Abstract

A large crossbar switch, which is a desirable building block for any low-latency interconnect network, is difficult to implement because of many practical problems associated with digital electronics. We propose a new method for implementing a large optoelectronic crossbar interconnect to take advantage of a unique principle of optics. Based on an emerging vertical-cavity surface-emitting laser (VCSEL) technology, a passive angle-multiplexed beam-steering architecture is proposed as a key component of the optoelectronic crossbar. Various optical system parameters are evaluated. Because there is no optical fan-out power loss, the interconnect capacity of the proposed system is determined by the diffraction-limited receiver power cutoff, and therefore interconnection of more than 1000 nodes with a per node bandwidth of 1 GHz is possible with today's technology. A 64-element VCSEL-array-based proof-of-principle optical system for studying the interconnect scalability has been built. Details of the features of the proposed system, its advantages and limitations, demonstration experimental results, and their analyses are presented.

© 1996 Optical Society of America

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  1. J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
  2. R. Barakat, J. Reif, “Lower bounds on the computational efficiency of optical computing systems,” Appl. Opt. 26, 1015–1018 (1987).
  3. N. Davidson, A. A. Friesem, E. Hasman, “On the limits of optical interconnects,” Appl. Opt. 31, 5426–5430 (1992).
  4. S. K. Tewksbury, “Interconnections within microelectronic systems,” in Microelectronic System Interconnections, Performance and Modeling, S. K. Tewksbury, ed. (Institute of Electrical and Electronic Engineers, New York, 1994), pp. 1–50.
  5. S. Kinoshita, K. Iga, “Circular buried heterodtructure (CBH) GaAlAs/GaAs surface emitting lasers” IEEE J. Quantum Electron. 23, 882–888 (1987).
  6. J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).
  7. A. Hartmann, S. Redfield, “Design sketches for optical crossbar switches intended for large-scale parallel processing applications” Opt. Eng. 28, 315–327 (1989).
  8. J. Ghosh, A. Varma, “Reduction of simultaneous-switching noise in large crossbar networks,” IEEE Trans. Circuits Syst. 38, 86–99 (1991).
  9. C. Clos, “A study of non-blocking switching networks” Bell Syst. Tech. J. 32, 406–424 (1953).
  10. V. E. Benes, “Heuristic remarks and mathematical problems regarding the theory of switching systems” Bell Syst. Tech. J. 41, 1201–1247 (1962).
  11. H. S. Stone, “Parallel processing with the perfect shuffle,” IEEE Trans. Comput. C-20, 153–161 (1971).
  12. Y. M. Yeh, T. Y. Feng, “On a class of rearrangeable networks,” IEEE Trans. Comput. 41, 1361–1379 (1992).
  13. A. Himeno, M. Kobayashi, “4 × 4 optical-gate matrix switch” J. Lightwave Technol. LT-3, 230–235 (1985).
  14. A. D. McAulay, “Optical crossbar signal processor,” in Real-Time Signal Processing VIII, K. Bromley, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.564, 131–138 (1985).
  15. A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).
  16. D. O. Harris, “Multichannel acousto-optic crossbar switch,” Appl. Opt. 30, 4245–4256 (1991).
  17. M. Fukui, K. Kitayama, “Design considerations of the optical image crossbar switch,” Appl. Opt. 31, 5542–5547 (1992).
  18. A. Chiou, P. Yeh, “Energy efficiency of optical interconnections using photorefractive holograms,” Appl. Opt. 29, 1111–1117 (1990).
  19. Y. Li, T. Wang, Z. G. Pan, J. Sharony, “Minimum-complexity free-space optical nonblocking networks for multicast interconnect applications” Opt. Lett. 19, 515–517 (1994).
  20. I. Redmond, E. Schenfeld, “Experimental results of a 64 channel, free-space optical interconnection network for massively parallel processing,” presented at The International Conference on Optical Computing, Edinburgh, U.K., 22–25 August 1994.
  21. V. Gupta, E. Schenfeld, “Performance analysis of a synchronous, circuit-switched interconnection cached network,” in Proceedings of the Eighth ACM International Conference on Supercomputing (ICS’94), (Association for Computing Machinery, New York, 1994), pp. 246–255.
  22. H. J. Caulfield, “Parallel N4 weighted optical interconnections,” Appl. Opt. 26, 4039–4040 (1987).
  23. J. W. Goodman, “Fan-in and fan-out with optical interconnects,” Opt. Acta 32, 1489–1492 (1985).
  24. C. W. Stirk, “Bit-error rate of optical logic: fan-in, threshold, and contrast,” Appl. Opt. 31, 5632–5641 (1992).
  25. B. L. Kasper, “Receiver design,” in Optical Fiber Telecommunications II, S. E. Miller, I. P. Kaminow, eds. (Academic, New York, 1988), Chap. 18, pp. 689–722.
  26. Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

1994 (1)

1992 (4)

1991 (2)

J. Ghosh, A. Varma, “Reduction of simultaneous-switching noise in large crossbar networks,” IEEE Trans. Circuits Syst. 38, 86–99 (1991).

D. O. Harris, “Multichannel acousto-optic crossbar switch,” Appl. Opt. 30, 4245–4256 (1991).

1990 (1)

1989 (2)

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

A. Hartmann, S. Redfield, “Design sketches for optical crossbar switches intended for large-scale parallel processing applications” Opt. Eng. 28, 315–327 (1989).

1987 (4)

S. Kinoshita, K. Iga, “Circular buried heterodtructure (CBH) GaAlAs/GaAs surface emitting lasers” IEEE J. Quantum Electron. 23, 882–888 (1987).

R. Barakat, J. Reif, “Lower bounds on the computational efficiency of optical computing systems,” Appl. Opt. 26, 1015–1018 (1987).

A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).

H. J. Caulfield, “Parallel N4 weighted optical interconnections,” Appl. Opt. 26, 4039–4040 (1987).

1985 (2)

J. W. Goodman, “Fan-in and fan-out with optical interconnects,” Opt. Acta 32, 1489–1492 (1985).

A. Himeno, M. Kobayashi, “4 × 4 optical-gate matrix switch” J. Lightwave Technol. LT-3, 230–235 (1985).

1984 (1)

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).

1971 (1)

H. S. Stone, “Parallel processing with the perfect shuffle,” IEEE Trans. Comput. C-20, 153–161 (1971).

1962 (1)

V. E. Benes, “Heuristic remarks and mathematical problems regarding the theory of switching systems” Bell Syst. Tech. J. 41, 1201–1247 (1962).

1953 (1)

C. Clos, “A study of non-blocking switching networks” Bell Syst. Tech. J. 32, 406–424 (1953).

Athale, R. A.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).

Barakat, R.

Benes, V. E.

V. E. Benes, “Heuristic remarks and mathematical problems regarding the theory of switching systems” Bell Syst. Tech. J. 41, 1201–1247 (1962).

Caulfield, H. J.

Chiou, A.

Clos, C.

C. Clos, “A study of non-blocking switching networks” Bell Syst. Tech. J. 32, 406–424 (1953).

Davidson, N.

Feng, T. Y.

Y. M. Yeh, T. Y. Feng, “On a class of rearrangeable networks,” IEEE Trans. Comput. 41, 1361–1379 (1992).

Florez, J. T.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Friesem, A. A.

Fukui, M.

Ghosh, J.

J. Ghosh, A. Varma, “Reduction of simultaneous-switching noise in large crossbar networks,” IEEE Trans. Circuits Syst. 38, 86–99 (1991).

Goodman, J. W.

J. W. Goodman, “Fan-in and fan-out with optical interconnects,” Opt. Acta 32, 1489–1492 (1985).

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).

Gupta, V.

V. Gupta, E. Schenfeld, “Performance analysis of a synchronous, circuit-switched interconnection cached network,” in Proceedings of the Eighth ACM International Conference on Supercomputing (ICS’94), (Association for Computing Machinery, New York, 1994), pp. 246–255.

Harbison, J. P.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Harris, D. O.

Hartmann, A.

A. Hartmann, S. Redfield, “Design sketches for optical crossbar switches intended for large-scale parallel processing applications” Opt. Eng. 28, 315–327 (1989).

Hasman, E.

Himeno, A.

A. Himeno, M. Kobayashi, “4 × 4 optical-gate matrix switch” J. Lightwave Technol. LT-3, 230–235 (1985).

Iga, K.

S. Kinoshita, K. Iga, “Circular buried heterodtructure (CBH) GaAlAs/GaAs surface emitting lasers” IEEE J. Quantum Electron. 23, 882–888 (1987).

Jenkins, B. K.

A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).

Jewell, J. L.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Kasahara, K.

Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

Kasper, B. L.

B. L. Kasper, “Receiver design,” in Optical Fiber Telecommunications II, S. E. Miller, I. P. Kaminow, eds. (Academic, New York, 1988), Chap. 18, pp. 689–722.

Kawai, S.

Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

Kinoshita, S.

S. Kinoshita, K. Iga, “Circular buried heterodtructure (CBH) GaAlAs/GaAs surface emitting lasers” IEEE J. Quantum Electron. 23, 882–888 (1987).

Kitayama, K.

Kobayashi, M.

A. Himeno, M. Kobayashi, “4 × 4 optical-gate matrix switch” J. Lightwave Technol. LT-3, 230–235 (1985).

Kosaka, H.

Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

Kung, S. Y.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).

Lee, Y. H.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Leonberger, F. I.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).

Li, Y.

Y. Li, T. Wang, Z. G. Pan, J. Sharony, “Minimum-complexity free-space optical nonblocking networks for multicast interconnect applications” Opt. Lett. 19, 515–517 (1994).

Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

McAulay, A. D.

A. D. McAulay, “Optical crossbar signal processor,” in Real-Time Signal Processing VIII, K. Bromley, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.564, 131–138 (1985).

McCall, S. L.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Pan, Z. G.

Raghavendra, C. S.

A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).

Redfield, S.

A. Hartmann, S. Redfield, “Design sketches for optical crossbar switches intended for large-scale parallel processing applications” Opt. Eng. 28, 315–327 (1989).

Redmond, I.

I. Redmond, E. Schenfeld, “Experimental results of a 64 channel, free-space optical interconnection network for massively parallel processing,” presented at The International Conference on Optical Computing, Edinburgh, U.K., 22–25 August 1994.

Reif, J.

Sawchuk, A. A.

A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).

Schenfeld, E.

I. Redmond, E. Schenfeld, “Experimental results of a 64 channel, free-space optical interconnection network for massively parallel processing,” presented at The International Conference on Optical Computing, Edinburgh, U.K., 22–25 August 1994.

V. Gupta, E. Schenfeld, “Performance analysis of a synchronous, circuit-switched interconnection cached network,” in Proceedings of the Eighth ACM International Conference on Supercomputing (ICS’94), (Association for Computing Machinery, New York, 1994), pp. 246–255.

Scherer, A.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Sharony, J.

Stirk, C. W.

Stone, H. S.

H. S. Stone, “Parallel processing with the perfect shuffle,” IEEE Trans. Comput. C-20, 153–161 (1971).

Tewksbury, S. K.

S. K. Tewksbury, “Interconnections within microelectronic systems,” in Microelectronic System Interconnections, Performance and Modeling, S. K. Tewksbury, ed. (Institute of Electrical and Electronic Engineers, New York, 1994), pp. 1–50.

Varma, A.

J. Ghosh, A. Varma, “Reduction of simultaneous-switching noise in large crossbar networks,” IEEE Trans. Circuits Syst. 38, 86–99 (1991).

A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).

Walker, S.

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

Wang, T.

Y. Li, T. Wang, Z. G. Pan, J. Sharony, “Minimum-complexity free-space optical nonblocking networks for multicast interconnect applications” Opt. Lett. 19, 515–517 (1994).

Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

Yeh, P.

Yeh, Y. M.

Y. M. Yeh, T. Y. Feng, “On a class of rearrangeable networks,” IEEE Trans. Comput. 41, 1361–1379 (1992).

Appl. Opt. (7)

Bell Syst. Tech. J. (2)

C. Clos, “A study of non-blocking switching networks” Bell Syst. Tech. J. 32, 406–424 (1953).

V. E. Benes, “Heuristic remarks and mathematical problems regarding the theory of switching systems” Bell Syst. Tech. J. 41, 1201–1247 (1962).

Electron. Lett. (1)

J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, J. T. Florez, “Low-threshold vertical-cavity surface emitting microlasers” Electron. Lett. 25, 1123–1124 (1989).

IEEE J. Quantum Electron. (1)

S. Kinoshita, K. Iga, “Circular buried heterodtructure (CBH) GaAlAs/GaAs surface emitting lasers” IEEE J. Quantum Electron. 23, 882–888 (1987).

IEEE Trans. Circuits Syst. (1)

J. Ghosh, A. Varma, “Reduction of simultaneous-switching noise in large crossbar networks,” IEEE Trans. Circuits Syst. 38, 86–99 (1991).

IEEE Trans. Comput. (3)

A. A. Sawchuk, B. K. Jenkins, C. S. Raghavendra, A. Varma, “Optical crossbar networks,” IEEE Trans. Comput. 36, 50–60 (1987).

H. S. Stone, “Parallel processing with the perfect shuffle,” IEEE Trans. Comput. C-20, 153–161 (1971).

Y. M. Yeh, T. Y. Feng, “On a class of rearrangeable networks,” IEEE Trans. Comput. 41, 1361–1379 (1992).

J. Lightwave Technol. (1)

A. Himeno, M. Kobayashi, “4 × 4 optical-gate matrix switch” J. Lightwave Technol. LT-3, 230–235 (1985).

Opt. Acta (1)

J. W. Goodman, “Fan-in and fan-out with optical interconnects,” Opt. Acta 32, 1489–1492 (1985).

Opt. Eng. (1)

A. Hartmann, S. Redfield, “Design sketches for optical crossbar switches intended for large-scale parallel processing applications” Opt. Eng. 28, 315–327 (1989).

Opt. Lett. (1)

Proc. IEEE (1)

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–866 (1984).

Other (6)

I. Redmond, E. Schenfeld, “Experimental results of a 64 channel, free-space optical interconnection network for massively parallel processing,” presented at The International Conference on Optical Computing, Edinburgh, U.K., 22–25 August 1994.

V. Gupta, E. Schenfeld, “Performance analysis of a synchronous, circuit-switched interconnection cached network,” in Proceedings of the Eighth ACM International Conference on Supercomputing (ICS’94), (Association for Computing Machinery, New York, 1994), pp. 246–255.

B. L. Kasper, “Receiver design,” in Optical Fiber Telecommunications II, S. E. Miller, I. P. Kaminow, eds. (Academic, New York, 1988), Chap. 18, pp. 689–722.

Y. Li, H. Kosaka, T. Wang, S. Kawai, K. Kasahara, “Applications of fiber image guides to bit-parallel optical interconnections,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper OThB5.

S. K. Tewksbury, “Interconnections within microelectronic systems,” in Microelectronic System Interconnections, Performance and Modeling, S. K. Tewksbury, ed. (Institute of Electrical and Electronic Engineers, New York, 1994), pp. 1–50.

A. D. McAulay, “Optical crossbar signal processor,” in Real-Time Signal Processing VIII, K. Bromley, W. J. Miceli, eds., Proc. Soc. Photo-Opt. Instrum. Eng.564, 131–138 (1985).

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Figures (19)

Fig. 1
Fig. 1

Connection diagram of a three-stage Clos network that contains three cascaded arrays of crossbar switches.

Fig. 2
Fig. 2

Functional diagrams of (a) a broadcast-and-select-type crossbar based on the use of fan-out and fan-in optics, and (b) a point-to-point active beam-steering-type crossbar. PE's, processing elements; A-O's, acousto-optic Bragg cells.

Fig. 3
Fig. 3

Functional diagram of a VCSEL-array-based point-to-point beam-steering-type crossbar.

Fig. 4
Fig. 4

1 out of N laser pixel selection scheme: a row–column addressing topology.

Fig. 5
Fig. 5

Schematic of the proposed angle-multiplexed beam-steering-type crossbar: (a) a general 3D view, (b) a top or side view of (a) with additional parameters defined. SLM, spatial light modulator.

Fig. 6
Fig. 6

Relations between the speeds of the macrolens and the microlens in a vignetting-free condition for the optical system of Fig. 5.

Fig. 7
Fig. 7

General relations between the dimensions of the microlenses and the macrolenses and the number of interconnected nodes.

Fig. 8
Fig. 8

Variations of the relations in Fig. 7 with respect to the change of other system parameters.

Fig. 9
Fig. 9

Minimum output diffraction spot dimension versus the number of interconnected nodes.

Fig. 10
Fig. 10

Propagating delay versus the number of interconnected nodes and its relation with β.

Fig. 11
Fig. 11

Typical relations between optical receiver sensitivity and the number of interconnected nodes for some fixed BER's.

Fig. 12
Fig. 12

Schematic of an experimental demonstration system for verifying the proposed beam-steering crossbar concept. The shaded area serves an optional monitoring purpose. LD, laser diode.

Fig. 13
Fig. 13

Typical variations of the output optical power versus driving current in a Bandgap Technology 8 × 8 VCSEL array.

Fig. 14
Fig. 14

Beam-collimation measurement results for the VCSEL–microlens module. The two correlation curves correspond to the two intensity profiles of the collimated beam at the two well-separated planes along the beam-propagation direction.

Fig. 15
Fig. 15

Results of macrolens focal-point shifts caused primarily by aberrations. The two curves represent an on-axis and a 2.5° off-axis scan results.

Fig. 16
Fig. 16

Two superimposed CCD images of aberration-limited receiver plane light distributions. A maximum image shift of less than 0.2 mm is observed.

Fig. 17
Fig. 17

(a) Photograph of our experimental arrangement, (b) the recorded eye pattern at 1 GHz during a full-aperture beam scan.

Fig. 18
Fig. 18

Experimentally measured BER versus the number of interconnected nodes for the system of Fig. 12.

Fig. 19
Fig. 19

Schematic O-E backplane for a crossbar interconnection of high-speed electronic components. Electronic boards and free-space optics are situated on opposite sides of the O-E backplanes. Guided-wave components serve as bridging components.

Equations (27)

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A ( x , y ) = g = 0 m / 2 h = 0 m / 2 k = 0 n / 2 l = 0 n / 2 ψ g , h , k , l × δ ( x ± g Δ x ± k p , y ± h Δ y ± l q ) ,
B ( x 1 , y 1 ) = A ( x , y ) × exp {   j 2 π [ x 1 λ f 1 ( x g Δ x ) + y 1 λ f 1 ( y h Δ y ) ] } ×  dxdy  = g = 0 m / 2 h = 0 m / 2 k = 0 n / 2 l = 0 n / 2 ψ g , h , k , l ×  rect ( x 1 ± g Δ x NA f 1 , y 1 ± h Δ y NA f 1 ) ×  exp [ j 2 π ( ± k p x 1 ± l q y 1 λ f 1 ) ] ,
C ( x 2 , y 2 ) = B ( x 1 , y 1 ) × exp [ j 2 π ( x 2 λ f 2 x 1 + y 2 λ f 2 y 1 ) ] d x 1 d y 1 = g = 0 m / 2 h = 0 m / 2 k = 0 n / 2 l = 0 n / 2 ψ g , h , k , l × { sinc  ( ρ x 2 ) sinc ( ρ y 2 ) × exp [ ± j 2 πρ ( g Δ x x 2 + h Δ y y 2 ) ] * δ [ ( x 2 ± k p f 2 f 1 , y 2 ± l q f 2 f 1 ) ] } ,
C ( x 2 , y 2 ) = g = 0 m / 2 h = 0 m / 2 k = 0 n / 2 l = 0 n / 2 ψ g , h , k , l × { sinc [ ρ ( x 2 ± k p f 2 f 1 ) ] × sinc [ ρ ( y 2 ± l q f 2 f 1 ) ] } ×  exp { j 2 πρ [ g Δ x ( x 2 ± k p f 2 f 1 ) + h Δ y ( y 2 ± l q f 2 f 1 ) ] } .
D 1 = n p + 2 NA f 1 .
D 1 = n p ( 1 2 NA F # 1 ) .
F # 1 = α F # 1 C = α 2 NA ,
D 2 = ( n 1 ) D 1 + n p ( d f 1 ) 2 f 1 .
D 2 = 2 F # 1 ( n 1 ) n p [ 2 F # 1 β F # 2 ( 1 2 NAF # 1 ) ] ( 1 2 NA F # 1 ) .
F # 2 F # 2 C = 2 F # 1 β ( 1 2 NA F # 1 ) .
D 2 = ( n 1 ) n p ( 1 2 NA F # 1 ) = ( n 1 ) D 1 .
δ x 2 = λ f 2 NA f 1 = 2 λ F # 2 ( n 1 ) NA [ 2 F # 1 β F # 2 ( 1 2 NA F # 1 ) ] .
T = d + f 1 + f 2 c = 2 F # 1 n p [ 2 F # 1 β F # 2 ( 1 2 NA F # 1 ) ] + 2 ( 1 + β ) F # 1 F # 2 n p ( n 1 ) c ( 1 2 NA F # 1 ) [ 2 F # 1 β F # 2 ( 1 2 NA F # 1 ) ] ,
T = n p [ 2 F # 1 + F # 2 ( n 1 ) ] c ( 1 2 NA F # 1 )
Δ x 2 = 2 p F # 2 ( n 1 ) 2 F # 1 ( 1 2 NA F # 1 ) β F # 1
Δ x 2 = p F # 2 ( n 1 ) F # 1
d ( Δ x 2 ) d p = 2 F # 2 ( n 1 ) 2 F # 1 ( 1 2 NA F # 1 ) β F # 1 ,
d ( Δ x 2 ) d p = F # 2 ( n 1 ) F # 1 ,
Δ x 2 δ x 2 = 2 p NA λ ,
BER = 1 2 [ 1 erf ( Q 2 ) ] ,
I s = η P ¯ q h v ,
I 2 N = 4 k T Γ g m ( 2 π C T ) 1 / 2 I 3 B 3 ,
C T = C g + C s + α d A d ,
P ¯ = 2 π h v q η ( 4 k T Γ I 3 g m ) 1 / 2 Q C T ( B 3 ) 1 / 2 .
P ¯ D 1 Q ( B 3 ) 1 / 2 [ 1 + D 2 ( N 1 ) 2 ] ,
D 1 = 2 π h v ( C g + C s ) q η ( 4 k T Γ I 3 g m ) 1 / 2 ,
D 2 = α d π 4 ( C g + C s ) { 2 λ F # 2 NA [ 2 F # 1 β F # 2 ( 1 2 NA F # 1 ) ] } 2

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