Abstract

Two iterative methods for the calculation of double-phase holograms (DPHs) are described. The calculation of DPHs by any of these methods allows one to qualitatively reconstruct the amplitude and phase of a wavefront on a format up to 50% of the spatial size of a reconstruction order (in the previous methods, this format was 35%). It is shown that the reconstruction can be realized in this case not only into the 0, +1, or 1 orders, as usual, but also into a half-order (+1/2  or  1/2). The diffraction efficiency of phase-only DPHs is also discussed. The results of numerical and optical experiments are presented.

© 2007 Optical Society of America

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References

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  1. L. P. Yaroslavsky and N. S. Merzlyakov, Methods of Digital Holography (Consultance Bureau, 1980).
  2. C. K. Hsueh and A. A. Sawchuk, "Computer-generated double-phase hologram," Appl. Opt. 17, 3874-3883 (1978).
    [Crossref] [PubMed]
  3. S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).
  4. J. M. Florence and R. D. Juday, "Full-complex spatial filtering with a phase mostly DMD," Proc. SPIE 1558, 487-498 (1991).
    [Crossref]
  5. V. Arrizón, "Improved double-phase computer-generated holograms implemented with phase-modulation devices," Opt. Lett. 27, 595-597 (2002).
    [Crossref]
  6. V. Arrizón and D. Sanchez-de-la-Llave, "Double-phase holograms implemented with phase-only spatial light modulators: performance evaluation and improvement," Appl. Opt. 27, 3436-3447 (2002).
    [Crossref]
  7. A. W. Lohmann and D. P. Paris, "Binary Fraunhofer holograms, generated by computer," Appl. Opt. 6, 1739-1748 (1967).
    [Crossref] [PubMed]
  8. H. Dammann, "Blazed synthetic phase-only holograms," Optik 31, 95-104 (1970).
  9. J. N. Mait and G. S. Himes, "Computer-generated holograms by means of a magnetooptic spatial light modulator," Appl. Opt. 28, 4879-4887 (1989).
    [Crossref] [PubMed]
  10. J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
    [Crossref]
  11. F. Wyrowski, "Diffractive optical elements: iterative calculation of quantized, blazed phase structures," J. Opt. Soc. Am. A 7, 961-969 (1990).
    [Crossref]
  12. A. V. Kuzmenko, "A method of the synthesis of a kinoform," UA patent 65295 (Ukraine), Byull. 1 (2006).
  13. F. Wyrowski, "Upper bound of the diffraction efficiency of diffractive phase elements," Opt. Lett. 16, 1915-1917 (1991).
    [Crossref] [PubMed]

2003 (1)

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

2002 (2)

V. Arrizón, "Improved double-phase computer-generated holograms implemented with phase-modulation devices," Opt. Lett. 27, 595-597 (2002).
[Crossref]

V. Arrizón and D. Sanchez-de-la-Llave, "Double-phase holograms implemented with phase-only spatial light modulators: performance evaluation and improvement," Appl. Opt. 27, 3436-3447 (2002).
[Crossref]

1991 (2)

J. M. Florence and R. D. Juday, "Full-complex spatial filtering with a phase mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[Crossref]

F. Wyrowski, "Upper bound of the diffraction efficiency of diffractive phase elements," Opt. Lett. 16, 1915-1917 (1991).
[Crossref] [PubMed]

1990 (2)

J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
[Crossref]

F. Wyrowski, "Diffractive optical elements: iterative calculation of quantized, blazed phase structures," J. Opt. Soc. Am. A 7, 961-969 (1990).
[Crossref]

1989 (1)

1978 (1)

1970 (1)

H. Dammann, "Blazed synthetic phase-only holograms," Optik 31, 95-104 (1970).

1967 (1)

Arrizón, V.

V. Arrizón, "Improved double-phase computer-generated holograms implemented with phase-modulation devices," Opt. Lett. 27, 595-597 (2002).
[Crossref]

V. Arrizón and D. Sanchez-de-la-Llave, "Double-phase holograms implemented with phase-only spatial light modulators: performance evaluation and improvement," Appl. Opt. 27, 3436-3447 (2002).
[Crossref]

Dammann, H.

H. Dammann, "Blazed synthetic phase-only holograms," Optik 31, 95-104 (1970).

Demoli, N.

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

Duerr, M.

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

Fagerholm, J.

J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
[Crossref]

Florence, J. M.

J. M. Florence and R. D. Juday, "Full-complex spatial filtering with a phase mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[Crossref]

Gruber, H.

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

Himes, G. S.

Hsueh, C. K.

Juday, R. D.

J. M. Florence and R. D. Juday, "Full-complex spatial filtering with a phase mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[Crossref]

Krueger, S.

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

Kuzmenko, A. V.

A. V. Kuzmenko, "A method of the synthesis of a kinoform," UA patent 65295 (Ukraine), Byull. 1 (2006).

Lohmann, A. W.

Mait, J. N.

Merzlyakov, N. S.

L. P. Yaroslavsky and N. S. Merzlyakov, Methods of Digital Holography (Consultance Bureau, 1980).

Paris, D. P.

Sanchez-de-la-Llave, D.

V. Arrizón and D. Sanchez-de-la-Llave, "Double-phase holograms implemented with phase-only spatial light modulators: performance evaluation and improvement," Appl. Opt. 27, 3436-3447 (2002).
[Crossref]

Sawchuk, A. A.

Taghizadeh, M. R.

J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
[Crossref]

Teiwes, S.

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

Turunen, J.

J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
[Crossref]

Vasara, A.

J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
[Crossref]

Wernicke, G.

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

Wyrowski, F.

Yaroslavsky, L. P.

L. P. Yaroslavsky and N. S. Merzlyakov, Methods of Digital Holography (Consultance Bureau, 1980).

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Turunen, J. Fagerholm, A. Vasara, and M. R. Taghizadeh, "Detour-phase kinoform interconnects: the concept and fabrication considerations," J. Opt. Soc. Am. 7, 1202-1208 (1990).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Optik (1)

H. Dammann, "Blazed synthetic phase-only holograms," Optik 31, 95-104 (1970).

Proc. SPIE (2)

S. Krueger, G. Wernicke, H. Gruber, N. Demoli, M. Duerr, and S. Teiwes, "New challenges for spatial modulators--laser beam splitting and beam shaping, reconstruction of digital holograms," Proc. SPIE 4291, 487-498 (2003).

J. M. Florence and R. D. Juday, "Full-complex spatial filtering with a phase mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[Crossref]

Other (2)

L. P. Yaroslavsky and N. S. Merzlyakov, Methods of Digital Holography (Consultance Bureau, 1980).

A. V. Kuzmenko, "A method of the synthesis of a kinoform," UA patent 65295 (Ukraine), Byull. 1 (2006).

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Figures (10)

Fig. 1
Fig. 1

Two-pixel SLM cell of a DPH.

Fig. 2
Fig. 2

Blazed Dammann-type grating playing the role of a spatial carrier in the DPH structure in the reconstruction into the 1 / 2 order (for τ 2 = δ ν / 2 ).

Fig. 3
Fig. 3

Structure of the object plane: S o , signal window; D, diffuser.

Fig. 4
Fig. 4

A, C, phase objects: apertures with transmittances exp [ i 2 π ( 0.016 + 0.0018 x 2 + 0.0009 y 2 ) ] and exp [ i 2 π ( 0.016 + 0.00214 x 2 + 0.00185 y 2 ) ] ; B, binary object.

Fig. 5
Fig. 5

Reconstructed phase for object A: (a) method 1 (size 63 × 63 samples, σ a = 4.7 × 10 3 , σ φ = 5.4 × 10 4 , η i h = 0.6 % , η o h = 2.7 % , Δ N = 33 , 100 iterations); (b) method 2 (size 57 × 63 samples, σ a = 3.7 × 10 3 , σ φ = 5.3 × 10 4 , η i h = 0.44 % , η o h = 52 % , α complex, 30 iterations); (c) iterationless method [5] (size 63 × 63 samples, σ a = 1.5 × 10 1 , σ φ = 6.8 × 10 1 , η i h = 0.7 % , η o h = 34.4 % ). The significant distortions of the reconstructed phase in (c) are clearly seen.

Fig. 6
Fig. 6

Method 1. Variance σ a versus number of iterations for various relative sizes of object A.

Fig. 7
Fig. 7

Method 1. Efficiency of a DPH versus number of iterations for various relative sizes of object A (solid curve, 10 % ; dashed curve, 25 % ; and dotted curve, 50 % ).

Fig. 8
Fig. 8

Method 1. Efficiency of a DPH versus relative size of object A.

Fig. 9
Fig. 9

Method 2. Efficiency of a DPH versus number of iterations (object A's size is 25 % ).

Fig. 10
Fig. 10

Reconstruction images in the 0 , 1 / 2 , and 1 orders: 1, first method; 2, second method; 3, Arrizón's method in view of Eq. (13).

Tables (1)

Tables Icon

Table 1 Wavefront Characteristics for the Reconstruction into a 1∕2 Order

Equations (37)

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H n m = A n m exp ( i Φ n m ) = exp ( i φ n m 1 ) + exp ( i φ n m 2 ) ,
φ n m 1 = ψ n m + Φ n m ,
φ n m 2 = ψ n m + Φ n m ,
ψ n m = arccos ( A n m 2 ) , 0 A n m 2 , 0 ψ n m π 2 .
φ 1 n m = ( 1 ) n + m ψ n m + Φ n m ,
φ n m 2 = ( 1 ) n + m ψ n m + Φ n m
H ( ν x , ν y ) = n , m rect ( ν y m δ ν τ 2 δ ν ) [ rect ( ν x ( n 1 / 4 ) δ ν τ 1 δ ν ) × exp ( i φ n m 1 ) + rect ( ν x ( n + 1 / 4 ) δ ν τ 1 δ ν ) × exp ( i φ n m 2 ) ] ,
rect ( z ) = { 1 , if   | z | 1 / 2 0 , otherwise ,
a ( x , y ) = H ( ν x , ν y ) exp { i 2 π [ ( x + x o ) ν x + y ν y ] } d ν x d ν y = c sinc 1 ( x ) sinc 2 ( y ) n , m ( exp { i [ ( x + x o ) ( n 1 / 4 ) + y m ] } exp ( i φ n m 1 ) + exp { i [ ( x + x o ) ( n + 1 / 4 ) + y m ] } exp ( i φ n m 2 ) ) ,
c = τ 1 τ 2 ( δ ν ) 2 ,   x = 2 π δ ν x ,   x o = 2 π δ ν x o ,
y = 2 π δ ν y ,   sinc 1 ( x ) = sinc [ τ 1 δ ν ( x + x 0 ) ] ,
sinc 2 ( y ) = sinc ( τ 2 δ ν y ) ,   sinc ( z ) = sin   π z π z ,
rect ( x Δ x ) rect ( y Δ x ) a ( x , y ) = c o u ( x , y ) ,
v ( x , y ) = u ( x , y ) [ sinc 1 ( x ) sinc 2 ( y ) ] 1
c o V p q = [ exp [ i ( φ p q 1 x o / 4 ) ] + exp [ i ( φ p q 2 + x o / 4 ) ] ] × exp ( i x o p ) + Q p q ,
Q p q = 1 4 n ( n p ) ( 1 ) n p [ exp [ i ( φ n q 1 x o / 4 ) ] n p 1 / 4 + exp [ i ( φ n q 2 + x o / 4 ) ] n p + 1 / 4 ] exp ( i x o n ) ,
p , q , n = N 2 ,   …   ,   N 2 .
W p q k = c o V p q Q p q ( φ n q k 1 1 , φ n q k 1 2 ) ,
φ p q 1 = 2 π P x p + π 2 P x + arccos ( 1 2 | W p q | ) + arg W p q ,
φ p q 2 = 2 π P x p π 2 P x arccos ( 1 2 | W p q | ) + arg W p q .
φ g r a t 1 = 2 π P x n + π 2 P x ,
φ g r a t 2 = 2 π P x n π 2 P x
π p + π 4 , π p π 4 ,   p = N 2 ,   …   ,   N 2 ,
φ n m 1 = 2 π P x n + π 2 P x + ( 1 ) n + m arccos ( A n m 2 ) + Φ n m ,
φ n m 2 = 2 π P x n π 2 P x ( 1 ) n + m arccos ( A n m 2 ) + Φ n m ,
u k ( x , y ) = α k ( x , y ) u o ( x , y ) ,
α k ( x , y ) = α k 1 ( x , y ) β k 1 ( x , y ) , ( k 2 ) ,
β k 1 ( x , y ) = | u o ( x , y ) | | a k 1 ( x , y ) | + ε ,
β k 1 ( x , y ) = u o ( x , y ) a k 1 ( x , y ) + ε .
u ( x , y ) = { α ( x , y ) u o ( x , y ) , if   x , y S o g ( x , y ) if   x , y D ,
v ( x , y ) = u ( x , y ) { sinc [ τ 1 δ ν ( x + x o ) ] sinc ( τ 2 δ ν ) } 1 ,
| x o | Δ x ;
φ n m 1 = 2 π P x n + π 2 P x + arccos ( 1 2 | V n m | ) + arg V n m ,
φ n m 2 = 2 π P x n π 2 P x arccos ( 1 2 | V n m | ) + arg V n m ,
n , m = N 2 ,   …   ,   N 2 ;
σ f = l , k | ξ l , k o b j e c t μ ξ l , k i m a g e | 2 l , k ( ξ l , k o b j e c t ) 2 ,
η upper = ( p q | V p q | ) 2 p q | V p q | 2 ,

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