Abstract

By direct numerical-integration comparisons, it is established that the Fresnel approximation for collimated propagation is quite good (within about 2% in amplitude and 0.02 rad in phase) in every case, including that with the limit of a high Fresnel number. Moreover, the Fresnel approximation begins to break down in phase for spherical-wave propagation for beams faster than about ƒ/12. It has been discovered, however, that if one also invokes the paraxial approximation, that is, replaces the spherical wave by a quadratic phase front, then the Fresnel approximation becomes valid for expanding (or diverging) beams as well. This result is substantiated through the use of stationary-phase arguments.

© 1981 Optical Society of America

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1979 (1)

J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974–980 (1979).

1978 (2)

1976 (1)

1975 (1)

E. A. Sziklas and A. E. Siegman, "Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method," Appl. Opt. 14, 1976 (1975).

1973 (1)

1968 (1)

1962 (1)

Abramowity, M.

M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.

Feiock, F. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 45–60.

Harvey, J. E.

J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974–980 (1979).

Horwitz, P.

Lalor, E.

Leith, E. N.

Siegman, A. E.

E. A. Sziklas and A. E. Siegman, "Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method," Appl. Opt. 14, 1976 (1975).

Southwell, W. H.

Stegun, I. A.

M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.

Sziklas, E. A.

E. A. Sziklas and A. E. Siegman, "Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method," Appl. Opt. 14, 1976 (1975).

Upatnieks, J.

Am. J. Phys. (1)

J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974–980 (1979).

Appl. Opt. (2)

P. Horwitz, "Modes in misaligned unstable resonators," Appl. Opt. 15, 167–178 (1976).

E. A. Sziklas and A. E. Siegman, "Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method," Appl. Opt. 14, 1976 (1975).

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 45–60.

M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.

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