Abstract

This paper describes a technique by which a large variation phase object may be visualized in an image having an irradiance which is directly proportional to the object phase. An image of the phase object derivative is formed in a coherent optical system and recorded on photographic film. This photograph is used as the input to a second coherent optical system, in which the image is an integral of the input. No limitations are imposed on the maximum object phase, only on its maximum slope. Experimental techniques for implementation of this system are discussed and results shown.

© 1972 Optical Society of America

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References

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  1. W. H. Steel, Interferometry (Cambridge U. P., London, 1967).
  2. J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 166–8.
  3. John W. Waddell, Jennie W. Waddell, Res. Develop. 21, 30 (1970).
  4. M. Françon, Progress in Microscopy (Pergamon, New York, 1961).
  5. L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).
  6. A. Bennett, H. Jupink, H. Osterberg, O. Richards, Phase Microscopy (Wiley, New York, 1951).
  7. M. Françon, Revue d’Optique (Paris, 1950).
  8. B. O. Payne, Res. Develop. 21, 47 (1963).
  9. J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 161–2.
  10. R. Eguchi, F. Carlson, Technical Report 127 (University of Washington, Department of Electrical Engineering, Seattle, Washington, 1968).
  11. J. C. Kent, Appl. Opt. 8, 2148 (1969).
    [CrossRef]
  12. C. E. Thomas, Appl. Opt. 7, 517 (1968).
    [CrossRef] [PubMed]
  13. R. A. Sprague, Ph.D. dissertation, University of Rochester (1971).
  14. B. P. Hildebrand, K. A. Haines, J. Opt. Soc. Am. 57, 155 (1967).
    [CrossRef]

1970 (1)

John W. Waddell, Jennie W. Waddell, Res. Develop. 21, 30 (1970).

1969 (1)

1968 (1)

1967 (1)

1963 (1)

B. O. Payne, Res. Develop. 21, 47 (1963).

Bennett, A.

A. Bennett, H. Jupink, H. Osterberg, O. Richards, Phase Microscopy (Wiley, New York, 1951).

Carlson, F.

R. Eguchi, F. Carlson, Technical Report 127 (University of Washington, Department of Electrical Engineering, Seattle, Washington, 1968).

DeVelis, J. B.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 161–2.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 166–8.

Eguchi, R.

R. Eguchi, F. Carlson, Technical Report 127 (University of Washington, Department of Electrical Engineering, Seattle, Washington, 1968).

Françon, M.

M. Françon, Progress in Microscopy (Pergamon, New York, 1961).

M. Françon, Revue d’Optique (Paris, 1950).

Haines, K. A.

Hildebrand, B. P.

Jupink, H.

A. Bennett, H. Jupink, H. Osterberg, O. Richards, Phase Microscopy (Wiley, New York, 1951).

Kent, J. C.

Martin, L. C.

L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).

Osterberg, H.

A. Bennett, H. Jupink, H. Osterberg, O. Richards, Phase Microscopy (Wiley, New York, 1951).

Payne, B. O.

B. O. Payne, Res. Develop. 21, 47 (1963).

Reynolds, G. O.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 161–2.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 166–8.

Richards, O.

A. Bennett, H. Jupink, H. Osterberg, O. Richards, Phase Microscopy (Wiley, New York, 1951).

Sprague, R. A.

R. A. Sprague, Ph.D. dissertation, University of Rochester (1971).

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. P., London, 1967).

Thomas, C. E.

Waddell, Jennie W.

John W. Waddell, Jennie W. Waddell, Res. Develop. 21, 30 (1970).

Waddell, John W.

John W. Waddell, Jennie W. Waddell, Res. Develop. 21, 30 (1970).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Res. Develop. (2)

John W. Waddell, Jennie W. Waddell, Res. Develop. 21, 30 (1970).

B. O. Payne, Res. Develop. 21, 47 (1963).

Other (9)

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 161–2.

R. Eguchi, F. Carlson, Technical Report 127 (University of Washington, Department of Electrical Engineering, Seattle, Washington, 1968).

W. H. Steel, Interferometry (Cambridge U. P., London, 1967).

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), pp. 166–8.

M. Françon, Progress in Microscopy (Pergamon, New York, 1961).

L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).

A. Bennett, H. Jupink, H. Osterberg, O. Richards, Phase Microscopy (Wiley, New York, 1951).

M. Françon, Revue d’Optique (Paris, 1950).

R. A. Sprague, Ph.D. dissertation, University of Rochester (1971).

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Figures (14)

Fig. 1
Fig. 1

Modulus of image amplitude produced in a finite-aperture differentiation system for a sloped phase edge object. Each vertical pair of traces represents the edge derivative for a given absolute value of b/Δ; the top diagram is calculated for a positive edge slope and the lower one is for a negative slope. For small b/Δ, the image is similar to the perfect derivative, which is a box function. For b/Δ greater than 0.75 Δ/f, however, this similarity no longer exists.

Fig. 2
Fig. 2

Integral of phase edge derivative taken across the edge position. A linear relation between this integral and the edge height b exists in the region −0.75 Δ/fb/Δ ≤ 0.75 Δ/f.

Fig. 3
Fig. 3

Sliding mask technique used to compensate for film nonlinearity in exposing the differentiation filter.

Fig. 4
Fig. 4

Experimental transmittance functions of differentiation filters.

Fig. 5
Fig. 5

Apparatus used to expose photographic plate in production of integration filter.

Fig. 6
Fig. 6

Experimental amplitude transmittance function of the integration filter. The left-hand diagram has an expanded horizontal scale to better show the transmittance function near axis. The right-hand diagram has an expanded vertical scale to better show the transmittance function in the high density region of the filter.

Fig. 7
Fig. 7

System used for optical differentiation. The rotating scatter is used to remove optical noise produced by scattering.

Fig. 8
Fig. 8

System used for optical integration. The optical flat is tilted continuously throughout the observation time to remove optical noise. The half-wave phase step is imaged onto the photographic portion of the spatial filter so that the two are effectively in the same Fourier transform plane.

Fig. 9
Fig. 9

Derivative of the phase resolution target. The x axis is oriented at 45° to the vertical. The random pattern in the broad clear areas is the actual derivative of the phase target surface, which was slightly irregular in these areas. The fringes inside the bars are due to multiple reflections between the front and back surfaces of the phase target.

Fig. 10
Fig. 10

Integral of the derivative of the phase resolution target. The two-dimensional numbers which label the individual bar groups have been almost completely reproduced, despite the one-dimensional differentiation and integration. The larger bars exhibit a distinct edge sharpening.

Fig. 11
Fig. 11

(a) Comparison of image amplitude with object phase for bar group 0–2 of the phase resolution target. (b) Comparison of image amplitude with object phase for bar group 2–2 of the phase resolution target.

Fig. 12
Fig. 12

(a) Comparison of image amplitude with object phase for small circular phase depression. (b) Comparison of image amplitude with object phase for large circular phase depression.

Fig. 13
Fig. 13

Comparison of image amplitude with object phase for set of phase steps of gradually increasing height.

Fig. 14
Fig. 14

Derivative of the phase resolution target formed in a low resolution differentiation system.

Equations (25)

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i 1 ( x 3 ) = [ ( x 0 / f ) + ϕ ( x 3 ) ] exp i k ϕ ( x 3 ) ,
t 3 ( x 3 ) = | ( x 0 / f ) + ϕ ( x 3 ) | 2 γ n γ p .
t 3 ( x 3 ) = [ | ( x 0 / f ) + ϕ ( x 3 ) | 2 ] 1 2 ;
t 3 ( x 3 ) = ( x 0 / f ) + ϕ ( x 3 ) .
t 4 ( x 4 ) = i B | x 4 | < D , = D / x 4 | x 4 | D ,
t 3 ( x 3 ) = ( x 0 / f ) + ϕ ( x 3 ) .
i 2 ( x 5 ) = ( c x 0 / f ) + ϕ ( x 5 ) ,
I 2 ( x 5 ) ( c 2 x 0 2 / f 2 ) + ( 2 c x 0 / f ) ϕ ( x 5 ) .
ϕ ( x 1 , y 1 ) = ϕ a ( x 1 , y 1 ) + ϕ b ( y 1 ) ,
I 1 ( x 3 , y 3 ) = | ( x 0 / f ) + ϕ a x ( x 3 , y 3 ) | 2 ,
i 2 ( x 5 , y 5 ) = ( c x 0 / f ) + ϕ a ( x 5 , y 5 ) .
ϕ ( x 1 ) = 0 x 1 < 0 , = b x 1 / Δ 0 x 1 Δ , = b x 1 > Δ ,
i 1 ( x 3 ) = ( α x 0 f / k ) [ π + 2 S i ( k A x 3 / f ) ] + ( α x 0 f / k ) ( exp i k b ) { π 2 S i [ k A ( x 3 + Δ ) / f ] } i α [ ( f x 0 / k ) + ( b f 2 / k Δ ) ] ( exp i k b x 3 / Δ ) [ C i { k [ ( b / Δ ) + ( A / f ) ] ( x 3 + Δ ) } C i { k [ ( b / Δ ) ( A / f ) ] ( x 3 + Δ ) } C i { k [ ( b / Δ ) + ( A / f ) ] x 3 } + C i { k [ ( b / Δ ) ( A / f ) ] x 3 } + i ( S i { k [ ( b / Δ ) + ( A / f ) ] ( x 3 + Δ ) } S i { k [ ( b / Δ ) ( A / f ) ] ( x 3 + Δ ) } S i { k [ ( b / Δ ) + ( A / f ) ] x 3 } + S i { k [ ( b / Δ ) ( A / f ) x 3 } ) ] ,
i 1 ( x 3 ) i α [ ( f x 0 / k ) + ( b f 2 / k Δ ) ] ( S i { k [ ( b / Δ ) + ( A / f ) ] ( x 3 + Δ ) } S i { k [ ( b / Δ ) ( A / f ) ] ( x 3 + Δ ) } S i { k [ ( b / Δ ) + ( A / f ) ] x 3 } + S i { k [ ( b / Δ ) ( A / f ) ] x 3 } ) exp i k b x 3 / Δ .
( f x 0 / k ) + ( b f 2 / k Δ ) 0 or x 0 b f / Δ .
x 0 0.75 A .
t 4 ( x 4 , y 4 ) = i B | x 4 | < D , = D / x 4 | x 4 | D .
t 4 a ( x 4 , y 4 ) = 1 | x 4 | < D , = D / x 4 | x 4 | D ,
t 4 p ( x 4 , y 4 ) = i B | x 4 | < D , = 1 | x 4 | D .
W ( x ) C | d E ( x ) d x | ] x = { [ ( a + b ) / a ] ( d 2 ) ( b x / a ) + ( δ / 4 ) , x > 0 [ ( a + b ) / a ] ( d 2 ) ( b x / a ) ( δ / 4 ) , x < 0 ,
δ = [ λ b ( a + b ) / 2 a ] 1 2 .
i ( x , y ) = [ c e i β d e exp i Δ k ϕ ( x , y ) ] exp i k ϕ ( x , y ) ,
Δ k ϕ ( x , y ) 2 π ,
i ( x , y ) = { C cos β d e + i [ C sin β d e Δ k ϕ ( x , y ) ] } exp i k ϕ ( x , y )
I ( x , y ) C 2 [ sin 2 β 2 sin β cos β Δ k ϕ ( x , y ) ] .

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