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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References

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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).
    [Crossref]
  3. J. Focke, “Wellenoptische Untersuchungen zum Öffnungsfehler,” Opt. Acta 3, 110–126 (1956).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  8. J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  11. A. 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).
    [Crossref]
  12. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. Leipzig 30, 755–776 (1909).
    [Crossref]
  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).
    [Crossref]
  15. B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys. Lett. 33, 511–513 (1978).
    [Crossref]
  16. S. K. Yao and D. E. Thompson, “Chirp-grating lens for guided-wave optics,” Appl. Phys. Lett. 33, 635–637 (1978).
    [Crossref]
  17. P. R. Ashley and W. S. C. Chang, “Fresnel lens in a thin-film waveguide,” Appl. Phys. Lett. 33, 490–492 (1978).
    [Crossref]
  18. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Sec. 8.3.2.
  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).
    [Crossref]
  20. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).
  21. L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves (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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  27. R. Courant and D. Hilbert, Methoden der Mathematische Physik2nd ed. (Springer-Verlag, Berlin, 1968).
    [Crossref]
  28. H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002 (1957).
    [Crossref]
  29. Ref. 18, Sec. 8.8.
  30. A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969).
    [Crossref]
  31. G. C. Sherman, “Generalization of the angular spectrum of plane waves and the diffraction transform,” J. Opt. Soc. Am. 59, 146–156 (1969).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]

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).
[Crossref]

A. 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).
[Crossref]

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

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

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

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).
[Crossref]

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

1973 (1)

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

1957 (1)

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002 (1957).
[Crossref]

1956 (1)

J. Focke, “Wellenoptische Untersuchungen zum Öffnungsfehler,” Opt. Acta 3, 110–126 (1956).
[Crossref]

1951 (1)

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Biswas, S. C.

A. 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).
[Crossref]

Boivin, A.

A. 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).
[Crossref]

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).
[Crossref]

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

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.

A. 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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Courant, R.

R. Courant and D. Hilbert, Methoden der Mathematische Physik2nd ed. (Springer-Verlag, Berlin, 1968).
[Crossref]

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).
[Crossref]

Debye, P.

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

Devaney, A. J.

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).
[Crossref]

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 Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

Focke, J.

J. Focke, “Wellenoptische Untersuchungen zum Öffnungsfehler,” Opt. Acta 3, 110–126 (1956).
[Crossref]

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).
[Crossref]

Hilbert, D.

R. Courant and D. Hilbert, Methoden der Mathematische Physik2nd ed. (Springer-Verlag, Berlin, 1968).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002 (1957).
[Crossref]

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).
[Crossref]

Marcuwitz, N.

L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves (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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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).
[Crossref]

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).
[Crossref]

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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

J. Math. Phys. (N.Y.) (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).
[Crossref]

J. Opt. Soc. Am. (7)

Opt. Acta (2)

J. Focke, “Wellenoptische Untersuchungen zum Öffnungsfehler,” Opt. Acta 3, 110–126 (1956).
[Crossref]

A. 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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Rep. Prog. Phys. (1)

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
[Crossref]

SIAM Rev. (1)

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).
[Crossref]

Other (11)

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

L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves (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 Physik2nd ed. (Springer-Verlag, Berlin, 1968).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

Geometry of the focusing problem. Only plane waves with directions between ka and kb give appreciable contributions to the field in the focal region.

Fig. 2
Fig. 2

Illustration of the difference between our phase-linearization procedure and the one based on a Taylor-series expansion (Hopkins’s procedure).

Fig. 3
Fig. 3

Symmetric geometry of the focusing problem used in the applications in Sections 5 and 6 and in the numerical computations in Section 7.

Fig. 4
Fig. 4

Plot of the quantity ERP:ERNP = sin x · ln[(1 + sin x)/cos x]/x2, where ERP is the ratio between the energy-per-unit length at the focal point and the average energy-per-unit length in the aperture in the case of a perfect incident wave, and ERNP is the same ratio in the case of an incident wave with a perfect phase but a nonperfect amplitude. The variable x along the horizontal axis is the angular aperture (θ0 in Fig. 3).

Fig. 5
Fig. 5

Plots of angular spectra in the case of a focusing geometry with an angular aperture of θ0 = 45° and a focal distance of z0 = 25λ (cf. Fig. 3) to illustrate the accuracy of the asymptotic method (Method 4 of Section 7). In (a), a plot is given of the spectral amplitude resulting from the asymptotic method. The difference in amplitude and phase between the asymptotic spectrum and the spectrum obtained by a FFT algorithm (Method 2 of Section 7) is plotted in (b) and (c), respectively. In (d)–(f), plots are given of the fields on the focal line. These plots are of the same kind as the plots in (a)–(c) for the angular spectrum. The scale on the horizontal axis is the same in (a)–(c) and in (d)–(f).

Fig. 6
Fig. 6

(a)–(e) Variations in the angular spectrum and in the field on the focal line with the spectral-sampling interval Δf. The spectral amplitudes of the asymptotic method are displayed on the left-hand side, together with the resulting field amplitudes on the focal line on the right-hand side, for various values of the spectral-sample interval Δf (and hence of the spatial-sample interval Δx = 1/NΔf). The number of samples is N = 1024, the geometry is the same as in Fig. 5, and the scale on the horizontal axis is the same in (a)–(e) on the left-hand side as well as on the right-hand side. (f) Plots of the field amplitudes on the focal line computed by means of the impulse-response method (Method 1 of Section 7) (solid curve) and by means of a FFT algorithm (Method 4 of Section 7) (dashed curve), showing the influence of aliasing in the spatial domain. The number of samples is N = 1024, and the geometry is the same as in Fig. 5. (g) and (h) Plots of the spectral amplitudes computed by means of a FFT algorithm using sampling intervals Δf = 0.0025/λ in (g) and Δf = 0.005/λ in (h) to demonstrate the influence of aliasing in the spatial-frequency domain. The number of samples is N = 1024, and the geometry is the same as in Fig. 5.

Fig. 7
Fig. 7

Plots of spectral amplitudes obtained using the Debye approximation and the Kirchhoff approximation (Methods 3 and 4 of Section 7) for various values of the angular aperture θ0 and the focal distance z0 (cf. Fig. 3). For each value of θ0, the spectral amplitudes are normalized so that the spectral amplitude in the Debye approximation does not vary with focal distance z0. The scale on the horizontal axis is the same in (a)–(c).

Fig. 8
Fig. 8

Plots of the field amplitudes on the focal line obtained using the Kirchhoff approximation and the Debye approximation (Methods 1 and 5 of Section 7) for various values of the angular aperture θ0 and the focal distance z0. The plots are scaled so that the curves for the field amplitudes in the Debye approximation look identical, irrespective of the values of θ0 and z0.

Fig. 9
Fig. 9

(a) Plots of the phase difference on the focal line between the Kirchhoff approximation and the Debye approximation (Methods 1 and 5 of Section 7) for various values of the angular aperture θ0. The focal distance is z0 = 10λ. (b) Plots of the same phase difference as (a) after subtracting the phase factor, given in Eq. (111), in the Kirchhoff approximation.

Fig. 10
Fig. 10

(a)–(e) Plots of the field amplitudes along the axis obtained using the Kirchhoff approximation (Method 1 of Section 7) (solid line) and the Debye approximation (Method 5 of Section 7) (dashed line) and of the phase difference between the two approximations (dot–dash line) for various values of the angular aperture θ0 at a fixed focal distance z0 = 10λ (cf. Fig. 3). The scale on the horizontal axis is the same in (a)–(e). In (d) and (e) the two amplitudes are the same within the accuracy of the plots. (f) and (g) Plots of the same quantities as in (a)–(e) at a fixed angular aperture (θ0 = 45°) for focal distances z0 = 10λ in (f) and z0 = 50λ in (g).

Fig. 11
Fig. 11

Plots of the field amplitudes on the focal line in the case of a perfect incident wave and in the case of an incident wave with perfect phase but nonperfect amplitude at a fixed focal distance z0 = 10λ for various angular apertures θ0. The field amplitudes, are normalized so that they all have the same value at the focal point.

Tables (1)

Tables Icon

Table 1 Values of the Parameters Δf, and Consequently of fmax and Δx, Used in the Computation of the Curves Presented in Figs. 6(a)–6(e).

Equations (122)

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

u ( x , z , t ) = 1 2 π 2 Re 0 ũ ( x , z , ω ) e - i ω t d ω ,
ũ ( x , z , ω ) = - u ( x , z , t ) e i ω t d ω .
u ( x , z ) = 1 2 π - U ( k x ) exp { i [ k x x + ( k 2 - k x 2 ) 1 / 2 z ] } d k x ,
U ( k x ) = - u ( x , 0 ) exp ( - i k x x ) d x .
( 2 x 2 + 2 z 2 + k 2 ) u ( x , z ) = 0 ,
u ( x , 0 ) = { u 0 ( x ) exp ( i k ϕ 0 ( x ) - ( - i k r 0 ) for a < x < b 0 for x < a or x > b ,
r 0 = [ ( x - x 0 ) 2 + z 0 2 ] 1 / 2
U ( α k ) = z 0 [ I * ( Ω , t a ) - I * ( Ω , t b ) ] ,
I ( Ω , z a ) = z a f ( t ) e i Ω q ( t ) d t ,
t a = a / z 0 ,             t b = b / z 0 ,             Ω = k z 0 ,
f ( t ) = u 0 ( z 0 t ) ,
q ( t ) = α t + [ 1 + ( t - t 0 ) 2 ] 1 / 2 - ϕ 0 ( z 0 t ) / z 0 ,             t 0 = x 0 / z 0 .
I S ( Ω , z a ) = ( π Ω q 2 ) 1 / 2 f ( t s ) exp [ i ( Ω q ( t s ) + β π / 4 ) ] [ 1 + O ( 1 / Ω 1 / 2 ) ] ,
q 2 = 1 2 q ( t s ) ; β = sgn ( q 2 ) .
I E ( Ω , z a ) = i f ( z a ) Ω q ( z a ) exp [ i Ω q ( z a ) ] [ 1 + O ( 1 / Ω ) ] .
I U ( Ω , z a ) ~ exp [ i ( Ω q ( t s ) + β π / 4 ) ] × { f ( t s ) Q ( Ω q 2 ) 1 / 2 + P 2 Ω exp [ i β ( Ω s a 2 + π / 4 ) ] } ,
s a = q ( z s ) - q ( z a ) 1 / 2 ,
P = - δ 2 f ( z a ) s a - f ( t s ) q ( z a ) / q 2 1 / 2 s a q ( z a ) ,
Q = ½ π { 1 - C ( ξ ) - S ( ξ ) + i β [ C ( ξ ) - S ( ξ ) ] } ,
ξ = - δ s a Ω ;             δ = sgn ( t s - z a ) .
C ( ξ ) = 2 2 π 0 ξ cos ( t 2 ) d t ,
S ( ξ ) = 2 2 π 0 ξ sin ( t 2 ) d t .
P = 1 q 2 [ C 1 + ( C 2 - 2 C 1 q 3 q 2 ) γ ] + O ( γ 2 ) ,
γ = t s - z a ,
C 1 = f 1 - f 0 q 3 q 2 ,
C 2 = f 2 + f 0 [ - 3 2 q 4 q 2 - 1 8 ( q 3 q 2 ) 2 ] + 1 2 f 1 q 3 q 2 ,
f n = 1 n ! f ( n ) ( t s ) ; q n = 1 n ! q ( n ) ( t s ) .
I U ( Ω , z a ) ~ I S ( Ω , z a ) H ( t s - t a ) + I E ( Ω , z a ) ,
k a k = sin θ a = x 0 - a [ ( x 0 - a ) 2 + z 0 2 ] 1 / 2 ,             a = a - ;
k b k = sin θ b = x 0 - b [ ( x 0 - b ) 2 + z 0 2 ] 1 / 2 ,             b = b + .
u ( x , z ) = 1 λ sin θ b sin θ a [ U ( α k ) exp ( i k z 1 - α 2 ) ] exp ( i k x α ) d α .
N = 2 W ( z ) λ sin θ a ,
u ( x , z ) = - u ( x , 0 ) h ( x - x , z ) d x ,
h ( x , z ) = 1 2 π - exp [ i ( k 2 - k x 2 ) 1 / 2 z ] exp ( i k x x ) d k x ,
h ( x , z ) = 1 2 π i d d z - exp [ i ( k 2 - k x 2 ) 1 / 2 z ] ( k 2 - k x 2 ) 1 / 2 exp ( i k x x ) d k x = 1 2 i d d z 1 π L exp ( i k r cos θ ) d θ ,
h ( x , z ) = 1 2 i d d z H 0 ( 1 ) ( k r ) = - k z 2 i r H 1 ( 1 ) ( k r ) .
u ( x , z ) = - u ( t , 0 ) - k z 2 i r H 1 ( 1 ) ( k r ) d t ,
r = [ ( x - t ) 2 + z 2 ] 1 / 2 .
u ( x , z ) = a b A ( t ) e i ψ ( t ) d t ,
A ( t ) = u 0 ( t ) 1 ( λ z ) 1 / 2 ( z r ) 3 / 2 [ P 1 2 ( k r ) + Q 1 2 ( k r ) ] 1 / 2 ,
ψ ( t ) = k Δ r - π / 4 + arctan [ Q 1 ( k r ) / P 1 ( k r ) ] + ϕ 0 ( t ) ,
Δ r = z + z 0 r + r 0 [ ( x - x 0 ) ( x + x 0 - 2 t ) / ( z + z 0 ) + z - z 0 ] ,
r = [ ( x - t ) 2 + z 2 ] 1 / 2 ;             r 0 = [ ( x 0 - t ) 2 + z 0 2 ] 1 / 2 .
ψ ( t ) ψ lin ( t ) = a n + b n t .
b n = 1 2 Δ n [ ψ ( t n U ) - ψ ( t n L ) ] ,
a n = 1 2 [ ψ ( t n M ) + ψ ( t n L ) - b n ( t n M + t n L ) ] ,
Δ n = 1 2 ( t n U - t n L ) ,             t n M = 1 2 ( t n U + t n L ) ,
t 0 L = a ,             t n U = b .
u ( x , z ) n = 0 N A ( t n M ) e i a n t n L t n U e i b n t d t ,
u ( x , z ) = 2 n = 0 N A ( t n M ) Δ n sinc ( b n Δ n ) e i ψ lin ( t n M ) .
U K ( α k ) = z 0 [ I * ( Ω , t a ) - I * ( Ω , t b ) ] .
U K ( α k ) = U Geom ( α k ) + U Diff ( α k ) .
U D ( α k ) = z 0 I S * ( Ω , z a ) [ H ( t s - t a ) - H ( t s - t b ) ] ,
U p ( k x ) = exp [ - i ( k 2 - k x 2 ) 1 / 2 z 0 ] ,
u p ( x , z 0 ) = k δ ( k x ) .
u p ( x , 0 ) = 1 2 π - exp [ - i ( k 2 - k x 2 ) 1 / 2 z 0 ] exp ( i k x x ) d k x .
u p ( t , 0 ) = - 1 2 i d d z 0 H 0 ( 2 ) ( k r 0 ) = k z 0 2 i r 0 H 1 ( 2 ) ( k r 0 ) ,
u p 0 ( t ) = 1 λ z 0 ( z 0 r 0 ) 3 / 2 [ P 1 2 ( k r 0 ) + Q 1 2 ( k r 0 ) ] 1 / 2 ,
ϕ p 0 ( t ) = π / 4 - arctan [ Q 1 ( k r 0 ) / P 1 ( k r 0 ) ] .
U ( α k ) = ( z 0 λ ) 1 / 2 e i π / 4 [ I * ( Ω , - t a ) - I * ( Ω , t a ) ] ,
t a = a / z 0 ,             Ω = k z 0 ,
f ( t ) = ( 1 + t 2 ) - 3 / 4 ,
q ( t ) = α t + 1 + t 2 .
q ( t a ) = t a / ( 1 + t a 2 ) 1 / 2 + α ,
t s = - α / ( 1 - α 2 ) 1 / 2 ,
f 0 = ( 1 - α 2 ) 3 / 4 ,
f 1 = ³ / α ( 1 - α 2 ) 5 / 4 ,
f 2 = - ³ / ( 2 - 7 α 2 ) ( 1 - α 2 ) 7 / 4 ,
q 0 = ( 1 - α 2 ) 1 / 2 ,
q 2 = ½ ( 1 - α 2 ) 3 / 2 ,
q 3 = ½ α ( 1 - α 2 ) 2 ,
q 4 = - ( 1 - 5 α 2 ) ( 1 - α 2 ) 5 / 2 .
U ( α k ) = ( z 0 λ ) 1 / 2 e i π / 4 I S * ( Ω , z a ) ,
U ( k x ) = exp [ - i z 0 ( k 2 - k x 2 ) 1 / 2 ] if k x < k , U ( k x ) = 0 if k x k .
u ( x , z ) = - a a A ( t ) e i ψ ( t ) d t ,
A ( t ) = 1 λ 1 z z 0 ( z z 0 r r 0 ) 3 / 2 ,
ψ ( t ) = k z + z 0 r + r 0 [ x ( x - 2 t ) / ( z + z 0 ) + z - z 0 ] .
u ( 0 , z 0 ) = z 0 2 λ - a a d t ( t 2 + z 0 2 ) 3 / 2 = 2 λ sin θ 0 ,
sin θ 0 = a / ( a 2 + z 0 2 ) 1 / 2 .
R = | u ( 0 , z 0 ) u ( 0 , 0 ) | = 2 z 0 λ sin θ 0 .
U D ( k x ) = exp [ - i z 0 ( k 2 - k x 2 ) 1 / 2 ] for t s < t a , U D ( k x ) = 0 for t s > t a .
u D ( x , z ) = 1 2 π × - k sin θ 0 k sin θ 0 exp { i [ k x x + ( k 2 - k x 2 ) 1 / 2 ( z - z 0 ) ] } d k x .
u D ( x , z 0 ) u D ( 0 , 0 ) = 2 z 0 λ sin θ 0 sinc ( k x sin θ 0 ) ,
u D ( 0 , 0 ) = k 2 π - θ 0 θ 0 exp ( - i k z 0 cos α ) cos α d α = k 2 π [ 2 π k z 0 exp ( - i k z 0 + i π / 4 ) + O ( 1 k z 0 ) ] .
u H ( x , z 0 ) u H ( 0 , 0 ) = 2 z 0 λ sinc ( k x ) .
u D ( 0 , z ) = k 2 π - θ 0 θ 0 exp ( i k Δ z cos α ) cos α d α ,
Δ z = z - z 0 .
u D ( 0 , z ) u D ( 0 , 0 ) = 2 z 0 λ [ sin θ 0 + i k Δ z ( θ 0 + sin θ 0 cos θ 0 ) ] + O [ ( k Δ z ) 2 ] .
u D ( 0 , z ) u D ( 0 , 0 ) = z 0 Δ z ( exp { i [ k Δ z - sgn ( Δ z ) π / 4 ] } ( 1 + i 2 k Δ z ) + i 2 cos θ 0 sgn ( Δ z ) 2 π sin θ 0 k Δ z exp ( i k Δ z cos θ 0 ) ) + O [ ( k Δ z ) - 2 ] .
u ( x , 0 ) = | k 2 H 0 ( 2 ) ( k r 0 ) | exp [ - i ( k r 0 - ϕ p 0 ( t ) ) ] ~ k 2 π ( λ r 0 ) 1 / 2 exp [ - i ( k r 0 - π / 4 ) ] ,
U ( α k ) = k 2 π ( z 0 λ ) 1 / 2 e i π / 4 [ I * ( Ω , - t a ) - I * ) ( Ω , t a ) ] ,
f ( t ) = ( 1 + t 2 ) - 1 / 4 ,
f 0 = ( 1 - α 2 ) 1 / 4 ,             f 1 = ½ α f 0 3 ,             f 2 = - ( 2 - 5 α 2 ) f 0 5 ,
A ( t ) = k 2 π z r ( r r 0 ) - 1 / 2 .
u ( 0 , z 0 ) = k z 0 2 π - a a d t t 2 + z 0 2 = k π arctan ( a / z 0 ) = k θ 0 π ,
R = | u ( 0 , z 0 ) u ( 0 , 0 ) | = 2 z 0 λ θ 0 .
E R = u ( 0 , z 0 ) 2 1 2 a - a a u ( x , 0 ) 2 d x .
E R P = 4 a λ sin θ 0 ,
E R N P = 4 z 0 λ θ 0 2 a / z 0 ln { a / z 0 + [ ( a / z 0 ) 2 + 1 ] 1 / 2 } = 4 z 0 λ θ 0 2 tan θ 0 ln ( sin θ 0 + 1 cos θ 0 ) .
E R P E R N P = sin θ 0 θ 0 2 ln ( sin θ 0 + 1 cos θ 0 ) f ( θ 0 ) .
U D ( α k ) = exp [ - i ( k 2 - k x 2 ) 1 / 2 z 0 ] ( k 2 - k x 2 ) 1 / 2 for k x < k sin θ 0 , U D ( α k ) = 0 for k x sin θ 0 ,
u D ( x , z ) = 1 2 π - k sin θ 0 k sin θ 0 exp [ i ( k 2 - k x 2 ) 1 / 2 ( z - z 0 ) ] ( k 2 - k x 2 ) 1 / 2 e i k x x d k x .
u D ( x , z 0 ) = 1 2 π - θ 0 θ 0 exp ( i k x sin α ) d α ,
u D ( 0 , z 0 ) = θ 0 π .
u D ( 0 , 0 ) = 1 2 π - θ 0 θ 0 exp ( - i k z 0 cos α ) d α = 1 2 π ( λ z 0 ) 1 / 2 exp ( - i k z 0 + i π / 4 ) ,
u D ( x , z 0 ) u D ( 0 , 0 ) = 2 z 0 λ θ 0 { 1 - ( k x 2 ) 2 [ 1 - sinc ( θ 0 ) cos θ 0 ] } + O [ ( k x ) 4 ] .
u D ( x , z 0 ) = 1 2 π - sin θ 0 sin θ 0 e i k x t 1 - t 2 d t ,
u D ( x , z 0 ) u D ( 0 , 0 ) = 2 z 0 λ tan θ 0 sinc ( k x sin θ 0 ) + O [ ( k x ) - 2 ] .
u D ( x , z 0 ) u D ( 0 , 0 ) = π z 0 λ J 0 ( k x ) .
u D ( 0 , z ) = 1 2 π - k sin θ 0 k sin θ 0 exp [ i ( k 2 - k x 2 ) 1 / 2 Δ z ] ( k 2 - k x 2 ) 1 / 2 d k x ;             Δ z = z - z 0 ,
u D ( 0 , z ) = 1 π 0 θ 0 exp ( i k Δ z cos α ) d α .
u D ( 0 , z ) u D ( 0 , 0 ) = 2 z 0 λ θ 0 { 1 - i k Δ z sinc ( θ 0 ) - ( k Δ z 2 ) 2 [ 1 + sinc ( θ 0 ) cos θ 0 ] } + O [ ( k Δ z ) 3 ] ,
u D ( 0 , z ) u D ( 0 , 0 ) = z 0 Δ z { exp [ i k Δ z - i sgn ( Δ z ) π / 4 ] + i ( 2 π ) 1 / 2 sgn ( Δ z ) π sin θ 0 ( k Δ z ) - 1 / 2 exp ( i k Δ z cos θ 0 ) } + O [ ( k z ) - 3 / 2 ] .
P 1 ( x ) = 1 + 0.1171875 / x 2 - 0.1441959 / x 4 ,
Q 1 ( x ) = - 0.375 / x - 0.10253906 / x 3
P 1 + i Q 1 1 + P + i Q ( 1 + P + ½ Q 2 ) e i arctan ( Q ) = 1.00019 e i 0.6 ° .
r - r 0 z - z 0 + x 2 2 z - x 0 2 2 z 0 + t 2 2 ( 1 z - 1 z 0 ) - 1 z x t + 1 z 0 x 0 t ,
u ( x , z 0 ) = 2 u ( 0 , 0 ) z 0 λ tan θ 0 e i ϕ 0 e - i X ( b + a ) / 2 sinc ( X ( b - a ) / 2 ) ,
ϕ 0 = k ( x 2 - x 0 2 ) / 2 z 0 ,
X = k ( x - x 0 ) / z 0 .
u ( x , z 0 ) = 2 u ( 0 , 0 ) 2 z 0 λ tan θ 0 e i k x 2 / 2 z 0 sinc ( k x tan θ 0 ) .
arg u ( x , z 0 ) = k cos 3 θ 0 [ x 2 + z 0 2 ) 1 / 2 - z 0 ]