Abstract

Holograms made of diffusely reflecting or diffusely illuminated objects can be scratched, spotted with dirt, and even broken into pieces without serious loss of information. This remarkable property is due to the redundancy introduced by diffuse illumination which, in effect, spreads information all over the hologram. Unfortunately, an unwanted by-product of diffuse illumination is speckle noise. This noise becomes more severe as hologram size is reduced. We show that, in order to obtain an acceptably high signal-to-noise ratio, the area of this type hologram must be more than 100 times the area needed to achieve a desired image resolution, making it prohibitively large for most data storage applications. In this paper we described a multibeam recording technique that produces redundant holograms that yield speckle free images of transparencies. We show that (1) achievable redundancy is equal to the ratio of actual hologram area to that area which just satisfies resolution requirements, (2) a two-dimensional phase grating provides a simple, efficient means for generating multiple beams, an optimized grating producing nine equally intense beams with 81% efficiency, (3) an optimized phase grating having a spatial period just equal to desired image resolution yields fourfold redundancy in holograms that are just large enough to encompass both side bands of the central (or zero order) beam, and (4) holograms having fourfold redundancy give reasonably good immunity to dust and scratches.

© 1968 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 53, 1377 (1963).
    [CrossRef]
  2. E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 54, 1295 (1964).
    [CrossRef]
  3. K. Exner, Sitzber. Akad. Wissensch. Wien 76-II, 522 (1877).
  4. L. I. Goldfischer, J. Opt. Soc. Amer. 55, 247 (1965).
    [CrossRef]
  5. J. A. Ratcliffe, Rep. Prog. Phys. 19, 188 (1956).
    [CrossRef]
  6. W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).
  7. J. Upatnieks, Appl. Opt. 6, 1905 (1967).
    [CrossRef] [PubMed]
  8. O. H. Schade, J. Soc. Motion Pic. Television Eng. 61, 97 (1953).
  9. R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
    [CrossRef]
  10. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964); see Eq. (45.29).
  11. L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
  12. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1963), pp. 115–120.

1967 (3)

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

J. Upatnieks, Appl. Opt. 6, 1905 (1967).
[CrossRef] [PubMed]

1965 (2)

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

L. I. Goldfischer, J. Opt. Soc. Amer. 55, 247 (1965).
[CrossRef]

1964 (1)

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 54, 1295 (1964).
[CrossRef]

1963 (1)

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 53, 1377 (1963).
[CrossRef]

1956 (1)

J. A. Ratcliffe, Rep. Prog. Phys. 19, 188 (1956).
[CrossRef]

1953 (1)

O. H. Schade, J. Soc. Motion Pic. Television Eng. 61, 97 (1953).

1877 (1)

K. Exner, Sitzber. Akad. Wissensch. Wien 76-II, 522 (1877).

Enloe, L. H.

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

Exner, K.

K. Exner, Sitzber. Akad. Wissensch. Wien 76-II, 522 (1877).

Goldfischer, L. I.

L. I. Goldfischer, J. Opt. Soc. Amer. 55, 247 (1965).
[CrossRef]

Leith, E. N.

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 54, 1295 (1964).
[CrossRef]

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 53, 1377 (1963).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964); see Eq. (45.29).

Martienssen, W.

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1963), pp. 115–120.

Powell, R. L.

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

Ratcliffe, J. A.

J. A. Ratcliffe, Rep. Prog. Phys. 19, 188 (1956).
[CrossRef]

Schade, O. H.

O. H. Schade, J. Soc. Motion Pic. Television Eng. 61, 97 (1953).

Spiller, S.

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

Stetson, K. A.

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

Upatnieks, J.

J. Upatnieks, Appl. Opt. 6, 1905 (1967).
[CrossRef] [PubMed]

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 54, 1295 (1964).
[CrossRef]

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 53, 1377 (1963).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

J. Opt. Soc. Amer. (4)

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 53, 1377 (1963).
[CrossRef]

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 54, 1295 (1964).
[CrossRef]

L. I. Goldfischer, J. Opt. Soc. Amer. 55, 247 (1965).
[CrossRef]

R. L. Powell, K. A. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
[CrossRef]

J. Soc. Motion Pic. Television Eng. (1)

O. H. Schade, J. Soc. Motion Pic. Television Eng. 61, 97 (1953).

Phys. Lett. (1)

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

Rep. Prog. Phys. (1)

J. A. Ratcliffe, Rep. Prog. Phys. 19, 188 (1956).
[CrossRef]

Sitzber. Akad. Wissensch. Wien (1)

K. Exner, Sitzber. Akad. Wissensch. Wien 76-II, 522 (1877).

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964); see Eq. (45.29).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1963), pp. 115–120.

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 (11)

Fig. 1
Fig. 1

Recording of speckle noise. A 1-mm2 diffuser was illuminated with a 6328-Å laser beam and the resultant speckle pattern was recorded on a photographic plate located 1 m from the diffuser.

Fig. 2
Fig. 2

Signal-to-noise ratio vs hologram size in diffused object beam holograms. I ¯ / ( I 2 ) ½ is the ratio of average intensity to rms fluctuation of intensity in the image. D/dmin is the ratio of the smallest dimension that can be resolved by the display to the smallest image dimension that can be reproduced by the hologram. H/hmin is the ratio of actual hologram size (i.e., the size that allows a resolution of dmin) to the size that will just allow a diffraction limited resolution of D to be realized.

Fig. 3
Fig. 3

Television images from diffused object beam holograms. Images from 1-mm and 2-mm holograms were projected directly on the photocathode of a standard vidicon TV camera. Photos shown here were taken directly from a TV monitor.

Fig. 4
Fig. 4

Setup for recording redundant, speckle free holograms. Diffraction grating generates multiple beams, thus enabling the same information to be recorded in different areas on the hologram.

Fig. 5
Fig. 5

Spatial distribution of sidebands in multiple beam hologram. The grating generates two first order carriers which are modulated by information contained on the transparency. By choosing a grating period equal to the resolution limit of the imaging system, redundancy is obtained without introducing background noise.

Fig. 6
Fig. 6

Spatial distribution of sidebands with a two-dimensional grating. Information is recorded on the hologram by nine spatial carriers, providing fourfold redundancy.

Fig. 7
Fig. 7

Images obtained from single object beam and multiple object beam holograms. Redundancy afforded by multiple beam recording tends to suppress the effects of imperfections of the hologram.

Fig. 8
Fig. 8

Effect of resolution of imaging system on background pattern. Image shown at left was photographed directly; resolution is limited by hologram size. Image shown at right was photographed from the face of a standard TV receiver; resolution of TV set was approximately equal to the resolution limit of the hologram.

Fig. 9
Fig. 9

Moiré effect introduced by grating. These images were photographed with a 1-mm × 1-mm aperture located where it obscured one-half of each first order image. Left photo was taken without the phase grating; right photo was taken with a 450-line pair/in. phase grating.

Fig. 10
Fig. 10

Coordinate systems.

Fig. 11
Fig. 11

Power spectrum of speckle noise for a square hologram.

Tables (2)

Tables Icon

Table I Performance of a Two-Dimensional Binary Absorption Grating

Tables Icon

Table II Performance of a Two-Dimensional Phase Grating Having Sinusoidal Modulation

Equations (47)

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

h min = λ z / D ,
D / d min = H / h min .
I ¯ / I ˜ 2 av 1 2 D / d min = H / h min .
I ¯ / I ˜ 2 av 1 2 = ( V W / N e v N e w ) 1 2 ( 1 / d min ) .
N e = 0 ( r ψ ¯ ) 2 d N ,
cos 2 [ ( π / 2 ) ( N v y / V ) ] ,             - V / N v < y < V / N v ,
[ r ψ ¯ ( N ) ] 2 = sin 2 π N / N v ( π N / N v ) 2 1 1 - ( N / N v ) 2 .
N e v = 0.541 N v ;             N e w = ( 4 / 3 ) N e v ,
I ¯ I 2 av 1 2 = 1.85 ( V / N v ) ( 1 / d min ) .
D = 1.85 ( V / N v ) .
H > ¯ 2 h min = 2 λ z / D = λ N / sin α ,
R = ( H / h min ) 2 .
T = n = - m = - T n T m exp ( 2 π i / Δ ) ( n x + m y ) ,
T n = 1 2 π 0 2 π 2 + e i x + e - i x 4 exp ( - i n x ) d x , T m = 1 2 π 0 2 π 2 + e i x + e - i x 4 exp ( - i m x ) d x .
T = p 2 n = - m = - ( sin n π p n π p ) exp ( i 2 n π Δ x ) × ( sin m π p m π p ) exp ( i 2 m π Δ y ) .
ϕ ( x ) = ϕ 0 + Δ ϕ cos [ 2 π ( x / Δ ) ] , ϕ ( y ) = ϕ 0 + Δ ϕ cos [ 2 π ( y / Δ ) ] ,
T = n = - m = - J n ( Δ ϕ ) exp ( i 2 n π Δ x ) J m ( Δ ϕ ) exp ( i 2 m π Δ y ) ,
t = ( 1.435 / 2 π ) λ / ( μ - 1 ) .
A ( x , y , z ) = - z 2 π A 0 ( x 0 , y 0 ) ( i k r 2 - 1 r 3 ) e i r k d x 0 d y 0 ,
r = [ ( x - x 0 ) 2 + ( y - y 0 ) 2 + z 2 ] 1 2 .
A ( x , y , z ) = - i k 2 π z A 0 ( x 0 , y 0 ) e i k r d x 0 d y 0 .
A 0 = 1 z u , v [ P 0 ( u , v ) Δ u Δ v ] 1 2 [ P ( x , y ) ] 1 2 exp ( i { k [ ( x - u ) 2 + ( y - v ) 2 + z 2 ] 1 2 + ϕ u v - 2 π ν t } ) .
A 1 = - i k 2 π z f u , v [ P 0 ( u , v ) Δ u Δ v ] 1 2 - x 0 x 0 d x - y 0 y 0 d y [ P ( x , y ) ] 1 2 exp ( i { k [ ( x - u ) 2 + ( y - v ) 2 + z 2 ] 1 2 + k [ ( ξ - x ) 2 + ( η - y ) 2 + f 2 ] 1 2 + ϕ u v - 2 π ν t } ) ,
A 2 = - ( i k ) 2 ( 2 π ) 2 f 2 z u , v [ P 0 ( u , v ) Δ u Δ v ] 1 2 - x 0 x 0 d x - y 0 y 0 d y × [ P ( x , y ) ] 1 2 - ξ 0 ξ 0 d ξ - η 0 η 0 d η exp ( i k { - [ ( x - u ) 2 + ( y - v ) 2 + z 2 ] 1 2 - [ ( ξ - x ) 2 + ( η - y ) 2 + f 2 ] 1 2 + [ ( ξ - x ) 2 + ( η - y ) 2 + f 2 ] 1 2 } + i ϕ u v - 2 π i ν t ) .
A 2 = k 2 4 π 2 f 2 z exp [ i ( k x 2 + y 2 2 f - 2 π ν t ) ] × u , v [ P 0 ( u , v ) Δ u Δ v ] 1 2 e i ϕ u v - ξ 0 ξ 0 d ξ - η 0 η 0 d η exp i k f { ξ [ u f f + z - x + z ξ 2 ( f + z ) ] + η [ v f f + z - y + z η 2 ( f + z ) ] } - x 0 x 0 d x - y 0 y 0 d y [ P ( x , y ) ] 1 2 exp { - i k ( f + z ) 2 f z × [ ( x - z ξ + f u f + z ) 2 + ( y - z η + f v f + z ) 2 ] } .
A 2 = k ( 1 - i ) 2 4 π f ( f + z ) u , v [ P 0 ( u , v ) Δ u Δ v ] 1 2 exp i × ( k x 2 + y 2 2 f - 2 π ν t + ϕ u v ) - ξ 0 ξ 0 d ξ - η 0 η 0 d η [ P ( z ξ + f u , f + z z η + f v f + z ) ] 1 2 × exp i k f { ξ [ u f f + z - x + z ξ 2 ( f + z ) ] + η × [ v f f + z - y + z η 2 ( f + z ) ] } .
B ( x , y ) = A 2 2 = B ¯ ( x , y ) + B ˜ ( x , y ) .
B ¯ ( x , y ) = C P ( x , y ) .
P ( x , y ) = 1 , x < x 0 and y < y 0 , P ( x , y ) = 0 , x > x 0 or y > y 0 .
B ¯ 4 ξ 0 η 0 / f 2
B ˜ 2 av = ( 4 ξ 0 η 0 / f 2 ) 2 = B ¯ 2 .
ω = ± ( k / f ) ( ξ - ξ ) ,             p = ± ( k / f ) ( ξ - ξ ) , Ω = ± ( k / f ) ( η - η ) ,             P = ± ( k / f ) ( η + η ) ,             for ξ ξ ,
B ˜ = 0 d ω - d Ω [ C ( ω , Ω ) cos ( ω x + Ω y ) + D ( ω , Ω ) sin ( ω x + Ω y ) ] .
B ˜ 2 av = 0 d ω - d Ω S ( ω , Ω ) ,
S ( ω , Ω ) = ( 2 π 2 / 4 x 0 y 0 ) [ C 2 ( ω , Ω ) + D 2 ( ω Ω ) ] .
ω 0 = ( 2 k ξ 0 / f ) , Ω 0 = ( 2 k η 0 / f ) ,
S ( ω , Ω ) d ω d Ω = 2 ( 4 ξ 0 η 0 f 2 ) 2 ( 1 - ω ω 0 ) ( 1 - Ω Ω 0 ) d ω ω 0 d Ω Ω 0 , for 0 < ω < ω 0 , - Ω 0 < Ω < Ω 0 , S ( ω , Ω ) = 0 for ω > ω 0 or Ω > Ω 0 .
ω 0 = 2 k sin θ x = 2 π / d min x , Ω 0 = 2 k sin θ y = 2 π / d min y .
q = ( ω 2 + Ω 2 ) 1 2 , y = q / ω 0 ,
S ( q ) d q = B ¯ 2 2 y ( π - 4 y + y 2 ) d y , 0 < y < 1 = B ¯ 2 2 y [ π - 2 - y 2 - 4 arcsec y + 4 ( y 2 - 1 ) 1 2 ] d y , 1 < y < ( 2 ) 1 2 .
S ( q ) d q = B ˜ 2 av 8 0 1 x ( 1 - x 2 ) 1 2 J 1 ( 2 π y x ) d x .
I ¯ = 4 X Y B ¯ = 16 X Y ξ 0 η 0 / f 2 .
I ˜ = - X X - Y Y B ˜ d x d y .
I ˜ 2 av = 256 X 2 Y 2 ξ 0 2 η 0 2 f 4 F ( 2 X ω 0 ) F ( 2 Y Ω 0 ) ,
F ( a ) = 4 { Si ( a ) a - 1 a 2 [ 1 - cos a - Ci ( a ) + ln ( γ a ) ] } = 1 - 4 a 2 3 · 4 · 4 ! + 4 a 4 5 · 6 · 6 ! - = 4 [ π 2 a - 1 + ln ( γ a ) a 2 + cos a a 4 + 4 sin a a 5 - ] ln ( γ ) = 0.5772
I ¯ I ˜ 2 av 1 2 = [ F ( 2 X ω 0 ) F ( 2 Y Ω 0 ) ] - 1 2 = [ F ( 4 π X d min x ) F ( 4 π Y d min y ) ] - 1 2 .
F ( 4 π X d min ) = 1 - 0.548 ( 2 X d min ) 2 + 0.288 ( 2 X d min ) 4 - 0.110 ( 2 X d min ) 4 + = d min 2 X [ 1 - 1 π 2 ( d min 2 X ) ( ln 2 X d min + 3.415 ) + 1 4 π 4 ( d min 2 X ) 3 cos ( 4 π X d min ) + ] .

Metrics