## Abstract

The position of the first minimum or stationary value which appears in the irradiance beyond the end of the geometrical image in the image plane of an objective with a narrow annular aperture when a line element of uniform radiance lies in the object plane is examined for its relation to the length of the line element. The line element is assumed to be self-radiant or its equivalent by scattering. A table of the relation between line length and the position of the first minimum or stationary value is presented for lengths 2*K* in the range 0≦2*K*≦5.728 Airy units. Line elements as short as 0.4 Airy unit appear measurable by this method.

© 1964 Optical Society of America

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### Equations (16)

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(1)
$${H}_{0}({x}_{a})={\mathit{\int}}_{-{\zeta}_{m}}^{{\zeta}_{m}}{{J}_{0}}^{2}(\beta {x}_{a}-t)dt,$$
(2)
$$\begin{array}{ll}{\zeta}_{m}=\beta K,\hfill & \beta =3.8317,\hfill \end{array}$$
(3)
$$2\pi {\rho}_{m}{r}_{a}=\beta ,$$
(4)
$$-2\beta {\mathit{\int}}_{-{\zeta}_{m}}^{{\zeta}_{m}}{J}_{0}(\beta {x}_{a}-t){J}_{1}(\beta {x}_{a}-t)dt=0.$$
(5)
$$\begin{array}{lll}{J}_{0}(b)=\pm {J}_{0}(h);\hfill & b=\beta {x}_{a}-{\zeta}_{m},\hfill & h=\beta {x}_{a}-{\zeta}_{m}.\hfill \end{array}$$
(6)
$$\begin{array}{ll}b={\gamma}_{n},\hfill & h={\gamma}_{\nu},\hfill \end{array}$$
(7)
$${x}_{a}=K+{\gamma}_{n}/\beta ,$$
(8)
$$K=({\gamma}_{\nu}-{\gamma}_{n})/(2\beta ).$$
(9)
$${J}_{0}(u+\upsilon )=\text{\u2211}_{\nu =0}^{\infty}{(-1)}^{\nu}{\u220a}_{\nu}{J}_{\nu}(u){J}_{\nu}(\upsilon ),$$
(10)
$${J}_{0}(u-\upsilon )=\text{\u2211}_{\nu =0}^{\infty}{\u220a}_{\nu}{J}_{\nu}(u){J}_{\nu}(\upsilon ),$$
(11)
$$\begin{array}{ll}\beta {x}_{a}-{\zeta}_{m}\hfill & =b\leqq {\gamma}_{1},\hfill \\ \beta {x}_{a}+{\zeta}_{m}\hfill & =h\leqq {\gamma}_{1}+2{\zeta}_{m}.\hfill \end{array}$$
(12)
$${H}_{0}({x}_{a})=-\beta {\mathit{\int}}_{h/\beta}^{b/\beta}{{J}_{0}}^{2}(\beta t)dt,$$
(13)
$$=\beta \left[{\mathit{\int}}_{0}^{2K+b/\beta}{{J}_{0}}^{2}(\beta t)dt-{\mathit{\int}}_{0}^{b/\beta}{{J}_{0}}^{2}(\beta t)dt\right],$$
(14)
$${H}_{0}({x}_{a})/\beta ={\mathit{\int}}_{0}^{2K}{{J}_{0}}^{2}(\beta t)dt-{\mathit{\int}}_{0}^{b/\beta}{{J}_{0}}^{2}(\beta t)dt.$$
(15)
$$K={x}_{a}-{\gamma}_{1}/\beta ={x}_{a}-0.6276.$$
(16)
$$0\leqq \mathrm{\Delta}K<0.005.$$