Abstract

The use of rays to construct fields is illustrated by finding the field in the region <i>z</i>>0 when the field is given on the plane <i>z</i>=0. This construction is valid for complex rays as well as real ones. The method is applied to a gaussian field in the plane <i>z</i>=0, in which case a gaussian beam results. The calculation involves only complex rays. Exactly the same results are also obtained by applying the method of stationary phase to an integral representation of the field. However, the ray method is simpler than the stationary-phase method, and it is also applicable to problems for which the stationary-phase method cannot be used because no integral representation of the field is known.

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  2. M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley-Interscience, New York, 1965).
  3. J. B. Keller, in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Math, edited by L. M. Graves (McGraw-Hill, New York, 1958), Vol. VIII, pp. 27โ€“52; J. Opt. Soc. Am. 52, 116 (1962).
  4. In the older literature, the asymptotic character of the field associated with geometrical optics was not appreciated. As a consequence, the splitting of the field into a phase factor and an amplitude factor was made without regard to K. The resulting phase is not appropriate for geometrical optics nor for use in the method of stationary phase. This becomes particularly clear when the field is the sum of two or more expressions of the form of Eq. (1). It is also illustrated by the two examples in the last section.

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley-Interscience, New York, 1965).

Keller, J. B.

J. B. Keller, in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Math, edited by L. M. Graves (McGraw-Hill, New York, 1958), Vol. VIII, pp. 27โ€“52; J. Opt. Soc. Am. 52, 116 (1962).

Kline, M.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley-Interscience, New York, 1965).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Other (4)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley-Interscience, New York, 1965).

J. B. Keller, in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Math, edited by L. M. Graves (McGraw-Hill, New York, 1958), Vol. VIII, pp. 27โ€“52; J. Opt. Soc. Am. 52, 116 (1962).

In the older literature, the asymptotic character of the field associated with geometrical optics was not appreciated. As a consequence, the splitting of the field into a phase factor and an amplitude factor was made without regard to K. The resulting phase is not appropriate for geometrical optics nor for use in the method of stationary phase. This becomes particularly clear when the field is the sum of two or more expressions of the form of Eq. (1). It is also illustrated by the two examples in the last section.

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