Abstract

With rapidly growing bandwidth demands in Local Area Networks, it is imperative to support next generation speeds beyond 40Gbit/s. Various holographic optimization techniques using spatial light modulators have recently been explored for adaptive channel impulse response improvement of MMF links. Most of these experiments are algorithmic-oriented. In this paper, a set of lenses and a spatial light modulator, acting as a binary amplitude filter, played the pivotal role in generating the input modal electric field into a graded-index MMF, rather than algorithms. By using a priori theoretical information to generate the incident modal electric field at the MMF, the bandwidth was increased by up to 3.4 times.

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References

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  1. S. Bois, Next Generation Fibers and Standards (Corning Optical Fiber 2009).
  2. R. J. Shapiro, "The Internet’s Capacity To Handle Fast-Rising Demand for Bandwidth," USIIA White Paper, (2007).
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  7. X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18(1), 244–246 (2006).
    [CrossRef]
  8. T. Shimada, N. Sakurai, and K. Kumozaki, “WDM access system based on shared demultiplexer and MMF links,” J. Lightwave Technol. 23(9), 2621–2628 (2005).
    [CrossRef]
  9. Z. Haas and M. A. Santoro, “A mode-filtering scheme for improvement of the bandwidth-distance product in multimode fiber systems,” J. Lightwave Technol. 11(7), 1125–1131 (1993).
    [CrossRef]
  10. S. Berdagué and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt. 21(11), 1950–1955 (1982).
    [CrossRef] [PubMed]
  11. F. Dubois, P. Emplit, and O. Hugon, “Selective mode excitation in graded-index multimode fiber by a computer-generated optical mask,” Opt. Lett. 19(7), 433–435 (1994).
    [CrossRef] [PubMed]
  12. D. Erni, M. Jungo, and W. Baechtol, "Segmented VCSEL contact geometry for active coupling efficiency enhancement," in Workshop on Compound Semiconductor Devices and Integrated Circuits (WOCSDICE), (Photonics Communication Group, Swiss Federal Institute of Technology (ETHZ), 2003), 67-68.
  13. P. Facq, F. De-Fornel, and F. Jean, “Tunable single-mode excitation in multimode fibres,” Electron. Lett. 20(15), 613–614 (1984).
    [CrossRef]
  14. L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Nowell, “An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links,” J. Lightwave Technol. 16(3), 324–331 (1998).
    [CrossRef]
  15. A. Amphawan, F. Payne, D. O'Brien, and N. Shah, “Derivation of an analytical expression for the power coupling coefficient for offset launch into multimode fiber,” J. Lightwave Technol. 28(6), 861–869 (2010).
    [CrossRef]
  16. K. Balemarthy, A. Polley, and S. E. Ralph, “Electronic equalization of multikilometer 10-Gb/s multimode fiber links: mode-coupling effects,” J. Lightwave Technol. 24(12), 4885–4894 (2006).
    [CrossRef]
  17. C. Xia, M. Ajgaonkar, and W. Rosenkranz, “On the performance of the electrical equalization technique in MMF links for 10-gigabit ethernet,” J. Lightwave Technol. 23(6), 2001–2011 (2005).
    [CrossRef]
  18. D. Lenz, B. Rankov, D. Erni, W. Bachtold, and A. Wittneben, "MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides," in International Zurich Seminar on Communications (IZS), (IEEE, 2004)
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  22. P. L. Neo, J. P. Freeman, and T. D. Wilkinson, "Modal Control of a 50μm core diameter Multimode Fiber Using a Spatial Light Modulator," in Optical Fiber Communication and the National Fiber Optic Engineers Conference, 2007. OFC/NFOEC 2007. Conference on, (Optical Society of America, 2007), 1-3.
  23. R. A. Panicker, J. M. Kahn, and S. P. Boyd, “Compensation of Multimode Fiber Dispersion Using Adaptive Optics via Convex Optimization,” J. Lightwave Technol. 26(10), 1295–1303 (2008).
    [CrossRef]
  24. R. A. Panicker and J. M. Kahn, “Algorithms for Compensation of Multimode Fiber Dispersion Using Adaptive Optics,” J. Lightwave Technol. 27(24), 5790–5799 (2009).
    [CrossRef]
  25. M. B. Shemirani and J. M. Kahn, “Compensation of Multimode Fiber Dispersion by Optimization of Launched Amplitude, Phase, and Polarization,” J. Lightwave Technol. 28(14), 2084–2095 (2010).
    [CrossRef]
  26. M. B. Shemirani, J. P. Wilde, and J. M. Kahn, “Adaptive Compensation of Multimode Fiber Dispersion by Control of Launched Amplitude, Phase, and Polarization,” J. Lightwave Technol. 28(18), 2627–2639 (2010).
    [CrossRef]
  27. G. Stepniak, L. Maksymiuk, and J. Siuzdak, "Increasing Multimode Fiber Transmission Capacity by Mode Selective Spatial Light Phase Modulation," in 36th European Conference on Optical Communications, 2010)
  28. A. Flatman, "In-Premises Optical Fibre Installed Base Analysis 2007," (LAN Technologies UK, Orlando, 2004).
  29. P. Bell, "Fiber Selectiion Guide for Premises Networks," Bell White Paper (2007).
  30. A. Palmentieri, and E. Verdonik, Thorlabs (personal communication, 2007).
  31. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
    [CrossRef] [PubMed]
  32. A. W. Snyder, and J. D. Love, Optical waveguide theory, Science paperbacks; 190 (Chapman and Hall, London, 1983), pp. viii, 734.
  33. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
    [CrossRef] [PubMed]
  34. G. T. Mase, and G. E. Mase, Continuum mechanics for engineers, 2nd ed. / G. Thomas Mase, George E. Mase. ed. (CRC Press, Boca Raton, Fla.; London, 1999), p. 377 p.
  35. G. B. Arfken, and H. J. Weber, Mathematical methods for physicists, 5th ed. ed. (Harcourt Academic, San Diego, Calif.; London, 2001), pp. xiv, 1112 p.
  36. R. Aris, Vectors, tensors, and the basic equations of fluid mechanics (Dover, 1989), pp. xiv,286.
  37. G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1983).
  38. R. Olshansky, “Effect of the cladding on pulse broadening in graded-index optical waveguides,” Appl. Opt. 16(8), 2171–2174 (1977).
    [CrossRef] [PubMed]
  39. A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
    [CrossRef]

2010 (3)

2009 (1)

2008 (2)

2006 (3)

K. Balemarthy, A. Polley, and S. E. Ralph, “Electronic equalization of multikilometer 10-Gb/s multimode fiber links: mode-coupling effects,” J. Lightwave Technol. 24(12), 4885–4894 (2006).
[CrossRef]

X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18(1), 244–246 (2006).
[CrossRef]

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

2005 (3)

2000 (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[CrossRef] [PubMed]

1998 (1)

1994 (1)

1993 (1)

Z. Haas and M. A. Santoro, “A mode-filtering scheme for improvement of the bandwidth-distance product in multimode fiber systems,” J. Lightwave Technol. 11(7), 1125–1131 (1993).
[CrossRef]

1984 (1)

P. Facq, F. De-Fornel, and F. Jean, “Tunable single-mode excitation in multimode fibres,” Electron. Lett. 20(15), 613–614 (1984).
[CrossRef]

1982 (1)

1977 (1)

1976 (1)

1975 (1)

J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibres,” Opt. Quantum Electron. 7(4), 297–303 (1975).
[CrossRef]

Abouraddy, A. F.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[CrossRef] [PubMed]

Ajgaonkar, M.

Amphawan, A.

Balemarthy, K.

Berdagué, S.

Boyd, S. P.

Capmany, J.

Cunningham, D. G.

De-Fornel, F.

P. Facq, F. De-Fornel, and F. Jean, “Tunable single-mode excitation in multimode fibres,” Electron. Lett. 20(15), 613–614 (1984).
[CrossRef]

Destrez, A.

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

Dubois, F.

Emplit, P.

Facq, P.

P. Facq, F. De-Fornel, and F. Jean, “Tunable single-mode excitation in multimode fibres,” Electron. Lett. 20(15), 613–614 (1984).
[CrossRef]

S. Berdagué and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt. 21(11), 1950–1955 (1982).
[CrossRef] [PubMed]

Fellows, D.

Fink, Y.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[CrossRef] [PubMed]

Gasulla, I.

Gholami, A.

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

Gu, X. J.

X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18(1), 244–246 (2006).
[CrossRef]

Haas, Z.

Z. Haas and M. A. Santoro, “A mode-filtering scheme for improvement of the bandwidth-distance product in multimode fiber systems,” J. Lightwave Technol. 11(7), 1125–1131 (1993).
[CrossRef]

Hugon, O.

Jean, F.

P. Facq, F. De-Fornel, and F. Jean, “Tunable single-mode excitation in multimode fibres,” Electron. Lett. 20(15), 613–614 (1984).
[CrossRef]

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[CrossRef] [PubMed]

Juskaitis, R.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[CrossRef] [PubMed]

Kahn, J. M.

Kumozaki, K.

Midwinter, J. E.

J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibres,” Opt. Quantum Electron. 7(4), 297–303 (1975).
[CrossRef]

Mohammed, W.

X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18(1), 244–246 (2006).
[CrossRef]

Neil, M. A. A.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[CrossRef] [PubMed]

Nowell, M. C.

O'Brien, D.

Olshansky, R.

Panicker, R. A.

Payne, F.

Pellevrault, S.

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

Pez, M.

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

Polley, A.

Quentel, F.

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

Raddatz, L.

Ralph, S. E.

Rosenkranz, W.

Sakurai, N.

Santoro, M. A.

Z. Haas and M. A. Santoro, “A mode-filtering scheme for improvement of the bandwidth-distance product in multimode fiber systems,” J. Lightwave Technol. 11(7), 1125–1131 (1993).
[CrossRef]

Shah, N.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[CrossRef] [PubMed]

Shemirani, M. B.

Shimada, T.

Smith, P. W.

X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18(1), 244–246 (2006).
[CrossRef]

Toffano, Z.

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

White, I. H.

Wilde, J. P.

Wilson, T.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[CrossRef] [PubMed]

Xia, C.

Zemon, S.

Appl. Opt. (3)

Electron. Lett. (1)

P. Facq, F. De-Fornel, and F. Jean, “Tunable single-mode excitation in multimode fibres,” Electron. Lett. 20(15), 613–614 (1984).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Gholami, Z. Toffano, A. Destrez, S. Pellevrault, M. Pez, and F. Quentel, “Optimization of VCSEL Spatiotemporal Operation in MMF Links for 10-Gb Ethernet,” IEEE J. Sel. Top. Quantum Electron. 12(4), 767–775 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18(1), 244–246 (2006).
[CrossRef]

J. Lightwave Technol. (10)

T. Shimada, N. Sakurai, and K. Kumozaki, “WDM access system based on shared demultiplexer and MMF links,” J. Lightwave Technol. 23(9), 2621–2628 (2005).
[CrossRef]

Z. Haas and M. A. Santoro, “A mode-filtering scheme for improvement of the bandwidth-distance product in multimode fiber systems,” J. Lightwave Technol. 11(7), 1125–1131 (1993).
[CrossRef]

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Nowell, “An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links,” J. Lightwave Technol. 16(3), 324–331 (1998).
[CrossRef]

A. Amphawan, F. Payne, D. O'Brien, and N. Shah, “Derivation of an analytical expression for the power coupling coefficient for offset launch into multimode fiber,” J. Lightwave Technol. 28(6), 861–869 (2010).
[CrossRef]

K. Balemarthy, A. Polley, and S. E. Ralph, “Electronic equalization of multikilometer 10-Gb/s multimode fiber links: mode-coupling effects,” J. Lightwave Technol. 24(12), 4885–4894 (2006).
[CrossRef]

C. Xia, M. Ajgaonkar, and W. Rosenkranz, “On the performance of the electrical equalization technique in MMF links for 10-gigabit ethernet,” J. Lightwave Technol. 23(6), 2001–2011 (2005).
[CrossRef]

R. A. Panicker, J. M. Kahn, and S. P. Boyd, “Compensation of Multimode Fiber Dispersion Using Adaptive Optics via Convex Optimization,” J. Lightwave Technol. 26(10), 1295–1303 (2008).
[CrossRef]

R. A. Panicker and J. M. Kahn, “Algorithms for Compensation of Multimode Fiber Dispersion Using Adaptive Optics,” J. Lightwave Technol. 27(24), 5790–5799 (2009).
[CrossRef]

M. B. Shemirani and J. M. Kahn, “Compensation of Multimode Fiber Dispersion by Optimization of Launched Amplitude, Phase, and Polarization,” J. Lightwave Technol. 28(14), 2084–2095 (2010).
[CrossRef]

M. B. Shemirani, J. P. Wilde, and J. M. Kahn, “Adaptive Compensation of Multimode Fiber Dispersion by Control of Launched Amplitude, Phase, and Polarization,” J. Lightwave Technol. 28(18), 2627–2639 (2010).
[CrossRef]

J. Microsc. (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibres,” Opt. Quantum Electron. 7(4), 297–303 (1975).
[CrossRef]

Phys. Rev. Lett. (1)

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[CrossRef] [PubMed]

Other (18)

G. T. Mase, and G. E. Mase, Continuum mechanics for engineers, 2nd ed. / G. Thomas Mase, George E. Mase. ed. (CRC Press, Boca Raton, Fla.; London, 1999), p. 377 p.

G. B. Arfken, and H. J. Weber, Mathematical methods for physicists, 5th ed. ed. (Harcourt Academic, San Diego, Calif.; London, 2001), pp. xiv, 1112 p.

R. Aris, Vectors, tensors, and the basic equations of fluid mechanics (Dover, 1989), pp. xiv,286.

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1983).

A. W. Snyder, and J. D. Love, Optical waveguide theory, Science paperbacks; 190 (Chapman and Hall, London, 1983), pp. viii, 734.

G. Stepniak, L. Maksymiuk, and J. Siuzdak, "Increasing Multimode Fiber Transmission Capacity by Mode Selective Spatial Light Phase Modulation," in 36th European Conference on Optical Communications, 2010)

A. Flatman, "In-Premises Optical Fibre Installed Base Analysis 2007," (LAN Technologies UK, Orlando, 2004).

P. Bell, "Fiber Selectiion Guide for Premises Networks," Bell White Paper (2007).

A. Palmentieri, and E. Verdonik, Thorlabs (personal communication, 2007).

E. Alon, V. Stojanovic, J. M. Kahn, S. Boyd, and M. Horowitz, "Equalization of modal dispersion in multimode fiber using spatial light modulators," in GLOBECOM '04. IEEE Global Telecommunications Conference, (IEEE, 2004), 1023- 1029.

P. L. Neo, J. P. Freeman, and T. D. Wilkinson, "Modal Control of a 50μm core diameter Multimode Fiber Using a Spatial Light Modulator," in Optical Fiber Communication and the National Fiber Optic Engineers Conference, 2007. OFC/NFOEC 2007. Conference on, (Optical Society of America, 2007), 1-3.

D. Lenz, B. Rankov, D. Erni, W. Bachtold, and A. Wittneben, "MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides," in International Zurich Seminar on Communications (IZS), (IEEE, 2004)

D. Erni, M. Jungo, and W. Baechtol, "Segmented VCSEL contact geometry for active coupling efficiency enhancement," in Workshop on Compound Semiconductor Devices and Integrated Circuits (WOCSDICE), (Photonics Communication Group, Swiss Federal Institute of Technology (ETHZ), 2003), 67-68.

S. Bois, Next Generation Fibers and Standards (Corning Optical Fiber 2009).

R. J. Shapiro, "The Internet’s Capacity To Handle Fast-Rising Demand for Bandwidth," USIIA White Paper, (2007).

S. Xirasagar, Traffic Management for Emerging (Networks Next Generation Networks , 2010).

Cisco, "Hyperconnectivity and the Approaching Zettabyte Era," Cisco White Paper (2010).

K. McCabe, IEEE Launches Next Generation of High-Rate Ethernet with New IEEE 802.3ba Standard (IEEE, 2010).

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Figures (9)

Fig. 1
Fig. 1

Experimental setup for holographic selective mode excitation.

Fig. 2
Fig. 2

Summary of technique used for generation of modal electric field of an infinite parabolic MMF.

Fig. 3
Fig. 3

The mapping of a complex field onto the required amplitude, adapted from [31].

Fig. 4
Fig. 4

Fourier transform of electric field of LP33 and corresponding binarized SLM hologram.

Fig. 5
Fig. 5

Measured intensity distribution of Fourier plane of L1 for LP41 mode and location of first diffraction order.

Fig. 6
Fig. 6

A set of four measured output intensity distributions for the noninterferometric modal decomposition of selective excitation of LP44 mode at the output of the MMF.

Fig. 7
Fig. 7

Modal decomposition of output field for selective excitation of various modes.

Fig. 8
Fig. 8

Channel transfer functions for selective excitation of modes (a) LP21 and (b) LP32 and corresponding channel transfer functions (c)-(d) at 633nm wavelength.

Fig. 9
Fig. 9

Estimated 3dB channel bandwidths of holographic selective excitation of various modes.

Equations (12)

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

b = F     sin     l ϕ ,
b = F cos     l ϕ .
F = R l L m 1 ( l ) ( V R 2 ) exp ( 0.5 V R 2 ) ,
f ( x , 1 y 1 ) = d ( x , 1 y 1 ) exp [ j ( τ x x + 1 τ y y ) 1 ] ,
g ( x , 1 y 1 ) = a o + 4 π     n = 1 a n     cos { n [ ξ ( x , 1 y 1 ) ​   + τ x x + 1 τ y y 1 ] } ,
G ( x 2 , y 2 ) = M o ( x 2 , y 2 )     +     n = 1 [ M n ( x 2 + n τ x ,     y 2 + n τ y ) + M n * ( n τ x x 2 , n τ y y 2 ) ] ,
c l m     i n = | A c o r e E i n ( x , y )     e l m * ( x , y ) d x d y | 2 / A c o r e | E i n ( x , y ) | 2 d x d y A c o r e | e l m ( x , y ) | 2 d x d y ,
Λ l 1 m 1 l 2 m 2 l 3 m 3 l 4 m 4 2 c l 3 m 3 c * l 4 m 4 = Λ l 1 m 1 l 2 m 2 l 3 m 3 .
Δ p , q = 1 N 2 l 1 m 1 l 2 m 2 l 3 m 3 l 4 m 4 c l 1 m 1 c * l 2 m 2 c l 3 m 3 c * l 4 m 4 Λ l 1 m 1 l 2 m 2 l 3 m 3 l 4 m 4                                         2 N l 1 m 1 l 2 m 2 c l 1 m 1 c * l 2 m 2 Γ l 1 m 1 l 2 m 2 p , q + P p , q ,
Γ l 1 m 1 l 2 m 2 p , q = c o r e I m e p , q [ e x l 1 m 1 ( x , y ) e x l 2 m 2 ( x , y )     + e y l 1 m 1 ( x , y ) e y l 2 m 2 ( x , y ) ]     d A ,
Λ l 1 m 1 l 2 m 2 l 3 m 3 l 4 m 4 = A     c o r e [ e x l 1 m 1 ( x , y ) e x l 2 m 2 ( x , y )     + e y l 1 m 1 ( x , y ) e y l 2 m 2 ( x , y ) ] ×                                                                                             [ e x l 3 m 3 ( x , y ) e x l 4 m 4 ( x , y )     + e y l 3 m 3 ( x , y ) e y l 4 m 4 ( x , y ) ]     d A ,
h ( t ) = q = 1 η q δ ( t t q ) ,

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