Abstract

X-ray shearing interferometers with three gratings are analyzed on the basis of generalized grating imaging theory. The result of this analysis is applied to an already proposed interferometer. The contrast of the imaging fringes in the interferometer is calculated quantitatively. It is also applied to explain the spatial resolution of the image.

© 2009 Optical Society of America

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References

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  1. W. R. Hendee and R. Ritenour, Medinal Imaging Physics, 3rd ed. (Mosby Year Book, 1992), p 1.
  2. A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometers,” Opt. Express 11, 2303-2314(2003).
    [CrossRef] [PubMed]
  3. K. Iwata, “Phase imaging and refractive index tomography for x-rays and visible rays,” in Progress in Optics, Vol. 47, E.Wolf, ed. (Elsevier, 2005), pp. 393-432.
    [CrossRef]
  4. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155-156 (1965).
    [CrossRef]
  5. K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.
  6. K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
    [CrossRef]
  7. K. Iwata, A. Kawasaki, and H. Kikuta, “Phase imaging with a phase-shifting x-ray shearing interferometer using an x-ray line source,” Opt. Rev. 7, 561-567.(2000).
    [CrossRef]
  8. K. Iwata, Y. Takeda, and H. Kikuta, “X-ray shearing interferometer and tomographic reconstruction of refractive index from its data,” Proc. SPIE 5535392-402 (2004).
  9. C. David, B. Nohammer, 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. 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, 6296-6304 (2005).
    [CrossRef] [PubMed]
  11. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258-261 (2006).
    [CrossRef]
  12. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
    [CrossRef] [PubMed]
  13. D. Crespo, J. Alonso, and E. Bernabeu, “Generalized grating imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231-1240 (2000).
    [CrossRef]
  14. D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223-1228 (2002).
    [CrossRef] [PubMed]
  15. K. Iwata, “Interpretation of generalized grating imaging,” J. Opt. Soc. Am. A 25, 2244-2250 (2008).
    [CrossRef]
  16. K. Iwata, “Interpretation of generalized grating imaging (further analysis and numerical calculation),” J. Opt. Soc. Am. A 25, 2939-2944 (2008).
    [CrossRef]

2008 (3)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

K. Iwata, “Interpretation of generalized grating imaging,” J. Opt. Soc. Am. A 25, 2244-2250 (2008).
[CrossRef]

K. Iwata, “Interpretation of generalized grating imaging (further analysis and numerical calculation),” J. Opt. Soc. Am. A 25, 2939-2944 (2008).
[CrossRef]

2006 (1)

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

2005 (2)

K. Iwata, “Phase imaging and refractive index tomography for x-rays and visible rays,” in Progress in Optics, Vol. 47, E.Wolf, ed. (Elsevier, 2005), pp. 393-432.
[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, 6296-6304 (2005).
[CrossRef] [PubMed]

2004 (1)

K. Iwata, Y. Takeda, and H. Kikuta, “X-ray shearing interferometer and tomographic reconstruction of refractive index from its data,” Proc. SPIE 5535392-402 (2004).

2003 (1)

2002 (2)

D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223-1228 (2002).
[CrossRef] [PubMed]

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

2000 (2)

K. Iwata, A. Kawasaki, and H. Kikuta, “Phase imaging with a phase-shifting x-ray shearing interferometer using an x-ray line source,” Opt. Rev. 7, 561-567.(2000).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabeu, “Generalized grating imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231-1240 (2000).
[CrossRef]

1999 (1)

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

1998 (1)

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

1992 (1)

W. R. Hendee and R. Ritenour, Medinal Imaging Physics, 3rd ed. (Mosby Year Book, 1992), p 1.

1965 (1)

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

Alonso, J.

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

Bernabeu, E.

Bonse, U.

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

Bronnimann, Ch.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

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

Cloetens, P.

Crespo, D.

David, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258-261 (2006).
[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, 6296-6304 (2005).
[CrossRef] [PubMed]

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

Diaz, A.

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

Grunsweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

Hagino, H.

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

Hart, M.

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

Hendee, W. R.

W. R. Hendee and R. Ritenour, Medinal Imaging Physics, 3rd ed. (Mosby Year Book, 1992), p 1.

Iwata, K.

K. Iwata, “Interpretation of generalized grating imaging (further analysis and numerical calculation),” J. Opt. Soc. Am. A 25, 2939-2944 (2008).
[CrossRef]

K. Iwata, “Interpretation of generalized grating imaging,” J. Opt. Soc. Am. A 25, 2244-2250 (2008).
[CrossRef]

K. Iwata, “Phase imaging and refractive index tomography for x-rays and visible rays,” in Progress in Optics, Vol. 47, E.Wolf, ed. (Elsevier, 2005), pp. 393-432.
[CrossRef]

K. Iwata, Y. Takeda, and H. Kikuta, “X-ray shearing interferometer and tomographic reconstruction of refractive index from its data,” Proc. SPIE 5535392-402 (2004).

K. Iwata, A. Kawasaki, and H. Kikuta, “Phase imaging with a phase-shifting x-ray shearing interferometer using an x-ray line source,” Opt. Rev. 7, 561-567.(2000).
[CrossRef]

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

Kawasaki, A.

K. Iwata, A. Kawasaki, and H. Kikuta, “Phase imaging with a phase-shifting x-ray shearing interferometer using an x-ray line source,” Opt. Rev. 7, 561-567.(2000).
[CrossRef]

Kikuta, H.

K. Iwata, Y. Takeda, and H. Kikuta, “X-ray shearing interferometer and tomographic reconstruction of refractive index from its data,” Proc. SPIE 5535392-402 (2004).

K. Iwata, A. Kawasaki, and H. Kikuta, “Phase imaging with a phase-shifting x-ray shearing interferometer using an x-ray line source,” Opt. Rev. 7, 561-567.(2000).
[CrossRef]

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

Kikuta, K.

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

Momose, A.

Nakano, T.

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

Nohammer, B.

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

Pfeiffer, F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258-261 (2006).
[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, 6296-6304 (2005).
[CrossRef] [PubMed]

Ritenour, R.

W. R. Hendee and R. Ritenour, Medinal Imaging Physics, 3rd ed. (Mosby Year Book, 1992), p 1.

Solak, H. H.

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

Stampanoni, M.

Tadano, K. H.

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

Takeda, Y.

K. Iwata, Y. Takeda, and H. Kikuta, “X-ray shearing interferometer and tomographic reconstruction of refractive index from its data,” Proc. SPIE 5535392-402 (2004).

Tandano, H.

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258-261 (2006).
[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, 6296-6304 (2005).
[CrossRef] [PubMed]

Ziegler, E.

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, 6296-6304 (2005).
[CrossRef] [PubMed]

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

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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

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

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

K. Iwata, H. Kikuta, H. Tandano, H. Hagino, and T. Nakano, “Phase-shifting x-ray shearing interferometer,” Jpn. J. Appl. Phys. 38, 6535-6539 (1999).
[CrossRef]

Nat. Mater. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunsweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134-137 (2008).
[CrossRef] [PubMed]

Nature Phys. (1)

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

Opt. Express (2)

Opt. Rev. (1)

K. Iwata, A. Kawasaki, and H. Kikuta, “Phase imaging with a phase-shifting x-ray shearing interferometer using an x-ray line source,” Opt. Rev. 7, 561-567.(2000).
[CrossRef]

Other (4)

K. Iwata, Y. Takeda, and H. Kikuta, “X-ray shearing interferometer and tomographic reconstruction of refractive index from its data,” Proc. SPIE 5535392-402 (2004).

K. Iwata, K. H. Tadano, K. Kikuta, H. Hagino, and T. Nakano, “X-ray shearing interferometer for non-destructive testing in Experimental Mechanics,” in: Advances in Design, Testing and Analysis, I.M.Allison, ed. (A. A. Balkema, 1998), p. 741.

W. R. Hendee and R. Ritenour, Medinal Imaging Physics, 3rd ed. (Mosby Year Book, 1992), p 1.

K. Iwata, “Phase imaging and refractive index tomography for x-rays and visible rays,” in Progress in Optics, Vol. 47, E.Wolf, ed. (Elsevier, 2005), pp. 393-432.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Optical system for x-ray shearing interferometer.

Fig. 2
Fig. 2

Optical system for general grating imaging.

Fig. 3
Fig. 3

Contrast C of the grating image as a function of the positions L 1 and L 2 of the gratings for a monochromatic ray and the opening ratio 0.5 for the grating 1.

Fig. 4
Fig. 4

Contrast C of the grating image along the diagonal line in Fig. 3 with a wavelength width δ of 0.02 nm and the various opening ratios γ for grating 1.

Fig. 5
Fig. 5

Contrast C B ( 1 , M I ) for phase gratings of rectangular shape with a duty cycle of 1 / 2 as a function of phase difference κ and F 0 . (a) For M I = 1 , (b)  M I = 2 .

Fig. 6
Fig. 6

Contrast C A for amplitude grating of rectangular shape with fill factor γ as a function of γ and E or F 0 ).

Fig. 7
Fig. 7

Optical system and interfering rays.

Fig. 8
Fig. 8

Deviation s of ray pairs on the second grating and their fringe amplitude amp for the system in Section 4 with γ = 1 / 2 .

Fig. 9
Fig. 9

Source size and the interferometer.

Equations (34)

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

p 1 = p 3 L 10 / L 20 , L 20 = p 2 2 / 8 λ .
a n = a n , b m = b m .
a n = | a n | exp ( i ϕ n ) , b m = | b m | exp ( i ψ m ) .
S ( L 1 + L 2 ) / ( L T p 1 ) 1 , S L 2 / ( L T p 2 ) 1 ,
I ( X ) = j = amp ( j , N I , M I ) cos [ 2 π X j μ 1 ( N I , M I ) + j Ψ 1 ( N I , M I ) ] ,
amp ( j , N I , M I ) = W ( j μ 2 ( N I , M I ) ) A ( j , N I , M I ) B ( j , N I , M I ) , A ( j , N I ) = n = | a n a n + j N I | cos [ ( 2 n + j N I ) j E + ( ϕ n ϕ n + j N I ) ] B ( j , M I ) = m = | b m b m j M I | cos [ ( 2 m j M I ) j F + ( ψ m ψ m j M I ) ] ,
W ( j μ 2 ) = S S exp [ i 2 π j μ 2 x ] d x = sin ( 2 π j S μ 2 ) / ( 2 π j S μ 2 ) ,
μ 1 ( N I , M I ) = [ L 0 N I / p 1 ( L 0 + L 1 ) M I / p 2 ] / L T ,
μ 2 ( N I , M I ) = [ ( L 1 + L 2 ) N I / p 1 L 2 M I / p 2 ] / L T ,
Ψ 1 ( N I , M I ) = 2 π { N I ε 1 / p 1 M I ε 2 / p 2 } ,
E = ( π λ L 0 / p 1 ) μ 2 ( N I , M I ) , F = ( π λ L 2 / p 2 ) μ 1 ( N I , M I ) .
1 / P = μ 1 ( N I , M I ) .
C ( j , N I , M I ) = 2 W ( j μ 2 ) C A ( j , N I ) C B ( j , M I ) , where     C A ( j , N I ) = A ( j , N I ) / A ( 0 , N I ) C B ( j , M I ) = B ( j , M I ) / B ( 0 , M I ) .
L 20 = L 1 p 2 / M I / ( p 1 / N I p 2 / M I ) .
L 1 λ N I / p 1 + L 20 λ ( N I / p 1 M I / p 2 ) = 0.
1 / P 0 = N I / p 1 + M I / p 2 .
L 20 = L 1 P 0 N I / p 1 .
A 0 ( j , N I ) = n = | a n a n + j N I | cos [ ϕ n ϕ n + j N I ] , B 0 ( j , M I ) = m = | b m b m j M I | cos [ ( 2 m j M I ) j F 0 + ( ψ m ψ m j M I ) ] ,
F 0 = π λ N I L I / ( p 1 p 2 ) .
B 0 δ ( j , M I ) = λ 0 δ λ 0 + δ B 0 ( j , M I ) d λ = m = | b m b m j M I | cos [ G λ 0 + ψ m ψ m j M I ] sin ( G δ ) / G δ ,
where     G = ( 2 m j M I ) j π L 1 N I / ( p 1 p 2 ) .
b 0 = 0 , b n = sin ( π n / 2 ) / ( π n / 2 ) , n = ± 1 , ± 2 .
B 0 ( 1 , 2 ) = 4 [ 1 ( 2 / 3 ) cos ( 4 F 0 ) ( 2 / 15 ) cos ( 8 F 0 ) ] / π 2 .
L 10 = p 1 p 2 / ( 4 λ 0 ) .
P 0 p 3 p 2 / 2.
a 0 = 1 , a n = sin ( π n γ ) / ( π n γ ) , n = ± 1 , ± 2 , .
b 0 = i * cos ( κ / 2 ) , b m = ( sin κ / 2 ) sin ( π m / 2 ) / ( π m / 2 ) , = ± 1 , ± 2 .
s = L 0 θ + L 10 ( θ + n λ / p 1 ) ,
L 0 θ + L 10 ( θ + n λ / p 1 ) + L 20 ( θ + n λ / p 1 + m λ / p 2 ) = 0.
s = [ L 0 n / p 1 + ( L 0 + L 10 ) m / p 2 ] ( λ L 20 / L T 0 ) .
amp ( 1 , N I , M I ) = | a n a n + N I b m b m M I |
Δ s = N I L 10 λ / p 1 .
w = S L 20 / L T 0 .
L T 0 = L 0 + ( F 0 / π ) P 0 p 1 / ( λ M I ) .

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