Abstract

If an object under a phase contrast microscope is illuminated with light coming from a point source in such a manner that the image of the source is projected upon one of the edges of the phase annulus, the image of the microscopic object shows certain peculiarities which cannot be accounted for under the hypothesis that the point source is imaged upon the annulus without aberrations. Experiment indicates that in cases where the edge effect is observed, the image of the light source is defocused with respect to the phase annulus. The theory of the edge effect may be adapted from the diffraction theory of the Foucault knife-edge test of Gascoigne and Linfoot.

© 1951 Optical Society of America

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References

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  1. L. I. Epstein, J. Opt. Soc. Am. 40, 291 (1950).
    [Crossref]
  2. K. Nienhuis, thesis, Groningen (1948). Reproduced by F. Zernike in Proc. Phys. Soc. (London)61, 158 (1948), Fig. 2.
    [Crossref]
  3. S. C. B. Gascoigne, M. N. R. A. S. 104, 326 (1944).
  4. E. H. Linfoot, Proc. Roy. Soc. (London) A186, 72 (1946).
    [Crossref]
  5. See, e.g., P. Drude, The Theory of Optics (Longmans, Green and Company, New York, 1902), pp. 192–195.

1950 (1)

1946 (1)

E. H. Linfoot, Proc. Roy. Soc. (London) A186, 72 (1946).
[Crossref]

1944 (1)

S. C. B. Gascoigne, M. N. R. A. S. 104, 326 (1944).

Drude, P.

See, e.g., P. Drude, The Theory of Optics (Longmans, Green and Company, New York, 1902), pp. 192–195.

Epstein, L. I.

Gascoigne, S. C. B.

S. C. B. Gascoigne, M. N. R. A. S. 104, 326 (1944).

Linfoot, E. H.

E. H. Linfoot, Proc. Roy. Soc. (London) A186, 72 (1946).
[Crossref]

Nienhuis, K.

K. Nienhuis, thesis, Groningen (1948). Reproduced by F. Zernike in Proc. Phys. Soc. (London)61, 158 (1948), Fig. 2.
[Crossref]

J. Opt. Soc. Am. (1)

M. N. R. A. S. (1)

S. C. B. Gascoigne, M. N. R. A. S. 104, 326 (1944).

Proc. Roy. Soc. (London) (1)

E. H. Linfoot, Proc. Roy. Soc. (London) A186, 72 (1946).
[Crossref]

Other (2)

See, e.g., P. Drude, The Theory of Optics (Longmans, Green and Company, New York, 1902), pp. 192–195.

K. Nienhuis, thesis, Groningen (1948). Reproduced by F. Zernike in Proc. Phys. Soc. (London)61, 158 (1948), Fig. 2.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Mouth epithelial cells, photographed with edge effect.

Fig. 2
Fig. 2

Laboratory set-up for observation of the edge effect.

Fig. 3
Fig. 3

Schematic diagram showing the interpretation of the edge effect as a diffraction phenomenon associated with the Foucault knife-edge test.

Fig. 4
Fig. 4

Light intensity (ordinate) at the point of greatest edge effect, as the image of a phase-retarding semi-infinite plane approaches from the left and sweeps across the point of observation. The abscissa is the distance of the image of the edge of the semi-infinite plane from the point of observation. The light intensity in the absence of an object is shown in the margin.

Fig. 5
Fig. 5

Similar to Fig. 3, except that the image of a phase-retarding semi-infinite plane initially covers the point of observation and then recedes towards the right.

Fig. 6
Fig. 6

Similar to Figs. 3 and 4, except that the object is a phase-retarding strip of finite width.

Equations (27)

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D ( x , y ) = 1 2 E ( x , y ) + ( i / 2 π ) - + E ( t , y ) d t / ( t - x ) .
D ( x , y ) = 1 2 E ( x , y ) - ( i / 2 π ) - + E ( t , y ) d t / ( t - x ) .
D ( x , y ) = 1 2 ( 1 + τ e i γ ) E ( x , y ) + ( i / 2 π ) ( 1 - τ e i γ ) - + [ E ( t , y ) d t / ( t - x ) ] .
E ( x , y ) = exp [ i a ( x 2 + y 2 ) ] .
E ( x , y ) = exp ( i a x 2 )
I ( x ) = - + exp ( i a t 2 ) d t / ( t - x ) ,
I ( x ) = 2 i exp ( i a x 2 ) × 0 [ exp ( i a z 2 ) / z ] sin 2 a x z d z .
I ( x ) = i π ( 1 + i ) exp ( i a x 2 ) [ C ( b x ) - i S ( b x ) ] ,
b = ( 2 a / π ) 1 2 ,
C ( z ) = 0 z cos 1 2 π t 2 d t ,             S ( z ) = 0 z sin 1 2 π t 2 d t
D ( x , y ) = 1 2 exp ( + i a x 2 ) × { 1 - ( 1 + i ) [ C ( b x ) - i S ( b x ) ] } .
D ( x , y ) 2 = 1 2 { [ 1 2 - C ( b x ) ] 2 + [ 1 2 - S ( b x ) ] 2 } .
D ( x , y ) 2 = 1 2 { [ 1 2 - C ( b x ) ] 2 + [ 1 2 - S ( b x ) ] 2 } + 1 2 τ 2 { [ 1 2 + C ( b x ) ] 2 + [ 1 2 + S ( b x ) ] 2 } + τ { S ( b x ) - C ( b x ) } .
E ( x , y ) = exp [ i ( a x 2 ± φ ) ] ,
D ( x , y ) = 1 2 ( 1 + τ e i γ ) exp [ i ( a x 2 - φ ) ] + ( i / 2 π ) ( 1 - τ e i γ ) [ e - i φ G - ( g x , g X ) + e - i φ L ( g x , g X ) + e + i φ G + ( g x , g X ) ] ,
g 2 = a = 1 2 π b 2 ,
G - ( g x , g X ) = - x - X exp ( i a t 2 ) d t / ( t - x ) ,
L ( g x , g X ) = x - X x + X exp ( i a t 2 ) d t / ( t - x ) ,
G + ( g x , g X ) = x + X + exp ( i a t 2 ) d t / ( t - x ) .
D ( x , y ) = 1 2 ( 1 + τ e i γ ) exp [ i ( a x 2 + φ ) ] + ( i / 2 π ) ( 1 - τ e i γ ) { e - i φ G - ( g x , - g X ) + e + i φ L ( g x , - g X ) + e + i φ G + ( g x , - g X ) } .
G + ( g x , g X ) = i exp [ i a ( x + X ) 2 ] 2 a ( x + X ) X + exp [ i a ( x + X ) 2 ] ( x + 2 X ) 4 a 2 ( x + X ) 3 X 2 + x + X exp ( i a t 2 ) 4 a 2 d d t [ 2 t - x t 3 ( t - x ) 2 ] d t .
| exp [ i a ( x + X ) 2 ] ( x + 2 X ) 4 a 2 ( x + X ) 3 X 2 + x + X exp ( i a t 2 ) 4 a 2 d d t [ 2 t - x t 3 ( t - x ) 2 ] d t | < | exp [ i a ( x + X ) 2 ] ( x + 2 X ) 4 a 2 ( x + X ) 3 X 2 | + x + X | exp ( i a t 2 ) 4 a 2 d d t [ 2 t - x t 3 ( t - x ) 2 ] | d t = | ( x + 2 X ) 2 a 2 ( x + X ) 3 X 2 | .
G - ( 1 2 π , 0.34 π ) = - 0.9727 - 0.3761 i G + ( 1 2 π , 0.34 π ) = - 0.4183 - 0.0765 i .
X 1 < X 2 < 0.
X 1 < 0 < X 2 .
0 < X 1 < X 2 .
D ( x , y ) = 1 2 ( 1 + i τ ) exp [ i ( a x 2 + φ ) ] + ( i / 2 π ) ( 1 - i τ ) { e + i φ G - ( g x , g X 1 ) + e + i φ L ( g x , g X 1 ) + e - i φ [ G + ( g x , g X 1 ) - G + ( g x , g X 2 ) ] + e + i φ G + ( g x , g X 2 ) } = 1 2 ( 1 + i τ ) exp [ i ( a x 2 + φ ) ] + ( i / 2 π ) ( 1 - i τ ) [ e + i φ G - ( g x , g X 1 ) + e + i φ L ( g x , g X 1 ) + e - i φ G + ( g x , g X 1 ) + 2 i sin φ G + ( g x , g X 2 ) ] .