Abstract

The adjustments of a variable phase microscope control two variable experimental parameters, specifically the phase difference δ between and the amplitude transmission ratio h of the conjugate and complementary areas of the diffraction plate or its polanret equivalent. It is shown that the center of the sharply focused diffraction image of a single unresolved particle can be brought to zero intensity by a definite choice of the parameters h and δ. The relations which exist among the required choice of h, δ, the area and the optical properties of the particle are derived and specialized to various classes of particles. These relations permit the computation of the area and one or more optical properties of the particle from the observed values of the parameters h, δ. Observations made upon opaque particles with a modified polanret microscope are described and found to be in accord with the theory. The radii of these particles ranged from 0.5 to 0.15 times the Airy limit of the test objective.

© 1950 Optical Society of America

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References

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  1. H. Osterberg, J. Opt. Soc. Am. 37, 726 (1947).
    [Crossref] [PubMed]
  2. H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 558 (1949), Eqs. (27) and (28).
    [Crossref]
  3. H. Osterberg and J. E. Wilkins, J. Opt. Soc. Am. 39, 553 (1949).
    [Crossref]
  4. H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 558 (1949).
    [Crossref]
  5. H. Osterberg, J. Opt. Soc. Am. 38, 685 (1948), Eq. (19).
    [Crossref] [PubMed]

1949 (3)

1948 (1)

1947 (1)

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Figures (3)

Fig. 1
Fig. 1

The essential elements of the phase microscope and their denotation. The refractive indices of the object and image spaces are, respectively, n0 and n=1. The conjugate area of the diffraction plate coincides with the geometrical image of the opening in the condenser diaphragm. The complementary comprises the remaining area of the diffraction plate. It is supposed that the diffraction plate is variable in the sense that the optical path difference between its conjugate and complementary areas and the light transmission ratio of these areas can be varied.

Fig. 2
Fig. 2

The preferred system. The opening in the condenser diaphragm and the conjugate area are circular and are centered upon the optical axis. For greatest ease in interpreting the polanret dark settings, the opening in the condenser diaphragm will be made so small that ρ1ρm.

Fig. 3
Fig. 3

The modified polanret system for measuring unresolved particles.

Tables (3)

Tables Icon

Table I Measurements of the radii R in wave-lengths for several mercury particles.

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Table II Measurement of a selected particle by six observers. The direct measurement of the radius of this particle is 1.57 wave-lengths.

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Table III Precision measurements of a selected particle.

Equations (75)

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F 0 ( x , y , p 0 , q 0 ) = F u ( x , y , p 0 , q 0 ) + F d ( x , y , p 0 , q 0 ) ,
M 2 F u ( x , y , p 0 , q 0 ) = f 1 exp [ 2 π i ( p 0 x + q 0 y ) / M ] P ( ( p 0 / M ) , ( q 0 / M ) )
F d = ( f 0 - f 1 ) exp [ 2 π i ( p 0 x 0 ) + q 0 y 0 ) ] U ( x - M x 0 , y - M y 0 ) d x 0 d y 0 .
U ( x , y ) = P ( p , q ) exp [ 2 π i ( p x + q y ) ] d p d q ;
p 2 + q 2 n 2 sin 2 ϑ m = sin 2 ϑ m ρ m 2 .
F d ( x , y , p 0 , q 0 ) = f A U ( x , y )
f f 0 - f 1 .
M 2 F 0 ( x , y , p 0 , q 0 ) = f 1 exp [ 2 π i ( p 0 x q 0 y ) / M ] + P ( ( p 0 / M ) , ( q 0 / M ) ) + f M 2 A U ( x , y ) .
M 2 F 0 ( x , y , p 0 , q 0 ) 2 f 1 2 P ( ( p 0 / M ) , ( q 0 / M ) ) 2
G ( x , y ) = ( S ( p 0 , q 0 ) ) / ( n 0 2 - p 0 2 - q 0 2 ) × F 0 ( x , y , p 0 , q 0 ) 2 d p 0 d q 0
U ( 0 , 0 ) = P ( p , q ) d p d q .
G ( 0 , 0 ) = 1 M 4 S ( p 0 , q 0 ) n 0 2 - p 0 2 - q 0 2 | f 1 P ( p 0 M , q 0 M ) + f M 2 A P ( p , q ) d p d q | 2 d p 0 d q 0 .
G ( ) = f 1 2 M 4 S ( p 0 , q 0 ) n 0 2 - p 0 2 - q 0 2 | P ( p 0 M , q 0 M ) | 2 d p 0 d q 0 .
f 1 P ( ( p 0 / M ) , ( q 0 / M ) ) + f M 2 A P ( p , q ) d p d q = 0.
P ( p , q ) d p d q 0.
P ( p , q ) = c ( p , q ) ψ ( p , q )
P ( p , q ) = c ( p , q ) .
f 1 0.
P 0 + M 2 A [ g exp ( - i Δ ) - 1 ] P ( p , q ) d p d q = 0 ,
P 0 P ( ( p 0 / M ) , ( q 0 / M ) )
g exp ( - i Δ ) f 0 / f 1 .
T c d p d q
P ( p , q ) d p d q = P 0 T c + P 1 ( P ( p , q ) / P 1 ) d p d q = P 0 T c + P 1 Q
Q [ P ( p , q ) / P 1 ] d p d q ,
P 0 + P 0 M 2 A T c [ g exp ( - i Δ ) - 1 ] + P 1 Q M 2 A [ g exp ( - i Δ ) - 1 ] = 0.
P 1 0.
h exp ( - i δ ) P 0 / P 1 .
h exp ( - i δ ) + M 2 A T c h exp ( - i δ ) [ g exp ( - i Δ ) - 1 ] + M 2 A Q [ g exp ( - i Δ ) - 1 ] = 0.
α h cos δ - γ h sin δ - M 2 A Q ( 1 - g cos Δ ) = 0 ; α h sin δ + γ h cos δ + M 2 A Q g sin Δ = 0 ;
α 1 - M 2 A T c ( 1 - g cos Δ ) ; γ M 2 A T c g sin Δ .
h = M 2 A Q ( g 2 - 2 g cos Δ + 1 ) 1 2 / ( α 2 + γ 2 ) 1 2 ; sin δ = - g sin Δ / ( g 2 - 2 g cos Δ + 1 ) 1 2 ( α 2 + γ 2 ) 1 2 ; cos δ = [ 1 - g cos Δ - M 2 A T c ( g 2 - 2 g cos Δ + 1 ) ] / [ ( g 2 - 2 g cos Δ + 1 ) ( α 2 + γ 2 ) ] 1 2 ;
α 2 + γ 2 = 1 - 2 M 2 A T c ( 1 - g cos Δ ) + M 4 A 2 T c 2 ( g 2 - 2 g cos Δ + 1 ) .
Q = d p d q ,
T d p d q ,
Q = T - T c ,
T c = 0 ρ 1 0 2 π ρ d ϕ d ρ = π ρ 1 2 ;
T = 0 ρ m 0 2 π ρ d ϕ d ρ = π ρ m 2 ;
N . A . = M ρ m ,
A m = π 3.8317 2 / 16 π 2 M 2 ρ m 2 .
( M 2 A T c ) m = ρ 1 2 / ρ m 2 .
( α 2 + γ 2 ) 1 2 = [ 1 - 2 M 2 A T c ( 1 - g cos Δ ) ] 1 2 1 - M 2 A T c ( 1 - g cos Δ )
T c T             or             ρ 1 2 ρ m 2 .
h = π A ( N . A . ) 2 ( g 2 - 2 g cos Δ + 1 ) 1 2 ;
sin δ = - g sin Δ / ( g 2 - 2 g cos Δ + 1 ) 1 2 ;
cos δ = ( 1 - g cos Δ ) / ( g 2 - 2 g cos Δ + 1 ) 1 2 .
G ( 0 , 0 ) : h exp ( - i δ ) + M 2 A T c h exp ( - i δ ) [ g exp ( - i Δ ) - 1 ] + M 2 A ( T - T c ) [ g exp ( - i Δ ) - 1 ] 2 .
h = π A ( N . A . ) 2 ;
δ = 0.
R 0 = K r a 0 = 0.6098 K / N . A .
h = 3.670 K 2 .
h = 2 π A ( N . A . ) 2 sin Δ / 2 ;
sin δ = - s g n ( sin Δ / 2 ) cos Δ / 2 ;
cos δ = sin Δ / 2 ;
δ = Δ / 2 - π / 2 ,             0 < Δ < 2 π ;
δ = Δ / 2 + π / 2 ,             - 2 π < Δ < 0.
h = 7.34 K 2 sin Δ / 2 .
h = π A ( N . A . ) 2 ( 1 - g cos Δ ) ;
δ = - g sin Δ .
h 2 = π A ( N . A . ) 2 sin Δ ; sin δ 2 = - s g n ( sin Δ ) cos Δ ;
δ 2 = Δ - π / 2 ,             0 < Δ < π / 2 ;
δ 2 = π / 2 - Δ ,             π / 2 < Δ < 0.
h 1 = π A ( N . A . ) 2 ( 1 - g 1 cos Δ ) ;
δ 1 = - g 1 sin Δ .
Δ 2 = Δ 1 ( n 1 - n 2 ) / ( n - n 1 ) .
h j = π A ( N . A . ) 2 ( g 2 - 2 g cos Δ j + 1 ) 1 2 ;
sin δ j = - g sin Δ j / ( g 2 - 2 g cos Δ j + 1 ) 1 2 ;
cos δ j = ( 1 - g cos Δ j ) / ( g 2 - 2 g cos Δ j + 1 ) 1 2 ;
h 2 = π A ( N . A . ) 2 g - 1 ;
δ 2 = 0 or π .
h 2 = π A ( N . A . ) 2 { ( 1 - g ) , δ 2 = 0 ; ( g - 1 ) , δ 2 = π .
a = h 2 / h 1 - cos δ 1 cos δ 2 ; b = ( h 2 / h 1 ) 2 - 2 ( h 2 / h 1 ) cos δ 1 cos δ 2 + 1.
sin Δ 1 = 2 a sin δ 1 cos δ 2 / b ;
1 / g = 1 - 2 a h 2 / h 1 b .
a cos δ 2 sin Δ / sin δ 1 + cos Δ = 1.
t = 2 π Δ 1 / ( n - n 1 ) ,