Abstract

A modeling and simulation methodology for digital optical computing systems is introduced in this paper. The methodology predicts maximum performance of a given optical computing architecture and evaluates its feasibility. As an application example, we apply this methodology to evaluate the feasibility and performance of the optical content-addressable parallel processor proposed in Appl. Opt. 31, 3241 (1992). The approach consists of two major phases. The first phase involves analytical studies of the effects of design parameters such as cross talk, diffraction-limited beam spot diameter, and pitch on system performance parameters such as signal packing density and skew time. In the second phase, a simulation model and a simulator are introduced by the use of glad (General Laser Analysis and Design, an optical software package developed by Applied Optics Research) to evaluate the combined effects of bit-error rate, bit rate, optical power efficiency, available source power, and signal contrast on the performance parameters such as signal packing density, misalignment tolerance, and distance between devices. The methodology presented here investigates the model, not on a component-by-component basis, but as a whole, which produces a more realistic representation of the actual laboratory prototype. The proposed methodology is intended to reduce the optical computing system design time as well as the design risk associated with building a prototype system.

© 1994 Optical Society of America

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References

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  1. A. Huang, “Architectural considerations involved in the design of an optical digital computer,” Proc. IEEE 72, 780–787 (1984).
    [CrossRef]
  2. A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
    [CrossRef]
  3. J. Taboury, J. M. Wang, P. Chavel, F. Devos, “Optical cellular processor architecture. 2: Illustration and system considerations,” Appl. Opt. 28, 3138–3147 (1989).
    [CrossRef] [PubMed]
  4. W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
    [CrossRef]
  5. P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
    [CrossRef]
  6. M. E. Prise, N. C. Craft, M. M. Downs, R. E. LaMarche, L. A. D’Asaro, L. M. F. Chirovshy, M. J. Murdocca, “Optical digital processor using arrays of symmetric self-electro-optic effect devices,” Appl. Opt. 30, 2287–2296 (1991).
    [CrossRef] [PubMed]
  7. J. R. Barry, “Knowledge-based environment for optical system design,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 346–358 (1990).
  8. J. P. Treptau, T. D. Milster, D. G. Flagello, “Laser beam modeling in optical storage systems,” in Modeling and Simulation of Laser Systems II, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1415, 317–321 (1991).
  9. T. Zhang, R. Hicks, N. S. Tucker, “Simulator models lightwave systems at microwave rates,” Microwaves Radio Freq. 30, 114–119 (1991).
  10. R. K. Kostuk, “Simulation of board-level free-space optical interconnects for electronic processing,” Appl. Opt. 31, 2438–2445 (1992).
    [CrossRef] [PubMed]
  11. B. Dhoedt, P. D. Dobbelaere, L. Buydens, P. Baets, B. Houssay, “Optical free-space board-to-board interconnect: options for optical pathways,” Appl. Opt. 31, 5508–5516 (1992).
    [CrossRef] [PubMed]
  12. F. E. Kiamilev, “Programmable optoelectronic multiprocessors: design, performance and CAD development,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1992).
  13. M. Murdocca, “Computer-aided design of digital optical computers using free-space interconnects,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 126–127 (1990).
  14. A. Louri, “Optical content-addressable parallel processor: architecture, algorithms, and design concepts,” Appl. Opt. 31, 3241–3258 (1992).
    [CrossRef] [PubMed]
  15. G. N. Lawrence, “Optical design with physical optics using glad,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 126–133 (1990).
  16. B. R. Irving, “A technical overview of code V version 7,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 285–293 (1987).
  17. D. C. Sinclair, “Super-Oslo optical design program,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 246–250 (1987).
  18. B. P. Zeigler, Object Oriented Simulation with Hierarchical, Modular Models: Intelligent Agents and Endomorphic Systems (Academic, San Diego, Calif., 1990), Chap. 1.
  19. P. J. Kiviat, R. Villanueva, H. M. Markowitz, The Simscript II Programming Language (Prentice-Hall, Englewood Cliffs, N.J., 1969).
  20. glad Theoretical Description Manual (Applied Optics Research, Tucson, Ariz., 1992).
  21. G. N. Lawrence, “Optical design and optimization with physical optics,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 15–22 (1990).
  22. G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Chap. 5.
  23. D. H. Hartman, “Digital high speed interconnects: a study of the optical alternative,” Opt. Eng. 25, 1086–1102 (1986).
  24. J. Shamir, H. J. Caulfield, R. B. Johnson, “Massive holographic interconnection networks and their limitations,” Appl. Opt. 28, 311–324 (1989).
    [CrossRef] [PubMed]
  25. A. S. Miller, A. A. Sawchuk, “Capabilities of simple lenses in a free space perfect shuffle,” in Optical Enhancements to Computing Technology, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1563, 81–92 (1991).
  26. J. Shamir, “Fundamental limits of optical computing,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 168–172 (1990).
  27. M. Govindarajan, S. R. Forrest, “Optically powered arrays for optoelectronic interconnection networks,” Appl. Opt. 30, 1335–1346 (1991).
    [CrossRef] [PubMed]
  28. T. V. Moui, “Receiver design for high-speed optical-fiber systems,” J. Lightwave Technol. LT-2, 243–267 (1984).
    [CrossRef]
  29. J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
    [CrossRef]

1992 (3)

1991 (3)

1990 (1)

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

1989 (4)

1986 (1)

D. H. Hartman, “Digital high speed interconnects: a study of the optical alternative,” Opt. Eng. 25, 1086–1102 (1986).

1984 (3)

A. Huang, “Architectural considerations involved in the design of an optical digital computer,” Proc. IEEE 72, 780–787 (1984).
[CrossRef]

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

T. V. Moui, “Receiver design for high-speed optical-fiber systems,” J. Lightwave Technol. LT-2, 243–267 (1984).
[CrossRef]

Baets, P.

Barry, J. R.

J. R. Barry, “Knowledge-based environment for optical system design,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 346–358 (1990).

Berra, P. B.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
[CrossRef]

Buydens, L.

Cathey, W. T.

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

Caulfield, H. J.

Chavel, P.

Chirovshy, L. M. F.

Craft, N. C.

D’Asaro, L. A.

Devos, F.

Dhoedt, B.

Dobbelaere, P. D.

Downs, M. M.

Flagello, D. G.

J. P. Treptau, T. D. Milster, D. G. Flagello, “Laser beam modeling in optical storage systems,” in Modeling and Simulation of Laser Systems II, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1415, 317–321 (1991).

Florez, L. T.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Forrest, S. R.

Ghafoor, A.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
[CrossRef]

Govindarajan, M.

Guizani, M.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
[CrossRef]

Harbison, J. P.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Hartman, D. H.

D. H. Hartman, “Digital high speed interconnects: a study of the optical alternative,” Opt. Eng. 25, 1086–1102 (1986).

Hicks, R.

T. Zhang, R. Hicks, N. S. Tucker, “Simulator models lightwave systems at microwave rates,” Microwaves Radio Freq. 30, 114–119 (1991).

Houssay, B.

Huang, A.

A. Huang, “Architectural considerations involved in the design of an optical digital computer,” Proc. IEEE 72, 780–787 (1984).
[CrossRef]

Irving, B. R.

B. R. Irving, “A technical overview of code V version 7,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 285–293 (1987).

Jewell, J. L.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Johnson, R. B.

Keiser, G.

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Chap. 5.

Kiamilev, F. E.

F. E. Kiamilev, “Programmable optoelectronic multiprocessors: design, performance and CAD development,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1992).

Kiviat, P. J.

P. J. Kiviat, R. Villanueva, H. M. Markowitz, The Simscript II Programming Language (Prentice-Hall, Englewood Cliffs, N.J., 1969).

Kostuk, R. K.

LaMarche, R. E.

Lawrence, G. N.

G. N. Lawrence, “Optical design with physical optics using glad,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 126–133 (1990).

G. N. Lawrence, “Optical design and optimization with physical optics,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 15–22 (1990).

Lee, Y. H.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Louri, A.

Marcinkowski, S. J.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
[CrossRef]

Markowitz, H. M.

P. J. Kiviat, R. Villanueva, H. M. Markowitz, The Simscript II Programming Language (Prentice-Hall, Englewood Cliffs, N.J., 1969).

McCall, S. L.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Miceli, W. J.

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

Miller, A. S.

A. S. Miller, A. A. Sawchuk, “Capabilities of simple lenses in a free space perfect shuffle,” in Optical Enhancements to Computing Technology, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1563, 81–92 (1991).

Milster, T. D.

J. P. Treptau, T. D. Milster, D. G. Flagello, “Laser beam modeling in optical storage systems,” in Modeling and Simulation of Laser Systems II, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1415, 317–321 (1991).

Mitkas, P. A.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
[CrossRef]

Moui, T. V.

T. V. Moui, “Receiver design for high-speed optical-fiber systems,” J. Lightwave Technol. LT-2, 243–267 (1984).
[CrossRef]

Murdocca, M.

M. Murdocca, “Computer-aided design of digital optical computers using free-space interconnects,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 126–127 (1990).

Murdocca, M. J.

Olsson, N. A.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Prise, M. E.

Sawchuk, A. A.

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

A. S. Miller, A. A. Sawchuk, “Capabilities of simple lenses in a free space perfect shuffle,” in Optical Enhancements to Computing Technology, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1563, 81–92 (1991).

Scherer, A.

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Shamir, J.

J. Shamir, H. J. Caulfield, R. B. Johnson, “Massive holographic interconnection networks and their limitations,” Appl. Opt. 28, 311–324 (1989).
[CrossRef] [PubMed]

J. Shamir, “Fundamental limits of optical computing,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 168–172 (1990).

Sinclair, D. C.

D. C. Sinclair, “Super-Oslo optical design program,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 246–250 (1987).

Strand, T. C.

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Taboury, J.

Treptau, J. P.

J. P. Treptau, T. D. Milster, D. G. Flagello, “Laser beam modeling in optical storage systems,” in Modeling and Simulation of Laser Systems II, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1415, 317–321 (1991).

Tucker, N. S.

T. Zhang, R. Hicks, N. S. Tucker, “Simulator models lightwave systems at microwave rates,” Microwaves Radio Freq. 30, 114–119 (1991).

Villanueva, R.

P. J. Kiviat, R. Villanueva, H. M. Markowitz, The Simscript II Programming Language (Prentice-Hall, Englewood Cliffs, N.J., 1969).

Wagner, K.

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

Wang, J. M.

Zeigler, B. P.

B. P. Zeigler, Object Oriented Simulation with Hierarchical, Modular Models: Intelligent Agents and Endomorphic Systems (Academic, San Diego, Calif., 1990), Chap. 1.

Zhang, T.

T. Zhang, R. Hicks, N. S. Tucker, “Simulator models lightwave systems at microwave rates,” Microwaves Radio Freq. 30, 114–119 (1991).

Appl. Opt. (7)

J. Lightwave Technol. (1)

T. V. Moui, “Receiver design for high-speed optical-fiber systems,” J. Lightwave Technol. LT-2, 243–267 (1984).
[CrossRef]

Microwaves Radio Freq. (1)

T. Zhang, R. Hicks, N. S. Tucker, “Simulator models lightwave systems at microwave rates,” Microwaves Radio Freq. 30, 114–119 (1991).

Opt. Eng. (2)

D. H. Hartman, “Digital high speed interconnects: a study of the optical alternative,” Opt. Eng. 25, 1086–1102 (1986).

J. L. Jewell, Y. H. Lee, A. Scherer, S. L. McCall, N. A. Olsson, J. P. Harbison, L. T. Florez, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29, 210–214 (1990).
[CrossRef]

Proc. IEEE (4)

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, P. A. Mitkas, “Optics and supercomputing,” Proc. IEEE 77, 1797–1815 (1989).
[CrossRef]

A. Huang, “Architectural considerations involved in the design of an optical digital computer,” Proc. IEEE 72, 780–787 (1984).
[CrossRef]

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Other (14)

F. E. Kiamilev, “Programmable optoelectronic multiprocessors: design, performance and CAD development,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1992).

M. Murdocca, “Computer-aided design of digital optical computers using free-space interconnects,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 126–127 (1990).

G. N. Lawrence, “Optical design with physical optics using glad,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 126–133 (1990).

B. R. Irving, “A technical overview of code V version 7,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 285–293 (1987).

D. C. Sinclair, “Super-Oslo optical design program,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 246–250 (1987).

B. P. Zeigler, Object Oriented Simulation with Hierarchical, Modular Models: Intelligent Agents and Endomorphic Systems (Academic, San Diego, Calif., 1990), Chap. 1.

P. J. Kiviat, R. Villanueva, H. M. Markowitz, The Simscript II Programming Language (Prentice-Hall, Englewood Cliffs, N.J., 1969).

glad Theoretical Description Manual (Applied Optics Research, Tucson, Ariz., 1992).

G. N. Lawrence, “Optical design and optimization with physical optics,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 15–22 (1990).

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Chap. 5.

A. S. Miller, A. A. Sawchuk, “Capabilities of simple lenses in a free space perfect shuffle,” in Optical Enhancements to Computing Technology, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1563, 81–92 (1991).

J. Shamir, “Fundamental limits of optical computing,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 168–172 (1990).

J. R. Barry, “Knowledge-based environment for optical system design,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 346–358 (1990).

J. P. Treptau, T. D. Milster, D. G. Flagello, “Laser beam modeling in optical storage systems,” in Modeling and Simulation of Laser Systems II, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1415, 317–321 (1991).

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Figures (13)

Fig. 1
Fig. 1

Architecture of the OCAPP.

Fig. 2
Fig. 2

Physical laboratory setup for implementing the first version of the OCAPP.

Fig. 3
Fig. 3

Simplified model of the OCAPP.

Fig. 4
Fig. 4

Intensity distribution of the Fresnel diffraction pattern of a square aperture of a SLM. The dimension of apixel is 356 μm × 356 μm.

Fig. 5
Fig. 5

Model of the detector aperture used for estimating the cross talk.

Fig. 6
Fig. 6

Cross talk for various pitches of the SLM array in the diffraction-limited OCAPP. Diffraction-limited beam spot diameter is set to 356 μm.

Fig. 7
Fig. 7

Diffraction-limited estimation of signal packing density of the SLM versus the pitch.

Fig. 8
Fig. 8

Simulation algorithm for the OCAPP.

Fig. 9
Fig. 9

Simulated detected optical signal power versus number of pixels of the optical data plane in the OCAPP.

Fig. 10
Fig. 10

Calculated required optical power P in versus the number of pixels of the optical data plane in the OCAPP.

Fig. 11
Fig. 11

Required optical power versus lateral misalignments applied.

Fig. 12
Fig. 12

Required optical power versus longitudinal misalignments applied.

Fig. 13
Fig. 13

Required power versus distance between SLM’s.

Tables (3)

Tables Icon

Table 1 List of the Parameters Used

Tables Icon

Table 2 Summary of Parameters Studied in the Analysis Phase

Tables Icon

Table 3 Summary of Parameters Studied in the Simulation Phase

Equations (19)

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

d D = 2 λ f a ,
d D = a = 2 λ f .
u 1 ( x , y ) = rect ( x / a , y / a ) = rect ( x / a ) rect ( y / a ) .
u 2 ( y ) = exp ( j k z ) exp ( j k x 2 / 2 z ) j λ z a sinc ( a y λ z ) .
u 2 ( x ) = exp ( j k z ) j λ z - a / 2 a / 2 exp [ j k 2 z ( x 1 - x ) 2 ] d x 1 .
χ = Σ P noise P signal ,
P signal = - d D / 2 d D / 2 - d D / 2 d D / 2 I ( x , y ) d x d y .
Σ P noise 2 P n 1 ,
P n 1 = - d D / 2 d D / 2 p - d D / 2 p + d D / 2 I ( x , y ) d x d y ,
ρ = 1 / p 2 .
T min = L / c .
T max = ( L - f ) + [ ( l / 2 ) 2 + f 2 ] 1 / 2 c .
T skew = T max - T min = [ ( l / 2 ) 2 + f 2 ] 1 / 2 - f c = l 2 c .
T skew = n p 2 c .
L = 2 d + 2 f ,
v = L l 2 = 2 ( d + f ) l 2 = 2 ( d + f ) ( n p ) 2 .
P crit = σ p ρ .
BER = 1 2 π Q exp ( - Q 2 2 ) ,
P in = ( 1 + r ) ( 1 - r ) Q h c λ e i NA 2 1 / 2 N η t ,

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