Abstract

The problem of phase contrast image formation with Köhler illumination is a special case of a more general problem which has been treated by Duffieux and his students: The Fraunhofer diffraction pattern due to the object is projected upon the phase pattern, and the diffraction pattern due to the latter constitutes the image. A semi-infinite plane covered with a uniform, transparent phase-shifting layer is to be regarded as a typical microscopic object. The variation in the appearance of the image as the light goes through different portions of the phase annulus is examined. The field is darkened on one side of the edge of the object and brightened up on the other side. If it is desired that the dark side shall be as dark as possible if a large portion of the annulus is filled with light, a quarter-wave phase shift must be introduced in the phase annulus. For light passing very close to the edge of the annulus, point-for-point image formation is no longer rigorously possible. As the annulus is made broader, the contrast becomes less pronounced.

© 1950 Optical Society of America

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References

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  1. Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.
  2. G. Toraldo di Francia, Nuovo Cimento [9],  6, 30 (1949).
    [CrossRef]
  3. M. Françon, Rev. optique 25, 257 (1946).
  4. J. Picht, Zeits. f. Instrumentenk 56, 363 (1936); Zeits. f. Instrumentenk 56, 481 (1936); Zeits. f. Instrumentenk 58, 1 (1938).
  5. Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
    [CrossRef]
  6. E. G. Ramberg, J. App. Phys. 20, 441 (1949).
    [CrossRef]
  7. F. Zernike, Physica 9, 686, 974 (1942).
    [CrossRef]
  8. R. Richter, Optik 2, 342 (1947).
  9. A. Rubinowicz, Ann. d. Physik 53, 257 (1917).
    [CrossRef]
  10. J. Magliozzi, J. Opt. Soc. Am. 37, 982A (1947).

1949 (2)

G. Toraldo di Francia, Nuovo Cimento [9],  6, 30 (1949).
[CrossRef]

E. G. Ramberg, J. App. Phys. 20, 441 (1949).
[CrossRef]

1947 (2)

R. Richter, Optik 2, 342 (1947).

J. Magliozzi, J. Opt. Soc. Am. 37, 982A (1947).

1946 (2)

M. Françon, Rev. optique 25, 257 (1946).

Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
[CrossRef]

1944 (1)

Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.

1942 (1)

F. Zernike, Physica 9, 686, 974 (1942).
[CrossRef]

1936 (1)

J. Picht, Zeits. f. Instrumentenk 56, 363 (1936); Zeits. f. Instrumentenk 56, 481 (1936); Zeits. f. Instrumentenk 58, 1 (1938).

1917 (1)

A. Rubinowicz, Ann. d. Physik 53, 257 (1917).
[CrossRef]

Bennett,

Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
[CrossRef]

Duffieux,

Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.

Françon, M.

M. Françon, Rev. optique 25, 257 (1946).

Guenoche,

Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.

Jupnik,

Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
[CrossRef]

Lansraux,

Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.

Magliozzi, J.

J. Magliozzi, J. Opt. Soc. Am. 37, 982A (1947).

Osterberg,

Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
[CrossRef]

Picht, J.

J. Picht, Zeits. f. Instrumentenk 56, 363 (1936); Zeits. f. Instrumentenk 56, 481 (1936); Zeits. f. Instrumentenk 58, 1 (1938).

Ramberg, E. G.

E. G. Ramberg, J. App. Phys. 20, 441 (1949).
[CrossRef]

Richards,

Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
[CrossRef]

Richter, R.

R. Richter, Optik 2, 342 (1947).

Rubinowicz, A.

A. Rubinowicz, Ann. d. Physik 53, 257 (1917).
[CrossRef]

Tirouflet,

Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento [9],  6, 30 (1949).
[CrossRef]

Zernike, F.

F. Zernike, Physica 9, 686, 974 (1942).
[CrossRef]

Ann. d. Physik (1)

A. Rubinowicz, Ann. d. Physik 53, 257 (1917).
[CrossRef]

Ann. de physique (1)

Duffieux, Tirouflet, Guenoche, and Lansraux, Ann. de physique 19, 355 (1944).Note the misprint on p. 369: Eq. (43) should not contain the factor of 12. Equation (41) is correct.

J. App. Phys. (1)

E. G. Ramberg, J. App. Phys. 20, 441 (1949).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Magliozzi, J. Opt. Soc. Am. 37, 982A (1947).

Nuovo Cimento [9] (1)

G. Toraldo di Francia, Nuovo Cimento [9],  6, 30 (1949).
[CrossRef]

Optik (1)

R. Richter, Optik 2, 342 (1947).

Physica (1)

F. Zernike, Physica 9, 686, 974 (1942).
[CrossRef]

Rev. optique (1)

M. Françon, Rev. optique 25, 257 (1946).

Trans. Am. Microscop. Soc. (1)

Bennett, Jupnik, Osterberg, and Richards, Trans. Am. Microscop. Soc. 65, 99 (1946).
[CrossRef]

Zeits. f. Instrumentenk (1)

J. Picht, Zeits. f. Instrumentenk 56, 363 (1936); Zeits. f. Instrumentenk 56, 481 (1936); Zeits. f. Instrumentenk 58, 1 (1938).

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Figures (3)

Fig. 1
Fig. 1

Pupil of objective with phase annulus, showing the notation.

Fig. 2
Fig. 2

Metastable light intensity on the bright and dark side, plotted against position of the pinpoint of light within the phase annulus.

Fig. 3
Fig. 3

Analogous to Fig. 2, but showing a misguided attempt to increase the light transmission of the annulus.

Equations (15)

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g ( μ , ν ) = - + - + f ( x , y ) e - 2 π i ( μ x + ν y ) d x d y .
u ( x , y ) = - + - + P g e + 2 π i ( μ x + ν y ) d μ d ν .
u = - + P e + 2 π i μ x - + f e - 2 π i μ x d x d μ ,
u = τ cos γ cos φ + ( 1 / π ) sin φ { - Ci [ 2 π x ( α 1 + t ) ] + ( 1 - τ cos γ ) Ci [ 2 π x ( α 1 + s ) ] + ( τ cos γ - 1 ) Ci [ 2 π x ( α 1 + q ) ] + ( 1 - τ cos γ ) Ci [ 2 π x ( α 1 - q ) ] + ( τ cos γ - 1 ) Ci [ 2 π x ( s - α 1 ) ] + Ci [ 2 π x ( t - α 1 ) ] - τ sin γ Si [ 2 π x ( α 1 + s ) ] + τ sin γ Si [ 2 π x ( α 1 + q ) ] - τ sin γ Si [ 2 π x ( α 1 - q ) ] - τ sin γ Si [ 2 π x ( s - α 1 ) ] } + i τ sin γ cos φ + ( i / π ) sin φ { - τ sin γ Ci [ 2 π x ( α 1 + s ) ] + τ sin γ Ci [ 2 π x ( α 1 + q ) ] - τ sin γ Ci [ 2 π x ( α 1 - q ) ] + τ sin γ Ci [ 2 π x ( s - α 1 ) ] + Si [ 2 π x ( α 1 + t ) ] - ( 1 - τ cos γ ) Si [ 2 π x ( α 1 + s ) ] - ( τ cos γ - 1 ) Si [ 2 π x ( α 1 + q ) ] - ( 1 - τ cos γ ) Si [ 2 π x ( α 1 - q ) ] + ( τ cos γ - 1 ) Si [ 2 π x ( s - α 1 ) ] + Si [ 2 π x ( t - α 1 ) ] } ,
Si ( z ) = 0 z ( sin μ / μ ) d μ
Ci ( z ) = - z ( cos μ / μ ) d μ
lim z Si ( z ) = 1 2 π
lim z Ci ( z ) = 0
lim z 0 [ Ci ( A z ) - Ci ( B z ) ] = ln A / B .
u τ cos γ cos φ + ( 1 / π ) sin φ ( 1 - τ cos γ ) × ln ( α 1 - q ) / ( s - α 1 ) + i { τ sin γ cos φ + ( 1 / π ) sin φ [ - τ sin γ ln ( α 1 - q ) / ( s - α 1 ) ± π ] } ,
u m 2 = ln 2 ( α 1 - q ) / ( s - α 1 ) × ( 1 / π 2 ) sin 2 φ ( 1 + τ 2 - 2 τ cos γ ) + ln ( α 1 - q ) / ( s - α 1 ) × ( 2 τ / π ) sin φ [ cos ( φ ± γ ) - τ cos φ ] + τ 2 cos 2 φ + sin 2 φ ± 2 τ sin φ cos φ sin γ .
ln ( α 1 - q ) / ( s - α 1 ) = - τ π [ cos ( φ ± γ ) - τ cos φ ] / { sin φ [ 1 + τ 2 - 2 τ cos γ ] } .
u m 2 min = [ τ sin ( γ φ ) ± sin φ ] 2 / ( 1 + τ 2 - 2 τ cos γ ) .
τ = sin φ / sin ( γ φ ) .
ln ( α 1 - q ) / ( s - α 1 ) = ± π cot γ