## Abstract

The physical limits of resolution are discussed for the following basic types of object specimens: periodic structures which exert no focusing action; periodic structures which do exert focusing action upon the incident light; two pin holes or two small particles against opaque or transmitting backgrounds; two narrow slits or object strips against opaque or transmitting backgrounds; and two, sub-microscopic, self-luminous particles. Abbe’s theory remains valid as applied to periodic structures. Generalized resolution formulas are given for periodic structures. Graphical data, based upon recent analysis, show that the resolution of two particles or slits is more complex than indicated by classical theory. The resolving power of the microscope is underestimated by the classical theory. Abnormally high resolutions can occur with properly chosen particles. The class of particles for which the phase microscope has superlative resolving power is described.

© 1950 Optical Society of America

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### Equations (20)

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(1)
$$\text{N}.\text{A}{.}_{\text{objective}}\equiv \mid M\mid \text{sin}{\vartheta}_{m}\equiv \mid M\mid {\rho}_{m}={n}_{0}{\rho}_{\text{om}}.$$
(2)
$$\text{N}.\text{A}{.}_{\text{condenser}}\equiv {n}_{0}\hspace{0.17em}\text{sin}{\vartheta}_{\text{cm}}\equiv {n}_{0}{\rho}_{\text{cm}}.$$
(3)
$$S\equiv \frac{\text{N}.\text{A}{.}_{\text{condenser}}}{\text{N}.\text{A}{.}_{\text{objective}}}={\rho}_{\text{cm}}/{\rho}_{\text{om}}.$$
(4)
$$\frac{4lh}{({\nu}^{2}{h}^{2}+{\mu}^{2}{l}^{2}{)}^{{\scriptstyle \frac{1}{2}}}}\geqq \frac{2}{\text{N}.\text{A}{.}_{\text{objective}}+\text{N}.\text{A}{.}_{\text{condenser}}}\text{wave-lengths}.$$
(5)
$$2l\geqq \frac{\mid \nu \mid}{\text{N}.\text{A}{.}_{\text{objective}}+\text{N}.\text{A}{.}_{\text{condenser}}}\text{wave-lengths}.$$
(6)
$$\frac{4lh}{({\nu}^{2}{h}^{2}+{\mu}^{2}{l}^{2}{)}^{{\scriptstyle \frac{1}{2}}}}\geqq \frac{2}{\text{N}.\text{A}{.}_{\text{objective}}+\text{N}.\text{A}{.}_{\text{condenser}}}\text{wave-lengths}.$$
(7)
$$\frac{4lh}{({m}^{2}{h}^{2}+{n}^{2}{l}^{2}{)}^{{\scriptstyle \frac{1}{2}}}}\geqq \frac{2}{\text{N}.\text{A}{.}_{\text{objective}}+\text{N}.\text{A}{.}_{\text{condenser}}}\leqq \frac{2}{\text{N}.\text{A}{.}_{\text{particles}}}.$$
(8)
$$\text{Airy}\hspace{0.17em}\text{limit}=\frac{0.6098}{\text{N}.\text{A}.}\text{wave-lengths}\equiv {{r}_{a}}^{0}$$
(9)
$$r=\frac{2\times 0.6098}{\text{N}.\text{A}{.}_{\text{objective}}+\text{N}.\text{A}{.}_{\text{condenser}}}\text{wave-lengths}.$$
(10)
$$r=\frac{1}{\text{N}.\text{A}{.}_{\text{objective}}+\text{N}.\text{A}{.}_{\text{condenser}}}\text{wave-lengths}.$$
(11)
$$G(x)/{\pi}^{2}{{\rho}_{m}}^{4}\equiv g(x)=[2{J}_{\text{i}}(2\pi {\rho}_{m}x)/2\pi {\rho}_{m}x{]}^{2}.$$
(12)
$$\mid M\mid {\rho}_{m}=\text{N}.\text{A}.$$
(13)
$${r}_{a}\equiv (0.6098)/{\rho}_{m}$$
(15)
$$[{d}^{2}G(x)]/d{x}^{2}=0$$
(16)
$$[{d}^{2}G(x)]/d{x}^{2}=0$$
(17)
$$f{k}_{1}=(g-1)(\text{N}.\text{A}.)W.$$
(18)
$$f{k}_{1}=-(g+1)(\text{N}.\text{A}.)\text{W}.$$
(19)
$$fk=-(1-g)\pi {(\text{N}.\text{A}.)}^{2}A.$$
(20)
$$fk=-(1+g)\pi {(\text{N}.\text{A}.)}^{2}A.$$