Abstract

Diffraction of light by periodic gratings is analyzed with a characteristic-matrix formalism based on a rigorous coupled-wave approach. This formalism is particularly convenient for modeling the diffraction by nonuniform periodic structures. In order to overcome numerical difficulties that are due to inhomogeneous eigenmodes, we propose a new algorithm that remains stable for gratings of any thickness. We obtain the stability by distinguishing in the computation the growing and the decaying inhomogeneous modes. Numerical examples and comparisons with previous results are given.

© 1994 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  2. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  3. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  5. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  7. M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
    [CrossRef]
  8. X. Y. Chen, “Using the finite element method to solve coupled wave equations in volume holograms,” J. Mod. Opt. 35, 1383–1391 (1988).
    [CrossRef]
  9. T. K. Gaylord, M. G. Moharam, “Coupled-wave analysis of reflection gratings,” Appl. Opt. 20, 240–244 (1981).
    [CrossRef] [PubMed]
  10. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  12. M. Nevière, “Sur un formalisme différentiel pour les problémes de diffraction dans le domaine de résonance: application à l’étude des réseaux optiques et de diverses structures périodiques,” thèse de doctorat (Université d’Aix-Marseille, Aix-Marseille, France, 1975).
  13. D. E. Tremain, K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,” J. Opt. Soc. Am. 68, 775–783 (1978).
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  14. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
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  15. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
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  16. A. K. Cousins, S. C. Gottschalk, “Application of the impedance formalism to diffraction gratings with multiple coating layers,” Appl. Opt. 29, 4268–4271 (1990).
    [CrossRef] [PubMed]
  17. D. Kermisch, “Non-uniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1413 (1969).
    [CrossRef]
  18. I. R. Redmond, “Holographic optical elements in dichromated gelatin,” Ph.D. dissertation (Heriot-Watt University, Edinburgh, UK, 1989).
  19. E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
    [CrossRef]
  20. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  21. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1987), pp. 36–70.

1991

1990

1988

X. Y. Chen, “Using the finite element method to solve coupled wave equations in volume holograms,” J. Mod. Opt. 35, 1383–1391 (1988).
[CrossRef]

1985

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983

1982

1981

1980

1978

1977

1969

D. Kermisch, “Non-uniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1413 (1969).
[CrossRef]

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

Awada, K. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1987), pp. 36–70.

Chang, K. C.

Chen, X. Y.

X. Y. Chen, “Using the finite element method to solve coupled wave equations in volume holograms,” J. Mod. Opt. 35, 1383–1391 (1988).
[CrossRef]

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Cousins, A. K.

Gaylord, T. K.

Glytsis, E. N.

Gottschalk, S. C.

Kermisch, D.

Kogelnik, H.

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

Kong, J. A.

Mei, K. K.

Moharam, M. G.

Nevière, M.

M. Nevière, “Sur un formalisme différentiel pour les problémes de diffraction dans le domaine de résonance: application à l’étude des réseaux optiques et de diverses structures périodiques,” thèse de doctorat (Université d’Aix-Marseille, Aix-Marseille, France, 1975).

Pai, D. M.

Redmond, I. R.

I. R. Redmond, “Holographic optical elements in dichromated gelatin,” Ph.D. dissertation (Heriot-Watt University, Edinburgh, UK, 1989).

Shah, V.

Solymar, L.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Soskin, M. S.

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Tamir, T.

Taranenko, V. B.

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Tremain, D. E.

Vasnetsov, M. V.

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1987), pp. 36–70.

Appl. Opt.

Bell Syst. Tech. J.

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

J. Mod. Opt.

X. Y. Chen, “Using the finite element method to solve coupled wave equations in volume holograms,” J. Mod. Opt. 35, 1383–1391 (1988).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Proc. IEEE

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

M. Nevière, “Sur un formalisme différentiel pour les problémes de diffraction dans le domaine de résonance: application à l’étude des réseaux optiques et de diverses structures périodiques,” thèse de doctorat (Université d’Aix-Marseille, Aix-Marseille, France, 1975).

I. R. Redmond, “Holographic optical elements in dichromated gelatin,” Ph.D. dissertation (Heriot-Watt University, Edinburgh, UK, 1989).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1987), pp. 36–70.

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Figures (14)

Fig. 1
Fig. 1

Grating and incident wave geometry.

Fig. 2
Fig. 2

Representation of a surface relief grating as a stack of planar volume elementary gratings with binary index.

Fig. 3
Fig. 3

Example of a compound periodic structure that can be represented by a characteristic-matrix formalism, including surface relief gratings, volume gratings, and uniform optical layers.

Fig. 4
Fig. 4

Surface relief grating with a rectangular profile; the spatial period is equal to the incident wavelength, and the angle of incidence is θ = 30°.

Fig. 5
Fig. 5

Comparison between the standard and new algorithms for rectangular surface profile: variations of the −1-order diffraction efficiency as a function of the groove depth in wavelengths with TE polarization.

Fig. 6
Fig. 6

Variations of the main diffracted orders as functions of the groove depth in wavelengths for the rectangular surface profile with TE polarization.

Fig. 7
Fig. 7

Variations of the main diffracted orders as functions of the groove depth in wavelengths for the rectangular surface profile with TM polarization.

Fig. 8
Fig. 8

Two possible geometries for a stairstep surface profile; the spatial period is equal to the incident wavelength, and the angle of incidence is θ = 30°.

Fig. 9
Fig. 9

Variations of the main diffracted orders as functions of the groove depth in wavelength units for two stairstep surface profiles with TE polarization. The zero reflected order of case B is superimposed upon that of case A; the other orders of case B are recognizable by their similarity with case A.

Fig. 10
Fig. 10

Holographic recording geometry of a planar volume grating.

Fig. 11
Fig. 11

Angular variations of diffraction efficiencies in + 1 and −1 orders for the phase volume grating with TE polarization.

Fig. 12
Fig. 12

Angular variations of diffraction efficiencies in +1 and −1 orders for the phase volume grating with TM polarization.

Fig. 13
Fig. 13

Angular variations of diffraction efficiencies in +1 and −1 orders for the absorption volume grating with TE polarization.

Fig. 14
Fig. 14

Angular variations of diffraction efficiencies in + 1 and −1 orders for the absorption volume grating with TM polarization.

Tables (1)

Tables Icon

Table 1 Example of Eigenvalue Distribution, Ordered by Growing Real Part, for Phase and Absorption Volume Gratings, Retaining Five Orders, with Wavelength λ= 500 nm and Incidence Angle θ = −30°

Equations (50)

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n 2 ( x , z ) = i = + ñ i exp [ j i ( K x x + K z z ) ] .
E y ( x , z ) = i = + E y ( i ) ( z ) exp ( j k x ( i ) x ) , h x ( x , z ) = i = + h x ( i ) ( z ) exp ( j k x ( i ) x ) .
k x ( 0 ) = k 0 n 0 sin ( θ ) ,
k x ( i ) = k x ( 0 ) + i K x , i .
h x ( x , z ) = j k 0 E y ( x , z ) z .
d E y ( i ) ( z ) d z = j k 0 h x ( i ) ( z ) , i .
2 E y ( x , z ) + k 0 2 n 2 ( x , z ) E y ( x . z ) = 0 ;
d h x ( i ) ( z ) d z = j { [ k z ( i ) ] 2 k 0 E y ( i ) ( z ) + k 0 l i ñ i l × exp [ j ( i l ) K z z ] E y ( l ) ( z ) } , i ,
[ k z ( i ) ] 2 = k 0 2 ñ 0 [ k x ( i ) ] 2 , i .
d [ h x ( i ) ( z ) exp ( j i K z z ) ] d z = j [ i K z h x ( i ) ( z ) exp ( j i K z z ) + [ k z ( i ) ] 2 k 0 E y ( i ) ( z ) exp ( ji K z z ) + k 0 l i ñ i l E y ( l ) ( z ) exp ( j l K z z ) ] , i ;
d [ E y ( i ) ( z ) exp ( ji K z z ) ] d z = j [ i K z E y ( i ) ( z ) exp ( j i K z z ) + k 0 h x ( i ) ( z ) exp ( j i K z z ) ] , i .
d U ( z ) dz = [ M ] U ( z ) ,
U ( z ) = [ E y ( i ν ) ( z ) exp [ j ( i ν ) K z z ] [ h x ( i ν ) ( z ) exp [ j ( i ν ) K z z ] .
[ M ] = j [ K z [ Δ ] k 0 [ I N ] k 0 [ Ω ] K z [ Δ ] ] .
Δ i , i = i ν , i { 0 , , N 1 } ;
Ω i , i = k z ( i ν ) 2 k 0 2 , i l : Ω i , l = ñ , i l , ( i , l ) { 0 , , N 1 } 2 .
U ( z 1 ) = exp { ( z 2 z 1 ) [ M ] } U ( z 2 ) .
[ M ] = [ P ] [ D ] [ P ] 1 ,
[ D ] = [ e 0 0 e 1 0 e 2 N 1 ] .
U ( z 1 ) = [ P ] exp { ( z 2 z 1 ) [ D ] } [ P ] 1 U ( z 2 ) .
i { 0 , , N 1 } : P i , l ( z ) = exp [ j ( i ν ) K z z ] P i , l , i { N , , 2 N 1 } : P i , l ( z ) = exp [ j ( i N ν ) K z z ] P i , l l { 0 , , 2 N 1 } .
[ E y ( i ν ) ( z 1 ) ] [ h x ( i ν ) ( z 1 ) ] = [ P ( z 1 ) ] [ exp [ e 0 ( z 2 z 1 ) ] 0 exp [ e 1 ( z 2 z 1 ) ] 0 exp [ e 2 N 1 ( z 2 z 1 ) ] ] [ P ( z 2 ) ] 1 [ E y ( i ν ) ( z 2 ) ] [ h x ( i ν ) ( z 2 ) ] .
l = 0 m 1 ( [ P l ( z l ) exp { ( z l + 1 z l ) [ D l ] } [ P l ( l + 1 ) ] 1 ) ,
E y ( x , z ) = i = + f F ( i ) exp { j [ k x ( i ) x + k F z ( i ) z ] } + i = + b F ( i ) exp { j [ k x ( i ) x k F z ( i ) ] } , h x ( x , z ) = 1 k 0 i = + k F z ( i ) f F ( i ) exp { j [ k x ( i ) x + k F z ( i ) z ] } + 1 k 0 i = + k F z ( i ) b F ( i ) exp { j [ k x ( i ) x k F z ( i ) z ] } ,
[ C ( z 0 ) ] = [ 0 exp [ j k F z ( i ν ) z 0 ] 0 ] [ 0 k F z ( i ν ) k 0 exp [ j k F z ( i ν ) z 0 0 ] × [ 0 exp [ j k F z ( i ν ) z 0 ] [ 0 k F z ( i ν ) k 0 exp [ j k F z ( i ν ) z 0 0 ] ,
[ E y ( i ν ) ( z 0 ) ] [ h x ( i ν ) ( z 0 ) ] = [ C ( z 0 ) ] [ f F ( i ν ) ] [ b F ( i ν ) ] .
[ f F ( i ν ) ] [ b F ( i ν ) ] = [ C ( z 0 ) ] 1 l = 0 m 1 ( [ P l ( z l ) ] exp { ( z l + 1 z l ) [ D ] } × [ P l ( z l + 1 ) ] 1 ) [ C ( z m ) ] [ f L ( i ν ) ] [ b L ( i ν ) ] .
[ f F ( i ν ) ] = [ 1 0 0 ] ;
[ b L ( i ν ) ] = [ 0 ] .
η B ( i ν ) = k F z ( i ν ) k F z ( 0 ) | b F ( i ν ) | 2 , η F ( i ν ) = k L z ( i ν ) k F z ( 0 ) | f L ( i ν ) | 2 .
[ I R ] = k = 0 m 1 [ A k ] [ T O ] ,
[ A ] k = [ [ A k 00 ] [ A k 01 ] A k 10 [ A k 11 ] ] , k { 0 , 1 , , m 1 } ;
k { 0 , 1 , , m 1 } : [ X k Y k ] = ( l = k m 1 [ A l ] ) [ T O ] , [ X m Y m ] = [ T O ] .
X k 1 = [ A k 1 00 ] X k + [ A k 1 10 ] Y k , Y k 1 = [ A k 1 10 ] X k + [ A k 1 11 ] Y k , k { 1 , 2 , , m } .
[ P k ] X k = Y k , [ Q k ] X k = T , k { 0 , 1 , m } .
[ P k 1 ] = ( [ A k 1 10 ] + [ A k 1 11 ] [ P k ] ) ( [ A k 1 00 ] + [ A k 1 01 ] [ P k ] ) 1 , [ Q k 1 ] = [ Q k ] ( [ A k 1 00 ] + [ A k 1 00 ] [ P k ] ) 1 , k { 1 , 2 , m } .
[ P m ] = N × N null matrix , [ Q m ] = N × N identity matrix .
[ P k 1 ] = [ A k 1 11 ] [ P k ] [ A k 1 00 ] 1 , [ Q k 1 ] = [ Q k ] [ A k 1 00 ] 1 , k { 1 , 2 , , m } .
R = [ P 0 ] I , T = [ Q 0 ] I .
E ( x , z ) = 2 z / d ( 1 + cos { ( 2 π n 0 / λ ) [ x ( sin θ 1 sin θ 2 ) + z ( cos θ 1 + cos θ 2 ) ] } ) .
n ( x , z ) = n 0 0.02 E ( x , z ) .
n ( x , z ) = n 0 + 0.01 j E ( x , z ) .
p ( x ) = det ( [ M ] x [ I 2 N ] ) = 0 .
p ( x * ) = det ( [ M ] + x * [ I 2 N ] ) = det ( [ j K z [ Δ ] + x * [ I N ] j k 0 [ I N ] j k 0 [ Ω ] j K z [ Δ ] + x * [ I N ] ] ) .
p ( x * ) = det ( [ K z j [ Δ ] + x * [ I N ] j k 0 [ Ω ] T j k 0 [ I N ] T j K z [ Δ ] + x * [ I N ] ] ) .
ñ i = ñ i * , i .
[ Ω ] T = [ Ω ] .
p ( x * ) = det ( [ ( j K z [ Δ ] ) * + x * [ I N ] * ( j k 0 [ Ω ] ) * ( j k 0 [ I N ] ) * ( j K z [ Δ ] ) * + x * [ I N ] * ] ) .
p ( x * ) = ( 1 ) 2 N × det ( [ ( j K z [ Δ ] ) * + x * [ I N ] * ( j k 0 [ I N ] ) * ( j k 0 [ Ω ] ) * ( j K z [ Δ ] ) * + x * [ I N ] * ] ) .
p ( x * ) = [ p ( x ) ] * = 0 .

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