Abstract

Diffraction of monochromatic light by rough apertures is analyzed. Correlation functions are derived for the electric field and for the intensity in the optical transform plane. The expectations calculated are over an ensemble of edges; their roughness is described in terms of a second-order density and associated characteristic function. It is shown that in-plane roughness causes a speckle pattern. The analytical details are markedly different from the more usual case in which speckle is caused by longitudinal phase delay across an extended aperture. Detailed solutions are presented for serrated gaps and edges. Both space and wavelength dependences are included and solutions for cross correlations of electric field and intensity are obtained. These are valid for arbitrary roughness and correlation coefficient. Experiments are described contrasting the optical transforms of serrated and sharp edges. Good qualitative agreement is obtained with the theory. The serration causes a damping of the major spike in the edge transform and it leads to considerable scattering of the radiation.

© 1980 Optical Society of America

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  1. See topical issue on Speckle for types of problems: J. Opt. Soc. Am., 66, 1145–1312 (1976).
  2. R. A. Shore, B. J. Thompson, and R. W. Whitney, "Diffraction by Apertures Illuminated with Partially Coherent Light," J. Opt. Soc. Am. 56, 733–738 (1966).
  3. E. Wolf and W. H. Carter, "Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources," J. Opt. Soc. Am. 68, 953–964 (1978).
  4. J. Carl Leader, "Far-zone range criteria for quasihomogeneous partially coherent sources," J. Opt. Soc. Am. 68, 1332–1338 (1978).
  5. P. J. Peters, "Aperture shaping—a technique for the control of the spatial distribution of diffracted energy," Proc. SPIE 107, 63–69 (1977).
  6. Yu. M. Polischuk, "Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].
  7. Yu. M. Polischuk, "Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].
  8. Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, "Statistical problems in diffraction theory," Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
  9. A. C. Livanos and Nicholas George, "Edge Diffraction of a Convergent Wave," Appl. Opt. 14, 608–613 (1975).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  11. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  14. W. Heitler, The Quantum Theory of Radiation (Oxford University, London, 1954), p. 69.
  15. G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).
  16. Nicholas George and Atul Jain, "Space and Wavelength Dependence of Speckle Intensity," Appl. Phys. 4, 201–212 (1974).
  17. I. S. Reed, "On a Moment Theorem for Complex Gaussian Processes," IRE Trans. Inf. Theory IT-8, 194–195 (1962).
  18. C. L. Mehta, "Coherence and Statistics of Radiation," in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).
  19. B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

1979 (1)

G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).

1978 (2)

1977 (1)

P. J. Peters, "Aperture shaping—a technique for the control of the spatial distribution of diffracted energy," Proc. SPIE 107, 63–69 (1977).

1975 (2)

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, "Statistical problems in diffraction theory," Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].

A. C. Livanos and Nicholas George, "Edge Diffraction of a Convergent Wave," Appl. Opt. 14, 608–613 (1975).

1974 (2)

Nicholas George and Atul Jain, "Space and Wavelength Dependence of Speckle Intensity," Appl. Phys. 4, 201–212 (1974).

Yu. M. Polischuk, "Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].

1971 (1)

Yu. M. Polischuk, "Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].

1966 (1)

1965 (1)

C. L. Mehta, "Coherence and Statistics of Radiation," in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).

1962 (1)

I. S. Reed, "On a Moment Theorem for Complex Gaussian Processes," IRE Trans. Inf. Theory IT-8, 194–195 (1962).

Carter, W. H.

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.

George, Nicholas

A. C. Livanos and Nicholas George, "Edge Diffraction of a Convergent Wave," Appl. Opt. 14, 608–613 (1975).

Nicholas George and Atul Jain, "Space and Wavelength Dependence of Speckle Intensity," Appl. Phys. 4, 201–212 (1974).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Heitler, W.

W. Heitler, The Quantum Theory of Radiation (Oxford University, London, 1954), p. 69.

Jain, Atul

Nicholas George and Atul Jain, "Space and Wavelength Dependence of Speckle Intensity," Appl. Phys. 4, 201–212 (1974).

Kratsov, Yu. A.

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, "Statistical problems in diffraction theory," Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].

Leader, J. Carl

Livanos, A. C.

Mehta, C. L.

C. L. Mehta, "Coherence and Statistics of Radiation," in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).

Morris, G. M.

G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Peters, P. J.

P. J. Peters, "Aperture shaping—a technique for the control of the spatial distribution of diffracted energy," Proc. SPIE 107, 63–69 (1977).

Polischuk, Yu. M.

Yu. M. Polischuk, "Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].

Yu. M. Polischuk, "Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].

Reed, I. S.

I. S. Reed, "On a Moment Theorem for Complex Gaussian Processes," IRE Trans. Inf. Theory IT-8, 194–195 (1962).

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.

Rytov, C. M.

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, "Statistical problems in diffraction theory," Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

Shore, R. A.

Tatarskii, V. I.

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, "Statistical problems in diffraction theory," Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].

Thompson, B. J.

Whitney, R. W.

Wolf, E.

Appl. Opt. (1)

Appl. Phys. (1)

Nicholas George and Atul Jain, "Space and Wavelength Dependence of Speckle Intensity," Appl. Phys. 4, 201–212 (1974).

IRE Trans. Inf. Theory (1)

I. S. Reed, "On a Moment Theorem for Complex Gaussian Processes," IRE Trans. Inf. Theory IT-8, 194–195 (1962).

J. Opt. Soc. Am. (3)

Radio Eng. Electron. Phys. (1)

Yu. M. Polischuk, "Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].

Radiotekh. Elektron. [Radio Eng. Electron. Phys. USSR] (1)

Yu. M. Polischuk, "Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities," Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].

Usp. Fiz. Nauk (1)

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, "Statistical problems in diffraction theory," Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].

Other (10)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

W. Heitler, The Quantum Theory of Radiation (Oxford University, London, 1954), p. 69.

G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).

C. L. Mehta, "Coherence and Statistics of Radiation," in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

See topical issue on Speckle for types of problems: J. Opt. Soc. Am., 66, 1145–1312 (1976).

P. J. Peters, "Aperture shaping—a technique for the control of the spatial distribution of diffracted energy," Proc. SPIE 107, 63–69 (1977).

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