Abstract

A theoretical method is described for reducing the diameter of the central bright disk in the diffraction pattern produced by an objective from an unresolved pinhole as object. On the basis of the Airy criterion the limit of resolution is therefore improved. The method offers a systematic approach for discovering how to coat the exit pupil of the objective with light absorbing and refracting materials so as to reduce the diameter of the central bright disk and yet obtain favorable energy distribution in the diffraction rings. The method selects that family of diffraction patterns whose amplitude and phase distribution can be expressed as a series of Sonine Diffraction Integrals. The coefficients of this series automatically determine both the characteristics of the diffraction pattern and of the corresponding coating which must be applied to the exit pupil. The numerical results obtained for a simple series involving Bessel Functions of the first and second orders are included. In this case the radius of the bright disk is 77 percent of that of the usual Airy disk when the energy density at the diffraction head is 21 percent of that of the classical Airy disk. The radius of the central bright disk can be reduced still further at the expense of serious reduction of the energy density at the diffraction head and a corresponding increase in the energy content of the diffraction rings.

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  1. R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), pp. 391–395.
  2. G. Lansraux, Revue D'Optique 26, 24–45 (1947).
  3. R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 386, Eq. (50.11).
  4. R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 51, Eq. (12.28).
  5. R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), pp. 373 and 375, Eqs. (47.58) and (48.33). When the wave front is spherical or very nearly spherical and when the index of the image space is unity (n12| LN - M2|) = (1 - sin2ϑ)=(1 - ρ2) Luneberg's quantity ø exp[ikW] is proportional to A(ρ) of this paper. The factors of proportionality are constants which may be set equal to unity without essential loss of generality.
  6. G. N. Watson, Bessel Functions (Cambridge University Press, London, 1945), p. 373.
  7. G. N. Watson, Bessel Functions (Cambridge University Press, London, 1945), p. 508.
  8. In fact, if one were willing to introduce Stieltjes integrals, one could permit ν to be a continuous variable.
  9. R. Courant and D. Hilbert, Methoden der Mathematisclien Physik (Verlag. Julius Springer, Berlin, 1931), Vol. I, p. 293.

Courant, R.

R. Courant and D. Hilbert, Methoden der Mathematisclien Physik (Verlag. Julius Springer, Berlin, 1931), Vol. I, p. 293.

Hilbert, D.

R. Courant and D. Hilbert, Methoden der Mathematisclien Physik (Verlag. Julius Springer, Berlin, 1931), Vol. I, p. 293.

Lansraux, G.

G. Lansraux, Revue D'Optique 26, 24–45 (1947).

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 386, Eq. (50.11).

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), pp. 391–395.

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 51, Eq. (12.28).

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), pp. 373 and 375, Eqs. (47.58) and (48.33). When the wave front is spherical or very nearly spherical and when the index of the image space is unity (n12| LN - M2|) = (1 - sin2ϑ)=(1 - ρ2) Luneberg's quantity ø exp[ikW] is proportional to A(ρ) of this paper. The factors of proportionality are constants which may be set equal to unity without essential loss of generality.

Watson, G. N.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1945), p. 373.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1945), p. 508.

Other (9)

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), pp. 391–395.

G. Lansraux, Revue D'Optique 26, 24–45 (1947).

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 386, Eq. (50.11).

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 51, Eq. (12.28).

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), pp. 373 and 375, Eqs. (47.58) and (48.33). When the wave front is spherical or very nearly spherical and when the index of the image space is unity (n12| LN - M2|) = (1 - sin2ϑ)=(1 - ρ2) Luneberg's quantity ø exp[ikW] is proportional to A(ρ) of this paper. The factors of proportionality are constants which may be set equal to unity without essential loss of generality.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1945), p. 373.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1945), p. 508.

In fact, if one were willing to introduce Stieltjes integrals, one could permit ν to be a continuous variable.

R. Courant and D. Hilbert, Methoden der Mathematisclien Physik (Verlag. Julius Springer, Berlin, 1931), Vol. I, p. 293.

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