Abstract

We present a theoretical analysis for multilayer gratings of arbitrary shape containing uniaxial materials in which all optic axes lie in the plane of incidence, which itself is perpendicular to the grating grooves. The analysis is based on the differential method with internal boundary conditions imposed by means of a scattering matrix. Our development allows for layer interfaces of different shape, although they must all share the same periodicity.

© 1995 Optical Society of America

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References

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  1. E. Glytsis, T. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
    [CrossRef]
  2. S. Mori, K. Mukai, J. Yamakita, K. Rokushima, “Analysis of dielectric lamellar gratings coated with anisotropic layers,” J. Opt. Soc. Am. A 7, 1661–1665 (1990).
    [CrossRef]
  3. R. A. Depine, M. E. Inchaussandague, “Corrugated diffraction gratings in uniaxial crystals,” J. Opt. Soc. Am. A 11, 173–180 (1994).
    [CrossRef]
  4. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  5. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  6. N. P. K Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  7. E. Popov, L. Mashev, D. Maystre, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
    [CrossRef]
  8. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
    [CrossRef]
  9. T. W. Preist, N. P. K Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
    [CrossRef]
  10. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  11. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1979).

1995 (2)

1994 (2)

1991 (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

1990 (2)

1988 (1)

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

1982 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1979).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1979).

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Chandezon, J.

Cornet, G.

Cotter, N. P. K

Depine, R. A.

Dupuis, M. T.

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Gaylord, T.

Glytsis, E.

Inchaussandague, M. E.

Ko, D. Y. K.

Li, L.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

Mori, S.

Mukai, K.

Popov, E.

E. Popov, L. Mashev, D. Maystre, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

Preist, T. W.

Rokushima, K.

Sambles, J. R.

Yamakita, J.

J. Opt. (Paris) (1)

E. Popov, L. Mashev, D. Maystre, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Phys. Rev. B (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Other (1)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1979).

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Figures (5)

Fig. 1
Fig. 1

Arrangement used in the theoretical description of the grating. The grating surface [described by y = a1(x) of period D] is coated by Q isotropic or uniaxial layers of mean thickness e1, e2, etc. The isotropic substrate permittivity is ɛ0, and the overlayer permittivities are described by the tensors ɛ ˜ 1 …, etc., with the top semi-infinite layer being isotropic with permittivity ɛQ+1.

Fig. 2
Fig. 2

Orientation of the optic axis within a uniaxial layer. Within any medium the optic axis is assumed to be in the xy plane and to make a fixed angle γj with the x axis.

Fig. 3
Fig. 3

Definitions of the field component G in neighboring media. For a given value of v along a particular interface, only the G component defined in the medium above it will be tangential to the groove profile, and we must allow for this when matching fields.

Fig. 4
Fig. 4

Optical response of an E7 liquid crystal waveguide (with the optic axis aligned in the direction of grating periodicity) surrounded by air and illuminated with light of 632.8-nm wavelength. Both interfaces were sinusoidal with 800-nm pitch. (a), (b) Reflectivity curves for A = 25 and 150 nm, respectively.

Fig. 5
Fig. 5

Comparison of reflectivity curves for an E7 liquid crystal waveguide (with the optic axis aligned in the plane of incidence 20° down from the direction of grating periodicity) surrounded by air and illuminated with light of 632.8-nm wavelength. Both interfaces were sinusoidal with 800-nm pitch and 25-nm amplitude. We compare the cases in which the two interfaces are (a) in phase (dotted curve) and (b) 180° out of phase (solid curve).

Tables (1)

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Table 1 Guided-Mode Positionsa

Equations (71)

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y = d j + a j ( x )             with             d Q + 1 = 0 , d j = - ( e Q + e Q - 1 + + e j ) ,             j = 1 , , Q + 1 ,
v = x , u = y - a j ( x ) , w = z .
e 1 = i + a j ,             e 2 = j ,             e 3 = k ,
A = A v e 1 + A u e 2 + A w e 3 ,
e 1 = i ,             e 2 = j - a i ,             e 3 = k ,
A = A 1 e 1 + A 2 e 2 + A 3 e 3 ,
A 2 = C A u + D A 1 ,             A v = C A 1 - D A u ,             A w = A 3 ,
C = 1 / ( 1 + a 2 ) ,             D = a / ( 1 + a 2 ) ,
curl A = γ Q ,
e 1 · curl A = γ e 1 · Q , giving [ A 3 u - w ( C A u + D A 1 ) ] ( 1 + a 2 ) + ( A 1 w - A 3 v ) a = γ Q 1 ;
e 3 · curl A = γ e 3 · Q , giving v ( C A u + D A 1 ) - A 1 u = γ Q 3 ;
e 2 · curl A = γ e 2 · Q , giving A 1 w - A 3 v = γ Q u .
D x = ɛ E x ,             D y = ɛ E y ,
ɛ a b = [ ɛ 0 0 ɛ ] .
ɛ i j = [ ɛ cos 2 γ + ɛ sin 2 γ sin γ cos γ ( ɛ - ɛ ) sin γ cos γ ( ɛ - ɛ ) ɛ sin 2 γ + ɛ cos 2 γ ] [ t 11 t 12 t 21 t 22 ] .
ɛ n m = x i q n x j q m ɛ i j ,             ɛ n m = q n x i x j q m ɛ i j , ɛ n m = q n x i q m x j ɛ i j ,             x i ( x , y ) ,             q n ( v , u ) ,
x i q n = [ 1 a 0 1 ]             q n x i = [ 1 0 - a 1 ] .
ɛ n m = [ x i q n ] [ t 11 t 12 t 21 t 22 ] [ x j q m ] T = [ 1 a 0 1 ] [ t 11 t 12 t 21 t 22 ] [ 1 0 a 1 ] = [ t 11 + 2 a t 12 + a 2 t 22 t 12 + a t 22 t 12 + a t 22 t 22 ] ,
ɛ n m = [ t 11 + a t 12 t 12 ( 1 - a 2 ) t 12 + a ( t 22 - t 11 ) t 22 - a t 12 ] ,             D n = ɛ n m E m ,
ɛ n m = [ t 11 t 12 - a t 11 t 12 - a t 11 - 2 a t 12 + a 2 t 11 + t 22 ] ,             D n = ɛ n m E m .
D 1 = ɛ 1 j E j = ɛ 11 E v + ɛ 12 E u = C ( ɛ 11 E 1 + ɛ 1 2 E u ) ,
D u = D 2 = ɛ 2 n E n = ɛ 2 1 E v + ɛ 2 2 E u = C ( ɛ 2 1 E 1 + ɛ 22 E u ) .
( 1 + a 2 ) H 3 u - a H 3 v = - k 0 Z 0 D 1 ,
v ( C E u + D E 1 ) - E 1 u = i k 0 Z 0 H 3 ,
H 3 v = i k 0 Z 0 D u ,             Z 0 = ( μ 0 / ɛ 0 ) 1 / 2 .
F u = X 1 F v + i k 0 X 2 G ,
G u = i k 0 F + G 2 v = i k 0 F + v ( X 1 G + i X 3 k 0 F v ) ,
X 1 = - ɛ 12 ɛ 22 ,             X 2 = ɛ ɛ ɛ 22 ,             X 3 = 1 ɛ 22 .
X μ ( v ) = n X n μ exp i n K v ,             μ = 1 , 2 , 3.
F ( v , u ) = m F m ( u ) exp i α m v , G ( v , u ) = m G m ( u ) exp i α m v ,
α m = ( ɛ Q + 1 ) 1 / 2 k 0 sin θ + m K .
- i F n u = m { α m X n - m 1 F m + k 0 X n - m 2 G m } ,
- i G n u = m [ ( - α n α m X n - m 1 k 0 + k 0 δ n m ) × F m + α n X n - m 3 G m ] ,
ξ ( u ) = ( F - N , , F N , G - N , , G N ) .
- i u ( F - N · F N - G - N · G N ) = [ α m X n - m 1 k 0 X n - m 2 - α n α m X n - m 1 k 0 + k 0 δ n m α n X n - m 3 ] × ( F - N · F N - G - N · G N )
- i ξ ( u ) u = T ( u ) ξ ( u ) ,
ξ ( u ) = q = 1 2 N + 1 b q V q exp i r q u ,
ξ j ( u ) = M j ϕ j ( u ) b j ,
G cos Θ + G u sin Θ ,
F j + 1 = F j ,
G j + 1 = G j + Δ j G 2 j ,
M n q j + 1 b q j + 1 = Ω n q j ϕ q j ( e j ) b q j ,
Ω ( + ) n q j = m M ( + ) m q j L n m q j ,
Ω ( - ) n q j = m M ( - ) m q j L n m q j + p m ( n - p - m ) K r q j × [ M ( - ) m q j X p 1 j - α m M ( + ) m q j X p 3 j ] L ( n - p ) m q j .
I j + 1 = [ ϕ j ( e j ) ] - 1 [ Ω j ] - 1 M j + 1 ,             [ ϕ j ( e j ) ] p q = ϕ q j ( e j ) δ p q .
( b + Q + 1 b - 0 ) = S ( 0 , Q + 1 ) ( b + 0 b - Q + 1 ) ,
b + 0 = 0
b + Q + 1 = S 12 b - Q + 1 ,             b - 0 = S 22 b - Q + 1 .
F s = exp i ( α s x + β s y ) = m L m - s ( - β s ) exp i ( α m v + β s u ) ,
G s = 1 k 0 m [ β s - ( m - n ) K α s β s ] L m - s ( - β s ) × exp i ( α m v + β s u ) ,
Incident field β s = - β 0 , Reflected field β s = + β n ,             n = 0 , ± 1 , ± 2 , , Transmitted field β s = - β n ,             n = 0 , ± 1 , ± 2 , , β n = ( k 2 - α n 2 ) 1 / 2 ,
ξ i = L exp - i β 0 u ( L L ) exp - i β 0 u ,
L m ( β 0 )             and             1 k 0 ( β 0 - α 0 β 0 m K ) L m ( β 0 ) ,             - N m N .
ξ d = M ϕ ( u ) B [ M M ] ϕ ( u ) B ,
L m - n ( - β n )             and             1 k 0 [ β n - ( m - n ) α n β n K ] × L m - n ( - β n ) ,             0 n P             - N m N
b ^ - Q + 1 = 0
M Q + 1 ϕ ( u ) ( b + Q + 1 0 ) + ( L L ) exp - i β 0 u + [ M M ] ϕ ( u ) R ,
M ^ Q + 1 ϕ Q + 1 ( u ) ( R ^ 0 ) + ( L L ) exp - i β 0 u ,             R ^ = ( R b ^ + Q + 1 ) ,
[ M 11 M 12 M 21 M 22 ] ( b + Q + 1 b - Q + 1 ) = [ M ^ 11 M ^ 12 M ^ 21 M ^ 22 ] ( R ^ 0 ) + ( L L ) .
R ^ = [ M ^ 11 - ( M 11 S 12 + M 12 ) ( M 21 S 12 + M 22 ) - 1 M ^ 21 ] - 1 × ( [ M 11 S 12 + M 12 ] [ M 21 S 12 + M 22 ] - 1 L - L ) ,
b - 0 = S 22 [ M 21 S 12 + M 22 ] - 1 ( M ^ 21 R ^ + L ) .
R p p = R n * R n cos θ n cos θ ,
M 0 ( 0 b ^ - 0 ) + [ M 0 M 0 ] ( 0 T ) M ^ 0 ( 0 T ^ ) ,
T ^ = ( T b ^ - 0 )
[ M 0 M 0 ] .
[ M 11 M 12 M 21 M 22 ] ( 0 b - 0 ) = [ M ^ 11 0 M ^ 12 0 M ^ 21 0 M ^ 22 0 ] ( 0 T ^ ) ,
( 0 T ^ ) = [ M ^ 11 0 M ^ 12 0 M ^ 21 0 M ^ 22 0 ] - 1 [ M 11 M 12 M 21 M 22 ] ( 0 b - 0 ) = [ I 0 0 Q ] ( 0 b - 0 ) .
T ^ = Q b - 0 ,
E t ( n ) = ( ɛ Q + 1 ɛ 0 ) 1 / 2 T n * T n cos θ n cos θ ,
k x = k mode - m K ,
sin θ = k mode k 0 - m λ D ,

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