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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References

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  1. R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, 1944), p. 364.
  2. Ramsay, Cleveland, and Koppius, J. Opt. Soc. Am. 31, 28 (1941).
  3. R. K. Luneberg, reference 1, p. 390.
  4. H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 561 (1949).
  5. H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 565 (1949).

1949 (2)

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 561 (1949).

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 565 (1949).

1941 (1)

Ramsay, Cleveland, and Koppius, J. Opt. Soc. Am. 31, 28 (1941).

Cleveland,

Ramsay, Cleveland, and Koppius, J. Opt. Soc. Am. 31, 28 (1941).

Koppius,

Ramsay, Cleveland, and Koppius, J. Opt. Soc. Am. 31, 28 (1941).

Luneberg, R. K.

R. K. Luneberg, reference 1, p. 390.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, 1944), p. 364.

Osterberg, H.

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 565 (1949).

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 561 (1949).

Ramsay,

Ramsay, Cleveland, and Koppius, J. Opt. Soc. Am. 31, 28 (1941).

Wissler, F. C.

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 565 (1949).

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 561 (1949).

J. Opt. Soc. Am. (3)

Ramsay, Cleveland, and Koppius, J. Opt. Soc. Am. 31, 28 (1941).

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 561 (1949).

H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 565 (1949).

Other (2)

R. K. Luneberg, reference 1, p. 390.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, 1944), p. 364.

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Figures (12)

Fig. 1
Fig. 1

Notation with either axial or oblique illumination.

Fig. 2
Fig. 2

Imagery with periodic object structures which exert no focusing action upon the incident light. The spectral orders ν, μ are focused in the second focal plane of the objective as images of the point in the source from which the incident wave front is radiated.

Fig. 3
Fig. 3

Imagery with periodic structures which exert focusing action upon the incident light. The location of the conjugate, ambient object and image planes X0, Y0 and X, Y, respectively, depends in general upon the inclination of the incident wave front.

Fig. 4
Fig. 4

Comparison of the Airy and the physical limits of resolution. The unbroken curve of the right-hand drawing represents the energy density G(x) in the image of two, like particles whose separation is greater than the physical limit. The broken curve represents G(x) when the separation of the particles is equal to the physical limit of resolution. Whereas the energy density g(x) is specialized, the energy density G(x) is not.

Fig. 5
Fig. 5

Comparison of the Abbe limits and the physical limits as functions of S in resolving two like pin holes in an opaque slide. 2L is the separation of the two particles in Airy units.

Fig. 6
Fig. 6

Separations 2L in Airy units at the physical limits of resolution of two unlike particles against an opaque background. Δ is the optical path difference between the two particles. The two particles are restricted to the class m=1 in which the product of the area and the amplitude transmission of one particle is equal to the same product for the second particle.

Fig. 7
Fig. 7

Comparison of diffraction curves obtained with S=0 and with S=1 in the case of two unlike particles (class m=1) whose optical path difference is 70° and whose separation is 0.9 Airy unit. When S=1, a maximum appears in the neighborhood of ±x/ra=0.45. The particles are unresolved at the condenser setting S=0.

Fig. 8
Fig. 8

Comparison of diffraction curves obtained with S=0 and S=1 for two particles (class m=1) whose optical path difference is 135° and whose separation is 0.9 Airy unit. When S=1, the first maxima occur near ±x/ra=0.45. When S=0, the first maxima are much more pronounced and occur near ±x/ra=0.59 Airy unit.

Fig. 9
Fig. 9

Comparison of the separations 2L in Airy units at the physical limits of resolution for two like pin holes and for two like slits in an opaque background.

Fig. 10
Fig. 10

Separations 2L in Airy units at the physical limits of resolution of two object slits against a transmitting background. See Eqs. (17) and (18) for the meaning of fk1.

Fig. 11
Fig. 11

Separations 2L in Airy units at the physical limits of resolution for two like particles against a transmitting background. See Eqs. (19) and (20) for the meaning of fk.

Fig. 12
Fig. 12

The separations 2L in Airy units at the physical limits of resolution as functions of fk at S=1 for two like particles against a transmitting background. The three comparison curves belong to special objectives whose exit pupils have been coated with refracting and absorbing materials for the purpose of controlling the diameter of the central or Airy diffraction disk.

Equations (20)

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N . A . objective M sin ϑ m M ρ m = n 0 ρ om .
N . A . condenser n 0 sin ϑ cm n 0 ρ cm .
S N . A . condenser N . A . objective = ρ cm / ρ om .
4 l h ( ν 2 h 2 + μ 2 l 2 ) 1 2 2 N . A . objective + N . A . condenser wave-lengths .
2 l ν N . A . objective + N . A . condenser wave-lengths .
4 l h ( ν 2 h 2 + μ 2 l 2 ) 1 2 2 N . A . objective + N . A . condenser wave-lengths .
4 l h ( m 2 h 2 + n 2 l 2 ) 1 2 2 N . A . objective + N . A . condenser 2 N . A . particles .
Airy limit = 0.6098 N . A . wave-lengths r a 0
r = 2 × 0.6098 N . A . objective + N . A . condenser wave-lengths .
r = 1 N . A . objective + N . A . condenser wave-lengths .
G ( x ) / π 2 ρ m 4 g ( x ) = [ 2 J i ( 2 π ρ m x ) / 2 π ρ m x ] 2 .
M ρ m = N . A .
r a ( 0.6098 ) / ρ m
d G ( x ) / d x = 0
[ d 2 G ( x ) ] / d x 2 = 0
[ d 2 G ( x ) ] / d x 2 = 0
f k 1 = ( g - 1 ) ( N . A . ) W .
f k 1 = - ( g + 1 ) ( N . A . ) W .
f k = - ( 1 - g ) π ( N . A . ) 2 A .
f k = - ( 1 + g ) π ( N . A . ) 2 A .