Abstract

We propose dynamic range compression deconvolution by a new nonlinear optical-limiter microelectromechanical system (NOLMEMS) device. The NOLMEMS uses aperturized, reflected coherent light from optically addressed, parabolically deformable mirrors. The light is collimated by an array of microlenses. The reflected light saturates as a function of optical drive intensity. In this scheme, a joint image of the blurred input information and the blur impulse response is captured and sent to a spatial light modulator (SLM). The joint information on the SLM is read through a laser beam and is Fourier transformed by a lens to the back of the NOLMEMS device. The output from the NOLMEMS is Fourier transformed to produce the restored image. We derived the input–output nonlinear transfer function of our NOLMEMS device, which relates the transmitted light from the pinhole to the light intensity incident on the back side of the device, and exhibits saturation. We also analyzed the deconvolution orders for this device, using a nonlinear transform method. Computer simulation of image deconvolution by the NOLMEMS device is also presented.

© 2009 Optical Society of America

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  1. Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
    [CrossRef]
  2. B. Haji-saeed, S. K. Sengupta, W. Goodhue, J. Khoury, C. L. Woods, and J. Kierstead, “Nonlinear dynamic range compression deconvolution,” Opt. Lett. 31, 1969-1971 (2006).
    [CrossRef] [PubMed]
  3. B. Haji-saeed, S. K. Sengupta, W. Goodhue, J. Khoury, C. L. Woods, and J. Kierstead, “Spectrally variable two-beam coupling nonlinear deconvolution,” Appl. Opt. 46, 8244-8249 (2007).
    [CrossRef] [PubMed]
  4. H. J. Eichler, P. Gunter, and D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, 1986).
  5. T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
    [CrossRef]
  6. P. Gunter and J. -P. Huignard, Photorefractive Materials and Their Applications, Vols. I and II (Springer-Verlag, 1988).
  7. B. Haji-saeed, R. Kolluru, D. Pyburn, R. Leon, S. K. Sengupta, M. Testorf, W. Goodhue, J. Khoury, A. Drehman, C. L. Woods, and J. Kierstead, “Photoconductive optically driven deformable membrane under high frequency bias: fabrication, characterization and modeling,” Appl. Opt. 45, 3226-3236 (2006).
    [CrossRef] [PubMed]
  8. J. Khoury, C. L. Woods, B. Haji-saeed, S. K. Sengupta, W. Goodhue, and J. Kierstead, “Optically driven microelectromechanical-system deformable mirror under high-frequency AC bias,” Opt. Lett. 31, 808-810 (2006).
    [CrossRef] [PubMed]
  9. J. Khoury, B. Haji-saeed, W. D. Goodhue, C. L. Woods, and J. Kierstead, “MEMS-based optical limiter,” Appl. Opt. 47, 5468-5472 (2008).
    [CrossRef] [PubMed]
  10. J. L. Horner and P. D. Gianino, “Effects of quadratic phase distortion on correlator performance,” Appl. Opt. 31, 3876-3878 (1992).
    [CrossRef] [PubMed]
  11. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill, 1958), Chaps. 12-13, pp. 255-311
  12. J. Van de Vegte, Fundamentals of Digital Signal Processing (Prentice-Hall, 2002), Chap. 15, pp. 659-660.
  13. J. L. Horner and M. A. Flavin, “Average amplitude matched filter paper,” Opt. Eng. 29, 31-37 (1990).
    [CrossRef]
  14. B. Haji-saeed, “Development of novel device assemblies and techniques for improving adaptive optics imaging systems,” Ph.D. dissertation (University of Massachusetts at Lowell, 2006).
  15. R. E. Hufnagel and N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54, 52-61 (1964).
    [CrossRef]
  16. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).
  17. G. Asimellis, J. Khoury, and C. Woods, “Effects of saturation on the nonlinear incoherent-erasure joint-transform correlator,” J. Opt. Soc. Am. A 13, 1345-1356 (1996).
    [CrossRef]
  18. J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transform correlators,” Opt. Eng. 37, 66-74 (1998).
    [CrossRef]
  19. J. Khoury, P. D. Gianino, and C. L. Woods “Engineering aspects of the two-beam coupling correlator,” Opt. Eng. 39, 1177-1183 (2000).
    [CrossRef]

2008 (1)

2007 (1)

2006 (3)

2004 (1)

Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
[CrossRef]

2000 (1)

J. Khoury, P. D. Gianino, and C. L. Woods “Engineering aspects of the two-beam coupling correlator,” Opt. Eng. 39, 1177-1183 (2000).
[CrossRef]

1998 (1)

J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transform correlators,” Opt. Eng. 37, 66-74 (1998).
[CrossRef]

1996 (1)

1992 (1)

1990 (1)

J. L. Horner and M. A. Flavin, “Average amplitude matched filter paper,” Opt. Eng. 29, 31-37 (1990).
[CrossRef]

1985 (1)

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
[CrossRef]

1964 (1)

Asimellis, G.

J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transform correlators,” Opt. Eng. 37, 66-74 (1998).
[CrossRef]

G. Asimellis, J. Khoury, and C. Woods, “Effects of saturation on the nonlinear incoherent-erasure joint-transform correlator,” J. Opt. Soc. Am. A 13, 1345-1356 (1996).
[CrossRef]

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
[CrossRef]

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill, 1958), Chaps. 12-13, pp. 255-311

Drehman, A.

Eichler, H. J.

H. J. Eichler, P. Gunter, and D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, 1986).

Flavin, M. A.

J. L. Horner and M. A. Flavin, “Average amplitude matched filter paper,” Opt. Eng. 29, 31-37 (1990).
[CrossRef]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
[CrossRef]

Gianino, P. D.

J. Khoury, P. D. Gianino, and C. L. Woods “Engineering aspects of the two-beam coupling correlator,” Opt. Eng. 39, 1177-1183 (2000).
[CrossRef]

J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transform correlators,” Opt. Eng. 37, 66-74 (1998).
[CrossRef]

J. L. Horner and P. D. Gianino, “Effects of quadratic phase distortion on correlator performance,” Appl. Opt. 31, 3876-3878 (1992).
[CrossRef] [PubMed]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).

Goodhue, W.

Goodhue, W. D.

Gunter, P.

P. Gunter and J. -P. Huignard, Photorefractive Materials and Their Applications, Vols. I and II (Springer-Verlag, 1988).

H. J. Eichler, P. Gunter, and D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, 1986).

Haji-saeed, B.

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
[CrossRef]

Horner, J. L.

Huang, Y.

Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
[CrossRef]

Hufnagel, R. E.

Huignard, J.-P.

P. Gunter and J. -P. Huignard, Photorefractive Materials and Their Applications, Vols. I and II (Springer-Verlag, 1988).

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
[CrossRef]

Khoury, J.

J. Khoury, B. Haji-saeed, W. D. Goodhue, C. L. Woods, and J. Kierstead, “MEMS-based optical limiter,” Appl. Opt. 47, 5468-5472 (2008).
[CrossRef] [PubMed]

B. Haji-saeed, S. K. Sengupta, W. Goodhue, J. Khoury, C. L. Woods, and J. Kierstead, “Spectrally variable two-beam coupling nonlinear deconvolution,” Appl. Opt. 46, 8244-8249 (2007).
[CrossRef] [PubMed]

B. Haji-saeed, R. Kolluru, D. Pyburn, R. Leon, S. K. Sengupta, M. Testorf, W. Goodhue, J. Khoury, A. Drehman, C. L. Woods, and J. Kierstead, “Photoconductive optically driven deformable membrane under high frequency bias: fabrication, characterization and modeling,” Appl. Opt. 45, 3226-3236 (2006).
[CrossRef] [PubMed]

B. Haji-saeed, S. K. Sengupta, W. Goodhue, J. Khoury, C. L. Woods, and J. Kierstead, “Nonlinear dynamic range compression deconvolution,” Opt. Lett. 31, 1969-1971 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, S. K. Sengupta, W. Goodhue, and J. Kierstead, “Optically driven microelectromechanical-system deformable mirror under high-frequency AC bias,” Opt. Lett. 31, 808-810 (2006).
[CrossRef] [PubMed]

J. Khoury, P. D. Gianino, and C. L. Woods “Engineering aspects of the two-beam coupling correlator,” Opt. Eng. 39, 1177-1183 (2000).
[CrossRef]

J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transform correlators,” Opt. Eng. 37, 66-74 (1998).
[CrossRef]

G. Asimellis, J. Khoury, and C. Woods, “Effects of saturation on the nonlinear incoherent-erasure joint-transform correlator,” J. Opt. Soc. Am. A 13, 1345-1356 (1996).
[CrossRef]

Kierstead, J.

Kolluru, R.

Leon, R.

Moharam, M. G.

Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
[CrossRef]

Pohl, D. W.

H. J. Eichler, P. Gunter, and D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, 1986).

Pyburn, D.

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill, 1958), Chaps. 12-13, pp. 255-311

Sengupta, S. K.

Siganakis, G.

Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
[CrossRef]

Stanley, N. R.

Testorf, M.

Van de Vegte, J.

J. Van de Vegte, Fundamentals of Digital Signal Processing (Prentice-Hall, 2002), Chap. 15, pp. 659-660.

Woods, C.

Woods, C. L.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).

Wu, S-T.

Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broadband optical limiter based on nonlinear photoinduced anisotropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85, 5445 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (3)

J. L. Horner and M. A. Flavin, “Average amplitude matched filter paper,” Opt. Eng. 29, 31-37 (1990).
[CrossRef]

J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transform correlators,” Opt. Eng. 37, 66-74 (1998).
[CrossRef]

J. Khoury, P. D. Gianino, and C. L. Woods “Engineering aspects of the two-beam coupling correlator,” Opt. Eng. 39, 1177-1183 (2000).
[CrossRef]

Opt. Lett. (2)

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect--a review,” Prog. Quantum Electron. 10, 77-146 (1985).
[CrossRef]

Other (6)

P. Gunter and J. -P. Huignard, Photorefractive Materials and Their Applications, Vols. I and II (Springer-Verlag, 1988).

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill, 1958), Chaps. 12-13, pp. 255-311

J. Van de Vegte, Fundamentals of Digital Signal Processing (Prentice-Hall, 2002), Chap. 15, pp. 659-660.

B. Haji-saeed, “Development of novel device assemblies and techniques for improving adaptive optics imaging systems,” Ph.D. dissertation (University of Massachusetts at Lowell, 2006).

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).

H. J. Eichler, P. Gunter, and D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, 1986).

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Figures (15)

Fig. 1
Fig. 1

A, original image; B, image recovery from the amplitude-only information; C, image recovery from the phase-only information; D, gray-level recovery via application of a lowpass filter to C.

Fig. 2
Fig. 2

Single-pixel MEMS optical-limiter architecture.

Fig. 3
Fig. 3

Plot of NOLMEMS transfer function.

Fig. 4
Fig. 4

Schematic diagram of a NOLMEMS joint Fourier processor.

Fig. 5
Fig. 5

H 1 ( v x , v y ) plots: A, absolute value; B, real value; C, imaginary value; and D, phase.

Fig. 6
Fig. 6

H 2 ( v x , v y ) plots: A, absolute value; B, real value; C, imaginary value; D, phase.

Fig. 7
Fig. 7

H 3 ( v x , v y ) plots: A, absolute value; B, real value; C, imaginary value; and D, phase.

Fig. 8
Fig. 8

Motion aberration indirect simulation results: A, noisy blurred input with SNR = 5 ; A′, corresponding recovered image. The B and C rows are sequentially the same as the A row for SNR = 1 and SNR = 0.1 , respectively.

Fig. 9
Fig. 9

Deconvolution result for the motion aberration compared to Wiener and inverse filters result: A, recovered image for SNR = 1 ; B, recovered image using the Wiener filter, with the expected value of F; C, recovered image using inverse filtering, and D,  recovered image using Wiener filter, with the exact value of F.

Fig. 10
Fig. 10

Atmospheric turbulence indirect simulation results: A, noisy blurred input with SNR = 5 ; A′, corresponding recovered image. The B and C rows are sequentially the same as the A row for SNR = 1 and SNR = 0.1 , respectively.

Fig. 11
Fig. 11

Deconvolution result for the atmospheric turbulence compared to the Wiener and inverse filters result: A, recovered image for SNR = 1 ; B, recovered image using the Wiener filter, with the expected value of F; C,  recovered image using inverse filtering; D, recovered image using the Wiener filter, with the exact value of F.

Fig. 12
Fig. 12

Misfocusing aberration indirect simulation results: A, noisy blurred input with SNR = 5 ; A′ the corresponding recovered image. The B and C rows are sequentially the same as the A row for SNR = 1 and SNR = 0.1 , respectively.

Fig. 13
Fig. 13

Deconvolution result for the misfocusing aberration compared to the Wiener and inverse filters result: A, recovered image for SNR = 1 ; B, recovered image using the Wiener filter, with the expected value of F; C, recovered image using inverse filtering; and D, recovered image using Wiener filter, with the exact value of F.

Fig. 14
Fig. 14

Phase-only aberration indirect simulation results: A, noisy blurred input with SNR = 5 ; A′, the corresponding recovered image. The B and C rows are sequentially the same as the A row for SNR = 1 and SNR = 0.1 , respectively.

Fig. 15
Fig. 15

Deconvolution result for the phase-only aberration compared to the Wiener and inverse filters result: A, recovered image for SNR = 1 ; B, recovered image using the Wiener filter, with the expected value of F; C, recovered image using inverse filtering; and D, recovered image using Wiener filter, with the exact value of F.

Equations (85)

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LPF ( u , ν ) = 1 1 + ( u + M / 2 ) 2 + ( ν + N / 2 ) 2 D 0 ,
G ( y ) ( A y A y + B ) 2 ,
A = 8 λ T s 2 ,
B = r 1 6 V 2 L 2 ω 2 π 3 ε 0 3 ,
y = ( a + b ( Δ n + n ) 2 ) ,
a = L 2 ω 2 π 2 r 1 4 ε 0 2 ,
b = q 2 μ n 2 A d 2 s 2 .
Δ n = α τ n I 0 h ν A d ,
h d = ε 0 r 1 2 V 2 32 T s 2 .
Δ n ( ν x , ν y ) = α τ n I 0 | R + S | 2 h ν A D ( λ f ) 2 ,
G ( y ) = k = 0 H k ( ν x , ν y ) cos [ 2 k y o ν y + k ( ϕ R ϕ S ) ] ,
H k ( v x , v y ) = 1 ε k i k + 1 { ( 2 m ( 0 ) J A × 1 f 2 + f 1 ) k [ B m 2 ( 0 ) A b ( 1 f 1 ) B 2 m 4 ( 0 ) 4 ( A b ) 2 ( f 2 + f 3 f 1 f 1 ) k B 2 m 4 ( 0 ) 4 ( A b ) 2 ( 2 f 2 ( 1 f 2 ) f 1 + f 1 f 1 ) ] ( 2 m ( 0 ) y 1 f 2 + e 1 ) k [ B m 2 ( 0 ) A b ( 1 e 1 ) B 2 m 4 ( 0 ) 4 ( A b ) 2 ( f 3 f 2 e 1 e 1 ) k B 2 m 4 ( 0 ) 4 ( A b ) 2 ( f 2 ( 1 f 2 ) e 1 + e 1 e 1 ) ] } ,
J A = ( R S ) 2 ,
J A + = ( R + S ) 2 ,
J A × = R S ,
J I = = R 2 + S 2 ,
f 2 = i m ( 0 ) ( J I + + n ) ,
f 3 = m 2 ( 0 ) ( J A + + n ) ( J A + n ) ,
f 1 = 1 2 f 2 f 3 ,
e 1 = 1 + 2 f 2 f 3 ,
m 2 ( 0 ) = A b a A + B .
H ( ν x , ν y ) = e β ( ν x 2 + ν y 2 ) 5 / 6
W ( u , v ) = [ H * ( u , v ) | F ( u , v ) | 2 | H ( u , v ) | 2 | F ( u , v ) | 2 + | N ( u , v ) | 2 ] ,
f ( E ) = 1 1 ± i G p ( E + E 0 ) ,
F ( ω ) = f ( E ) exp ( i ω E ) d E ,
f ( E ) = 1 2 π F ( ω ) exp ( i ω E ) d ω .
E = | S ( ν x , ν y ) + R ( ν x , ν y ) | 2 = S 2 ( ν x , ν y ) + S ( ν x , ν y ) R ( ν x , ν y ) exp [ i ϕ S ( ν x , ν y ) i ϕ R ( ν x , ν y ) ] exp ( i 2 y 0 ν y ) + S ( ν x , ν y ) R ( ν x , ν y ) exp [ i ϕ S ( ν x , ν y ) + i ϕ R ( ν x , ν y ) ] exp ( + i 2 y 0 ν y ) + R 2 ( ν x , ν y ) ,
f ( E ) = 1 2 π F ( ω ) exp [ i ω ( R 2 + S 2 ) ] exp [ i 2 ω R S cos ( 2 y 0 ν y + ϕ R ϕ S ) ] d ω .
f ( E ) = 1 2 π F ( ω ) exp [ i ω ( R 2 + S 2 ) ] ( k = i k J k ( 2 ω R S ) cos [ 2 k y 0 ν y + k ( ϕ R ϕ S ) ] ) d ω ,
f ( E ) = k = 0 H k ( ν x , ν y ) cos [ 2 k y 0 ν y + k ( ϕ R ϕ S ) ] ,
H k ( ν x , ν y ) = ε k 2 π i k F ( ω ) exp [ i ω ( R 2 + S 2 ) ] J k ( 2 ω R S ) d ω ,
F ( ω ) = i 2 π ± G p e ω / ± G p e i ω E 0 .
H k ( v x , v y ) = ε k 2 π i k 0 i 2 π ± G p e ω / ± G p e i ω ( J I + ) e i ω E 0 J k ( 2 ω J A × ) d ω = ε k i k + 1 [ 0 e r cos ( r G p ( J I + + E 0 ) ) J k ( 2 r G p J A × ) d r + i 0 e r sin ( r G p ( J I + 2 + E 0 ) ) J k ( 2 r G p J A × ) d r ] ,
± r = ω ± G p ,
J A = ( R S ) 2 , J A + = ( R + S ) 2 , J A × = R S , J I + = R 2 + S 2 .
{ 0 e p x sin b x 0 e p x cos b x } J ν ( a x ) d x = 1 2 { i 1 } ( u + v + ν u v ν ) ,
u ± = [ ( p ± i b ) 2 + a 2 ] 1 2 ,
ν ± = a 1 [ p ± i b + ( ( p ± i b ) 2 + a 2 ) ] .
u ± = 1 ( 1 ± i G p ( J I + + E 0 ) ) 2 + 4 G p 2 J A × 2 ,
ν ± = 1 ± i G p ( J I + + E 0 ) + ( 1 ± i G p 2 ( J I + + E 0 ) ) 2 + 4 G p 2 J A × 2 2 G p J A × .
H C k ( v x , v y ) = ε k i k + 1 [ 1 { ( 1 i ( ± G p ) ( J A + + E 0 ) ) ( 1 i ( ± G p ) ( J A + E 0 ) ) } 1 / 2 × ( 2 ( ± G p ) J A × 1 i ( ± G p ) ( J I + + x 0 ) + { ( 1 i ( ± G p ) ( J A + + E 0 ) ) ( 1 i ( ± G p ) ( J A + E 0 ) ) } 1 / 2 ) k ] .
H C 0 ( v x , v y ) = 1 { ( 1 i ( ± G p ) ( J A + + E 0 ) ) ( 1 i ( ± G p ) ( J A + E 0 ) ) } 1 / 2 ,
P C k ( v x , v y ) = ( 2 ( ± G p ) J A × 1 i ( ± G p ) ( J I + + x 0 ) + { ( 1 i ( ± G p ) ( J A + + E 0 ) ) ( 1 i ( ± G p ) ( J A + E 0 ) ) } 1 / 2 ) k
H C k ( v x , v y ) = ε k i k + 1 H C 0 ( v x , v y ) P C k ( v x , v y ) .
I out = A 2 y 2 ( A y + B ) 2 I in = G ( y ) I in ,
y = a + b ( x + x 0 ) 2 ,
a = L 2 ω 2 π 2 r 1 4 ε 0 2 ,
b = q 2 μ n 2 A 2 s 2 ,
A = 8 λ T s 2 ,
B = r 1 6 V 2 L 2 ω 2 π 3 ε 0 3 .
[ a A [ A a + B ] + b A [ A a + B ] ( x + x 0 ) 2 ] 2 [ A a + B ] 2 [ 1 + b A A a + B ( x + x 0 ) 2 ] 2 = [ α 1 β ( 0 ) + γ 2 ( 0 ) ( x + x 0 ) 2 ] 2 β 2 ( 0 ) [ 1 + γ 2 ( 0 ) ( x + x 0 ) 2 ] 2 = [ γ 1 ( 0 ) + m 2 ( 0 ) ( x + x 0 ) 2 ] 2 β 2 ( 0 ) [ 1 + m 2 ( 0 ) ( x + x 0 ) 2 ] 2 ,
α 1 = a A ,
α 2 = A b ,
β ( 0 ) = a A + B ,
γ 1 ( 0 ) = α 1 β ( 0 ) = A a a A + B ,
γ 2 ( 0 ) = α 2 β ( 0 ) = A b a A + B = m 2 ( 0 ) .
x ( ν x , ν y ) = α τ n I 0 | R + S | 2 h ν A D ( λ f ) 2 ,
[ γ 1 ( 0 ) + m 2 ( 0 ) ( α τ n h ν A D ( λ f ) 2 ) 2 ( I 0 | R + S | 2 + x 0 h ν A D ( λ f ) 2 α τ n ) 2 ] 2 β 2 ( 0 ) [ 1 + m 2 ( 0 ) ( α τ n h ν A D ( λ f ) 2 ) 2 ( I 0 | R + S | 2 + x 0 h ν A D ( λ f ) 2 α τ n ) 2 ] 2 = [ γ 1 ( 0 ) + m 2 ( 0 ) B f 2 ( I 0 | R + S | 2 + x 0 B f ) 2 ] 2 β 2 ( 0 ) [ 1 + m 2 ( 0 ) B f 2 ( I 0 | R + S | 2 + x 0 B f ) 2 ] 2 = [ γ 1 ( 0 ) + C m ( I 0 | R + S | 2 + I e q ) 2 ] 2 β 2 ( 0 ) [ 1 + C m ( I 0 | R + S | 2 + I e q ) 2 ] 2 ,
B f = α τ n h ν A D ( λ f ) 2 ,
C m = m ( 0 ) B f ,
I eq = x 0 B f .
G ( y ) = A 2 y 2 ( A y + B ) 2 = 1 2 A B y + B 2 ( A y + B ) 2 .
2 A B y + B 2 ( A y + B ) 2 = lim ε 0 2 A B y + B 2 ( A y + B + ε ) ( A y + B ε ) .
1 g 1 ( ε ) A y + B + ε + g 1 ( ε ) A y + B ε ,
g 1 ( ± ε ) = B [ 1 + B ± 2 ε ]
g 1 ( ± ε ) A y + B ± ε = g 1 ( ± ε ) β ( ± ε ) · 1 1 + γ 2 ( ± ε ) ( x + x 0 ) 2 ,
β ( ± ε ) = a A + B ± ε ,
γ 2 ( ± ε ) = α 2 β ( ± ε ) = A b a A + B ± ε = m 2 ( ± ε ) .
g 1 ( ± ε ) 2 β ( ± ε ) · 1 γ 2 ( ± ε ) x + i + γ 2 ( ± ε ) x 0 + g 1 ( ± ε ) 2 β ( ± ε ) · 1 γ 2 ( ± ε ) x i + γ 2 ( ± ε ) x 0 = i g 2 ( ± ε ) 1 i m ( ± ε ) ( x + x 0 ) + i g 2 ( ± ε ) 1 + i m ( ± ε ) ( x + x 0 ) ,
g 2 ( ± ε ) = g 1 ( ± ε ) 2 β ( ± ε ) .
H C k ( v x , v y , m ( ± ε ) ) = ε k i k + 1 g 2 ( ± ε ) [ 1 { ( 1 i m ( ± ε ) ( J A + + E 0 ) ) ( 1 i m ( ± ε ) ( J A + E 0 ) ) } 1 / 2 × ( 2 m ( ± ε ) J A x 1 i m ( ± ε ) ( J I + + x 0 ) + { ( 1 i m ( ± ε ) ( J A + + E 0 ) ) ( 1 i m ( ± ε ) ( J A + E 0 ) ) } 1 / 2 ) 1 { ( 1 + i m ( ± ε ) ( J A + + E 0 ) ) ( 1 + i m ( ± ε ) ( J A + E 0 ) ) } 1 / 2 × ( 2 m ( ± ε ) J A x 1 i m ( ± ε ) ( J I + + x 0 ) + { ( 1 + i m ( ± ε ) ( J A + + E 0 ) ) ( 1 + i m ( ± ε ) ( J A + E 0 ) ) } 1 / 2 ) k ] ,
J A = ( R S ) 2 , J A + = ( R + S ) 2 , J A × = R S , J I = = R 2 + S 2 .
H C k ( v x , v y , m ( ± ε ) ) = 1 { g 1 ( + ε ) ε k i k + 1 2 β ( + ε ) [ 1 { ( 1 i m ( + ε ) ( J A + + E 0 ) ) ( 1 i m ( + ε ) ( J A + E 0 ) ) } 1 / 2 ( 2 m ( + ε ) J A x 1 i m ( + ε ) ( J I + + x 0 ) + { ( 1 i m ( + ε ) ( J A + + E 0 ) ) ( 1 i m ( + ε ) ( J A + E 0 ) ) } 1 / 2 ) k 1 { ( 1 + i m ( + ε ) ( J A + + E 0 ) ) ( 1 + i m ( + ε ) ( J A + E 0 ) ) } 1 / 2 ( 2 m ( + ε ) J A x 1 i m ( + ε ) ( J I + + x 0 ) + { ( 1 + i m ( + ε ) ( J A + + E 0 ) ) ( 1 + i m ( + ε ) ( J A + E 0 ) ) } 1 / 2 ) k ] + g 1 ( ε ) ε k i k + 1 2 β ( ε ) [ 1 { ( 1 i m ( ε ) ( J A + + E 0 ) ) ( 1 i m ( ε ) ( J A + E 0 ) ) } 1 / 2 ( 2 m ( ε ) J A x 1 i m ( ε ) ( J I + + x 0 ) + { ( 1 i m ( ε ) ( J A + + E 0 ) ) ( 1 i m ( ε ) ( J A + E 0 ) ) } 1 / 2 ) k 1 { ( 1 + i m ( ε ) ( J A + + E 0 ) ) ( 1 + i m ( ε ) ( J A + E 0 ) ) } 1 / 2 ( 2 m ( ε ) J A x 1 i m ( ε ) ( J I + + x 0 ) + { ( 1 + i m ( ε ) ( J A + + E 0 ) ) ( 1 + i m ( ε ) ( J A + E 0 ) ) } 1 / 2 ) k ] } .
H C k ( v x , v y ) = lim ε 0 [ H C k ( v x , v y , m ( ε ) ) + H C k ( v x , v y , m ( ε ) ) ] .
H k ( v x , v y ) = 1 ε k i k + 1 { ( 2 m ( 0 ) J A × 1 f 2 + f 1 ) k [ B m 2 ( 0 ) A b ( 1 f 1 ) B 2 m 4 ( 0 ) 4 ( A b ) 2 ( f 2 + f 3 f 1 f 1 ) k B 2 m 4 ( 0 ) 4 ( A b ) 2 ( 2 f 2 ( 1 f 2 ) f 1 + f 1 f 1 ) ] ( 2 m ( 0 ) y 1 f 2 + e 1 ) k [ B m 2 ( 0 ) A b ( 1 e 1 ) B 2 m 4 ( 0 ) 4 ( A b ) 2 ( f 3 f 2 e 1 e 1 ) k B 2 m 4 ( 0 ) 4 ( A b ) 2 ( f 2 ( 1 f 2 ) e 1 + e 1 e 1 ) ] } ,
f 2 = i m ( 0 ) ( J I + + x 0 ) ,
f 3 = m 2 ( 0 ) ( J A + + x 0 ) ( J A + x 0 ) ,
f 1 = 1 2 f 2 f 3 ,
e 1 = 1 + 2 f 2 f 3 .
H k ( v x , v y ) = 1 ε k i k + 1 { ( 2 C m J A × I 0 B f ( 1 f 2 + f 1 ) ) k [ B C m 2 B f 2 A b ( 1 f 1 ) B 2 C m 4 4 B f 4 ( A b ) 2 ( f 2 + f 3 f 1 f 1 ) k B 2 C m 4 4 B f 4 ( A b ) 2 ( 2 f 2 ( 1 f 2 ) f 1 + f 1 f 1 ) ] ( 2 C m J A × I 0 B f ( 1 f 2 + e 1 ) ) k [ B C m 2 B f 2 A b ( 1 e 1 ) B 2 C m 4 4 B f 4 ( A b ) 2 ( f 3 f 2 e 1 e 1 ) k B 2 C m 4 4 B f 4 ( A b ) 2 ( f 2 ( 1 f 2 ) e 1 + e 1 e 1 ) ] } ,
f 2 = i C m B f ( J I + + x 0 ) ,
f 3 = ( C m B f ) 2 ( J A + + x 0 ) ( J A + x 0 ) ,
f 1 = 1 2 f 2 f 3 ,
e 1 = 1 + 2 f 2 f 3 ,
C m = m ( 0 ) B f .

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