Abstract

A new approach is developed to calculate diffraction efficiency for a dielectric grating with an arbitrary refractive index profile. By treating a one-dimensional grating as a segment of a virtual two-dimensional (2D) photonic crystal, we exploit a rigorous theory of photonic crystal refraction and calculate the diffraction efficiencies. We expand, analytically in many cases, the dielectric function of the grating into 2D Fourier series. We find the eigenmodes for the virtual photonic crystal, and then use these eigenmodes to match the boundary conditions by solving a set of linear equations. In two such simple steps, the diffraction efficiencies can be computed rigorously without slicing the grating into thin layers.

© 2006 Optical Society of America

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References

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  1. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
  2. M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982).
    [CrossRef]
  3. L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2581-2591 (1993).
    [CrossRef]
  4. N. Chateau and J. P. Hugonin, "Algorithm for the rigorous coupled-wave analysis of grating diffraction," J. Opt. Soc. Am. A 11, 1321-1331 (1994).
    [CrossRef]
  5. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995).
    [CrossRef]
  6. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  7. R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
    [CrossRef]
  8. M. R. Wang, G. J. Sonek, R. T. Chen, and T. Jannson, "Large fanout optical interconnects using thick holographic gratings and substrate wave propagation," Appl. Opt. 31, 236-249 (1992).
    [CrossRef] [PubMed]
  9. L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
    [CrossRef]
  10. L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
    [CrossRef]
  11. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
    [CrossRef]
  12. W. Jiang, R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
    [CrossRef]
  13. P. Lancaster, Lambda-Matrices and Vibrating Systems (Pergamon, 1966).
  14. W. Jiang, "Wavelength-selective micro- and nano-photonic devices for wavelength division multiplexing networks," Ph.D. dissertation (University of Texas at Austin, 2005).
  15. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, 1965).
  16. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968).

2005 (4)

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
[CrossRef]

W. Jiang, R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

W. Jiang, "Wavelength-selective micro- and nano-photonic devices for wavelength division multiplexing networks," Ph.D. dissertation (University of Texas at Austin, 2005).

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
[CrossRef]

1982 (1)

1969 (1)

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

1968 (1)

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968).

1966 (1)

P. Lancaster, Lambda-Matrices and Vibrating Systems (Pergamon, 1966).

1965 (1)

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, 1965).

Chateau, N.

Chen, R. T.

L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
[CrossRef]

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

W. Jiang, R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

M. R. Wang, G. J. Sonek, R. T. Chen, and T. Jannson, "Large fanout optical interconnects using thick holographic gratings and substrate wave propagation," Appl. Opt. 31, 236-249 (1992).
[CrossRef] [PubMed]

R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
[CrossRef]

Chen, X.

L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
[CrossRef]

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Gu, L.

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
[CrossRef]

Howley, B.

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

Hugonin, J. P.

Jannson, T.

M. R. Wang, G. J. Sonek, R. T. Chen, and T. Jannson, "Large fanout optical interconnects using thick holographic gratings and substrate wave propagation," Appl. Opt. 31, 236-249 (1992).
[CrossRef] [PubMed]

R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
[CrossRef]

Jiang, W.

W. Jiang, "Wavelength-selective micro- and nano-photonic devices for wavelength division multiplexing networks," Ph.D. dissertation (University of Texas at Austin, 2005).

L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
[CrossRef]

W. Jiang, R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kogelnik, H.

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

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Lancaster, P.

P. Lancaster, Lambda-Matrices and Vibrating Systems (Pergamon, 1966).

Li, L.

Liu, J.

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

Lu, H.

R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
[CrossRef]

Lu, X.

W. Jiang, R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Moharam, M. G.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Pommet, D. A.

Robinson, D.

R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
[CrossRef]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968).

Shi, Z.

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

Sonek, G. J.

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Wang, M. R.

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, 1965).

Appl. Opt. (1)

Appl. Phys. Lett. (3)

R. T. Chen, H. Lu, D. Robinson, and T. Jannson, "Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing," Appl. Phys. Lett. 59, 1144-1146 (1991).
[CrossRef]

L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, "Bandwidth-enhanced volume grating for dense wavelength-division multiplexer using a phase-compensation scheme," Appl. Phys. Lett. 86, 181103 (2005).
[CrossRef]

L. Gu, X. Chen, W. Jiang, and R. T. Chen, "A solution to the fringing-field effect in liquid crystal based high-resolution switchable gratings," Appl. Phys. Lett. 87, 201106 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Phys. Rev. B (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

W. Jiang, R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Other (4)

P. Lancaster, Lambda-Matrices and Vibrating Systems (Pergamon, 1966).

W. Jiang, "Wavelength-selective micro- and nano-photonic devices for wavelength division multiplexing networks," Ph.D. dissertation (University of Texas at Austin, 2005).

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, 1965).

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968).

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Figures (4)

Fig. 1
Fig. 1

Concept of a virtual photonic crystal, where we treat a grating as a single layer of a 2D photonic crystal. (a) A surface-relief grating with an arbitrary profile. The origin of the coordinate system is located on the farthest extrusion of the surface. Three regions are marked by I, II, and III. (b) The corresponding virtual 2D photonic crystal, where a unit cell is indicated by dotted lines. For a rectangular lattice a 1 = Λ , a 2 = d . The layer actually present in the grating problem is enclosed by dashed lines.

Fig. 2
Fig. 2

Diffraction efficiencies for a sinusoidal grating, where ϵ I = 1 , ϵ III = 2.5 , λ = Λ , and θ = 30 ° . One reflection order R 0 and three transmission orders T 1 , T 0 , T 1 are shown. The crosses represent the published results from Fig. 4 of Ref. [2]. Our results are in excellent agreement with those of Ref. [2].

Fig. 3
Fig. 3

Diffraction efficiencies for a sawtooth grating, where ϵ I = 1 , ϵ III = 2.5 , λ = Λ , and θ = 30 ° . One reflection order and three transmission orders are shown. The crosses represent the data from Fig. 7 of Ref. [2]. The agreement is excellent.

Fig. 4
Fig. 4

Alternative choice of the virtual photonic crystal. (a) Region II has to be redefined (now between y = 0 and y = a 2 ). And the backside boundary is now located at y = a 2 . (b) The corresponding 2D photonic crystal. A unit cell is indicated by dotted lines. The layer actually present in the grating problem is enclosed by dashed lines.

Tables (1)

Tables Icon

Table 1 Comparison of Diffraction Efficiencies with Those in Ref. [2]

Equations (31)

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ε ( G ) = 1 A d x d y exp ( i G x ) ϵ ( x ) ,
G l , m = l b 1 + m b 2 .
2 E ( x ) + ω 2 ε ( x ) E ( x ) = 0 ,
E ( x ) = exp ( i k x ) G E ( G ) exp ( i G x ) ,
[ ( k x + G x ) 2 + ( k y + G y ) 2 ] E ( G ) + ω 2 G ε ( G G ) E ( G ) = 0 .
( k y 2 [ I ] + 2 k y [ B ] + [ C ] ) [ E ] = 0 ,
[ B I D B ] [ E Z ] = k y ( E Z ) ,
B μ ν = δ μ ν ( G μ ) y ,
C μ ν = δ μ ν { ( G μ ) y 2 + [ k x + ( G μ ) x ] 2 } ω 2 ε ( G μ ν ) ,
D μ ν = ω 2 ε ( G μ ν ) [ k x + ( G μ ) x ] 2 δ μ ν ,
b 2 2 Re k y < b 2 2 ,
k y ± i [ k x + ( G l 0 ) x ] 2 ε ( 0 ) ω 2 ,
l = ± L 0 , ± ( L 0 + 1 ) , , ± L ,
E I ( x ) = exp ( i q 0 x ) + l r l exp ( i p l x ) ,
E II ( x ) = s l , m c s E s ( G l m ) exp [ i ( k x + l b 1 ) x + i ( k y ( s ) + m b 2 ) y ] ,
E III ( x ) = l t l exp ( i v l x ) ,
p l = ( q 0 x + l b 1 ) e x ε I ω 2 ( q 0 x + l b 1 ) 2 e y ,
v l = ( q 0 x + l b 1 ) e x + ε III ω 2 ( q 0 x + l b 1 ) 2 e y ;
δ l , 0 + r l = s c s m E s ( G l m ) ,
q 0 y δ l , 0 + p l , y r l = s c s m [ k y ( s ) + m b 2 ] E s ( G l m ) ,
exp ( i v l , y d ) t l = s c s exp [ i k y ( s ) d ] m E s ( G l m ) ,
v l , y exp ( i v l , y d ) t l = s c s exp [ i k y ( s ) d ] m [ k y ( s ) + m b 2 ] E s ( G l m ) .
c ̃ s = { c s exp [ i k y ( s ) d ] Im k y ( s ) < 0 c s otherwise } .
δ l , 0 + r l = s + c ̃ s + m E s + ( G l m ) + s c ̃ s exp [ i k y ( s ) d ] m E s ( G l m ) ,
q 0 y δ l , 0 + p l , y r l = s + c ̃ s + m [ k y ( s + ) + m b 2 ] E s + ( G l m ) + s c ̃ s exp [ i k y ( s ) d ] m [ k y ( s ) + m b 2 ] E s ( G l m ) ,
t ̃ l = s + c ̃ s + exp [ i k y ( s + ) d ] m E s + ( G l m ) + s c ̃ s m E s ( G l m ) ,
v l , y t ̃ l = s + c ̃ s + exp [ i k y ( s + ) d ] m [ k y ( s + ) + m b 2 ] E s + ( G l m ) + s c ̃ s m [ k y ( s ) + m b 2 ] E s ( G l m ) ,
reflection : r l 2 p l , y q 0 y ,
transmission : t l 2 v l , y q 0 y .
ϵ ( G l m ) = { 1 2 ( ϵ I + ϵ III ) δ l , 0 + 1 4 ( ϵ III ϵ I ) ( δ l , 1 + δ l , 1 ) m = 0 ϵ III ϵ I 2 π m [ i δ l , 0 + ( 1 ) m i l 1 J l ( m π ) ] m 0 } ,
ϵ ( G l m ) = { 1 2 ( ϵ I + ϵ III ) l = m = 0 ( ϵ III ϵ I ) ( 2 π i l ) l 0 , m = 0 ( ϵ III ϵ I ) 2 π i m [ δ l , 0 δ l + m , 0 ] m 0 } .

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