Abstract

Fourier ptychographic microscopy (FPM), simply replacing the traditional light source of a microscope with a LED array, can obtain high resolution and large field of view images simultaneously. A symmetrical LED illumination FPM is presented to improve the quality of high resolution images for defocus or thick samples in this paper. It is proven theoretically that symmetrical LED illumination had higher defocus tolerance than that of single LED illumination at first. Then, it is confirmed that symmetrical LED illumination FPM has better low-resolution images and reconstruction effect for defocus samples than that of single LED illumination by simulated and experimental results. Compared with the single illumination FPM, it is demonstrated that the depth of field of symmetrical FPM is more than doubled.

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References

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  1. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
    [Crossref]
  2. X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
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    [Crossref]
  8. X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
    [Crossref] [PubMed]
  9. J. Sun, Q. Chen, Y. Zhang, and C. Zuo, “Efficient positional misalignment correction method for Fourier ptychographic microscopy,” Biomed. Opt. Express 7, 1336–1350 (2016).
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  11. R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
    [Crossref] [PubMed]
  12. Y. Zhang, W. Jiang, and Q. Dai, “Nonlinear optimization approach for Fourier ptychographic microscopy,” Opt. Express 23, 33822–33835 (2015).
    [Crossref]
  13. L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
    [Crossref]
  14. M. Born and E. Wolf, Principles of Optics(Cambridge University Press, 1999), 7th ed.
    [Crossref]
  15. L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).
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  17. G. Zheng, Fourier Ptychographic Imaging (Morgan & Claypool Publishers, 2016).
  18. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and systems(Prentice-Hall International, Inc, 1997), 2nd ed.

2017 (1)

L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).

2016 (1)

2015 (5)

2014 (4)

2013 (3)

Ames, B.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

Bian, L.

Bian, Z.

Born, M.

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

Chen, F.

Chen, Q.

Chen, R. Y.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

Cox, M. E.

J. W. Goodman and M. E. Cox, Introduction to Fourier Optics (The McGraw-Hill Companies, Inc, 1968).

Dai, Q.

Dong, S.

Goodman, J. W.

J. W. Goodman and M. E. Cox, Introduction to Fourier Optics (The McGraw-Hill Companies, Inc, 1968).

Guo, K.

Horstmeyer, R.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
[Crossref] [PubMed]

Jiang, W.

Li, X.

Li, Z.

Liang, Y.

L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).

Liao, J.

Liu, D.

Nanda, P.

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and systems(Prentice-Hall International, Inc, 1997), 2nd ed.

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and systems(Prentice-Hall International, Inc, 1997), 2nd ed.

Ou, X.

Ramchandran, K.

Shiradkar, R.

Sun, J.

Suo, J.

Tang, L.

L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).

Tian, L.

Tropp, J. A.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

Waller, L.

Wang, X.

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and systems(Prentice-Hall International, Inc, 1997), 2nd ed.

Wolf, E.

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

Yang, C.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

Zhang, J.

Zhang, L.

L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).

Zhang, M.

L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).

Zhang, Y.

Zheng, G.

Zuo, C.

Acta Phys. Sin. (1)

L. Zhang, L. Tang, M. Zhang, and Y. Liang, “Symmetric illumination in Fourier ptychography,” Acta Phys. Sin. 66, 211–217 (2017).

Biomed. Opt. Express (2)

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

New J. Phys. (1)

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Optica (1)

Photon. Res. (1)

Other (4)

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

J. W. Goodman and M. E. Cox, Introduction to Fourier Optics (The McGraw-Hill Companies, Inc, 1968).

G. Zheng, Fourier Ptychographic Imaging (Morgan & Claypool Publishers, 2016).

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and systems(Prentice-Hall International, Inc, 1997), 2nd ed.

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Figures (5)

Fig. 1
Fig. 1 Simulated results of LR images based on Taylor expansion and Fourier transform. (a) is the original intensity with grid-like pattern. (b) is the LR image obtained by LED (-7, 7) illumination, and is calculated according Eq. (1) to show the result of Fourier transform with no defocus and can be seen as the standard image. (c) is the LR image obtained by LED (-7, 7) illumination and is calculated according to Eq. (1) to show the result of Fourier transform with no defocus and can be seen as the standard image and (d) is calculated by Eq. (25). (e1)-(e2) and (f1)-(f2) are results with defocus distance 5   μ m . (g1)-(g2) and (h1)-(h2) are results with defocus distance 35   μ m . (e1) and (g1) are results by using Fourier transform (Eq. (1)). (e2) and (g2) are results by using Taylor expansion (Eq. (25)). (f1) and (h1) are results by using Fourier transform. (f2) and (h2) are results by using Taylor expansion. (e1)-(e2) and (g1)-(g2) are results by LED (-7, 7) illumination. (f1)-(f2) and (h1)-(h2) are results by LEDs (-7, 7) and (7, -7) illumination. All images are normalized to 1.
Fig. 2
Fig. 2 Simulated results of 1951 USAF resolution target. (a) is the LR image obtained at the focal plane and LED (-1, 0) illumination, and is regarded as the ground truth. (b) and (c) are the results of LED (-1, 0) and symmetrical LEDs (-1, 0) and (1, 0) illumination with 35   μ m defocus distance, respectively, and (d) is the profiles of intensity distribution pointed by yellow arrows in (a), (b) and (c). Colors of the curves, red, green and blue, are corresponding to those of lines in (a), (b) and(c), respectively.
Fig. 3
Fig. 3 Experimental results of single and symmetrical LED illumination, respectively. (a1)-(e1) are LR images where only the center LED was turned on, whose defocus are + 80   μ m , + 40   μ m , 0   μ m , 40   μ m and 80   μ m , respectively.(a2)-(e2) and (a3)-(e3) are the enlarged reconstructed HR images under single and symmetrical LED illumination corresponding to yellow boxes in (a1)-(e1), respectively.
Fig. 4
Fig. 4 The experimental results of using the wing of housefly as the sample. (a) is the LR image with only the center LED turned on. (b1) is the reconstructed HR image with single LED illumination. (c1) is the reconstructed HR image with symmetrical LEDillumination. (b2) and (c2) are enlarged images of the red boxes in (b1) and (c1), respectively. (d1) and (d2) are images of two positions along optical axis captured with a 20 × , 0.5 N A objective lens, whose distance between the two positions is 10   μ m . Bars in all images represent 10   μ m .
Fig. 5
Fig. 5 The phase image of the thick sample. (a) is the phase image of Fig. 4(b1). (b) is the phase image of Fig. 4(c1).

Equations (27)

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I g , h = | F 1 { O ^ ( f x f 1 , f y f 2 ) P ( f x , f y , z ) } | 2 ,
O ^ ( f x f 1 , f y f 2 ) = F { O ( x , y ) exp  [ i 2 π ( f 1 x + f 2 y ) ] } .
P ( f x , f y , z ) = { exp  [ i z ( 2 π λ ) 2 f x 2 f y 2 ] , f x 2 + f y 2 N A 2 π λ 0 , o t h e r s .
I = I g , h + I g , h .
I g , h = | F 1 { O ^ ( f x + f 1 , f y + f 2 ) P ( f x , f y , z ) } | 2 .
O ( x , y ) = | O ( x , y ) | e i ϕ ( x , y ) = | O ( x , y ) | ( 1 + i ϕ ( x , y ) ) = | O ( x , y ) | + i T ( x , y ) .
| O ( x , y ) | = j = 0 + A 1 ( j ) cos  { F x 1 ( j ) [ x + x c 1 ( j ) ] + F y 1 ( j ) [ y + y c 1 ( j ) ] } .
T ( x , y ) = j = 0 + A 2 ( j ) cos  { F x 2 ( j ) [ x + x c 2 ( j ) ] + F y 2 ( j ) [ y + y c 2 ( j ) ] } .
O ^ ( f x f 1 , f y f 2 ) = j = 0 + 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x f 1 F x 1 ( j ) , f y f 2 F y 1 ( j ) ) + j = 0 + 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x f 1 + F x 1 ( j ) , f y f 2 + F y 1 ( j ) ) + i j = 0 + 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x f 1 F x 2 ( j ) , f y f 2 F y 2 ( j ) ) + i j = 0 + 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x f 1 + F x 2 ( j ) , f y f 2 + F y 2 ( j ) ) ,
O ^ ( f x + f 1 , f y + f 2 ) = j = 0 + 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x + f 1 F x 1 ( j ) , f y + f 2 F y 1 ( j ) ) + j = 0 + 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x + f 1 + F x 1 ( j ) , f y + f 2 + F y 1 ( j ) ) + i j = 0 + 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x + f 1 F x 2 ( j ) , f y + f 2 F y 2 ( j ) ) + i j = 0 + 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x + f 1 + F x 2 ( j ) , f y + f 2 + F y 2 ( j ) ) .
O ^ ' ( f x f 1 , f y f 2 ) = j = m 1 n 1 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x f 1 + F x 1 ( j ) , f y f 2 + F y 1 ( j ) ) + j = p 1 q 1 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x f 1 F x 1 ( j ) , f y f 2 F y 1 ( j ) ) + i j = m 2 n 2 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x f 1 + F x 2 ( j ) , f y f 2 + F y 2 ( j ) ) + i j = p 2 q 2 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x f 1 F x 2 ( j ) , f y f 2 F y 2 ( j ) ) ,
O ^ ' ( f x + f 1 , f y + f 2 ) = j = m 1 n 1 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x + f 1 F x 1 ( j ) , f y + f 2 F y 1 ( j ) ) + j = p 1 q 1 2 π 2 A 1 ( j ) e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] δ ( f x + f 1 + F x 1 ( j ) , f y + f 2 + F y 1 ( j ) ) + i j = m 2 n 2 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x + f 1 F x 2 ( j ) , f y + f 2 F y 2 ( j ) ) + i j = p 2 q 2 2 π 2 A 2 ( j ) e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] δ ( f x + f 1 + F x 2 ( j ) , f y + f 2 + F y 2 ( j ) ) .
I g , h = | j = m 1 n 1 A 1 ( j ) 2 e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] e i z ( 2 π λ ) 2 [ F x 1 ( j ) f 1 ] 2 [ F y 1 ( j ) f 2 ] 2 e i [ ( F x 1 ( j ) f 1 ) x + ( F y 1 ( j ) f 2 ) y ] + j = p 1 q 1 A 1 ( j ) 2 e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] e i z ( 2 π λ ) 2 [ F x 1 ( j ) + f 1 ] 2 [ F y 1 ( j ) + f 2 ] 2 e i [ ( F x 1 ( j ) + f 1 ) x + ( F y 1 ( j ) + f 2 ) y ] + i j = m 2 n 2 A 2 ( j ) 2 e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] e i z ( 2 π λ ) 2 [ F x 2 ( j ) f 1 ] 2 [ F y 2 ( j ) f 2 ] 2 e i [ ( F x 2 ( j ) f 1 ) x + ( F y 2 ł e f t ( j ) f 2 ) y + i j = p 2 q 2 A 2 ( j ) 2 e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] e i z ( 2 π λ ) 2 [ F x 2 ( j ) + f 1 ] 2 [ F y 2 ( j ) + f 2 ] 2 e i [ ( F x 2 ( j ) + f 1 ) x + ( F y 2 t ( j ) + f 2 ) y | 2 ,
I g , h = | j = m 1 n 1 A 1 ( j ) 2 e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] e i z ( 2 π λ ) 2 [ F x 1 ( j ) f 1 ] 2 [ F y 1 ( j ) f 2 ] 2 e i [ ( F x 1 ( j ) f 1 ) x + ( F y 1 ( j ) f 2 ) y ] + j = p 1 q 1 A 1 ( j ) 2 e i [ F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) ] e i z ( 2 π λ ) 2 [ F x 1 ( j ) + f 1 ] 2 [ F y 1 ( j ) + f 2 ] 2 e i [ ( F x 1 ( j ) + f 1 ) x + ( F y 1 ( j ) + f 2 ) y ] + i j = m 2 n 2 A 2 ( j ) 2 e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] e i z ( 2 π λ ) 2 [ F x 2 ( j ) f 1 ] 2 [ F y 2 ( j ) f 2 ] 2 e i [ ( F x 2 ( j ) f 1 ) x + ( F y 2 ( j ) f 2 ) y ] + i j = p 2 q 2 A 2 ( j ) 2 e i [ F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) ] e i z ( 2 π λ ) 2 [ F x 2 ( j ) + f 1 ] 2 [ F y 2 ( j ) + f 2 ] 2 e i [ ( F x 2 ( j ) + f 1 ) x + ( F y 2 ( j ) + f 2 ) y ] | 2 .
B 1 ( j ) = F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) + [ F x 1 ( j ) f 1 ] x + [ F y 1 ( j ) f 2 ] y D 1 ( j ) = F x 1 ( j ) x c 1 ( j ) + F y 1 ( j ) y c 1 ( j ) + [ F x 1 ( j ) + f 1 ] x + [ F y 1 ( j ) + f 2 ] y B 2 ( j ) = F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) + [ F x 2 ( j ) f 1 ] x + [ F y 2 ( j ) f 2 ] y D 2 ( j ) = F x 2 ( j ) x c 2 ( j ) + F y 2 ( j ) y c 2 ( j ) + [ F x 2 ( j ) + f 1 ] x + [ F y 2 ( j ) + f 2 ] y ,
G 1 ( j ) = 2 π ( 1 λ ) 2 [ F x 1 ( j ) f 1 ] 2 [ F y 1 ( j ) f 2 ] 2 H 1 ( j ) = 2 π ( 1 λ ) 2 [ F x 1 ( j ) + f 1 ] 2 [ F y 1 ( j ) + f 2 ] 2 G 2 ( j ) = 2 π ( 1 λ ) 2 [ F x 2 ( j ) f 1 ] 2 [ F y 2 ( j ) f 2 ] 2 H 2 ( j ) = 2 π ( 1 λ ) 2 [ F x 2 ( j ) + f 1 ] 2 [ F y 2 ( j ) + f 2 ] 2 .
I g , h = | j = m 1 n 1 A 1 ( j ) e i [ z G 1 ( j ) B 1 ( j ) ] + j = p 1 q 1 A 1 ( j ) e i [ z H 1 ( j ) + D 1 ( j ) ] + i j = m 2 n 2 A 2 ( j ) e i [ z G 2 ( j ) B 2 ( j ) ] + i j = p 2 q 2 A 2 ( j ) e i [ z H 2 ( j ) + D 2 ( j ) ] | 2 ,
I g , h = | j = m 1 n 1 A 1 ( j ) e i [ z G 1 ( j ) + B 1 ( j ) ] + j = p 1 q 1 A 1 ( j ) e i [ z H 1 ( j ) D 1 ( j ) ] + i j = m 2 n 2 A 2 ( j ) e i [ z G 2 ( j ) + B 2 ( j ) ] + i j = p 2 q 2 A 2 ( j ) e i [ z H 2 ( j ) D 2 ( j ) ] | 2 .
I g , h = | j = m n A 1 ( j ) cos   [ z X 1 ( j ) + Y 1 ( j ) ] j = p q A 2 ( j ) sin [ z X 2 ( j ) + Y 2 ( j ) ] + i j = m n A 1 ( j ) sin   [ z X 1 ( j ) + Y 1 ( j ) ] + i j = p q A 2 ( j ) cos [ z X 2 ( j ) + Y 2 ( j ) ] | 2 ,
I g , h = | j = m n A 1 ( j ) cos   [ z X 1 ( j ) Y 1 ( j ) ] j = p q A 2 ( j ) sin [ z X 2 ( j ) Y 2 ( j ) ] + i j = m n A 1 ( j ) sin   [ z X 1 ( j ) Y 1 ( j ) ] + i j = p q A 2 ( j ) cos   [ z X 2 ( j ) Y 2 ( j ) ] | 2 .
I g , h = j = m n A 1   2 ( j ) cos  2 [ z X 1 ( j ) + Y 1 ( j ) ] + j = p q A 2   2 ( j ) cos  2 [ z X 2 ( j ) + Y 2 ( j ) ] + j = m n A 1   2 ( j ) sin  2 [ z X 1 ( j ) + Y 1 ( j ) ] + j = p q A 2   2 ( j ) sin  2 [ z X 2 ( j ) + Y 2 ( j ) ] + 2 t , s [ m , n ] t s A 1 ( s ) A 1 ( t ) cos  [ z X 1 ( s ) + Y 1 ( s ) ] cos  [ z X 1 ( t ) + Y 1 ( t ) ] + 2 t , s [ m , n ] t s A 1 ( s ) A 1 ( t ) sin  [ z X 1 ( s ) + Y 1 ( s ) ] sin  [ z X 1 ( t ) + Y 1 ( t ) ] + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) cos  [ z X 2 ( k ) + Y 2 ( k ) ] cos  [ z X 2 ( l ) + Y 2 ( l ) ] + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) sin  [ z X 2 ( k ) + Y 2 ( k ) ] sin  [ z X 2 ( l ) + Y 2 ( l ) ] 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) cos  [ z X 1 ( b ) + Y 1 ( b ) ] sin  [ z X 2 ( d ) + Y 2 ( d ) ] + 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) sin  [ z X 1 ( b ) + Y 1 ( b ) ] cos  [ z X 2 ( d ) + Y 2 ( d ) ] ,
I g , h = j = m n A 1   2 ( j ) cos   2 [ z X 1 ( j ) Y 1 ( j ) ] + j = p q A 2   2 ( j ) cos   2 [ z X 2 ( j ) Y 2 ( j ) ] + j = m n A 1   2 ( j ) sin   2 [ z X 1 ( j ) Y 1 ( j ) ] + j = p q A 2   2 ( j ) sin   2 [ z X 2 ( j ) Y 2 ( j ) ] + 2 t , s [ m , n ] t s A 1 ( s ) A 1 ( t ) cos   [ z X 1 ( s ) Y 1 ( s ) ] cos   [ z X 1 ( t ) Y 1 ( t ) ] + 2 t , s [ m , n ] t s A 1 ( s ) A 1 ( t ) sin   [ z X 1 ( s ) Y 1 ( s ) ] sin   [ z X 1 ( t ) Y 1 ( t ) ] + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) cos   [ z X 2 ( k ) Y 2 ( k ) ] cos   [ z X 2 ( l ) Y 2 ( l ) ] + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) sin   [ z X 2 ( k ) Y 2 ( k ) ] sin   [ z X 2 ( l ) Y 2 ( l ) ] 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) cos   [ z X 1 ( b ) Y 1 ( b ) ] sin   [ z X 2 ( d ) Y 2 ( d ) ] + 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) sin   [ z X 1 ( b ) Y 1 ( b ) ] cos   [ z X 2 ( d ) Y 2 ( d ) ] .
I g , h = j = m n A 1   2 ( j ) + j = p q A 2   2 ( j ) + 2 t , s [ m , n ] t s A 1 ( t ) A 1 ( s ) { cos   [ z X 1 ( s ) z X 1 ( t ) ] cos   [ Y 1 ( s ) Y 1 ( t ) ] sin   [ z X 1 ( s ) z X 1 ( t ) ] sin   [ Y 1 ( s ) Y 1 ( t ) ] } + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) { cos   [ z X 2 ( k ) z X 2 ( l ) ] cos   [ Y 2 ( k ) Y 2 ( l ) ] sin   [ z X 2 ( k ) z X 2 ( l ) ] sin   [ Y 2 ( k ) Y 2 ( l ) ] } + 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) { sin   [ z X 1 ( b ) z X 2 ( d ) ] cos   [ Y 1 ( b ) + Y 2 ( d ) ] + cos   [ z X 1 ( b ) z X 2 ( d ) ] sin   [ Y 1 ( b ) + Y 2 ( d ) ] } ,
I g , h = j = m n A 1   2 ( j ) + j = p q A 2   2 ( j ) + 2 t , s [ m , n ] t s A 1 ( t ) A 1 ( s ) { cos   [ z X 1 ( s ) z X 1 ( t ) ] cos   [ Y 1 ( s ) Y 1 ( t ) ] + sin   [ z X 1 ( s ) z X 1 ( t ) ] sin   [ Y 1 ( s ) Y 1 ( t ) ] } + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) { cos   [ z X 2 ( k ) z X 2 ( l ) ] cos   [ Y 2 ( k ) Y 2 ( l ) ] + sin   [ z X 2 ( k ) z X 2 ( l ) ] sin   [ Y 2 ( k ) Y 2 ( l ) ] } + 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) { sin   [ z X 1 ( b ) z X 2 ( d ) ] cos   [ Y 1 ( b ) + Y 2 ( d ) ] cos   [ z X 1 ( b ) z X 2 ( d ) ] sin   [ Y 1 ( b ) + Y 2 ( d ) ] } .
I g , h = j = m n A 1   2 ( j ) + j = p q A 2   2 ( j ) + 2 t , s [ m , n ] t s A 1 ( t ) A 1 ( s ) { [ 1 1 2 z 2 ( X 1 ( s ) X 1 ( t ) ) 2 ] cos   [ Y 1 ( s ) Y 1 ( t ) ] z [ X 1 ( s ) X 1 ( t ) ] sin   [ Y 1 ( s ) Y 1 ( t ) ] } + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) { [ 1 1 2 z 2 ( X 2 ( k ) X 2 ( l ) ) 2 ] cos   [ Y 2 ( k ) Y 2 ( l ) ] z [ X 2 ( k ) X 2 ( l ) ] sin   [ Y 2 ( k ) Y 2 ( l ) ] } + 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) { z [ X 1 ( b ) X 2 ( d ) ] cos   [ Y 1 ( b ) + Y 2 ( d ) ] + [ 1 1 2 z 2 ( X 1 ( b ) X 2 ( d ) ) 2 ] sin   [ Y 1 ( b ) + Y 2 ( d ) ] } + O ( z 3 ) ,
I g , h = j = m n A 1   2 ( j ) + j = p q A 2   2 ( j ) + 2 t , s [ m , n ] t s A 1 ( t ) A 1 ( s ) { [ 1 1 2 z 2 ( X 1 ( s ) X 1 ( t ) ) 2 ] cos   [ Y 1 ( s ) Y 1 ( t ) ] + z [ X 1 ( s ) X 1 ( t ) ] sin   [ Y 1 ( s ) Y 1 ( t ) ] } + 2 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) { [ 1 1 2 z 2 ( X 2 ( k ) X 2 ( l ) ) 2 ] cos   [ Y 2 ( k ) Y 2 ( l ) ] + z [ X 2 ( k ) X 2 ( l ) ] sin   [ Y 2 ( k ) Y 2 ( l ) ] } + 2 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) { z [ X 1 ( b ) X 2 ( d ) ] cos   [ Y 1 ( b ) + Y 2 ( d ) ] [ 1 1 2 z 2 ( X 1 ( b ) X 2 ( d ) ) 2 ] sin   [ Y 1 ( b ) + Y 2 ( d ) ] } + O ( z 3 ) .
I = I g , h + I g , h = 2 j = m n A 1   2 ( j ) + 2 j = p q A 2   2 ( j ) + 4 t , s [ m , n ] t s A 1 ( t ) A 1 ( s ) [ 1 1 2 z 2 ( X 1 ( s ) X 1 ( t ) ) 2 ] cos   [ Y 1 ( s ) Y 1 ( t ) ] + 4 k , l [ p , q ] k l A 2 ( k ) A 2 ( l ) [ 1 1 2 z 2 ( X 2 ( k ) X 2 ( l ) ) 2 ] cos [ Y 2 ( k ) Y 2 ( l ) ] + 4 b [ m , n ] , d [ p , q ] A 1 ( b ) A 2 ( d ) z [ X 1 ( b ) X 2 ( d ) ] cos [ Y 1 ( b ) + Y 2 ( d ) ] + O ( z 3 ) .

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