Abstract

Two different methods are presented for efficient computation of two-dimensional wave fields in focal regions. Both methods are valid for arbitrarily large relative apertures. One method is based on the impulse-response integral and the other on the angular-spectrum representation. The latter method is used to analyze the discrepancy between applying the Kirchhoff or the Debye assumption to obtain an approximation for the field in the aperture. Two cases of idealized incident waves are analyzed in detail. First, we treat the case of a perfect incident wave, i.e., a wave that, in the limit of an infinitely large aperture, would produce a δ-function field distribution on the focal line if account were taken of evanescent waves. Second, the incident wave is taken to be the field radiated by a point source and subsequently focused by a lens that delays the phase of the incoming wave in a perfect manner without influencing its amplitude. The latter wave has the same phase distribution over the aperture as the perfect wave, but a different amplitude distribution.

© 1981 Optical Society of America

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  1. For a review of the literature up to 1951, see Ref. 2.
  2. E. Wolf, "The diffraction theory of aberrations," Rep. Prog. Phys. 14, 95–120 (1951).
  3. J. Focke, "Wellenoptische Untersuchungen zum Öffnungsfehler," Opt. Acta 3, 110–126 (1956).
  4. E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253, 349–357 (1959).
  5. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 358–379 (1959).
  6. A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561–1565 (1965).
  7. A. Boivin, J. Dow, and E. Wolf, "Energy flow in the neighborhood of the focus of a coherent beam," J. Opt. Soc. Am. 57, 1171–1175 (1967).
  8. J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176–178 (1969).
  9. M. A. Gusinov, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quart. Electron. 9, 465–471 (1977).
  10. A. Yoshida and T. Asakura, "Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma," Opt. Commun. 25, 133–136 (1978).
  11. cal mirror under oblique illumination: An integral representation of the fieldA. Boivin, N. Brousseau, and S. C. Biswas, "Electromagnetic diffraction in the focal region of a wide-angle spherical mirror under oblique illumination: An integral representation of the field," Opt. Acta 25, 415–444 (1978).
  12. P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. Leipzig 30, 755–776 (1909).
  13. J. Picht, "Über den Schwingungsvorgang der einem beliebigen (astigmatischen) Strahlenbündel," Ann. Phys. Leipzig 77, 685–782 (1925).
  14. W. H. Southwell, "Index profiles for generalized Luneburg lenses and their use in planar optical waveguides," J. Opt. Soc. Am. 67, 1010–1014 (1977).
  15. B. Chen, E. Marom, and R. J. Morrison, "Diffraction-limited geodesic lens for integrated optics circuits," Appl. Phys. Lett. 33, 511–513 (1978).
  16. S. K. Yao and D. E. Thompson, "Chirp-grating lens for guided-wave optics," Appl. Phys. Lett. 33, 635–637 (1978).
  17. P. R. Ashley and W. S. C. Chang, "Fresnel lens in a thin-film waveguide," Appl. Phys. Lett. 33, 490–492 (1978).
  18. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Sec. 8.3.2.
  19. 19. For an account of angular-spectrum representations of three-dimensional waves, see, for example, A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973).
  20. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).
  21. L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waues (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 4.
  22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1965).
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).
  24. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  25. G. C. Sherman, J. J. Stamnes, and É. Lalor, "Asymtotic approximations to angular-spectrum representations," J. Math. Phys. (N.Y.) 17, 760–776 (1976).
  26. 26. For a three-dimensional example, see, for example, G. C. Sherman, "Application of the convolution theorem to Rayleigh’s integral formulas," J. Opt. Soc. Am. 57, 546–547 (1967).
  27. R. Courant and D. Hilbert, Methoden der Mathematische Physik 2nd ed. (Springer-Verlag, Berlin, 1968).
  28. H. H. Hopkins, "The numerical evaluation of the frequency response of optical systems," Proc. Phys. Soc. B 70, 1002 (1957).
  29. Ref. 18, Sec. 8.8.
  30. A. Walther, "Lenses, wave optics and eikonal functions," J. Opt. Soc. Am. 59, 1325–1333 (1969).
  31. G. C. Sherman, "Generalization of the angular spectrum of plane waves and the diffraction transform," J. Opt. Soc. Am. 59, 146–156 (1969).
  32. Ref. 18, Sec. 9.1.
  33. Ref. 18, Sec. 8.6.2.
  34. J. Gasper, G. C. Sherman, and J. J. Stamnes, "Reflection and refraction of an arbitrary electromagnetic wave at a plane interface," J. Opt. Soc. Am. 66, 955–961 (1976).
  35. J. J. Stamnes and G. C. Sherman, "Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals," J. Opt. Soc. Am. 67, 683–695 (1977).

1978 (5)

A. Yoshida and T. Asakura, "Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma," Opt. Commun. 25, 133–136 (1978).

cal mirror under oblique illumination: An integral representation of the fieldA. Boivin, N. Brousseau, and S. C. Biswas, "Electromagnetic diffraction in the focal region of a wide-angle spherical mirror under oblique illumination: An integral representation of the field," Opt. Acta 25, 415–444 (1978).

B. Chen, E. Marom, and R. J. Morrison, "Diffraction-limited geodesic lens for integrated optics circuits," Appl. Phys. Lett. 33, 511–513 (1978).

S. K. Yao and D. E. Thompson, "Chirp-grating lens for guided-wave optics," Appl. Phys. Lett. 33, 635–637 (1978).

P. R. Ashley and W. S. C. Chang, "Fresnel lens in a thin-film waveguide," Appl. Phys. Lett. 33, 490–492 (1978).

1977 (3)

1976 (2)

J. Gasper, G. C. Sherman, and J. J. Stamnes, "Reflection and refraction of an arbitrary electromagnetic wave at a plane interface," J. Opt. Soc. Am. 66, 955–961 (1976).

G. C. Sherman, J. J. Stamnes, and É. Lalor, "Asymtotic approximations to angular-spectrum representations," J. Math. Phys. (N.Y.) 17, 760–776 (1976).

1973 (1)

19. For an account of angular-spectrum representations of three-dimensional waves, see, for example, A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973).

1969 (3)

1967 (2)

1965 (1)

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561–1565 (1965).

1959 (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253, 349–357 (1959).

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 358–379 (1959).

1957 (1)

H. H. Hopkins, "The numerical evaluation of the frequency response of optical systems," Proc. Phys. Soc. B 70, 1002 (1957).

1956 (1)

J. Focke, "Wellenoptische Untersuchungen zum Öffnungsfehler," Opt. Acta 3, 110–126 (1956).

1951 (1)

E. Wolf, "The diffraction theory of aberrations," Rep. Prog. Phys. 14, 95–120 (1951).

1925 (1)

J. Picht, "Über den Schwingungsvorgang der einem beliebigen (astigmatischen) Strahlenbündel," Ann. Phys. Leipzig 77, 685–782 (1925).

1909 (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. Leipzig 30, 755–776 (1909).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Asakura, T.

A. Yoshida and T. Asakura, "Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma," Opt. Commun. 25, 133–136 (1978).

Ashley, P. R.

P. R. Ashley and W. S. C. Chang, "Fresnel lens in a thin-film waveguide," Appl. Phys. Lett. 33, 490–492 (1978).

Biswas, S. C.

cal mirror under oblique illumination: An integral representation of the fieldA. Boivin, N. Brousseau, and S. C. Biswas, "Electromagnetic diffraction in the focal region of a wide-angle spherical mirror under oblique illumination: An integral representation of the field," Opt. Acta 25, 415–444 (1978).

Boivin, A.

cal mirror under oblique illumination: An integral representation of the fieldA. Boivin, N. Brousseau, and S. C. Biswas, "Electromagnetic diffraction in the focal region of a wide-angle spherical mirror under oblique illumination: An integral representation of the field," Opt. Acta 25, 415–444 (1978).

A. Boivin, J. Dow, and E. Wolf, "Energy flow in the neighborhood of the focus of a coherent beam," J. Opt. Soc. Am. 57, 1171–1175 (1967).

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561–1565 (1965).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Sec. 8.3.2.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Brousseau, N.

cal mirror under oblique illumination: An integral representation of the fieldA. Boivin, N. Brousseau, and S. C. Biswas, "Electromagnetic diffraction in the focal region of a wide-angle spherical mirror under oblique illumination: An integral representation of the field," Opt. Acta 25, 415–444 (1978).

Chang, W. S. C.

P. R. Ashley and W. S. C. Chang, "Fresnel lens in a thin-film waveguide," Appl. Phys. Lett. 33, 490–492 (1978).

Chen, B.

B. Chen, E. Marom, and R. J. Morrison, "Diffraction-limited geodesic lens for integrated optics circuits," Appl. Phys. Lett. 33, 511–513 (1978).

Courant, R.

R. Courant and D. Hilbert, Methoden der Mathematische Physik 2nd ed. (Springer-Verlag, Berlin, 1968).

Dainty, J. C.

J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176–178 (1969).

Debye, P.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. Leipzig 30, 755–776 (1909).

Devaney, A. J.

19. For an account of angular-spectrum representations of three-dimensional waves, see, for example, A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973).

Dow, J.

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).

Felsen, L. B.

L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waues (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

Focke, J.

J. Focke, "Wellenoptische Untersuchungen zum Öffnungsfehler," Opt. Acta 3, 110–126 (1956).

Gasper, J.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1965).

Gusinov, M. A.

M. A. Gusinov, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quart. Electron. 9, 465–471 (1977).

Hilbert, D.

R. Courant and D. Hilbert, Methoden der Mathematische Physik 2nd ed. (Springer-Verlag, Berlin, 1968).

Hopkins, H. H.

H. H. Hopkins, "The numerical evaluation of the frequency response of optical systems," Proc. Phys. Soc. B 70, 1002 (1957).

Lalor, É.

G. C. Sherman, J. J. Stamnes, and É. Lalor, "Asymtotic approximations to angular-spectrum representations," J. Math. Phys. (N.Y.) 17, 760–776 (1976).

Marcuwitz, N.

L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waues (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

Marom, E.

B. Chen, E. Marom, and R. J. Morrison, "Diffraction-limited geodesic lens for integrated optics circuits," Appl. Phys. Lett. 33, 511–513 (1978).

Morrison, R. J.

B. Chen, E. Marom, and R. J. Morrison, "Diffraction-limited geodesic lens for integrated optics circuits," Appl. Phys. Lett. 33, 511–513 (1978).

Palmer, M. A.

M. A. Gusinov, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quart. Electron. 9, 465–471 (1977).

Picht, J.

J. Picht, "Über den Schwingungsvorgang der einem beliebigen (astigmatischen) Strahlenbündel," Ann. Phys. Leipzig 77, 685–782 (1925).

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 358–379 (1959).

Riley, M. E.

M. A. Gusinov, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quart. Electron. 9, 465–471 (1977).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1965).

Sherman, G. C.

Southwell, W. H.

Stamnes, J. J.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Thompson, D. E.

S. K. Yao and D. E. Thompson, "Chirp-grating lens for guided-wave optics," Appl. Phys. Lett. 33, 635–637 (1978).

Walther, A.

Wolf, E.

A. Boivin, J. Dow, and E. Wolf, "Energy flow in the neighborhood of the focus of a coherent beam," J. Opt. Soc. Am. 57, 1171–1175 (1967).

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561–1565 (1965).

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253, 349–357 (1959).

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 358–379 (1959).

E. Wolf, "The diffraction theory of aberrations," Rep. Prog. Phys. 14, 95–120 (1951).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Sec. 8.3.2.

Yao, S. K.

S. K. Yao and D. E. Thompson, "Chirp-grating lens for guided-wave optics," Appl. Phys. Lett. 33, 635–637 (1978).

Yoshida, A.

A. Yoshida and T. Asakura, "Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma," Opt. Commun. 25, 133–136 (1978).

Ann. Phys. Leipzig (2)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. Leipzig 30, 755–776 (1909).

J. Picht, "Über den Schwingungsvorgang der einem beliebigen (astigmatischen) Strahlenbündel," Ann. Phys. Leipzig 77, 685–782 (1925).

Appl. Phys. Lett. (3)

B. Chen, E. Marom, and R. J. Morrison, "Diffraction-limited geodesic lens for integrated optics circuits," Appl. Phys. Lett. 33, 511–513 (1978).

S. K. Yao and D. E. Thompson, "Chirp-grating lens for guided-wave optics," Appl. Phys. Lett. 33, 635–637 (1978).

P. R. Ashley and W. S. C. Chang, "Fresnel lens in a thin-film waveguide," Appl. Phys. Lett. 33, 490–492 (1978).

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, and É. Lalor, "Asymtotic approximations to angular-spectrum representations," J. Math. Phys. (N.Y.) 17, 760–776 (1976).

J. Opt. Soc. Am. (7)

Opt. Acta (2)

cal mirror under oblique illumination: An integral representation of the fieldA. Boivin, N. Brousseau, and S. C. Biswas, "Electromagnetic diffraction in the focal region of a wide-angle spherical mirror under oblique illumination: An integral representation of the field," Opt. Acta 25, 415–444 (1978).

J. Focke, "Wellenoptische Untersuchungen zum Öffnungsfehler," Opt. Acta 3, 110–126 (1956).

Opt. Commun. (2)

J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176–178 (1969).

A. Yoshida and T. Asakura, "Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma," Opt. Commun. 25, 133–136 (1978).

Opt. Quart. Electron. (1)

M. A. Gusinov, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quart. Electron. 9, 465–471 (1977).

Phys. Rev. B (1)

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561–1565 (1965).

Proc. Phys. Soc. B (1)

H. H. Hopkins, "The numerical evaluation of the frequency response of optical systems," Proc. Phys. Soc. B 70, 1002 (1957).

Proc. R. Soc. London Ser. A (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253, 349–357 (1959).

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 358–379 (1959).

Rep. Prog. Phys. (1)

E. Wolf, "The diffraction theory of aberrations," Rep. Prog. Phys. 14, 95–120 (1951).

SIAM Rev. (1)

19. For an account of angular-spectrum representations of three-dimensional waves, see, for example, A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973).

Other (11)

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).

L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waues (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Sec. 8.3.2.

For a review of the literature up to 1951, see Ref. 2.

Ref. 18, Sec. 8.8.

Ref. 18, Sec. 9.1.

Ref. 18, Sec. 8.6.2.

R. Courant and D. Hilbert, Methoden der Mathematische Physik 2nd ed. (Springer-Verlag, Berlin, 1968).

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