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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 57, Iss. 7 — Jul. 1, 1967
  • pp: 932–939

Optical Systems with Resolving Powers Exceeding the Classical Limit. II

W. LUKOSZ  »View Author Affiliations

JOSA, Vol. 57, Issue 7, pp. 932-939 (1967)

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The fundamental invariant of an optical system is the number N of degrees of freedom of the message it can transmit. The spatial bandwidth of the system can be increased over the classical limit by reducing one of the other constituent factors of N. As examples of this invariance theorem N= const. established in Part I of this series [J. Opt. Soc. Am. 56, 1463 (1966)], we discuss (a) a system whose spatial-bandwidth increase is achieved by a proportional reduction of its temporal bandwidth, and (b) the airborne synthetic-aperture, terrain-mapping radar, whose spatial resolution comes from exploitation of the temporal degrees of freedom of the received signal. The increase of the spatial bandwidth beyond the classical limit is, however, limited by the appearance of evanescent waves.

The number of degrees of freedom of the object wave field stored in a hologram is discussed. The storage capacity of the photographic plate, which is proportional to its size times its spatial cutoff frequency, is fully exploited only by single-sideband Fraunhofer but not by single-sideband Fresnel holograms.

W. LUKOSZ, "Optical Systems with Resolving Powers Exceeding the Classical Limit. II," J. Opt. Soc. Am. 57, 932-939 (1967)

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  1. W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
  2. In this paper kx, ky denote radian spatial frequencies, kx/2π, ky/2π the corresponding proper spatial frequencies (in lines/mm); ω denotes a radian temporal frequency, and v=ω/2 the corresponding temporal frequency. In Eq. (1.2) both the object field and the aperture of the system are assumed to be rectangular.
  3. A. Bachl and W. Lukosz, J. Opt. Soc. Am. 57, 163 (1967).
  4. W. Lukosz, Z. Naturforsch. 18a, 436 (1963); W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963).
  5. L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 5, 127 (1961).
  6. C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
  7. L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Oplo-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).
  8. E. N. Leith, L. J. Cutrona, and L. J. Porcello, J. Opt. Soc. Am. 56, 1419A (1966).
  9. In the experiments reported in Ref. 4, no coherent background was used. The receiver integrated the intensity in the image plane. In this case the gratings M and M' have to be inserted in or very near to the object and image planes, respectively.
  10. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); 53, 1377 (1963); and 54, 1295 (1964).
  11. In quantum mechanics this is the condition for the validity of the WKB approximation [cf. E. E. Merzbacher, Quantum Mechianics (John Wiley & Sons, New York, 1961), Ch. 7].
  12. If the emitting antenna has the diameter Dx, the width of the illuminated ground field is Lx,=r0λ0/Dx. The resolution is Δx0=Dx/2, according to L. J. Cutrona and G. O. Hall, IRE Trans. Military Electron. 6, 119 (1962).
  13. J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
  14. G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
  15. This assumption involves no loss of generality of our subsequent considerations. The effect of an off-axis angle of the reference beam is an apparent lateral shift of the object and of the light source illuminating the object.
  16. A. Lohmann, Opt. Acta 3, 97 (1956).
  17. G. B. Parrent and G. O. Reynolds, J. Opt. Soc. Am. 56, 1400 (1966).

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