Abstract

A simple and economical method of recording and reconstructing the Fraunhofer subhologram array is presented. In the readout process a collecting lens is placed in the plane of the subhologram array. The field distribution in some planes of the readout system is found. The reconstruction processes with different types of waves are theoretically described and discussed.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Vander Lugt, Appl. Opt. 12, 1675 (1973).
    [CrossRef]
  2. Y. Takeda, Jpn. J. Appl. Phys. 11, 656 (1972).
    [CrossRef]
  3. D. C. J. Reid, P. Waterworth, in Proceedings of the International Symposium of Holography, Besançon, July (1970), paper
  4. P. Graf, M. Lang, Appl. Opt. 11, 1382 (1972).
    [CrossRef] [PubMed]
  5. K. Chałasińska-Macukow, T. Szoplik, in print.
  6. K. Chałasińska-Macukow, T. Szoplik, in Proceedings of the Third Polish-Czechoslovak Conference on Applied Optics, Nove Mesto, September (1976).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  9. C. S. Williams, Appl. Opt. 12, 872 (1973).
    [CrossRef] [PubMed]
  10. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series and Transformations. (Moscow, 1962), Eq. 3.937 (1,2).

1973

1972

1966

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Chalasinska-Macukow, K.

K. Chałasińska-Macukow, T. Szoplik, in print.

K. Chałasińska-Macukow, T. Szoplik, in Proceedings of the Third Polish-Czechoslovak Conference on Applied Optics, Nove Mesto, September (1976).

Goodman, J. W.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series and Transformations. (Moscow, 1962), Eq. 3.937 (1,2).

Graf, P.

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Lang, M.

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Reid, D. C. J.

D. C. J. Reid, P. Waterworth, in Proceedings of the International Symposium of Holography, Besançon, July (1970), paper

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series and Transformations. (Moscow, 1962), Eq. 3.937 (1,2).

Szoplik, T.

K. Chałasińska-Macukow, T. Szoplik, in Proceedings of the Third Polish-Czechoslovak Conference on Applied Optics, Nove Mesto, September (1976).

K. Chałasińska-Macukow, T. Szoplik, in print.

Takeda, Y.

Y. Takeda, Jpn. J. Appl. Phys. 11, 656 (1972).
[CrossRef]

Vander Lugt, A.

Waterworth, P.

D. C. J. Reid, P. Waterworth, in Proceedings of the International Symposium of Holography, Besançon, July (1970), paper

Williams, C. S.

Appl. Opt.

Jpn. J. Appl. Phys.

Y. Takeda, Jpn. J. Appl. Phys. 11, 656 (1972).
[CrossRef]

Proc. IEEE

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other

D. C. J. Reid, P. Waterworth, in Proceedings of the International Symposium of Holography, Besançon, July (1970), paper

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series and Transformations. (Moscow, 1962), Eq. 3.937 (1,2).

K. Chałasińska-Macukow, T. Szoplik, in print.

K. Chałasińska-Macukow, T. Szoplik, in Proceedings of the Third Polish-Czechoslovak Conference on Applied Optics, Nove Mesto, September (1976).

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

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

Recording the Fraunhofer hologram: O is the object, L is the lens of the focal length f placed in the object plane z = 0, H is the hologram, Δf is the distance from the hologram plane to the len’s focal plane, and θ is the angle between the wave vector of the reference wave and the z axis.

Fig. 2
Fig. 2

Configuration for the subhologram array readout with the collecting lens: H is the storage plate placed at the plane z = 0, L is the collecting lens just behind the hologram, O is the first image plane of the hologram–lens system a distance z1 from the lens, O′ is the second image plane, which is a Fourier transform plane, placed at z2 = f1 behind the collecting lens, and θ is the angle between the wave vector of the readout wave and the z axis.

Equations (21)

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

U ( x H y H ) = A B a x a y exp [ i k ( f Δ f ) ] f Δ f exp [ i k Δ f d 2 2 f ( f Δ f ) ] exp [ i k ( x H 2 + y H 2 ) 2 ( f Δ f ) ] m = M / 2 M / 2 n = N / 2 N / 2 exp [ i k Δ f ( m 2 a x 2 + n 2 a y 2 ) 2 f ( f Δ f ) ] exp [ i k sin θ x H ] exp [ i k ( f Δ f ) ( m a x x H + n a y y H ) ] [ l = 1 ( i Δ f d f ) l J l ( 2 π d ρ m n λ ( f Δ f ) ) ρ m n l ] = U ( x H y H ) exp [ i k ( x H 2 + y H 2 ) 2 ( f Δ f ) ] exp [ i k ( f Δ f ) ( m a x x H + n a y y H ) ] ,
C ( x H p , y H q ) = C exp [ i k sin θ ( x H p ) ] .
U p q ( x 0 y 0 ) = exp ( i k z ) i λ z exp [ i k 2 z ( x 0 2 + y 0 2 ) ] [ ( x H p ) 2 + ( y H q ) 2 ] 1 / 2 t U ( x H p , y H q ) S ( x H y H ) × C ( x H p , y H q ) exp [ i k ( x H 2 + y H 2 ) 2 z ] exp [ i k z ( x H x 0 + y H y 0 ) ] d x H d y H ,
1 / ( f Δ f ) + 1 / f 1 = 1 / z 1 ,
z 1 = f 1 ( f Δ f ) f 1 + f Δ f .
U p q ( x 0 y 0 z 1 ) = exp ( i k z 1 ) i λ z 1 exp [ i k ( x 0 2 + y 0 2 ) 2 z 1 ] exp [ i k ( p 2 + q 2 ) 2 ( f Δ f ) ] × exp [ i k ( m a x p + n a y q ) ( f Δ f ) ] circ [ ( x H p ) 2 + ( y H q ) 2 ] 1 / 2 t U ( x H p , y H q ) × exp { i k z 1 [ x H ( x 0 p z 1 f Δ f m a x z 1 f Δ f ) + y H ( y 0 q z 1 f Δ f n a y z 1 f Δ f ) ] } d x H d y H .
m Δ f a x f x H = ρ m n cos α ; n Δ f a y f y H = ρ m n sin α
x 0 p z 1 f Δ f m a x z 1 f Δ f = r m n cos β ; y 0 q z 1 f Δ f n a y z 1 f Δ f = r m n sin β ,
U p q ( r m n β ) = 2 π A B C a x a y exp [ i k ( z 1 f + Δ f ) ] f i λ z 1 Δ f exp [ i k Δ f d 2 2 f ( f Δ f ) ] exp [ i k ( p 2 + q 2 ) 2 ( f Δ f ) ] × m = M / 2 M / 2 n = N / 2 N / 2 exp [ i δ 1 ( r m n , β , Δ f , m , n , a x , a y , p , q ) ] [ l = 1 ( i Δ f d f ) l 0 t J l ( 2 π d ρ m n λ ( f Δ f ) ) ρ m n l ρ m n J 0 ( 2 π r m n ρ m n λ z 1 ) d ρ m n ] ,
δ 1 ( r m n , β , Δ f , m , n , a x , a y , p , q ) = k z 1 [ p ( x 0 p z 1 f Δ f ) + q ( y 0 q z 1 f Δ f ) ] + k 2 z 1 [ ( r m n cos β + p z 1 f Δ f + m a x z 1 f Δ f ) 2 + ( r m n sin β + q z 1 f Δ f + n a y z 1 f Δ f ) 2 ] k Δ f ( m 2 a x 2 + n 2 a y 2 ) 2 f ( f Δ f ) k z 1 ( m Δ f a x r m n cos β f + n Δ f a y r m n sin β f ) .
U p q ( r m n β ) = 2 π exp [ i k ( f 1 f + Δ f ) ] A B C a x a y f i λ f 1 Δ f m = M / 2 M / 2 n = N / 2 N / 2 exp [ i δ 2 ( r m n , β , Δ f , m , n , a x , a y , p , q ) ] × [ l = 1 ( i Δ f d f ) l 0 t exp [ i k ρ m n 2 2 ( f Δ f ) ] · J 1 ( 2 π d ρ m n λ ( f Δ f ) ) ρ m n l × ρ m n J 0 ( 2 π ρ m n r m n λ f 1 ) d ρ m n ] ,
x 0 ( f 1 / f ) m a x = r m n cos β y 0 ( f 1 / f ) n a y = r m n sin β
δ 2 ( r m n , β , Δ f , m , n , a x , a y , p , q ) = k 2 f 1 r m n 2 + k ( f 1 Δ f ) 2 f 2 ( m 2 a x 2 + n 2 a y 2 ) + k ( f 1 Δ f ) f f 1 r m n ( m a x cos β + n a y sin β ) k f 1 [ ( r m n cos β + f 1 f m a x ) p + ( r m n sin β + f 1 f n a y ) q ] .
D ( x H p , y H q ) = D exp [ i k sin θ ( x H p ) ] exp [ i k ( x H p ) 2 + ( y H q ) 2 2 R ] .
U p q ( r m n , β , f 1 ) = 2 A B D a x a y f π i λ f 1 Δ f exp [ i k ( f Δ f f 1 ) ] exp [ i k Δ f d 2 2 f ( f Δ f ) ] m = M / 2 M / 2 n = N / 2 N / 2 exp [ i δ 3 ( r m n , β , m , n , a x , a y , Δ f , p , q ) ] × [ l = 1 ( i Δ f d f ) l 0 t J 1 ( 2 π d ρ m n λ ( f Δ f ) ) ρ m n l ρ m n J 0 ( 2 π ρ m n r m n λ f 1 ) d ρ m n ] ,
δ 3 ( r m n , β , m , n , a x , a y , Δ f , p , q ) = k r m n 2 2 f 1 k f 1 [ p ( r m n cos β + m a x f 1 f Δ f ) + q ( r m n sin β + n a y f 1 f Δ f ) ] + k 2 [ f 1 f Δ f ( f Δ f ) f ( f Δ f ) 2 ( m 2 a x 2 + n 2 a y 2 ) ] + k r m n [ f 1 f Δ f ( f Δ f ) ] f 1 f ( f Δ f ) ( m a x cos β + n a y sin β ) ,
x 0 m a x f 1 f Δ f = r m n cos β ; y 0 n a y f 1 f Δ f = r m n sin β .
U G p q ( x H p , y H q , z H ) = U ( z H ) exp [ i Φ ( z H ) ] exp [ i k sin θ ( x H p ) ] exp [ i k ( x H p ) 2 + ( y H q ) 2 2 R ( z H ) ] exp [ ( x H p ) 2 + ( y H q ) 2 ω 2 ( z H ) ] ,
R ( z H ) = z H { z H 2 + ( 1 + z H R ) 2 π 2 ω 4 λ 2 z H 2 + [ ( z H R ) 2 + z H R ] π 2 ω 4 λ 2 } , ω 2 ( z H ) = ω 2 [ ( 1 + z H R ) 2 + z H 2 λ 2 π 2 ω 4 ] ,
U G p q ( r m n β f 1 ) = 2 π A B a x a y f U ( z H ) i λ f 1 Δ f exp [ i k ( f Δ f f 1 ) ] exp [ i Φ ( z H ) ] exp [ i k Δ f d 2 2 f ( f Δ f ) ] m = M / 2 M / 2 n = N / 2 N / 2 exp [ i δ 3 ( r m n β , m , n , a x , a y , Δ f , p , q ) ] exp [ 1 ω 2 ( z H ) ( m 2 Δ f 2 a x 2 f 2 + n 2 Δ f 2 a y 2 f 2 ) ] × [ l = 1 ( i Δ f d f ) l 0 t exp [ ρ m n 2 ω 2 ( z H ) ] J l ( 2 π d ρ m n λ ( f Δ f ) ) ρ m n l ρ m n J 0 [ i ( C m n + i D m n ) 1 / 2 ] d ρ m n ,
C m n = 4 Δ f 2 ρ m n 2 ω 4 ( z H ) f 2 ( m 2 a x 2 + n 2 a y 2 ) k 2 ρ m n 2 r m n 2 f 1 2 , D m n = 4 Δ f ρ m n 2 r m n k ω 2 ( z H ) f f 1 ( m a x cos β + n a y sin β ) .

Metrics