Abstract

A model for designing and analyzing complicated surface relief diffractive elements in the resonance domain is developed. It is based on subdividing the complicated diffractive element into many highly efficient local diffraction gratings whose surface relief modulations can be effectively characterized as slanted volume gratings for which closed form analytic solutions exist. The model is illustrated by finding in the resonance domain the local period, effective slant angle, and groove depth at each location on an off-axis cylindrical diffractive lens.

© 2007 Optical Society of America

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  1. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969).
    [CrossRef]
  2. E. Hasman and A. A. Friesem, "Analytic optimization for holographic optical elements," J. Opt. Soc. Am. A 6, 62-72 (1989).
    [CrossRef]
  3. M. A. Golub, I. N. Sissakian, and V. A. Soifer, "Infra-red radiation focusators," Opt. Lasers Eng. 15, 297-309 (1991).
    [CrossRef]
  4. M. A. Golub, "Generalized conversion from the phase function to the blazed surface-relief profile of diffractive optical elements," J. Opt. Soc. Am. A 16, 1194-1201 (1999).
    [CrossRef]
  5. M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982).
    [CrossRef]
  6. S. Peng and G. M. Morris, "Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings," J. Opt. Soc. Am. A 12, 1087-1096 (1995).
    [CrossRef]
  7. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 38, 304-313 (1999).
    [CrossRef]
  8. M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).
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  10. Y. Sheng, D. Feng, and S. Larochelle, "Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled-wave theory," J. Opt. Soc. Am. A 14, 1562-1568 (1997).
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  14. A. Schilling and H. P. Herzig, "Phase function encoding of diffractive structures," Appl. Opt. 39, 5273-5279 (2000).
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  15. J. M. Miller, N. Beaucoudrey, P. Chavel, J. Turunen, and Edmond Cambril, "Design and fabrication of binary slanted surface-relief gratings for a planar optical interconnection," Appl. Opt. 36, 5717-5727 (1997).
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  16. T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, "Electromagnetic field computation in semiconductor laser resonators," J. Opt. Soc. Am. A 23, 906-911 (2006).
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  17. D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," Appl. Opt. 32, 1154-1167 (1993).
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  18. M. A. Golub and A. A. Friesem, "Effective grating theory for the resonance domain surface relief diffraction gratings," J. Opt. Soc. Am. A 22, 1115-1126 (2005).
    [CrossRef]
  19. M. A. Golub, A. A. Friesem, and L. Eisen, "Bragg properties of efficient surface relief gratings in the resonance domain," Opt. Commun. 235, 261-267 (2004).
    [CrossRef]
  20. M. A. Golub and A. A. Friesem, "Analytical theory for efficient surface relief gratings in the resonance domain," in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H.John Caulfield, ed. (SPIE Press, 2004) Chap. 19, pp. 307-328.
  21. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
  22. M. G. Moharam, T. K. Gaylord, and R. Magnusson, "Criteria for Bragg regime diffraction by phase gratings," Opt. Commun. 32, 14-18 (1980).
    [CrossRef]

2006

2005

2004

M. A. Golub, A. A. Friesem, and L. Eisen, "Bragg properties of efficient surface relief gratings in the resonance domain," Opt. Commun. 235, 261-267 (2004).
[CrossRef]

2002

2001

2000

1999

1997

1995

1993

1992

1991

M. A. Golub, I. N. Sissakian, and V. A. Soifer, "Infra-red radiation focusators," Opt. Lasers Eng. 15, 297-309 (1991).
[CrossRef]

1989

1982

1980

M. G. Moharam, T. K. Gaylord, and R. Magnusson, "Criteria for Bragg regime diffraction by phase gratings," Opt. Commun. 32, 14-18 (1980).
[CrossRef]

1969

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969).
[CrossRef]

Beaucoudrey, N.

Cambril, E.

Cambril, Edmond

Chandezon, J.

Chavel, P.

Eisen, L.

M. A. Golub, A. A. Friesem, and L. Eisen, "Bragg properties of efficient surface relief gratings in the resonance domain," Opt. Commun. 235, 261-267 (2004).
[CrossRef]

Feng, D.

Friesem, A. A.

M. A. Golub and A. A. Friesem, "Effective grating theory for the resonance domain surface relief diffraction gratings," J. Opt. Soc. Am. A 22, 1115-1126 (2005).
[CrossRef]

M. A. Golub, A. A. Friesem, and L. Eisen, "Bragg properties of efficient surface relief gratings in the resonance domain," Opt. Commun. 235, 261-267 (2004).
[CrossRef]

E. Hasman and A. A. Friesem, "Analytic optimization for holographic optical elements," J. Opt. Soc. Am. A 6, 62-72 (1989).
[CrossRef]

M. A. Golub and A. A. Friesem, "Analytical theory for efficient surface relief gratings in the resonance domain," in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H.John Caulfield, ed. (SPIE Press, 2004) Chap. 19, pp. 307-328.

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, "Criteria for Bragg regime diffraction by phase gratings," Opt. Commun. 32, 14-18 (1980).
[CrossRef]

Golub, M. A.

M. A. Golub and A. A. Friesem, "Effective grating theory for the resonance domain surface relief diffraction gratings," J. Opt. Soc. Am. A 22, 1115-1126 (2005).
[CrossRef]

M. A. Golub, A. A. Friesem, and L. Eisen, "Bragg properties of efficient surface relief gratings in the resonance domain," Opt. Commun. 235, 261-267 (2004).
[CrossRef]

M. A. Golub, "Generalized conversion from the phase function to the blazed surface-relief profile of diffractive optical elements," J. Opt. Soc. Am. A 16, 1194-1201 (1999).
[CrossRef]

M. A. Golub, I. N. Sissakian, and V. A. Soifer, "Infra-red radiation focusators," Opt. Lasers Eng. 15, 297-309 (1991).
[CrossRef]

M. A. Golub and A. A. Friesem, "Analytical theory for efficient surface relief gratings in the resonance domain," in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H.John Caulfield, ed. (SPIE Press, 2004) Chap. 19, pp. 307-328.

Granet, G.

Hamamoto, T.

Hasman, E.

Herzig, H. P.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969).
[CrossRef]

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Lalanne, P.

Larochelle, S.

Lee, M. L.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969).
[CrossRef]

Li, L.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, "Criteria for Bragg regime diffraction by phase gratings," Opt. Commun. 32, 14-18 (1980).
[CrossRef]

Miller, J. M.

Moharam, M. G.

M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, "Criteria for Bragg regime diffraction by phase gratings," Opt. Commun. 32, 14-18 (1980).
[CrossRef]

Morris, G. M.

Nevière, M.

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).

Noponen, E.

Peng, S.

Plumey, J.-P.

Popov, E.

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).

Prather, D. W.

Raguin, D. H.

Rodier, J.

Schilling, A.

Sheng, Y.

Shi, S.

Shiono, T.

Sissakian, I. N.

M. A. Golub, I. N. Sissakian, and V. A. Soifer, "Infra-red radiation focusators," Opt. Lasers Eng. 15, 297-309 (1991).
[CrossRef]

Soifer, V. A.

M. A. Golub, I. N. Sissakian, and V. A. Soifer, "Infra-red radiation focusators," Opt. Lasers Eng. 15, 297-309 (1991).
[CrossRef]

Taghizadeh, M. R.

Takahara, K.

Tervo, J.

Turunen, J.

Vahimaa, P.

Vallius, T.

Vasara, A.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

IBM J. Res. Dev.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, "Criteria for Bragg regime diffraction by phase gratings," Opt. Commun. 32, 14-18 (1980).
[CrossRef]

M. A. Golub, A. A. Friesem, and L. Eisen, "Bragg properties of efficient surface relief gratings in the resonance domain," Opt. Commun. 235, 261-267 (2004).
[CrossRef]

Opt. Lasers Eng.

M. A. Golub, I. N. Sissakian, and V. A. Soifer, "Infra-red radiation focusators," Opt. Lasers Eng. 15, 297-309 (1991).
[CrossRef]

Opt. Lett.

Other

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).

M. A. Golub and A. A. Friesem, "Analytical theory for efficient surface relief gratings in the resonance domain," in The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, H.John Caulfield, ed. (SPIE Press, 2004) Chap. 19, pp. 307-328.

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Figures (10)

Fig. 1
Fig. 1

Relevant parameters and geometry of a beam-transforming DOE.

Fig. 2
Fig. 2

Diffraction efficiency as a function of grating period/wavelength when groove depth is optimized for the scalar domain, TE polarization. A diffraction order with highest diffraction efficiency is + 1 st for the resonance domain and 1 st in the scalar domain. The graph is calculated by a rigorous coupled-wave analysis[5] for a sawtooth surface relief grating with incidence angle 11.7 ° , λ = 1 μ m , n M = 1.45042 , and groove depth 2.17 μ m . (a) Full range of grating periods, (b) magnified part showing mainly the resonance domain.

Fig. 3
Fig. 3

Diffraction efficiency as a function of grating period/wavelength when groove depth is optimized for the resonance domain, TE polarization. Magnified part showing mainly the resonance domain. The graph is calculated by a rigorous coupled-wave analysis (RCWA) for a sawtooth surface relief grating with incidence angle 11.7 ° , λ = 1 μ m , n M = 1.45042 , and groove depth 2.47 μ m .

Fig. 4
Fig. 4

Geometric and optical parameters of a single groove of the surface relief grating.

Fig. 5
Fig. 5

Bragg TE and TM diffraction efficiency for surface relief gratings with triangular groove profiles as functions of the normalized groove depth h λ for the resonance domain and for the scalar domain. Calculations were done by the effective grating model. Effective slant angle is φ s = 15 ° , Λ λ = 1.1 , refractive index of grooves n M = 1.46 , n i = 1 .

Fig. 6
Fig. 6

Geometrical parameters of the off-axis one-dimensional focusing diffractive lens. (a) entire lens, (b) magnified one-dimensional cross section.

Fig. 7
Fig. 7

Variations of the normalized local grating period Λ λ of the diffractive off-axis cylindrical lens in the resonance domain as a function of the relative lateral coordinate 2 x D on the lens. Also shown are the lower and upper period bounds Λ l o w and Λ u p . Parameters are λ = 0.633 mm , n M = 1.457 , α = 45 ° , θ i n c = 15 ° , F = 50 mm , D = 25 mm , η B d = 100 % .

Fig. 8
Fig. 8

Variations of the local effective slant angle φ s of the resonance domain diffractive off-axis cylindrical lens, which provide the Bragg condition locally at each coordinate x point of the cylindrical lens, plotted as a function of the relative lateral coordinate 2 x D on the lens. Parameters are the same as in Fig. 7.

Fig. 9
Fig. 9

Variations of the relative groove peak position q c of the resonance domain diffractive off-axis cylindrical lens, which provide Bragg condition locally at each coordinate x point of the cylindrical lens, plotted as a function of the relative lateral coordinate 2 x D on the lens. Parameters are the same as in Fig. 7.

Fig. 10
Fig. 10

Variations of the required normalized local groove depths h o p t ( η B d ) λ of the resonance domain diffractive off-axis cylindrical lens, which provide the Bragg condition with η B d = 100 % locally at each coordinate x point of the cylindrical lens, plotted as a function of the relative lateral coordinate 2 x D on the lens. Also shown is the scalar depth h o p t S c a l a r . Parameters are λ = 0.633 mm , n M = 1.457 , α = 45 ° , θ i n c = 15 ° , F = 50 mm , D = 25 mm .

Tables (1)

Tables Icon

Table 1 Design Data for the Local Diffraction Gratings of the Resonance Domain Off-Axis One-Dimensional Focusing Diffractive Lens a

Equations (33)

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cos θ i n c = N i n c N n , cos θ o u t = N o u t N n .
N i n c = S i n c n i ,
N o u t = S o u t n i .
ν = 1 2 π φ = n i λ ( N o u t , N i n c , ) ,
Λ = 1 ν , s = ν ν , s = 1 ,
φ = k ( S o u t S i n c ) ,
n i ( sin θ i n c + sin θ o u t ) = λ Λ .
h o p t S c a l a r λ = 1 n M n i .
tan φ s = Λ h ( q c 0.5 ) = Λ h p s ,
n ¯ 2 = n i 2 + Δ n M 2 g ¯ ,
G 1 s = 0 1 g s ( χ ) exp ( i 2 π χ ) d χ ,
g ¯ = 0 1 g ( χ ) d χ , Δ n M 2 = n M 2 n i 2 .
n i sin θ i n c , B = λ 2 Λ tan φ s [ n ¯ 2 1 + tan 2 φ s ( λ 2 Λ ) 2 ] 1 2
η B = sin 2 ( 2 π h λ n ¯ κ ¯ 01 cos φ s c 0 s B ) ,
c 0 s B = ( 1 sin 2 θ s 1 ) 1 2 , sin θ s 1 = λ 2 n ¯ Λ cos φ s ,
κ ¯ 01 T E = δ n M 2 G 1 s , δ n M 2 = Δ n M 2 2 n ¯ 2 ,
κ ¯ 01 T M = κ ¯ 01 T E ( 1 2 sin 2 θ s 1 ) .
η S c a l a r = sin c 2 ( h h o p t S c a l a r 1 ) ,
Λ S W λ = 1 min ( n ¯ , n i ) + n i sin θ i n c , B .
κ ¯ 01 c 0 s B = γ max ,
Λ B λ = ε h i g h 1 4 cos φ s ( n ¯ Δ n s ) 1 2 ,
p = p s Δ p l o w = h tan φ s Λ Δ p l o w ,
n ¯ ( Q A tan φ s cos 2 φ s Q A 2 ) = λ 2 Λ A Λ A P h [ n ¯ 2 1 + ( Λ A p h ) 2 ( λ 2 Λ A ) 2 ] 1 2 ,
Q A = λ 2 n ¯ Λ A ξ m i s cos 3 φ s π Λ A h c 0 s B , ξ mis = 0.507 .
tan φ s = Λ λ [ ( n ¯ 2 n i 2 sin 2 θ i n c ) 1 2 ( n ¯ 2 n i 2 sin 2 θ o u t ) 1 2 ] ,
P s = h Λ tan φ s , q c = 0.5 + p s .
h o p t ( η B d ) λ = c 0 s B cos φ s 2 π n ¯ κ ¯ 01 [ π 2 ± ( π 2 arcsin η B d ) ] ,
φ ( x ) = 2 π λ { F [ 1 γ ( x ) ] x sin θ i n c } ,
γ ( x ) = 1 + 2 sin α x F + x 2 F 2 .
φ ( x ) 2 π λ { ( sin θ i n c + sin α ) x x 2 2 F } .
ν x = 1 λ [ sin θ i n c + sin θ o u t ] ,
Λ = 1 ν x ,
sin θ o u t = sin α + x F γ ( x ) .

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