Abstract

Systems of cascaded circular apertures may be built to focus electromagnetic waves if the difference (<i>d</i>) between the Fresnel numbers of any two apertures pertaining to two points (<i>A</i> and <i>B</i>) is an even number. With a point source at <i>A</i>, the spherical incident wave will be focused to <i>B</i>. In general, the irradiance at B (principal focus) is roughly proportional to (n±1)<sup>2</sup> where <i>n</i> is the number of apertures; the plus sign is used when the Fresnel numbers are odd integers, and minus when even. A system of a few apertures with <i>d = 0</i> can focus waves over a wide range of frequencies, though the actual irradiance at the principal focus is a function of frequency. A condition to maximize the irradiance at the principal focus has been found. Theoretically such systems have been analyzed by the boundary-diffraction-wave theory generalized by Miyamoto and Wolf. Analytical expressions for the diffraction wave amplitude have been obtained for axial and off-axial points, when the iteration method of Fox and Li is also applicable, the two theoretical results agree very well with each other. Experimental data obtained agree well with the theoretical results.

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  1. M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).
  2. R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).
  3. M. De, R. Tremblay and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).
  4. T. Young, A Course of Lectures an Natural Philosophy and Mechanical Arts (London, 1807).
  5. A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).
  6. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
  7. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).
  8. The subscript n in Un(j) (Q) means the number of diffractions the wave has undergone, while the superscript (j) indicates the last aperture which diffracts the wave and from which the wave propagates to the point Q.
  9. Since the scalar wave theory is the basis of the calculations, the point Q must be far enough from the edge so that the theory is applicable. Also, Q must be off the axis by a distance great enough so that the contribution from the points with stationary phase represents reasonably well the boundary wave.
  10. E. T. Copson, Asymptotic Expansions (Cambridge University Press, Cambridge, 1965).
  11. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).
  12. A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec, 1964).
  13. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).
  14. G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).
  15. R. W. Wood, Phil. Mag. 45, 511 (1898).
  16. J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.
  18. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1958).
  19. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).
  20. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.

Boivin, A.

A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec, 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).

Copson, E. T.

E. T. Copson, Asymptotic Expansions (Cambridge University Press, Cambridge, 1965).

De, M.

M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).

R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).

M. De, R. Tremblay and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).

di Francia, G. Toraldo

G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).

Fox, A. G.

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

Li, T.

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Lit, J. W. Y.

M. De, R. Tremblay and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).

M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).

Marchand, E. W.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).

Miyamoto, K.

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).

Rubinowicz, A.

A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).

Soret, J. L.

J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.

Tremblay, R.

R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).

M. De, R. Tremblay and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).

M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1958).

Wolf, E.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Wood, R. W.

R. W. Wood, Phil. Mag. 45, 511 (1898).

Young, T.

T. Young, A Course of Lectures an Natural Philosophy and Mechanical Arts (London, 1807).

Other

M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).

R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).

M. De, R. Tremblay and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).

T. Young, A Course of Lectures an Natural Philosophy and Mechanical Arts (London, 1807).

A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).

The subscript n in Un(j) (Q) means the number of diffractions the wave has undergone, while the superscript (j) indicates the last aperture which diffracts the wave and from which the wave propagates to the point Q.

Since the scalar wave theory is the basis of the calculations, the point Q must be far enough from the edge so that the theory is applicable. Also, Q must be off the axis by a distance great enough so that the contribution from the points with stationary phase represents reasonably well the boundary wave.

E. T. Copson, Asymptotic Expansions (Cambridge University Press, Cambridge, 1965).

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).

A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec, 1964).

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).

R. W. Wood, Phil. Mag. 45, 511 (1898).

J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1958).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).

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