Abstract

Diffractive optics allows the incorporation of several optical functions, e.g., wave shaping and focusing, in one element. A method suitable to calculate a diffractive phase element with this feature is described. Coding and quantization effects are analyzed. As an example an array generator with integrated focal power is designed.

© 1991 Optical Society of America

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References

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  1. O. Bryngdahl, F. Wyrowski, “Digital holography— computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
    [CrossRef]
  2. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  3. F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219 (1989).
  4. A. M. Steane, H. N. Rutt, “Diffraction calculations in the near field and the validity of the Fresnel approximation,” J. Opt. Soc. Am. A 6, 1809–1814 (1989).
    [CrossRef]
  5. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. A 71, 7–14 (1981).
    [CrossRef]
  6. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  7. M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital Fresnel holography,” Opt. Commun. 77, 4–8 (1990).
    [CrossRef]
  8. D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
    [CrossRef]
  9. A. VanderLugt, “Optimum sampling of Fresnel transforms,” Appl. Opt. 29, 3352–3361 (1990).
    [CrossRef] [PubMed]
  10. J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [CrossRef]
  11. W. J. Dallas, A. Lohmann, “Phase quantization in holograms,” Appl. Opt. 11, 192–194 (1972).
    [CrossRef] [PubMed]
  12. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
    [CrossRef]
  13. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  14. F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
    [CrossRef]
  15. R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578 (1991).
    [CrossRef]
  16. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  17. F. Wyrowski, “Digital holography as a useful model in diffractive optics,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507 (to be published).

1991

1990

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

A. VanderLugt, “Optimum sampling of Fresnel transforms,” Appl. Opt. 29, 3352–3361 (1990).
[CrossRef] [PubMed]

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital Fresnel holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography— computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

1989

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

A. M. Steane, H. N. Rutt, “Diffraction calculations in the near field and the validity of the Fresnel approximation,” J. Opt. Soc. Am. A 6, 1809–1814 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
[CrossRef]

1988

1984

D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
[CrossRef]

1981

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. A 71, 7–14 (1981).
[CrossRef]

1972

1970

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

1968

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

Bernhardt, M.

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital Fresnel holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

Bräuer, R.

Bryngdahl, O.

R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578 (1991).
[CrossRef]

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital Fresnel holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography— computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

Dallas, W. J.

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

Lohmann, A.

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
[CrossRef]

Rutt, H. N.

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Southwell, W. H.

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. A 71, 7–14 (1981).
[CrossRef]

Steane, A. M.

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

VanderLugt, A.

Weissbach, S.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Wyrowski, F.

R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578 (1991).
[CrossRef]

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital Fresnel holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography— computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

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

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

F. Wyrowski, “Digital holography as a useful model in diffractive optics,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507 (to be published).

F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219 (1989).

Appl. Opt.

Commun. ACM

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

IBM J. Res. Dev.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

J. Mod. Opt.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Coding and binarization in digital Fresnel holography,” Opt. Commun. 77, 4–8 (1990).
[CrossRef]

Prog. Opt.

D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography— computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

Other

F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219 (1989).

F. Wyrowski, “Digital holography as a useful model in diffractive optics,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507 (to be published).

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Figures (8)

Fig. 1
Fig. 1

Optical setup to generate the Fourier transform of a DPE(FT): lens L1 provides the focusing and lens L2 provides the phase correction.

Fig. 2
Fig. 2

Illustration of the substitution of a Fresnel lens and a DPE(FT) by a DPE(FRT).

Fig. 3
Fig. 3

Signal with the extent Δf = (Δfx, Δfy) positioned at xf = (xfyf) in an area δu−1 x δυ−1.

Fig. 4
Fig. 4

Flow diagram of an iterative quantization algorithm.

Fig. 5
Fig. 5

Row out of the calculated diffraction pattern generated by DPE(FRT)'s that were coded and quantized by different methods: (a) directly coded and quantized; (b) iteratively coded and directly quantized; (c) iteratively coded and quantized. Parameters: z = zmin, Δf = 64δx, xf = 0, Δx = 256δx, and Z = 3.

Fig. 6
Fig. 6

Row out of the amplitude of the quantization noise generated by DPE(FRT)'s that were calculated by different methods: (a) iteratively coded and directly quantized; (b) iteratively coded and quantized with noise reduction β = 1/10. For parameters see Fig. 5.

Fig. 7
Fig. 7

Part of the calculated diffraction pattern of an iteratively quantized (Z = 3) DPE(FRT). A row out of the pattern is shown in Fig. 5(c).

Fig. 8
Fig. 8

SNR of DPE(FRT)'s with three and four phase levels dependent on focal length z.

Tables (2)

Tables Icon

Table I Theoretical and Calculated Diffraction Efficiencies η ̅ th and η ̅ ca of DPE(FRT)'s Quantized into Z Levels

Tables Icon

Table II SNR of DPE(FRT)'s that were Quantized into Z Levels by Different Methodsa

Equations (51)

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g ̅ ( x ) α f ( x x f ) , x F ,
| g ̅ ( x ) | 2 α 2 | f ( x x f ) | 2 , x F ,
η ̅ = | g ̅ ( x ) | 2 x F | g ̅ ( x ) | 2 x D ,
SNR = | f ( x x 0 ) | 2 x F ( | g ̅ ( x ) | c | f ( x x f ) | ) 2 x F ,
c = | g ̅ ( x ) | | f ( x x f ) | x F | f ( x x f ) | 2 x F ,
W ( z , u ) = exp ( j π λ z u 2 ) ,
G ( z , u ) = G ( u ) W ( z , u )
w ( z , x ) = exp [ j π ( λ z ) 1 x 2 ]
g ( z , x ) = w ( z , x ) g ( x ) .
w * ( z , x ) = exp [ j π ( λ z ) 1 x 2 ]
f ( x x f ) = f ( x x f ) w * ( z , x ) .
F ( u ) = 1 f ( x x f )
G ( u ) = exp [ j Γ ( u ) ] = C F ( u ) .
G ( z , u ) = W ( z , u ) G ( u )
G ( z , u ) = exp { j [ Γ ( u ) π λ z u 2 ] } .
g ( z , x ) = G ( z , u ) = g ( x ) .
G r ( z , u ) = [ G ( z , u ) comb ( u , δ u ) ] * rect ( u , δ u )
min { δ x ̂ , δ ŷ } > 5 λ
π λ z ( Δ x ̂ / 2 ) 2 π λ z ( Δ x ̂ / 2 δ x ̂ ) 2 π
δ x ̂ max = δ ŷ max = max [ Δ x ̂ 2 , Δ ŷ 2 ] { max [ ( Δ x ̂ 2 ) 2 , ( Δ ŷ 2 ) 2 ] λ z } ½ .
z min = 1 λ Δ x ̂ · δ x ̂
z min > 5 ( Δ x ̂ + Δ ŷ ) .
F # = z | Δ x ̂ | ,
F # > 5 Δ x ̂ + Δ ŷ | Δ x ̂ | ,
F # > 5 .
Γ ( z , u ) = 2 π n Z for | Γ ( z , u ) 2 π n Z | 1 2 Z
G ̅ ( z , u ) = sinc ( 1 Z ) G ( z , u ) + υ 0 sinc ( 1 Z + ν ) [ G ( z , u ) ] Z ν + 1 ,
g ̅ ( z , x ) = sinc ( 1 Z ) g ( z , x ) + υ 0 q ν ( x ) .
q ν ( x ) = { sinc ( 1 Z + ν ) [ G ( z , u ) ] Z ν + 1 } ,
x ν = ( Z ν + 1 ) x f .
G ̅ ( z , u ) = Q K k = 1 K { C 1 X Q k } J G ( z , u ) .
G ( z , u ) = W ( z , u ) G ( u )
Q k G ( z , u ) = exp [ j Γ ̅ ( z , u ) ] ,
Γ ̅ ( z , u ) = 2 π n Z for | Γ ( z , u ) 2 π n Z | < ( k ) 2 Z ,
X g ̅ ( z , x ) = { | c f ( x x f ) | exp [ j γ ̅ ( x ) ] , x F , g ̅ ( z , x ) , x F ,
C F ( z , u ) = exp [ j arg { F ( z , u ) } ]
= w ( z , x ) W * ( z , u ) ,
1 = W ( z , u ) 1 w ( z , x ) .
G ̅ ( z , u ) = Q K k = 1 K { C W ( z , u ) 1 w * ( z , x ) × X w ( z , x ) W * ( z , u ) Q k } J G ( z , u ) .
G ̅ ( z , u ) = Q K k = 1 K { W ( z , u ) C 1 w * ( z , x ) w ( z , z ) X w * ( z , u ) Q k } J G ( z , u ) .
G ̅ ( z , u ) = Q k W ( z , u ) k = 1 K { C 1 X W * ( z , u ) Q k W ( z , u ) } J G ( u )
G d ( u ) = G ( u ) * comb ( u , Δ u ) ,
g d ( x ) = g ( x ) comb ( x , δ x )
G d ( z , u ) = W ( z , u ) G d ( u ) .
q ( x ) = ν 0 sinc ( 1 Z + ν ) [ G d ( z , u ) ] Z ν + 1
g ̅ ( z , x ) = sinc ( 1 Z ) g d ( z , x ) + q ( x ) .
F d = { x | x = ( n δ x , m δ y ) n , m | x | Δ f x 2 , | y | Δ f y 2 } .
x g ̅ d ( z , x ) = { | c f d ( x x f ) | exp [ j γ ̅ ( x ) ] , x F d , β g ̅ d ( z , x ) , x F F d , g ̅ d ( z , x ) , x F ,
η ̅ th
η ̅ ca
SNR x F

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