Abstract

A theory of diffraction tomography using incoherent radiation is presented. Diffraction gratings are used in the formation of Fourier components of the object distribution.

© 1986 Optical Society of America

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References

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  1. H. P. Baltes, Ed., Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  2. B. J. Chang, “Grating Based Interferometer,” Ph.D. Thesis, U. Michigan (1974);available from University Microfilms, Ann Arbor,MI, 74-25-170.
  3. G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
    [CrossRef]
  4. F. O. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
    [CrossRef]
  5. E. N. Leith, G. J. Swanson, “Optical Processing Techniques in Incoherent Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 38 (1983).
  6. G. D. Collins, “Achromatic Fourier Transformation Holography,” Appl. Opt. 20, 3109 (1981).
    [CrossRef] [PubMed]
  7. G. D. Collins, “Temporally and Spatially Incoherent Methods for Fourier Transform Holography and Optical Information Processing,” Ph.D. Dissertation, U. Michigan (1983);available from University Microfilms, Ann Arbor, MI.
  8. K. M. Jauch, H. P. Baltes, “Coherence of Radiation Scattered by Gratings Covered by a Diffuser,” Opt. Acta 28, 1013 (1981).
    [CrossRef]
  9. A. S. Glass, H. P. Baltes, “The Significance of Far-Zone Coherence for Sources or Scatters with Hidden Periodicity,” Opt. Acta 29, 169 (1982).
    [CrossRef]
  10. K. M. Jauch, H. P. Baltes, “Reversing-Wave-Front Interferometry of Radiation from a Diffusely Illuminated Phase Grating,”; Opt. Lett. 7, 127 (1982).
    [CrossRef] [PubMed]

1983 (1)

E. N. Leith, G. J. Swanson, “Optical Processing Techniques in Incoherent Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 38 (1983).

1982 (3)

1981 (2)

G. D. Collins, “Achromatic Fourier Transformation Holography,” Appl. Opt. 20, 3109 (1981).
[CrossRef] [PubMed]

K. M. Jauch, H. P. Baltes, “Coherence of Radiation Scattered by Gratings Covered by a Diffuser,” Opt. Acta 28, 1013 (1981).
[CrossRef]

1959 (1)

F. O. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Baltes, H. P.

A. S. Glass, H. P. Baltes, “The Significance of Far-Zone Coherence for Sources or Scatters with Hidden Periodicity,” Opt. Acta 29, 169 (1982).
[CrossRef]

K. M. Jauch, H. P. Baltes, “Reversing-Wave-Front Interferometry of Radiation from a Diffusely Illuminated Phase Grating,”; Opt. Lett. 7, 127 (1982).
[CrossRef] [PubMed]

K. M. Jauch, H. P. Baltes, “Coherence of Radiation Scattered by Gratings Covered by a Diffuser,” Opt. Acta 28, 1013 (1981).
[CrossRef]

Chang, B. J.

B. J. Chang, “Grating Based Interferometer,” Ph.D. Thesis, U. Michigan (1974);available from University Microfilms, Ann Arbor,MI, 74-25-170.

Collins, G. D.

G. D. Collins, “Achromatic Fourier Transformation Holography,” Appl. Opt. 20, 3109 (1981).
[CrossRef] [PubMed]

G. D. Collins, “Temporally and Spatially Incoherent Methods for Fourier Transform Holography and Optical Information Processing,” Ph.D. Dissertation, U. Michigan (1983);available from University Microfilms, Ann Arbor, MI.

Glass, A. S.

A. S. Glass, H. P. Baltes, “The Significance of Far-Zone Coherence for Sources or Scatters with Hidden Periodicity,” Opt. Acta 29, 169 (1982).
[CrossRef]

Jauch, K. M.

K. M. Jauch, H. P. Baltes, “Reversing-Wave-Front Interferometry of Radiation from a Diffusely Illuminated Phase Grating,”; Opt. Lett. 7, 127 (1982).
[CrossRef] [PubMed]

K. M. Jauch, H. P. Baltes, “Coherence of Radiation Scattered by Gratings Covered by a Diffuser,” Opt. Acta 28, 1013 (1981).
[CrossRef]

Leith, E. N.

E. N. Leith, G. J. Swanson, “Optical Processing Techniques in Incoherent Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 38 (1983).

G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
[CrossRef]

Swanson, G. J.

E. N. Leith, G. J. Swanson, “Optical Processing Techniques in Incoherent Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 38 (1983).

G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
[CrossRef]

Weinberg, F. O.

F. O. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Wood, N. B.

F. O. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

F. O. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Opt. Acta (2)

K. M. Jauch, H. P. Baltes, “Coherence of Radiation Scattered by Gratings Covered by a Diffuser,” Opt. Acta 28, 1013 (1981).
[CrossRef]

A. S. Glass, H. P. Baltes, “The Significance of Far-Zone Coherence for Sources or Scatters with Hidden Periodicity,” Opt. Acta 29, 169 (1982).
[CrossRef]

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

E. N. Leith, G. J. Swanson, “Optical Processing Techniques in Incoherent Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 38 (1983).

Other (3)

G. D. Collins, “Temporally and Spatially Incoherent Methods for Fourier Transform Holography and Optical Information Processing,” Ph.D. Dissertation, U. Michigan (1983);available from University Microfilms, Ann Arbor, MI.

H. P. Baltes, Ed., Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
[CrossRef]

B. J. Chang, “Grating Based Interferometer,” Ph.D. Thesis, U. Michigan (1974);available from University Microfilms, Ann Arbor,MI, 74-25-170.

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Figures (9)

Fig. 1
Fig. 1

Two-grating interferometer.

Fig. 2
Fig. 2

Three-grating interferometer with the test object.

Fig. 3
Fig. 3

Object and its sampling.

Fig. 4
Fig. 4

Frequency domain and transformation.

Fig. 5
Fig. 5

Optical configuration of the system.

Fig. 6
Fig. 6

Intensity transmittance of the test object: (A) low contrast square wave grating of spatial frequency of 1.97 line/mm, (B) overlay of two low contrast square wave gratings with spatial frequencies of 3.15 and 3.94 lines/mm.

Fig. 7
Fig. 7

Fourier transform of the fringe recording (at 15° to the axis).

Fig. 8
Fig. 8

Reconstructed slice B of the object by the dominant Fourier components recorded (with phase difference π/10).

Fig. 9
Fig. 9

Reconstruction of B by a similar dominant Fourier components after Fourier decomposition by the computer.

Equations (29)

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H = { h 1 , h 2 , , h n }
F = { f 1 , f 2 , , f m }
f i = E i ( h 1 , h 2 , , h n ) ,
E : H F .
E 1 : F H .
½ + ½ cos ( 2 π f 1 x ) ,
u = exp ( i { 2 π ( f 0 + k f 1 + m f 2 ) x π λ [ d 1 ( f 0 + k f 1 ) 2 + d 2 ( f 0 + k f 1 + m f 2 ) 2 ] } ) ,
I = 2 + 2 cos { 2 π [ ( k k ] f 1 + ( m m ) f 2 ] x 2 π λ [ ( k k ) d 1 f 1 + ( k k ) d 2 f 1 + ( m m ) d 2 f 2 ] f 0 π λ [ ( k 2 k 2 ) d 1 f 1 2 + d 2 f 2 2 ( k 2 k 2 ) + d 2 f 2 2 ( m 2 m 2 ) + 2 d d 2 f 1 f 2 ( k m k m ) ] } ,
I = 2 + 2 cos [ 4 π ( f 1 f 2 ) x 4 π λ d 1 f 0 f 1 4 π λ d 2 f 0 ( f 1 f 2 ) ] .
d 2 = d 1 1 + f 2 ( m m ) f 1 ( k k ) .
1 d 1 + 1 d 2 = 1 F g ,
F g = f 1 ( k k ) f 2 ( m m ) d 1 ,
d 2 = d 1 1 + ( m 2 m 2 ) ( k 2 k 2 ) [ f 2 f 1 ] 2 + 2 ( k m k m ) ( k 2 k 2 ) [ f 2 f 1 ] .
M = ( k k ) f 1 ( k k ) f 1 + ( m m ) f 2 = d 2 d 1 .
u upper = exp ( i 2 π f 0 x ) exp ( i 2 π f x ) exp [ i 2 π λ d 1 ( f 0 + f ) 2 ] × exp ( i 2 π f x ) exp [ i π λ ( d 2 + d 3 + Δ d ) ( f 0 + f f ) 2 ] ,
u lower = exp ( i 2 π f 0 x ) exp [ i π λ ( d 1 + d 2 ) f 0 2 ] × exp ( i 2 π f x ) exp [ i π λ ( d 3 + Δ d ) ( f 0 + f ) 2 ] .
u upper = exp ( i 2 π f 0 x ) exp [ i π λ d ( f 0 + f ) 2 ] × exp [ i π λ ( 2 d + Δ d ) f 0 2 ] ,
u lower = exp ( i 2 π f 0 x ) exp ( i π λ 2 d f 0 2 ) exp ( i 2 π f x ) × exp [ i π λ ( d + Δ d ) ( f 0 + f ) 2 ] .
I = | u upper + u lower | 2 = | exp ( i π λ Δ d f 0 2 ) + exp ( i 2 π f x ) exp [ i π λ Δ d ( f 0 + f ) 2 ] | 2 = 2 + 2 cos [ 2 π f x π λ Δ d ( f 2 + 2 f 0 f ) ] ,
T a ( x , z ) = a b + n = 0 N [ a n cos ( 2 π n f s x ) + a n sin ( 2 π n f s x ) ]
T a ( x , z ) = a b + n = 0 N a n cos ( 2 π f n x + ϕ n ) ,
a n { 1 + cos [ 2 π ( f ± f n ) x + 2 π λ f 0 ( z f n z f D f n ± z f n d f n ) π λ ( z + D z + d ) f n 2 π λ z f 2 2 π λ ( z + d ) f n f + ϕ n ] } ,
z = ( D z + d ) f n f ± f n .
a n m 2 cos ( 2 π n f s x + ϕ n m ) ,
a n m 2 { 1 + cos [ 2 π ( f ± f n ) x + 2 π λ f 0 ( z f n z f D f n ± z m f n d f n ) π λ ( z + D z m + d ) f n 2 π λ z f 2 2 π λ ( z + d ) f n f + ϕ n ] } ,
z = ( D z m + d ) f n f + f n .
b w 1 k w 11 + b w 2 k w 12 + + b w m k w 1 m + + b w M k w 1 M = D w ( x 1 ) , b w 1 k w j 1 + b w 2 k w j 2 + + b w m k w m j + + b w M k w j M = D w ( x j ) , b w 1 k w J 1 + b w 2 k w J 2 + + b w m k w J m + + b w M k w J M = D w ( x j ) ,
k w 11 k w 12 k w 1 M b w 1 D w ( x 1 ) k w 21 k w 22 k w 2 M b w 2 D w ( x 2 ) = k w j 1 k w j 2 k wjM b w j D w ( x j ) k w J 1 k w J 2 k wJM b w J D w ( x J ) ,
k wjm ( z w , z m , f nwm , x j ) = 1 + cos [ 2 π ( f ± f nwm ) x j + 2 π λ f 0 ( z w f nwm ± z w f D f nwm ± z m f nwm d f nwm ) ± π λ ( z w D ± z m d ) f nwm 2 ± π λ z w f 2 + 2 π λ ( z w + d ) f nwm f + ϕ nwm ] ,

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