Abstract

Applications such as optical data storage, optical computing, and optical interconnects require optical systems that manipulate binary-valued images. Such an optical system can be viewed as a two-dimensional array of binary communication channels. This perspective is used to motivate the use of pagewise mutual information as a metric for optical system analysis and design. Fresnel propagation and coherent imaging both are analyzed in terms of mutual-information transmission. An information-based space–bandwidth product is used to analyze the trade-off between the numerical aperture and the number of image pixels in a coherent 4f system. We propose a new merit function to facilitate information-based optical system design. Information maximization and bit-error-rate minimization both are possible with the new radially weighted encircled-energy merit function. We demonstrate the use of this new merit function through a design example and show that the information throughput is increased by 8% and the bit-error rate is reduced by 36% when compared with systems designed with traditional criteria.

© 2000 Optical Society of America

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F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. ur Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

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[CrossRef]

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X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Select. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

M. A. Neifeld, “Information, resolution, and space–bandwidth product,” Opt. Lett. 23, 1477–1479 (1998).
[CrossRef]

M.-P. Bernal, G. W. Burr, H. Coufal, M. Quintanilla, “Balancing interpixel cross talk and detector noise to optimize areal density in holographic storage systems,” Appl. Opt. 37, 5377–5385 (1998).
[CrossRef]

C. Le Brun, E. Guillard, J. Citerne, “Communication systems interactive software (comsis): modeling of components and its application to the simulation of optical communication systems,” Appl. Opt. 37, 6059–6065 (1998).
[CrossRef]

W.-C. Chou, M. A. Neifeld, “Interleaving and error correction in volume holographic memory systems,” Appl. Opt. 37, 6951–6968 (1998).
[CrossRef]

G. W. Burr, W.-C. Chou, M. A. Neifeld, H. Coufal, J. A. Hoffnagle, C. M. Jefferson, “Experimental evaluation of user capacity in holographic data-storage systems,” Appl. Opt. 37, 5431–5443 (1998).
[CrossRef]

S. P. Levitan, T. P. Kurzweg, P. J. Marchand, M. A. Rempel, D. M. Chiarulli, J. A. Martinez, J. M. Bridgen, C. Fan, F. B. McCormick, “Chatoyant: a computer-aided-design tool for free-space optoelectronic systems,” Appl. Opt. 37, 6078–6092 (1998).
[CrossRef]

G. A. Betzos, M. S. Porter, J. F. Hutton, P. A. Mitkas, “Optical storage interactive simulator (oasis): an interactive tool for the analysis of page-oriented optical memory,” Appl. Opt. 37, 6115–6126 (1998).
[CrossRef]

1997 (2)

M. A. Neifeld, W.-C. Chou, “Information-theoretic limits to the capacity of volume holographic optical memory,” Appl. Opt. 36, 514–517 (1997).
[CrossRef] [PubMed]

H. M. Ozaktas, K.-H. Brenner, A. W. Lohmann, “Interpretation of the space–bandwidth product as the entropy of distinct connection patterns in multifacet optical inter-connection architectures,” J. Opt. Soc. Am. 10, 418–422 (1997).
[CrossRef]

1996 (5)

C. Berrou, A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” IEEE Trans. Commun. 44, 1261–1271 (1996).
[CrossRef]

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 429–445 (1996).
[CrossRef]

Y. Ichioka, T. Iwaki, K. Matsuoka, “Optical information processing and beyond,” Proc. IEEE 84, 694–719 (1996).
[CrossRef]

I. McMichael, W. Christian, D. Pletcher, T. Y. Chang, J. H. Hong, “Compact holographic storage demonstration with rapid access,” Appl. Opt. 35, 2375–2379 (1996).
[CrossRef] [PubMed]

A. Pu, D. Psaltis, “High-density recording in photopolymer-based holographic three-dimensional disks,” Appl. Opt. 35, 2389–2398 (1996).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (2)

R. Torroba, H. Rabal, B. Ruiz, “An entropy approach to light propagation,” J. Mod. Opt. 39, 1939–1946 (1992).
[CrossRef]

C. W. Stirk, “Bit error rate of optical logic: fan-in, threshold, and contrast,” Appl. Opt. 31, 5632–5641 (1992).
[CrossRef] [PubMed]

1991 (1)

C. L. Fales, “An information theory of image gathering,” Inf. Sci. (New York) 57–58, 245–285 (1991).

1988 (1)

1986 (3)

I. J. Cox, C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3, 1152–1158 (1986).
[CrossRef]

P. R. Haugen, S. Rychnovsky, A. Husain, L. D. Hutcheson, “Optical interconnects for high speed computing,” Opt. Eng. 25, 1076–1085 (1986).
[CrossRef]

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

1985 (1)

1984 (2)

1980 (1)

J. Jahns, “Concepts of optical digital computing—a survey,” Optik 57, 429–449 (1980).

1979 (1)

1975 (1)

D. Gabor, “The transmission of information by coherent light. Parts I–III. Classical theory,” J. Phys. 8, 73–78, 161–163, 253–255 (1975).

1972 (2)

1969 (1)

D. Gabor, “Information processing with coherent light,” Opt. Acta 16, 519–533 (1969).
[CrossRef]

1968 (1)

1962 (1)

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

1961 (2)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

1955 (1)

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

1946 (1)

D. Gabor, “Theory of communication,” Proc. IEEE 93, 429–457 (1946).

Alter-Gartenberg, R.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. ur Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

R. Alter-Gartenberg, S. K. Park, “Information as a quality metric for high-resolution imaging,” in Very High Resolution and Quality Imaging III, V. R. Algazi, A. G. Tescher, eds., Proc. SPIE3308, 16–27 (1998).
[CrossRef]

Beléndez, A.

Bergman, L. A.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Bernal, M.-P.

Berrou, C.

C. Berrou, A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” IEEE Trans. Commun. 44, 1261–1271 (1996).
[CrossRef]

C. Berrou, A. Glavieux, P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: turbo-codes(1),” in IEEE International Conference on Communications, Geneva, Switzerland (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 1064–1070.

Betzos, G. A.

Brenner, K.-H.

H. M. Ozaktas, K.-H. Brenner, A. W. Lohmann, “Interpretation of the space–bandwidth product as the entropy of distinct connection patterns in multifacet optical inter-connection architectures,” J. Opt. Soc. Am. 10, 418–422 (1997).
[CrossRef]

Bridgen, J. M.

Burr, G. W.

Carretero, L.

Casasent, D.

Chang, T. Y.

Chen, X.

X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Select. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

Chiarulli, D. M.

Chou, W.-C.

Christian, W.

Chugg, K. M.

X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Select. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

Citerne, J.

Coufal, H.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley, New York, 1991).
[CrossRef]

Cox, I. J.

Drabik, T. J.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Esener, S. C.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Fales, C. L.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. ur Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

C. L. Fales, “An information theory of image gathering,” Inf. Sci. (New York) 57–58, 245–285 (1991).

F. O. Huck, C. L. Fales, J. A. McCormick, S. K. Park, “Image-gathering system design for information and fidelity,” J. Opt. Soc. Am. A 5, 285–299 (1988).
[CrossRef]

F. O. Huck, C. L. Fales, N. Halyo, R. W. Samms, K. Stacy, “Image gathering and processing: information and fidelity,” J. Opt. Soc. Am. A 2, 1644–1666 (1985).
[CrossRef] [PubMed]

C. L. Fales, F. O. Huck, R. W. Samms, “Imaging system design for improved information capacity,” Appl. Opt. 23, 872–888 (1984).
[CrossRef] [PubMed]

F. O. Huck, C. L. Fales, Z. ur Rahman, Visual Communication: An Information Theory Approach (Kluwer Academic, Boston, Mass., 1997).

Fan, C.

Feldman, M.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Fimia, A.

Frieden, B. R.

Gabor, D.

D. Gabor, “The transmission of information by coherent light. Parts I–III. Classical theory,” J. Phys. 8, 73–78, 161–163, 253–255 (1975).

D. Gabor, “Information processing with coherent light,” Opt. Acta 16, 519–533 (1969).
[CrossRef]

D. Gabor, “Theory of communication,” Proc. IEEE 93, 429–457 (1946).

D. Gabor, “Information and light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1961), Vol. 1.
[CrossRef]

Glavieux, A.

C. Berrou, A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” IEEE Trans. Commun. 44, 1261–1271 (1996).
[CrossRef]

C. Berrou, A. Glavieux, P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: turbo-codes(1),” in IEEE International Conference on Communications, Geneva, Switzerland (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 1064–1070.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Guest, C. C.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Guillard, E.

Hagenauer, J.

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 429–445 (1996).
[CrossRef]

Halyo, N.

Haugen, P. R.

P. R. Haugen, S. Rychnovsky, A. Husain, L. D. Hutcheson, “Optical interconnects for high speed computing,” Opt. Eng. 25, 1076–1085 (1986).
[CrossRef]

Hoffnagle, J. A.

Hong, J. H.

Huck, F. O.

Husain, A.

P. R. Haugen, S. Rychnovsky, A. Husain, L. D. Hutcheson, “Optical interconnects for high speed computing,” Opt. Eng. 25, 1076–1085 (1986).
[CrossRef]

Hutcheson, L. D.

P. R. Haugen, S. Rychnovsky, A. Husain, L. D. Hutcheson, “Optical interconnects for high speed computing,” Opt. Eng. 25, 1076–1085 (1986).
[CrossRef]

Hutton, J. F.

Ichioka, Y.

Y. Ichioka, T. Iwaki, K. Matsuoka, “Optical information processing and beyond,” Proc. IEEE 84, 694–719 (1996).
[CrossRef]

Iwaki, T.

Y. Ichioka, T. Iwaki, K. Matsuoka, “Optical information processing and beyond,” Proc. IEEE 84, 694–719 (1996).
[CrossRef]

Jahns, J.

J. Jahns, “Concepts of optical digital computing—a survey,” Optik 57, 429–449 (1980).

Jefferson, C. M.

Johnston, A. R.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Kurzweg, T. P.

Landau, H. J.

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Le Brun, C.

Lee, S. H.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Levitan, S. P.

Linfoot, E. H.

Lohmann, A. W.

H. M. Ozaktas, K.-H. Brenner, A. W. Lohmann, “Interpretation of the space–bandwidth product as the entropy of distinct connection patterns in multifacet optical inter-connection architectures,” J. Opt. Soc. Am. 10, 418–422 (1997).
[CrossRef]

Marchand, P. J.

Martinez, J. A.

Matsuoka, K.

Y. Ichioka, T. Iwaki, K. Matsuoka, “Optical information processing and beyond,” Proc. IEEE 84, 694–719 (1996).
[CrossRef]

McCormick, F. B.

McCormick, J. A.

McMichael, I.

Michael, B. V.

Mitkas, P. A.

Neifeld, M. A.

Nixon, R.

L. A. Bergman, W. H. Wu, A. R. Johnston, R. Nixon, S. C. Esener, C. C. Guest, P. Yu, T. J. Drabik, M. Feldman, S. H. Lee, “Holographic optical interconnects for VLSI,” Opt. Eng. 25, 1109–1118 (1986).
[CrossRef]

Offer, E.

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 429–445 (1996).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, K.-H. Brenner, A. W. Lohmann, “Interpretation of the space–bandwidth product as the entropy of distinct connection patterns in multifacet optical inter-connection architectures,” J. Opt. Soc. Am. 10, 418–422 (1997).
[CrossRef]

Papke, L.

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 429–445 (1996).
[CrossRef]

Park, S. K.

F. O. Huck, C. L. Fales, R. Alter-Gartenberg, S. K. Park, Z. ur Rahman, “Information-theoretic assessment of sampled imaging systems,” Opt. Eng. 38, 742–762 (1999).
[CrossRef]

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[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

An input page with pixel X is transmitted through a blurry optical system. The output pixel Y experiences both interpixel interference and postdetection Gaussian noise.

Fig. 2
Fig. 2

Example conditional PDF’s of the intensity signal at output pixel Y in the presence of IPI and Gaussian noise with the assumption of equal prior signals. The curve labeled Th (threshold) indicates the position of the optimal fixed-threshold decision rule. The shaded area below the curve indicates the BER that is associated with this rule.

Fig. 3
Fig. 3

Fresnel free-space propagation: It is assumed that a unit-amplitude plane wave normally illuminates the input plane. The propagation distance z is normalized with respect to the input pixel pitch s and the wavelength λ.

Fig. 4
Fig. 4

Mutual information, as determined by use of intensity detection, that passes between input pixel X and output pixel Y plotted as a function of the normalized propagation distance z for several FF’s and CR’s. The SNR = 5 at z = 0.

Fig. 5
Fig. 5

Mutual information, as determined by use of phase detection, that passes between input pixel X and output pixel Y for binary [U(α, β, 0) = 0, 1] and bipolar [U(α, β, 0) = -1, + 1] signaling schemes. The SNR = 5 at z = 0.

Fig. 6
Fig. 6

Phase and intensity information, as determined by use of equal prior ternary encoding, that is transmitted between input pixel X and output pixel Y plotted as a function of the propagation distance z. The SNR = 5 at z = 0, and U(α, β, 0) = -1, 0, 1.

Fig. 7
Fig. 7

System setup for a 4f coherent imaging system with a low-pass aperture A at the Fourier plane.

Fig. 8
Fig. 8

I(X; Y) plotted as a function of the normalized defocus position z for three Fourier plane apertures A. The system’s Nyquist aperture (for normalization) is defined as A Nyquist = (2λf)/s, and the SNR = 5 at z = 0.

Fig. 9
Fig. 9

DOF plotted as a function of the normalized aperture size A/ A Nyquist from both the IT and the conventional geometric-optics points of view. DOF = (s/2)/tan(θ) = (A Nyquist/A)(s 2/2λ).

Fig. 10
Fig. 10

Given a fixed input–output page size and optical illumination power constraints, SBPIT is plotted as a function of the number of pixels N 2 on the page. N 2 is normalized with respect to the page area, the system F/#, and the wavelength. The SNR = 5 at N 2 = 1. Inset: The degradation of the page-average pixel mutual information as N increases.

Fig. 11
Fig. 11

Optical system design setup comprising two off-the-shelf doublets to image the input plane onto the output plane with a pixel size w, a pixel pitch s, and a field size H.

Fig. 12
Fig. 12

Spot diagrams along the upper-right diagonal of an output page for the use of (a) the minimum worst rms spot size and (b) the radially weighted encircled-energy merit functions.

Fig. 13
Fig. 13

(a) SBPIT plotted as a function of N for two merit functions and two field sizes. (b) SBPIT plotted as a function of N for H = 7.4 mm and two FF’s.

Fig. 14
Fig. 14

Optimum SBPIT plotted as a function of H for the use of different merit functions given FF = 1.

Fig. 15
Fig. 15

Ĩ(X; Y)〉 and 〈BER〉 plotted as functions of the detection-plane position z.

Fig. 16
Fig. 16

SBPBER=10-6 plotted as a function of H and w for several merit functions.

Fig. 17
Fig. 17

Normalized difference between block and single-pixel mutual information for several aperture sizes and initial SNR’s.

Equations (7)

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

IX; Y=i=0,1 πi  fy|X=ilog2fy|X=ifydy,
Ux, y, z=expikziλz  dαdβUα, β, 0×expi k2zx-α2+y-β21/2,
IX; Y=i=12N2    pxifylog2fy|xifydy,
IX; YI˜X; Y=j=1N2 IXj; Yj,IXj; Yj=xjXj  pxjfyjlog2fyj|xjfyjdyj,
SBPIT=N2ĨX; Y,
IˆmX; Y=j=1N2/m IXj; Yj,
IXj; Yj=xjXj    pxjfyjlogfyj|xjfyjdyj,

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