Abstract

Far-field in-line holography has been studied in detail when one or both of the beams used for recording and reconstruction are Gaussian. The contrast of the high frequency interference fringes and hence their recordability have been investigated with a specific example of the object having circular cross section. For reconstruction the effects of uniform as well as Gaussian beams are studied and compared.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. J. Thompson, P. Dunn, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 102 (1980).
  2. S. L. Cartwright, P. Dunn, B. J. Thompson, Opt. Eng. 19, 727 (1980).
    [CrossRef]
  3. P. Dunn, J. M. Walls, Appl. Opt. 18, 263 (1979).
    [CrossRef] [PubMed]
  4. P. Dunn, B. J. Thompson, Opt. Eng. 21, 327 (1982).
    [CrossRef]
  5. I. Prikryl, C. M. Vest, Appl. Opt. 21, 2541 (1982).
    [CrossRef] [PubMed]
  6. J. S. Crane, P. Dunn, B. J. Thompson, J. Z. Knapp, J. Zeiss, Appl. Opt. 21, 2548 (1982).
    [CrossRef] [PubMed]
  7. C. S. Vikram, M. L. Billet, Optik 61, 427 (1982).
  8. C. S. Vikram, M. L. Billet, Optik 63, 109 (1983).
  9. G. Molesini, D. Bertani, M. Cetica, Opt. Acta 29, 479 (1982).
    [CrossRef]
  10. G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
    [CrossRef]
  11. Obviously, by particle here we mean particle, bubble, aerosol, etc., depending on the situation.
  12. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 6.
  13. Data sheet from Agfa-Gevaert;see also, S. Johansson, K. Biedermann, Appl. Opt. 13, 2288 (1974).
    [CrossRef] [PubMed]
  14. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 755.
  15. B. J. Thompson, J. Phys. E 7, 781 (1974).
    [CrossRef]

1983

C. S. Vikram, M. L. Billet, Optik 63, 109 (1983).

1982

G. Molesini, D. Bertani, M. Cetica, Opt. Acta 29, 479 (1982).
[CrossRef]

P. Dunn, B. J. Thompson, Opt. Eng. 21, 327 (1982).
[CrossRef]

I. Prikryl, C. M. Vest, Appl. Opt. 21, 2541 (1982).
[CrossRef] [PubMed]

J. S. Crane, P. Dunn, B. J. Thompson, J. Z. Knapp, J. Zeiss, Appl. Opt. 21, 2548 (1982).
[CrossRef] [PubMed]

C. S. Vikram, M. L. Billet, Optik 61, 427 (1982).

1980

B. J. Thompson, P. Dunn, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 102 (1980).

S. L. Cartwright, P. Dunn, B. J. Thompson, Opt. Eng. 19, 727 (1980).
[CrossRef]

1979

1976

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

1974

Bertani, D.

G. Molesini, D. Bertani, M. Cetica, Opt. Acta 29, 479 (1982).
[CrossRef]

Biedermann, K.

Billet, M. L.

C. S. Vikram, M. L. Billet, Optik 63, 109 (1983).

C. S. Vikram, M. L. Billet, Optik 61, 427 (1982).

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 755.

Cartwright, S. L.

S. L. Cartwright, P. Dunn, B. J. Thompson, Opt. Eng. 19, 727 (1980).
[CrossRef]

Cetica, M.

G. Molesini, D. Bertani, M. Cetica, Opt. Acta 29, 479 (1982).
[CrossRef]

Crane, J. S.

Dunn, P.

J. S. Crane, P. Dunn, B. J. Thompson, J. Z. Knapp, J. Zeiss, Appl. Opt. 21, 2548 (1982).
[CrossRef] [PubMed]

P. Dunn, B. J. Thompson, Opt. Eng. 21, 327 (1982).
[CrossRef]

S. L. Cartwright, P. Dunn, B. J. Thompson, Opt. Eng. 19, 727 (1980).
[CrossRef]

B. J. Thompson, P. Dunn, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 102 (1980).

P. Dunn, J. M. Walls, Appl. Opt. 18, 263 (1979).
[CrossRef] [PubMed]

Johansson, S.

Knapp, J. Z.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 6.

Molesini, G.

G. Molesini, D. Bertani, M. Cetica, Opt. Acta 29, 479 (1982).
[CrossRef]

Prikryl, I.

Thompson, B. J.

J. S. Crane, P. Dunn, B. J. Thompson, J. Z. Knapp, J. Zeiss, Appl. Opt. 21, 2548 (1982).
[CrossRef] [PubMed]

P. Dunn, B. J. Thompson, Opt. Eng. 21, 327 (1982).
[CrossRef]

B. J. Thompson, P. Dunn, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 102 (1980).

S. L. Cartwright, P. Dunn, B. J. Thompson, Opt. Eng. 19, 727 (1980).
[CrossRef]

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

B. J. Thompson, J. Phys. E 7, 781 (1974).
[CrossRef]

Tyler, G. A.

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

Vest, C. M.

Vikram, C. S.

C. S. Vikram, M. L. Billet, Optik 63, 109 (1983).

C. S. Vikram, M. L. Billet, Optik 61, 427 (1982).

Walls, J. M.

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 755.

Zeiss, J.

Appl. Opt.

J. Phys. E

B. J. Thompson, J. Phys. E 7, 781 (1974).
[CrossRef]

Opt. Acta

G. Molesini, D. Bertani, M. Cetica, Opt. Acta 29, 479 (1982).
[CrossRef]

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

Opt. Eng.

P. Dunn, B. J. Thompson, Opt. Eng. 21, 327 (1982).
[CrossRef]

S. L. Cartwright, P. Dunn, B. J. Thompson, Opt. Eng. 19, 727 (1980).
[CrossRef]

Optik

C. S. Vikram, M. L. Billet, Optik 61, 427 (1982).

C. S. Vikram, M. L. Billet, Optik 63, 109 (1983).

Proc. Soc. Photo-Opt. Instrum. Eng.

B. J. Thompson, P. Dunn, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 102 (1980).

Other

Obviously, by particle here we mean particle, bubble, aerosol, etc., depending on the situation.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 6.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 755.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Schematic diagram (not to scale) of the recording and reconstruction arrangement. Collimated beam for recording and reconstruction with the same wavelength is considered such that object and image distances are both equal (z).

Fig. 2
Fig. 2

Visibility of the fine interference fringes for the case ζ0 = η0 = 0 as a function of K for different values of r/ω. The dashed curve corresponds to the case when a uniform beam is used for the recording.

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

B exp ( ζ 2 + η 2 ω 2 ) ,
ψ ( x , y ) = i B λ z exp ( ikz ) + [ 1 A ( ζ , η ) ] exp ( ζ 2 + η 2 ω 2 ) × exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } d ζ d η ,
ψ ( x , y ) = i B λ z exp ( ikz ) + exp ( ζ 2 + η 2 ω 2 ) × exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } d ζ d η + i B λ z exp ( i k z ) exp ( ζ 0 2 + η 0 2 ω 2 ) + A ( ζ , η ) × exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } d ζ d η .
+ exp ( ζ 2 + η 2 ω 2 ) exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } d ζ d η = 2 π ω 2 2 i k ω 2 / z exp [ 2 ikz ( x 2 + y 2 ) 4 z 2 + ( k ω 2 ) 2 ] exp [ k 2 ω 2 ( x 2 + y 2 ) 4 z 2 + ( k 2 ω 2 ) 2 ] .
i λ z exp ( x 2 + y 2 ω 2 ) .
π ( ζ 2 + η 2 ) λ z 1
ψ ( x , y ) B exp ( i k z ) exp ( x 2 + y 2 ω 2 ) + i B λ z exp ( i k z ) exp ( ζ 0 2 + η 0 2 ω 2 ) × exp [ π i λ z ( ζ 0 2 + η 0 2 ) ] Ã ( x λ z , y λ z ) exp [ π i ( x 2 + y 2 ) λ z ] ,
à ( x λ z , y λ z ) = + A ( ζ , η ) exp { 2 π i [ ( x λ z ) ζ + ( y λ z ) η ] } d ζ d η
I ( x , y ) = | ψ ( x , y ) | 2 = B 2 ( exp ( 2 r 2 / ω 2 ) 2 λ z exp ( r 2 / ω 2 ) exp ( ζ 0 2 + η 0 2 ω 2 ) × { sin [ π ( r 2 + ζ 0 2 + η 0 2 ) / λ z ] Re à ( x λ z , y λ z ) + cos [ π ( r 2 + ζ 0 2 + η 0 2 ) / λ z ] Im à ( x λ z , y λ z ) } + 1 λ 2 z 2 exp ( 2 ζ 0 2 + 2 η 0 2 ω 2 ) à ( x λ z , y λ z ) à * ( x λ z , y λ z ) ) ,
r 2 = x 2 + y 2 .
A ( ζ , η ) = 1 for ( ζ 2 + η 2 ) 1 / 2 a = 0 for ( ζ 2 + η 2 ) 1 / 2 > a ,
à ( r λ z ) = π a 2 [ 2 J 1 ( 2 π a r λ z ) 2 π a r λ z ] .
I ( r ) B 2 = exp ( 2 r 2 / ω 2 ) 2 π a 2 λ z sin ( π r 2 λ z ) exp ( r 2 / ω 2 ) [ 2 J 1 ( k a r z ) k a r z ] + π 2 a 4 λ 2 z 2 [ 2 J 1 ( k a r z ) k a r z ] 2 .
I max ( r ) B 2 = exp ( 2 r 2 / ω 2 ) + 2 C r exp ( r 2 / ω 2 ) | 2 J 1 ( kar z ) kar z | + C r 2 | 2 J 1 ( kar z ) k a r z | 2 ,
I min ( r ) B 2 = exp ( 2 r 2 / ω 2 ) 2 C r exp ( r 2 / ω 2 ) | 2 J 1 ( kar z ) kar z | + C r 2 | 2 J 1 ( kar z ) kar z | 2 ,
V ( r ) = I max ( r ) I min ( r ) I max ( r ) + I min ( r ) = 2 C r exp ( r 2 / ω 2 ) | 2 J 1 ( kar z ) kar z | exp ( 2 r 2 / ω 2 ) + C r 2 | 2 J 1 ( kar z ) kar z | 2 .
V ( 0 ) = 2 C r 1 + C r 2 ,
K = C r | 2 J 1 ( kar z ) kar z | .
V ( r ) = 2 K exp ( r 2 / ω 2 ) K 2 + exp ( 2 r 2 / ω 2 ) .
V ( r ) | uniform beam = 2 K 1 + K 2 .
K < exp ( r 2 2 ω 2 ) ,
ψ ( μ , ν ) = i c λ z exp ( ikz ) + I ( x , y ) × exp { i k 2 z [ ( μ x ) 2 + ( ν y ) 2 ] } dxdy = i C B 2 λ z exp ( ikz ) ( I 1 + I 2 + I 3 + I 4 ) ,
I 1 = + exp [ 2 ( x 2 + y 2 ) ω 2 ] × exp { i k 2 z [ ( μ x ) 2 + ( ν y ) 2 ] } dxdy ,
I 2 = i λ z + exp ( x 2 + y 2 ω 2 ) A * ( ζ , η ) × exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } × exp { i k 2 z [ ( μ x ) 2 + ( ν y ) 2 ] } d ζ d η dxdy ,
I 3 = i λ z + exp ( x 2 + y 2 ω 2 ) A ( ζ , η ) × exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } × exp { i k 2 z [ ( μ x ) 2 + ( ν y ) 2 ] } d ζ d η dxdy ,
I 4 = 1 λ 2 z 2 + ( + A ( ζ , η ) × exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } d ζ d η × + A * ( ζ , η ) exp { i k 2 z [ ( x ζ ) 2 + ( y η ) 2 ] } d ζ d η ) × exp { i k 2 z [ ( μ x ) 2 + ( ν y ) 2 ] } dxdy .
I 1 = i λ z exp ( 2 R 2 ω 2 ) .
I 2 = i λ z A * ( μ , ν ) ,
I 3 = 1 2 exp ( R 2 ω 2 ) exp ( i k R 2 4 z ) Ã ( μ 2 λ z , ν 2 λ z ) ,
I 4 = i λ z à ( μ λ z , ν λ z ) à * ( μ λ z , ν λ z ) .
ψ ( μ , ν ) = C B 2 exp ( ikz ) [ exp ( 2 R 2 ω 2 ) Ã * ( μ , ν ) + i 2 λ z exp ( R 2 / ω 2 ) exp ( i k R 2 4 z ) Ã ( μ 2 λ z , ν 2 λ z ) + 1 λ 2 z 2 Ã ( μ λ z , ν λ z ) Ã * ( μ λ z , ν λ z ) ] .
ψ ( μ , ν ) = i c λ z exp ( i k z ) + exp ( x 2 + y 2 ω R 2 ) I ( x , y ) × exp { i k 2 z [ ( μ x ) 2 + ( ν y ) 2 ] } dxdy ,
ω e = ω / ( 1 + ω / ω R ) 1 / 2 .
I 4 exp ( R 2 / ω R 2 ) I 4 .
I 2 = i λ z + A * ( ζ , η ) exp { i k 2 z [ ( μ 2 ζ 2 ) + ( ν 2 η 2 ) ] } × exp { i k 2 z [ 2 ( ζ μ ) x + 2 ( η ν ) y ] } exp ( x 2 + y 2 ω 2 ) dxdyd ζ d η .
I 2 = π i ω 2 λ z + A * ( ζ , η ) exp { i k 2 z [ ( μ 2 ζ 2 ) + ( ν 2 η 2 ) ] } × exp { k 2 ω 2 4 z 2 [ ( μ ζ ) 2 + ( ν η ) 2 ] } d ζ d η .
k ω 2 z π 1 / 2 exp ( k 2 ω 2 μ 2 4 z 2 ) ,
k ω 2 z π 1 / 2 exp ( k 2 ω 2 ν 2 4 z 2 )
I 2 = i λ z A * ( μ , ν ) .
I 2 radius  c = π i ω 2 λ z 0 c 2 π exp ( k 2 ω 2 l 2 4 z 2 ) l d l = i λ z [ 1 exp ( k ω c 2 z ) ] .
exp ( k ω c 2 z )
exp ( k ω c 2 z ) = exp ( 1.2 π 2 × 0.74 ) 0.08 .
I 3 = i λ z exp ( i k R 2 4 z ) + A ( ζ , η ) ( + exp ( x 2 + y 2 ω 2 ) × exp { 2 i k 2 z [ ( x + ζ + μ 2 ) 2 + ( y + η + ν 2 ) 2 ] } dxdy ) × exp [ i k 2 z ( ζ μ + η ν ) ] d ζ d η ,
I 3 = 1 2 exp ( i k R 2 4 z ) + A ( ζ , η ) × exp { i λ z 8 π ω 2 [ ( ζ + μ ) 2 + ( η + ν ) 2 ] } × exp [ ( ζ + μ ) 2 + ( η + ν ) 2 ω 2 ] exp [ i k 2 z ( ζ μ + η ν ) ] d ζ d η .
z max = d 4 λ .
i λ z 8 π ω 4 [ ( ζ + μ ) 2 + ( η + ν ) 2 ] max i d 3 32 π ω 4 0 ,
exp [ ( ζ + μ ) 2 + ( η + ν ) 2 ω 2 ] exp ( μ 2 + ν 2 ω 2 ) .
I 3 = 1 2 exp ( R 2 / ω 2 ) exp ( i k R 2 4 z ) Ã ( μ 2 λ z , ν 2 λ z ) .

Metrics