Abstract

The theory and experimental evidence of a shearing interferometer based on the Talbot effect are presented. Multiple-shearing interferences are obtained that can be reduced to triple-shearing or double-shearing interferences by the addition of simple spatial filtering. When the shear is less than the width of the details in the object, these interferences become either the second or first derivative of the object under test, respectively. Either lateral or constant radial shear can be introduced by choosing Ronchi rulings or circular gratings. Thus both lateral and radial derivatives are easily obtained. If white light is used as a source, color fringes of high contrast are observed.

© 1972 Optical Society of America

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References

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  1. O. Bryngdahl, Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 39–83.
    [Crossref]
  2. E. Waetzmann, Ann. Phys. 39, 1042 (1912).
    [Crossref]
  3. O. Bryngdahl, J. Opt. Soc. Am. 61, 169 (1971).
    [Crossref]
  4. A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971); S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971); D. Silva, Appl. Opt. 10, 1980 (1971); A. Lohmann, D. Silva, Opt. Commun. 4, 326 (1972).
    [Crossref] [PubMed]
  5. J. Cowley, A. Moodie, Proc. Phys. Soc. B70, 486 (1957); W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967); J. Winthrop, C. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [Crossref]
  6. F. Talbot, Phil. Mag. 9, 401 (1836).
  7. A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971).
    [Crossref]
  8. R. F. Edgar, Optica Acta 16, 281 (1969).
    [Crossref]
  9. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48.
  10. W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).
    [Crossref]
  11. A. Papoulis, Systems and Transformations with Applications in Optics (McGraw Hill, New York, 1968), p. 234.
  12. E. Lau, Optik, 12, 23 (1955).
  13. Product of Edmund Scientific, Barrington, N.J.
  14. H. Dammann, G. Groh, M. Kock, Appl. Opt. 10, 1454 (1971).
    [Crossref] [PubMed]

1971 (4)

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971); S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971); D. Silva, Appl. Opt. 10, 1980 (1971); A. Lohmann, D. Silva, Opt. Commun. 4, 326 (1972).
[Crossref] [PubMed]

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

O. Bryngdahl, J. Opt. Soc. Am. 61, 169 (1971).
[Crossref]

H. Dammann, G. Groh, M. Kock, Appl. Opt. 10, 1454 (1971).
[Crossref] [PubMed]

1969 (1)

R. F. Edgar, Optica Acta 16, 281 (1969).
[Crossref]

1967 (1)

1957 (1)

J. Cowley, A. Moodie, Proc. Phys. Soc. B70, 486 (1957); W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967); J. Winthrop, C. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
[Crossref]

1955 (1)

E. Lau, Optik, 12, 23 (1955).

1912 (1)

E. Waetzmann, Ann. Phys. 39, 1042 (1912).
[Crossref]

1836 (1)

F. Talbot, Phil. Mag. 9, 401 (1836).

Bryngdahl, O.

O. Bryngdahl, J. Opt. Soc. Am. 61, 169 (1971).
[Crossref]

O. Bryngdahl, Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 39–83.
[Crossref]

Cowley, J.

J. Cowley, A. Moodie, Proc. Phys. Soc. B70, 486 (1957); W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967); J. Winthrop, C. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
[Crossref]

Dammann, H.

Edgar, R. F.

R. F. Edgar, Optica Acta 16, 281 (1969).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48.

Groh, G.

Kock, M.

Lau, E.

E. Lau, Optik, 12, 23 (1955).

Lohmann, A.

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971); S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971); D. Silva, Appl. Opt. 10, 1980 (1971); A. Lohmann, D. Silva, Opt. Commun. 4, 326 (1972).
[Crossref] [PubMed]

Montgomery, W. D.

Moodie, A.

J. Cowley, A. Moodie, Proc. Phys. Soc. B70, 486 (1957); W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967); J. Winthrop, C. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
[Crossref]

Papoulis, A.

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw Hill, New York, 1968), p. 234.

Silva, D.

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971); S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971); D. Silva, Appl. Opt. 10, 1980 (1971); A. Lohmann, D. Silva, Opt. Commun. 4, 326 (1972).
[Crossref] [PubMed]

Talbot, F.

F. Talbot, Phil. Mag. 9, 401 (1836).

Waetzmann, E.

E. Waetzmann, Ann. Phys. 39, 1042 (1912).
[Crossref]

Ann. Phys. (1)

E. Waetzmann, Ann. Phys. 39, 1042 (1912).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Commun. (2)

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971); S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971); D. Silva, Appl. Opt. 10, 1980 (1971); A. Lohmann, D. Silva, Opt. Commun. 4, 326 (1972).
[Crossref] [PubMed]

A. Lohmann, D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

Optica Acta (1)

R. F. Edgar, Optica Acta 16, 281 (1969).
[Crossref]

Optik (1)

E. Lau, Optik, 12, 23 (1955).

Phil. Mag. (1)

F. Talbot, Phil. Mag. 9, 401 (1836).

Proc. Phys. Soc. (1)

J. Cowley, A. Moodie, Proc. Phys. Soc. B70, 486 (1957); W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967); J. Winthrop, C. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
[Crossref]

Other (4)

O. Bryngdahl, Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 39–83.
[Crossref]

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48.

Product of Edmund Scientific, Barrington, N.J.

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw Hill, New York, 1968), p. 234.

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Figures (12)

Fig. 1
Fig. 1

Talbot interferometer setup. Telecentric system is focused onto the object. The spatial filter plane is used to simplify the multiple-shearing interferences.

Fig. 2
Fig. 2

A lens under test. The gratings were adjusted to give a moiré fringe pattern in the background causing the fringes to rotate. The fringes indicate regions of equal phase derivative and are approximately spaced by df/z apart, where f is its focal length.

Fig. 3
Fig. 3

(a) Candle flame as object with zero-order filtering. Note that triple-shearing interferences are observed by counting three wicks. (b) Candle flame with first-order filtering. Double-shearing interferences are obtained as evidenced by two wicks of equal contrast.

Fig. 4
Fig. 4

Talbot interferometer with circular gratings.

Fig. 5
Fig. 5

Two-dimensional drawing of the first diffracted orders beyond grating G1. Positive orders diffract away from the axis; negative orders diffract toward the axis. Dotted lines represent rays from the lower half of the grating. The crosshatched region represents the inner cone to be neglected, and the shaded regions are the areas where self-imaging occurs.

Fig. 6
Fig. 6

The grating G2 is shifted to the left, and the self-image, now distorted by the object O, is shifted right. The rings overlap, and the deviation δ from the y axis is a measure of the aberrations.

Fig. 7
Fig. 7

Talbot interferometer modified to do spatial filtering. The telecentric system provides a filter plane for the insertion of the binary filter, and the image plane is conjugate to the object.

Fig. 8
Fig. 8

A diagram showing the m + n = +1 and m + n = −1 order intermingling in the back focal plane of our lens L2. m correspond to the diffracted order from G1 and n to those from G2. The positive order diffract away from the axis, and the negative diffract toward the axis. The parentheses refer to the order combinations (m,n).

Fig. 9
Fig. 9

Bifocal lens as seen in the field of observation. No spatial filtering is used.

Fig. 10
Fig. 10

Experiment using a test lens of 38-cm focal length in collimation setup. The deviations from a straight vertical line are a measure of the aberrations.

Fig. 11
Fig. 11

A solid ring object placed 973 mm from G2. Zero-order filtering was used, and one half of the field was blocked by an opaque screen.

Fig. 12
Fig. 12

A solid ring object placed 973 mm from G2. First-order filtering was used, and one half of the field was blocked by an opaque screen.

Equations (46)

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u ( x , y , - z 1 ) = exp ( - i k z 1 ) n = - C n exp ( 2 π i n x / d ) ,
u ( x , y , z ) = exp ( - i k z 1 ) ( n ) C n × exp ( i k { n x λ / d + ( z 1 + z ) [ 1 - ( n λ / d ) ] 1 2 } ) ,
u ( x , y , 0 - ) = exp ( - i k z 1 ) ( n ) C n × exp ( i k { n x λ / d + z 1 [ 1 - ( n λ / d ) 2 ] 1 2 } ) .
u ( x , y , 0 + ) = exp ( - i k z 1 ) ( n ) C n u 0 ( x , y ) × exp ( i k { n x λ / d + z 1 [ 1 - ( n λ / d ) 2 ] 1 2 } ) ,
u ˜ ( ν , μ ; ) + ) = u ( x , y , 0 + ) exp [ - 2 π i ( ν x + μ y ) ] d x d y ,
u ˜ ( ν , μ ; z 2 - ) = u ˜ ( ν , μ ; 0 + ) exp { i k z 2 [ 1 - λ 2 ( ν 2 + μ 2 ) ] 1 2 } = exp ( - i k z 1 ) ( n ) C n u ˜ 0 ( ν - n / d , μ ) × exp ( i k { z 1 [ 1 - ( n λ / d ) 2 ] 1 2 + z 2 [ 1 - λ 2 ( ν 2 + μ 2 ) ] 1 2 } ) ,
u ( x , y , z 2 + ) = u ( x , y , z 2 - ) ( m ) C m exp ( i π m ) exp ( 2 π i m x / d ) ,
u ˜ ( ν , μ ; z 2 + ) = exp ( - i k z 1 ) ( n ) ( m ) C n C m × exp ( i π m ) u ˜ 0 ( ν - ( n + m ) / d , μ ) exp [ i k ( z 1 [ 1 - ( n λ / d ) ] ) 2 ] 1 2 + z 2 { 1 - λ 2 [ ( ν - m / d ) 2 + μ 2 ] } 1 2 ) ] .
v ˜ ( ν , μ ) = u ˜ ( ν , μ ; z 2 + ) exp { - i k z 2 [ 1 - λ 2 ( ν 2 + μ 2 ) ] 1 2 } .
v ˜ ( ν , μ ) = ( n ) ( m ) C n C m exp ( i π m ) u ˜ 0 [ ν - ( n + m ) / d , μ ] × exp { i k [ z 2 λ 2 ν m / d - z 1 ( n λ / d ) 2 + z 2 ( m λ / d ) 2 ] } .
v ˜ ( ν ˜ , μ ) = ( n ) ( m ) C n C m exp ( i π m ) u ˜ 0 [ ν - ( n + m ) / d , μ ] × exp { 2 π i [ λ z 2 ν m / d - z 2 λ ( m 2 - n 2 ) / 2 d 2 ] } .
v ( x , y ) = ( n ) ( m ) C n C m u 0 ( x + m z 2 λ / d , y ) × exp { 2 π i [ m / 2 + ( m + n ) 2 z 2 / z T + ( n + m ) x / d ] } .
v ( x , y ) = ( m ) C m C - m exp ( i π m ) u 0 ( x + m Δ , y ) ,
v ( x , y ) = C 0 2 { u 0 ( x , y ) + 2 m 1 ( C m / C 0 ) 2 exp ( i π m ) × [ u 0 ( x + m Δ , y ) + u 0 ( x - m Δ , y ) ] } .
v ( x , y ) C 0 2 { u 0 ( x , y ) - ( C 1 / C 0 ) 2 [ u 0 ( x + Δ , y ) + u 0 ( x - Δ , y ) ] } .
v ( x , y ) u 0 ( x , y ) ( C 0 2 - 2 C 1 2 ) - 2 C 1 2 p = 2 u 0 ( p ) ( x , y ) Δ p / p ! ,
v ( x , y ) ( 1 / 20 ) u 0 ( x , y ) - ( 1 / 10 ) ( z λ / d ) 2 2 u 0 ( x , y ) / x 2 .
v ( x , y ) = C 0 C 1 exp { 2 π i [ x / d + z / z T ] } { u 0 ( x , y ) - u 0 ( x + z λ / d , y ) } .
{ } = p 1 u 0 ( p ) ( x , y ) ( z λ / d ) p 1 / p ! u 0 ( 1 ) ( x , y ) ( z λ / d ) .
u ( r , z ) = m = 0 m max C m exp ( i k γ m z ) J 0 ( α m r ) ,
C m = 2 / J 1 2 ( α m ) 0 1 u 0 ( r ) J 0 ( α m r ) r d r .
u ( r , 0 ) = u 0 ( r ) = m = 0 m max C m J 0 ( α m r )
G ( r ) = { ( m ) C m exp ( 2 π i m r / a ) , 4 a < r A , 0 , otherwise ,
J 0 ( α n r ) 1 / ( π α n r ) 1 2 cos ( α n r - π / 4 ) ; α n r > 25.
v ( r , φ ) ~ ( m ) ( n ) C m C n exp ( i π m ) × exp [ 2 π i ( m + n ) r / a ] u 0 ( r - m Δ , φ ) ,
v ( r , φ ) ~ C 0 2 { u 0 ( r , φ ) - ( C 1 / C 0 ) 2 [ u 0 ( r + Δ , φ ) + u 0 ( r - Δ , φ ) ] } .
( x + s / 2 ) 2 + y 2 = r m 2 ;             ( x - s / 2 ) 2 + y 2 = r m 2 .
v ( r , φ ) ~ ( m ) C m 2 exp ( i π m ) u 0 ( r - m Δ , φ ) .
v ( r , φ ) ~ C 0 2 { u 0 ( r , φ ) - ( C 1 / C 0 ) 2 [ u 0 ( r + δ , φ ) + u 0 ( r - Δ , φ ) ] } .
v ( r , φ ) 1 2 ( m ) ( n ) C m C n exp ( i π m ) × exp [ 2 π i ( m + n ) r / a ] u 0 ( r - m Δ , φ ) .
v ( r , φ ) ~ 1 2 C 0 C 1 [ u 0 ( r , φ ) - u 0 ( r - Δ , φ ) ] exp ( 2 π i r / a ) .
v ( r , φ ) 2 ~ C 0 4 u 0 ( r , φ ) 2 + C 1 4 [ u 0 ( r + Δ , φ ) 2 + u 0 ( r - Δ , φ ) 2 ] .
u ( r , φ , z ) ~ 1 λ z 0 0 2 π u ( r , z + ) × exp { i π [ r 2 + r 2 - 2 r r cos ( φ - φ ) ] / λ z } r d r d φ ,
u ( r , z ) ~ 2 π λ z ( m ) 4 a A r J 0 ( 2 π r r / λ z ) × exp { i π [ 2 r m / a + ( r 2 + r 2 ) / λ z ] d r .
r J 0 ( 2 π r r / λ z ) ~ ( z r / r ) 1 2 { exp [ i π ( 2 r r / λ z - 1 4 ) ] + exp [ - i π ( 2 r r / λ z - 1 4 ) ] }
u ( r , z ) ~ ( m ) C m 4 a A { exp i π [ ( r - r ) 2 λ z + 2 m r a ] + exp i π [ ( r + r ) 2 λ z + 2 m r a ] } d r .
u ( r , z ) ( m ) C m exp [ - i π λ z ( m / a ) 2 ] exp ( 2 π i m r / a ) .
u ( r , 0 - ) ( m ) C m exp [ i π λ z 1 ( m / a ) 2 ] exp ( 2 π i m r / a ) u 0 ( r , φ ) ,
u ( r , φ , z ) ~ f l ( z ) f u ( z ) 0 2 π u ( r , φ , 0 + ) × exp { i π [ r 2 + r 2 - 2 r r cos ( φ - φ ) ] / λ z } r d r d φ ,
u ( r , φ , z ) ~ 2 π ( m ) ( p ) C m exp [ i π ( m / a ) 2 λ z 1 ] exp [ i p ( φ - π / 2 ) ] × f l ( z ) f u ( z ) r J p ( 2 π r r / λ z ) u p ( r ) exp { i π [ ( r 2 + r 2 ) / λ z + ( 2 m r / a ) ] } d r .
2 π r J p ( 2 π r r / λ z ) ~ ( λ z r / r ) 1 2 cos [ ( 2 π r r / λ z ) - ( π / 4 ) - ( p π / 2 ) ] ,
u ( r , φ , z ) ~ ( m ) ( p ) C m exp [ i π ( m / a ) 2 λ z 1 ] exp [ i p ( φ - π / 2 ) ] × f l ( z ) f u ( z ) u p ( r ) { exp i π [ ( r - r ) 2 / λ z + 1 4 + p / 2 + 2 m r / a ] + exp i π [ ( r + r ) 2 / λ z - 1 4 - p / 2 + 2 m r / a ] } d r .
u ( r , φ , z ) ~ ( m ) C m exp [ - i π ( m / a ) 2 λ ( z - z 1 ) ] × u 0 ( r - m λ / a , φ ) exp ( 2 π i m r / a ) .
u ( r , φ , z 2 + ) = ( m ) ( n ) C m C n exp [ - i π λ ( m / a ) 2 ( z 2 - z 1 ) × u 0 ( r - m λ z 2 / a , φ ) ] exp [ 2 π i ( m + n ) r / a ] .
v ( r , φ ) u ( r , φ , z 2 + ) ~ ( m ) ( n ) C m C n × exp ( i π m ) exp [ 2 π i ( m + n ) r / a ] u 0 ( r - m λ 2 / a , φ ) ,
v ( r , φ ) ~ C 0 2 u 0 ( r , φ ) - C 1 2 [ u 0 ( r + Δ , φ ) + u 0 ( r - Δ , φ ) ] .

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