (2006).
[CrossRef]

T.  Weitkamp, C.  David, C.  Kottler, O.  Bunk, and F.  Pfeiffer, "Tomography with grating interferometers at low-brilliance sources," Proc. SPIE 6318, 63180S (2006).
[CrossRef]

A.  Momose, W.  Yashiro, Y.  Takeda, Y.  Suzuki, and T.  Hattori, "Phase Tomography by X-ray Talbot Interferometry for Biological Imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

T. E.  Gureyev, Ya. I.  Nesterets, D. M.  Paganin, A.  Pogany, and S. W.  Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Ya. I.  Nesterets, P.  Coan, T. E.  Gureyev, A.  Bravin, P.  Cloetens, and S. W.  Wilkins, "On qualitative and quantitative analysis in analyser-based imaging," Acta Cryst. A 62, 296-308 (2006).
[CrossRef]

2005 (2)

T.  Weitkamp, B.  Nöhammer, A.  Diaz, C.  David, and E.  Ziegler, "X-ray wavefront analysis and optics characterization with a grating interferometer," Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

T.  Weitkamp, A.  Diaz, C.  David, F.  Pfeiffer, M.  Stampanoni, P.  Cloetens, and E.  Ziegler, "X-ray phase imaging with a grating interferometer," Opt. Express 13, 6295-6304 (2005).
[CrossRef]

2003 (1)

A.  Momose, S.  Kawamoto, I.  Koyama, Y.  Hamaishi, K.  Takai, and Y.  Suzuki, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
[CrossRef]

2002 (1)

C.  David, B.  Nöhammer, H. H.  Solak, and E.  Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

1996 (2)

S. W.  Wilkins, T. E.  Gureyev, D.  Gao, A.  Pogany, and A. W.  Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-338 (1996).
[CrossRef]

P.  Cloetens, R.  Barrett, J.  Baruchel, J.-P.  Guigay, and M.  Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

1995 (4)

A.  Snigirev, I.  Snigireva, V.  Kohn, S.  Kuznetsov, and I.  Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

V. N.  Ingal and E. A.  Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D: Appl. Phys. 28, 2314-2317 (1995).
[CrossRef]

T. J.  Davis, D.  Gao, T. E.  Gureyev, A. W.  Stevenson, and S. W.  Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature 373, 595-598 (1995).
[CrossRef]

A.  Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods A 352, 622-628 (1995).
[CrossRef]

1991 (1)

V. A.  Somenkov, A. K.  Tkalich, and S. Sh.  Shil’shtein, "Refraction contrast in x-ray introscopy," Sov. Phys. Tech. Phys. 36, 1309-1311 (1991).

1979 (1)

K.  Goetz, M. P.  Kalashnikov, Yu. A.  Mikhailov, G. V.  Sklizkov, S. I.  Fedotov, E.  Foerster, and P.  Zaumseil, "Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method," Sov. J. Quantum Electron. 9, 607-610 (1979).
[CrossRef]

1971 (1)

M. V.  Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26, 65-115 (1971).
[CrossRef]

1965 (1)

U.  Bonse and M.  Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

Acta Cryst. A (1)

Ya. I.  Nesterets, P.  Coan, T. E.  Gureyev, A.  Bravin, P.  Cloetens, and S. W.  Wilkins, "On qualitative and quantitative analysis in analyser-based imaging," Acta Cryst. A 62, 296-308 (2006).
[CrossRef]

Appl. Phys. Lett. (3)

U.  Bonse and M.  Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
[CrossRef]

C.  David, B.  Nöhammer, H. H.  Solak, and E.  Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

T.  Weitkamp, B.  Nöhammer, A.  Diaz, C.  David, and E.  Ziegler, "X-ray wavefront analysis and optics characterization with a grating interferometer," Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

J. Phys. D: Appl. Phys. (2)

V. N.  Ingal and E. A.  Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D: Appl. Phys. 28, 2314-2317 (1995).
[CrossRef]

P.  Cloetens, R.  Barrett, J.  Baruchel, J.-P.  Guigay, and M.  Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996).
[CrossRef]

Jpn. J. Appl. Phys. (3)

A.  Momose, S.  Kawamoto, I.  Koyama, Y.  Hamaishi, K.  Takai, and Y.  Suzuki, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003).
[CrossRef]

A.  Momose, W.  Yashiro, Y.  Takeda, Y.  Suzuki, and T.  Hattori, "Phase Tomography by X-ray Talbot Interferometry for Biological Imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

Y.  Takeda, W.  Yashiro, Y.  Suzuki, S.  Aoki, T.  Hattori, and A.  Momose, "X-Ray Phase Imaging with Single Phase Grating," Jpn. J. Appl. Phys. 46, L89-L91 (2007).
[CrossRef]

Nat. Phys. (1)

F.  Pfeiffer, T.  Weitkamp, O.  Bunk, and C.  David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nat. Phys. 2, 258-261 (2006).
[CrossRef]

Nature (2)

S. W.  Wilkins, T. E.  Gureyev, D.  Gao, A.  Pogany, and A. W.  Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-338 (1996).
[CrossRef]

T. J.  Davis, D.  Gao, T. E.  Gureyev, A. W.  Stevenson, and S. W.  Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature 373, 595-598 (1995).
[CrossRef]

Nucl. Instrum. Methods A (1)

A.  Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods A 352, 622-628 (1995).
[CrossRef]

Opt. Express (1)

T.  Weitkamp, A.  Diaz, C.  David, F.  Pfeiffer, M.  Stampanoni, P.  Cloetens, and E.  Ziegler, "X-ray phase imaging with a grating interferometer," Opt. Express 13, 6295-6304 (2005).
[CrossRef]

Opt. Commun. (1)

T. E.  Gureyev, Ya. I.  Nesterets, D. M.  Paganin, A.  Pogany, and S. W.  Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Proc. SPIE (1)

T.  Weitkamp, C.  David, C.  Kottler, O.  Bunk, and F.  Pfeiffer, "Tomography with grating interferometers at low-brilliance sources," Proc. SPIE 6318, 63180S (2006).
[CrossRef]

Rev. Sci. Instrum. (1)

A.  Snigirev, I.  Snigireva, V.  Kohn, S.  Kuznetsov, and I.  Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Russ. Math. Surveys (1)

M. V.  Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26, 65-115 (1971).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

V. A.  Somenkov, A. K.  Tkalich, and S. Sh.  Shil’shtein, "Refraction contrast in x-ray introscopy," Sov. Phys. Tech. Phys. 36, 1309-1311 (1991).

Sov. J. Quantum Electron. (1)

K.  Goetz, M. P.  Kalashnikov, Yu. A.  Mikhailov, G. V.  Sklizkov, S. I.  Fedotov, E.  Foerster, and P.  Zaumseil, "Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method," Sov. J. Quantum Electron. 9, 607-610 (1979).
[CrossRef]

Other (5)

M.  Ando and S.  Hosoya, "An attempt at x-ray phase-contrast microscopy," in Proc. 6th Intern. Conf. On X-ray Optics and Microanalysis, G. Shinoda, K. Kohra and T. Ichinokawa Eds. (Univ. of Tokyo Press, Tokyo, 1972) pp. 63-68.

J. F.  Clauser, "Ultrahigh resolution interferometric X-ray imaging," US patent No. 5,812,629 (1998).

C.  David, "Apparatus and method to obtain phase contrast x-ray images," Europ. patent No. EP 1,447,046-A1; Internat. publ. No. WO 2004/071298-A1; Aust. patent No. AU 2003/275964-A1 (2004).

C.  David, T.  Weitkamp, and F.  Pfeiffer, "Interferometer for quantitative phase contrast imaging and tomography with an incoherent polychromatic x-ray source," Europ. patent No. EP 1,731,099-A1; Internat. publ. No. WO 2006/131235-A1 (2006).

L.  Mandel and E.  Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic representation of the scanning double-screen imaging system.

Fig. 2.
Fig. 2.

Simulated SDG images with Δx=d/4 of a model object described in the text using geometrical parameters from Table 2:λ=3Å and d 2=48µm (a), 24µm (b) and 12µm (c); λ=1.5Å and d 2=12µm (d). A finite source, wS =d/8, and an ideal detector have been used in our calculations. White (black) corresponds to the maximum (minimum) displayed intensity; the difference between the maximum and minimum displayed intensity is 0.16 in all images.

Fig. 3.
Fig. 3.

Simulated SDG images with Δx=d/4 of a model object described in the text using geometrical parameters corresponding to wS =5µm, d 2 8µm and two values of λ: 1Å (a) and 0.5Å (b). The difference between the minimum and maximum displayed intensity is 0.06 for both the images (unit intensity of the incident wave is assumed).

Fig. 4.
Fig. 4.

(a) Model spectral density distributions in X-ray energy (Gaussian distributions with different FWHM, Δλ, were used for λ); (b) simulated ‘reflectivity’ curves (in the differential contrast regime with Δx=d/4) corresponding to monochromatic radiation with 3Å wavelength and using the spectra shown in Fig. 4(a); (c) first derivative of the simulated ‘reflectivity’ curves shown in Fig. 4(b); (d) second derivative of the simulated ‘reflectivity’ curves shown in Fig. 4(b).

Fig. 5.
Fig. 5.

Dark-field images of a spherical feature of diameter 0.5mm smeared with 100µm (FWHM) Gaussian function, for different values of Δλ/λ: 0 (a), 10% (b), 20% (c), 80% (d). The displayed range of intensities in the images is equal to [0, 0.2] (unit intensity of the incident wave is assumed).

Tables (4)

Tables Icon

Table 1. Some characteristics of the direct phase-contrast imaging techniques

Tables Icon

Table 2. Geometrical parameters of the double-grating imaging system with R=250mm and satisfying the fractional Talbot condition, eq.(45), with m=1, for different values of d 2 and λ.

Tables Icon

Table 3. Geometrical parameters of the double-grating imaging system with m=1.

Tables Icon

Table 4. Some characteristics of the images of the model object described in the text.

Equations (73)

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

E OE 1 ( x , y ; x 0 ) = q ( x , y ) t 1 ( x x 1 ) ,
E det ( x , y ; x 1 , Δ x ) = ( E OE 1 * P R 2 ) ( x , y ; x 1 ) t 2 ( x x 1 Δ x ) ,
I det ( x , y ; x 1 , Δ x ) = ( E OE 1 * P R 2 ) ( x , y ; x 1 ) 2 T 2 ( x x 1 Δ x ) ,
I det ( x , y ; Δ x ) L 1 L dx 1 I det ( x , y ; x 1 , Δ x ) ,
I det ( x , y ; Δ x ) = dx dx q R 2 ( x x , y ) q R 2 * ( x x , y ) T x ( x , x ; Δ x ) ,
q R 2 ( x , y ) d y q ( x , y y ) P R 2 ( y ) ,
T x ( x , x ; Δ x ) P R 2 ( x ) P R 2 * ( x ) G ( x Δ x , x Δ x ) ,
G ( x , x ) L 1 L dX t 1 ( X x ) t 1 * ( X x ) T 2 ( X ) .
I ̂ det ( 1 ) ( u , y ; Δ x ) = du q ̂ R 2 ( 1 ) ( u + u , y ) [ q ̂ R 2 ( 1 ) ( u , y ) ] * T ̂ x ( u + u , u ; Δ x ) ,
T ̂ x ( u , u ; Δ x ) = dwdw P ̂ R 2 ( u w ) P ̂ R 2 * ( u w ) G ̂ ( w , w ) exp [ 2 π i ( w + w ) Δ x ] .
T ̂ x ( u , u ; Δ x ) = P ̂ R 2 ( u ) P ̂ R 2 * ( u ) dwdw P ̂ R 2 ( w ) P ̂ R 2 * ( w ) G ̂ ( w , w )
× exp { 2 π i [ w ( Δ x + λ R 2 u ) + w ( Δ x λ R 2 u ) ] } .
Γ ~ in ( x , y , x , y , λ ) = W in ( x , y , x , y , λ ) exp [ i π ( x 2 + y 2 x 2 y 2 ) ( λ R 1 ) ] ,
S det ( x , y , λ ; x 1 , x 2 ) = T 2 ( x x 2 ) dXdX dYdY W in ( X , Y , X , Y , λ ) ( λ R 1 ) 2 P R 1 ( X , Y ) P R 1 * ( X , Y )
× q ( X , Y ) q * ( X , Y ) t 1 ( X x 1 ) t 1 * ( X x 1 ) P R 2 ( x X , y Y ) P R 2 * ( x X , y Y ) ,
M 2 S det ( M x , M y , λ ; x 1 , x 2 ) = T 2 ( M x x 2 ) d X d X d Y d Y W in ( X , Y , X , Y , λ ) q ( X , Y ) q * ( X , Y )
× t 1 ( X x 1 ) t 1 * ( X x 1 ) P R ( x X , y Y ) P R * ( x X , y Y ) ,
W in ( x , y , x , y , λ ) = S in 1 2 ( x , y , λ ) S in 1 2 ( x , y , λ ) g in ( x x , y y , λ )
× exp { i [ φ in ( x , y , λ ) φ in ( x , y , λ ) ] } ,
M 2 S det ( Mx , My , λ ; x 1 , x 2 ) = T 2 ( Mx x 2 ) dXdX dYdY g in ( X X , Y Y , λ ) Q ( x X , y Y )
× Q * ( x X , y Y ) t 1 ( x X x 1 ) t 1 * ( x X x 1 ) P R ( X , Y ) P R * ( X , Y ) ,
M 2 S det ( Mx , My , λ ; Δ x ) = dXdX dYdY T sys ( X , Y , X , Y ; λ , Δ x ) Q ( x X , y Y )
× Q * ( x X , y Y )
T sys ( x , y , x ' , y ' ; λ , Δ x ) g in ( x ' x , y ' y , λ ) P R ' ( x , y ) P R ' * ( x ' , y ' ) G ( x Δ x , x ' Δ x ) ,
G ( x , x ) L 1 L dX t 1 ( X x ) t 1 * ( X x ) T 1 ( MX ) ,
S ̂ det ( u M , v M , λ ; Δ x ) = dUdV T ̂ sys ( u + U , v + V , U , V ; λ , Δ x ) Q ̂ ( u + U , v + V ) Q ̂ * ( U , V ) .
T ̂ sys ( u , v , u , v ; λ , Δ x ) = dUdV g ̂ in ( U , V , λ ) T ̂ id ( u + U , v + V , u U , v V ; λ , Δ x ) .
g in ( Δ x , Δ y , λ ) = S ̂ src [ Δ x ( λ R 1 ) , Δ y ( λ R 1 ) , λ ] .
I det ( x , y ; Δ x ) = d λ S det ( x , y , λ ; Δ x ) .
M 2 S det ( Mx , My , λ ; Δ x ) = dXdX dYdY Q ( X , Y ) Q * ( X , Y )
× dUdU exp { 2 π i [ U ( x X ) + U ( x X ) ] }
× dVdV exp { 2 π i [ V ( y Y ) + V ( y Y ) ] }
× T ̂ sys ( U , V , U , V ; λ , Δ x ) .
M 2 S det ( Mx , My , λ ; Δ x ) S 0 ( x , y , λ ) T ̂ sys ( u 0 , u 0 , u 0 , u 0 ; λ , Δ x ) ,
T ̂ sys ( u 0 , u 0 , u 0 , u 0 ; λ , Δ x ) r sys [ ( Δ x R ' ) + λ u 0 ] ,
α x ( x , y ; λ ) α x ( x , y ; λ 0 ) ( 1 + 2 ε ) , b ( x , y ; λ ) b ( x , y ; λ 0 ) [ 1 + k ( x , y ) ε ] ,
r sys [ θ 0 + α ( x , y ; λ ) ; λ ] r sys [ θ 0 + α ( x , y ; λ 0 ) ; λ ] ,
+ 2 ε α ( x , y ; λ 0 ) r sys ' [ θ 0 + α ( x , y ; λ 0 ) ; λ ] ,
exp [ 2 b ( x , y ; λ ) ] exp [ 2 b ( x , y ; λ 0 ) ] [ 1 2 b ( x , y ; λ 0 ) k ( x , y ) ε ] .
M 2 I det ( M x , M y , Δ x ) S in , spat ( x , y ) exp [ 2 b ( x , y ; λ 0 ) ] ( r sys , poly [ θ ( x , y ; λ 0 ) ]
+ 2 d λ S in , spec ( λ ) ε { α ( x , y ; λ 0 ) r sys ' [ θ ( x , y ; λ 0 ) ; λ ] b ( x , y ; λ 0 ) k ( x , y ) r sys [ θ ( x , y ; λ 0 ) ; λ ] } )
M 2 I det ( Mx , My ; Δ x ) S in , spat ( x , y ) exp [ 2 b ( x , y ; λ 0 ) ] r sys , poly [ θ 0 + α ( x , y , λ 0 ) ] ,
r sys , poly ( θ ) r sys ( θ ; λ ) S in , spec ( λ ) .
t 1 ( x ) = n a n exp ( 2 π inx d ) , T 2 ( x ) = n b n exp [ 2 π inx ( Md ) ] ,
a n = d 1 0 d dx exp ( 2 π inx d ) t 1 ( x ) ,
b n = ( Md ) 1 0 Md dx exp [ 2 π i nx ( Md ) ] T 2 ( x ) = d 1 0 d dx exp ( 2 π i nx d ) T 2 ( Mx ) .
G ( x , x ) = n m a n a m * b m n exp [ 2 π i ( nx mx ) d ] .
T ̂ id ( u , v , u , v ; λ , Δ x ) = n m a n a m * b m n exp [ 2 π i ( m n ) Δ x d ] P ̂ R ( u + n d , v )
× P ̂ R * ( u m d , v ) .
T ̂ sys ( u , v , u , v ; λ , Δ x ) = n m a n a m * b m n exp [ 2 π i ( m n ) Δ x d ] P ̂ R ( u + n d , v )
× P ̂ R * ( u m d , v ) g in [ λ R ( u + n d + u m d ) , λ R ( v + v ) , λ ] .
g i n ( Δ x , Δ y , λ ) = g i n , x ( Δ x , λ ) g i n , y ( Δ y , λ ) .
S ̂ det ( u M , v M , λ ; Δ x ) = g in , y ( λ R v , λ ) S ̂ ( u , v , λ ; Δ x ) ,
S ̂ ( 1 ) ( u , y , λ ; Δ x ) dU Q ̂ R ( 1 ) ( u + U , y ) [ Q ̂ R ( 1 ) ( U , y ) ] * T ̂ x ( u + U , U ; λ , Δ x ) ,
T ̂ x ( u , u ; λ , Δ x ) = n m a n a m * b m n exp [ 2 π i ( m n ) Δ x d ] P ̂ R ( u + n d )
× P ̂ R * ( u m d ) g in , x [ λ R ( u + n d + u m d ) , λ ] .
Q R ( x , y ) dY Q ( x , y Y ) P R ( Y ) ,
w S , eff = w S ( 1 M 1 ) .
w S , eff nd ,
w S nd .
z m = md 1 2 ( 2 η 2 λ ) ,
R = M 2 ( M 1 ) R .
M = 1 + md 2 2 ( 2 λ R ) .
d R = d 2 R + 2 λ ( md 2 ) .
( d 2 ) opt = ( 2 λ R m ) 1 2 , ( d R ) min = 2 [ 2 λ ( mR ) ] 1 2 .
d max = ( d 2 ) opt 2 .
w S > n d .
M max = ( 1 nd w S ) 1 .
M max = 1 + n d 2 w S .
R = md 2 2 [ 2 λ ( M 1 ) ] , d R = 2 λ M ( md 2 ) .
R = md 2 w S ( 2 n λ ) , d R = ( 2 λ m ) [ ( 1 d 2 ) + ( n w S ) ] .
Δ λ λ < 1 8 .
Δ λ λ < 1 ( 2 m 1 ) ,

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