Abstract

This paper concerns a general theory of linear optical data-processing systems. Well-known basic ideas are critically reviewed, fundamental constraints upon the system behavior are investigated, and a new operational notation is proposed. Every system is considered as a bidirectional processor and every two-dimensional signal is equivalently described in the space and frequency domain. Without referring to field-theoretical notation, the constraints of “reciprocity” and “losslessness” are introduced. In rough terms, reciprocity states equality of transmissions A → B and B → A, where A and B are points in the two reference planes, losslessness states equality of signal energies in the two planes. “Symmetry” has to be independently defined in the space and the frequency domain, and “shift invariance” and “spreadlessness” appear to be dual constraints. The important constraint of “time reversibility” is obtained when a system simultaneously satifies reciprocity and losslessness. In the second part, the general theory is applied to lenses, sections of free space, Fourier transformers, and magnifiers. With the aid of a suitable shorthand notation, the main results of Fourier optics can be easily derived.

© 1977 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6.
  2. E. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  3. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 11.
  4. A. W. Lohmann and D. P. Paris, "Space-Variant Image Formation, " J. Opt. Soc. Am. 55, 1007–1013 (1965).
  5. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).
  6. E. L. O'Neill, "Spatial Filtering in Optics," IRE Trans. Inf. Theory IT-2, 56–65 (1956).
  7. A. A. Sawchuk, "Space-Variant System Analysis of Image Motion, " J. Opt. Soc. Am. 63, 1052–1063 (1973).
  8. J. L. Harris, "Image Evaluation and Restoration, "A J. Opt. Soc. Am. 56, 569–575 (1966).
  9. K. G. Birch, "Spatial filtering in optical data-processing," Rep. Prog. Phys. 35, 1265–1314 (1972).
  10. A. Papoulis, "Dual Optical Systems, " J. Opt. Soc. Am. 58, 653–654 (1968).
  11. H. B. G. Casimir, "Reciprocity Theorems and Irreversible Processes," Proc. IEEE 51, 1570–1573 (1963).
  12. R. P. Williams, "A Convolution and Fouriertransform Approach to Paraxial Optics, " Contemp. Phys. 15, 537–553 (1974).
  13. A. Van der Lugt, "Operational Notation for the Analysis and Synthesis of Optical Data-Processing Systems," Proc. IEEE 54, 1055–1063 (1966).
  14. The time-harmonic field quantities are assumed to vary according to e-ωt. The scalar treatment of optical phenomena implies the neglection of possible interactions between the x and y components of the electromagnetic field vectors inside the optical system.
  15. In mathematical terms ø satisfies the homogeneous Helmholtz equation in this region with a radiation condition at infinity.
  16. If not stated otherwise, all integrations extend from -∞ to +∞. The double sign convention in (1) which is discussed further down, was suggested by M. J. Bastiaans (private communication).
  17. Also "mixed" relations can be formulated where the excitation is described in the space domain and the response in the frequency domain (or vice versa).
  18. To the best of our present knowledge all (finite) optical weighting functions are realizable. This fact strongly contrasts with the situation in time-domain filtering, where the principle of causality puts severe limitations on realizability.
  19. At this stage, we take advantage of the double sign convention in (1). If the same signs had been chosen in (1) for either direction of transmission, Eq. (10a) would be transformed into G21(R2, R1)= G12(-R2, -R1), according to (4). This asymmetry with respect to space and frequency description would prohibit us to develop the fundamental notion of duality in the further course of the paper.
  20. g21* denotes the complex conjugate of g21.
  21. In a more restricted sense, two systems are dual if the associated functions H21(R) and m21(r) have the same mathematical structure (this is the case, for instance, with lens and free space, see Sec. VI).
  22. In this light the Fourier relationship (1) can be viewed as a unitary coordinate transformation.
  23. Here "temporal" frequencies are meant and not "spatial" frequencies. Furthermore, t denotes the time variable.
  24. While a spreadless system is completely specified by the function m (r) and the choice of the coordinate system r(x, y), we need for the specification of a shift-invariant system besides H(R) and the coordinate system an agreement about the signs in (1). Clearly, if the sign convention is altered, H(R) is transformed into H(-R).
  25. In other words, the "powers" (=reciprocal focal distances) have to be added.
  26. In all these cascades the overall system properties are independent of the order of arrangement.
  27. Reciprocity of all systems yields the profit that all proofs need only be given for one direction of transmission.

1974 (1)

R. P. Williams, "A Convolution and Fouriertransform Approach to Paraxial Optics, " Contemp. Phys. 15, 537–553 (1974).

1973 (1)

1972 (1)

K. G. Birch, "Spatial filtering in optical data-processing," Rep. Prog. Phys. 35, 1265–1314 (1972).

1968 (1)

1966 (1)

J. L. Harris, "Image Evaluation and Restoration, "A J. Opt. Soc. Am. 56, 569–575 (1966).

1965 (1)

Harris, J. L.

J. L. Harris, "Image Evaluation and Restoration, "A J. Opt. Soc. Am. 56, 569–575 (1966).

Van der Lugt, A.

A. Van der Lugt, "Operational Notation for the Analysis and Synthesis of Optical Data-Processing Systems," Proc. IEEE 54, 1055–1063 (1966).

Birch, K. G.

K. G. Birch, "Spatial filtering in optical data-processing," Rep. Prog. Phys. 35, 1265–1314 (1972).

Casimir, H. B. G.

H. B. G. Casimir, "Reciprocity Theorems and Irreversible Processes," Proc. IEEE 51, 1570–1573 (1963).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).

Lohmann, A. W.

O’Neill, E.

E. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

O’Neill, E. L.

E. L. O'Neill, "Spatial Filtering in Optics," IRE Trans. Inf. Theory IT-2, 56–65 (1956).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).

Papoulis, A.

A. Papoulis, "Dual Optical Systems, " J. Opt. Soc. Am. 58, 653–654 (1968).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 11.

Paris, D. P.

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).

Sawchuk, A. A.

Williams, R. P.

R. P. Williams, "A Convolution and Fouriertransform Approach to Paraxial Optics, " Contemp. Phys. 15, 537–553 (1974).

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

J. L. Harris, "Image Evaluation and Restoration, "A J. Opt. Soc. Am. 56, 569–575 (1966).

Contemp. Phys. (1)

R. P. Williams, "A Convolution and Fouriertransform Approach to Paraxial Optics, " Contemp. Phys. 15, 537–553 (1974).

J. Opt. Soc. Am. (3)

Rep. Prog. Phys. (1)

K. G. Birch, "Spatial filtering in optical data-processing," Rep. Prog. Phys. 35, 1265–1314 (1972).

Other (21)

H. B. G. Casimir, "Reciprocity Theorems and Irreversible Processes," Proc. IEEE 51, 1570–1573 (1963).

A. Van der Lugt, "Operational Notation for the Analysis and Synthesis of Optical Data-Processing Systems," Proc. IEEE 54, 1055–1063 (1966).

The time-harmonic field quantities are assumed to vary according to e-ωt. The scalar treatment of optical phenomena implies the neglection of possible interactions between the x and y components of the electromagnetic field vectors inside the optical system.

In mathematical terms ø satisfies the homogeneous Helmholtz equation in this region with a radiation condition at infinity.

If not stated otherwise, all integrations extend from -∞ to +∞. The double sign convention in (1) which is discussed further down, was suggested by M. J. Bastiaans (private communication).

Also "mixed" relations can be formulated where the excitation is described in the space domain and the response in the frequency domain (or vice versa).

To the best of our present knowledge all (finite) optical weighting functions are realizable. This fact strongly contrasts with the situation in time-domain filtering, where the principle of causality puts severe limitations on realizability.

At this stage, we take advantage of the double sign convention in (1). If the same signs had been chosen in (1) for either direction of transmission, Eq. (10a) would be transformed into G21(R2, R1)= G12(-R2, -R1), according to (4). This asymmetry with respect to space and frequency description would prohibit us to develop the fundamental notion of duality in the further course of the paper.

g21* denotes the complex conjugate of g21.

In a more restricted sense, two systems are dual if the associated functions H21(R) and m21(r) have the same mathematical structure (this is the case, for instance, with lens and free space, see Sec. VI).

In this light the Fourier relationship (1) can be viewed as a unitary coordinate transformation.

Here "temporal" frequencies are meant and not "spatial" frequencies. Furthermore, t denotes the time variable.

While a spreadless system is completely specified by the function m (r) and the choice of the coordinate system r(x, y), we need for the specification of a shift-invariant system besides H(R) and the coordinate system an agreement about the signs in (1). Clearly, if the sign convention is altered, H(R) is transformed into H(-R).

In other words, the "powers" (=reciprocal focal distances) have to be added.

In all these cascades the overall system properties are independent of the order of arrangement.

Reciprocity of all systems yields the profit that all proofs need only be given for one direction of transmission.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6.

E. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 11.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).

E. L. O'Neill, "Spatial Filtering in Optics," IRE Trans. Inf. Theory IT-2, 56–65 (1956).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.