Abstract

The full optical response of multilayer gratings containing anisotropic layers until now has been the domain of the experimentalist with no suitable theoretical model available to investigate such systems. The introduction of such a model would clearly benefit this relatively unexplored area of diffractive optics. To this end we present a differential theory based on the work of Chandezon et al. [ J. Opt. Soc. Am. A 72, 839 ( 1982)], extending our previous analysis to explore not only the in-plane diffraction with anisotropic layers but also the twisted grating, or conical diffraction, case. In this case there are now two possible mechanisms for transverse magnetic to transverse electric conversion, those being the twisted grating and the anisotropic, uniaxial layer. To illustrate the modeling, results of the new theory are compared with experimental data for a twisted grating anisotropic liquid-crystal system. Resonance mode position and intensity are in good agreement, showing the validity of the new mathematical procedure.

© 1996 Optical Society of America

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References

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  1. G. P. Bryan-Brown, J. R. Sambles, M. C. Hutley, “Polarisation conversion through the excitation of surface plasmons on a metallic grating,” J. Mod. Opt. 37, 1227–1232 (1990).
    [CrossRef]
  2. E. Popov, L. Mashev, “Diffraction anomalies of coated dielectric gratings in conical mounting,” Opt. Commun. 59, 323–330 (1986).
    [CrossRef]
  3. R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic material,” in Periodic Structures, Diffraction Gratings, and Moiré Phenomena, J. Lerner, ed., Proc. SPIE815, 11–16 (1987).
    [CrossRef]
  4. M. L. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1994).
    [CrossRef]
  5. 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]
  6. 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]
  7. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  8. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  9. E. Popov, L. Mashev, “Conical diffraction mounting generalisation of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
    [CrossRef]
  10. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
    [CrossRef]
  11. 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]
  12. J. B. Harris, T. W. Preist, J. R. Sambles, “A differential method for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
    [CrossRef]
  13. B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
    [CrossRef]

1995 (3)

1994 (2)

M. L. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1994).
[CrossRef]

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

1991 (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

1990 (1)

G. P. Bryan-Brown, J. R. Sambles, M. C. Hutley, “Polarisation conversion through the excitation of surface plasmons on a metallic grating,” J. Mod. Opt. 37, 1227–1232 (1990).
[CrossRef]

1988 (1)

1986 (2)

E. Popov, L. Mashev, “Diffraction anomalies of coated dielectric gratings in conical mounting,” Opt. Commun. 59, 323–330 (1986).
[CrossRef]

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

1982 (1)

1981 (1)

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

G. P. Bryan-Brown, J. R. Sambles, M. C. Hutley, “Polarisation conversion through the excitation of surface plasmons on a metallic grating,” J. Mod. Opt. 37, 1227–1232 (1990).
[CrossRef]

Chandezon, J.

Cornet, G.

Cotter, N. P. K.

Depine, R. A.

M. L. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1994).
[CrossRef]

Dupuis, M. T.

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Gigli, M. L.

M. L. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1994).
[CrossRef]

Harris, J. B.

Hutley, M. C.

G. P. Bryan-Brown, J. R. Sambles, M. C. Hutley, “Polarisation conversion through the excitation of surface plasmons on a metallic grating,” J. Mod. Opt. 37, 1227–1232 (1990).
[CrossRef]

Ko, D. Y. K.

Laks, B.

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Li, L.

Maradudin, A. A.

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Mashev, L.

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

E. Popov, L. Mashev, “Diffraction anomalies of coated dielectric gratings in conical mounting,” Opt. Commun. 59, 323–330 (1986).
[CrossRef]

Maystre, D.

Mills, D. L.

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Petit, R.

R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic material,” in Periodic Structures, Diffraction Gratings, and Moiré Phenomena, J. Lerner, ed., Proc. SPIE815, 11–16 (1987).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, “Diffraction anomalies of coated dielectric gratings in conical mounting,” Opt. Commun. 59, 323–330 (1986).
[CrossRef]

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

Preist, T. W.

Sambles, J. R.

Tayeb, G.

R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic material,” in Periodic Structures, Diffraction Gratings, and Moiré Phenomena, J. Lerner, ed., Proc. SPIE815, 11–16 (1987).
[CrossRef]

J. Mod. Opt. (2)

M. L. Gigli, R. A. Depine, “Conical diffraction from uniaxial gratings,” J. Mod. Opt. 42, 1281–1299 (1994).
[CrossRef]

G. P. Bryan-Brown, J. R. Sambles, M. C. Hutley, “Polarisation conversion through the excitation of surface plasmons on a metallic grating,” J. Mod. Opt. 37, 1227–1232 (1990).
[CrossRef]

J. Opt. (Paris) (1)

E. Popov, L. Mashev, “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 (5)

Opt. Commun. (1)

E. Popov, L. Mashev, “Diffraction anomalies of coated dielectric gratings in conical mounting,” Opt. Commun. 59, 323–330 (1986).
[CrossRef]

Phys. Rev. B (2)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

B. Laks, D. L. Mills, A. A. Maradudin, “Surface polaritons on large amplitude gratings,” Phys. Rev. B 23, 4965–4976 (1981).
[CrossRef]

Other (1)

R. Petit, G. Tayeb, “About the electromagnetic theory of gratings made with anisotropic material,” in Periodic Structures, Diffraction Gratings, and Moiré Phenomena, J. Lerner, ed., Proc. SPIE815, 11–16 (1987).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Arrangement used in the theoretical description of the grating. The grating surface [described by y = s1(x) of period λg] 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 oriented at an angle Θ to the y axis and makes an azimuthal angle Φ with the xy plane.

Fig. 3
Fig. 3

Cell configuration for comparison with experimental data. A 1.48-μm liquid-crystal layer, aligned homogeneously so that the nematic director lies along the grating grooves, is confined between a silver-coated diffraction grating of pitch 554.02 nm and groove depth 26 nm and a flat glass top plate of refractive index 1.512.

Fig. 4
Fig. 4

Experimental data from the grating shown in Fig. 3 compared with our theoretical calculation when the grating is illuminated with p-polarized monochromatic (632.8-nm) light at azimuthal angle 30°.

Fig. 5
Fig. 5

Theoretical calculation of Rpp using the same parameters as in Fig. 4 but across a range of both polar and azimuthal angles: note that we have reversed the polar angle range with respect to Fig. 4 to obtain a more pleasing perspective. Variations in the position of the surface-plasmon minima as a function of azimuthal angle can be seen clearly and in Fig. 6 are compared with a k-space coupling diagram.

Fig. 6
Fig. 6

k-Space representation of the +1-order guided mode and SPP resonances compared with experimental data. Agreement is best for the ordinary TM1 mode and departs slightly for the extraordinary TE modes as their direction of propagation tends towards the optic axis. This behavior can be attributed to a small (< 5°) twist within the liquid-crystal layer, and we conclude that the theory is providing a good description of the real system.

Fig. 7
Fig. 7

Results of a theoretical calculation using the same parameters as in Fig. 4 but for Rps across a range of both polar and azimuthal angles. As expected, the overall intensity of p-to-s conversion increases as the incident wave vector twists away from the symmetry plane of the grating, and localized maxima are found at angles corresponding to the excitation of surface and guided modes.

Equations (56)

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y = d j + s j ( x ) ,             d Q + 1 = 0 , d j = - ( e Q + e Q - 1 + + e j ) ,             j = 1 , , Q + 1 ,
v = x ,             u = y - s j ( x ) ,             w = z ,
e 1 = i + s j ,             e 2 = j ,             e 3 = k ,
e 1 = i ,             e 2 = j - s i ,             e 3 = k
[ E 3 u - w ( C E 2 + D E 1 ) ] ( 1 + s 2 ) + ( E 1 w - E 3 v ) s = i H 1 ,
v ( C E 2 + D E 1 ) - E 1 u = i H 3 ,
E 1 w - E 3 v = i H 2 ,
[ H 3 u - w ( C H 2 + D H 1 ) ] ( 1 + s 2 ) + ( H 1 w - H 3 v ) s = - i D 1 ,
v ( C H 2 + D H 1 ) - H 1 u = - i D 3 ,
H 1 w - H 3 v = - i D 2 ,
A 2 = C A 2 + D A 1 A 1 = C A 1 + D A 2 , A 3 = A 3 ,
C = 1 1 + s 2 ,             D = s 1 + s 2 .
ɛ i j = ɛ i j = ɛ j i = [ a b c b d e c e f ] , a = ɛ sin 2 Θ cos 2 Φ + ɛ ( cos 2 Θ cos 2 Φ + sin 2 Φ ) , b = ( ɛ - ɛ ) sin Θ cos Θ cos Φ , c = ( ɛ - ɛ ) sin 2 Θ sin Φ cos Φ , d = ɛ cos 2 Θ + ɛ sin 2 Θ , e = ( ɛ - ɛ ) sin Θ cos Θ sin Φ , f = ɛ sin 2 Θ sin 2 Φ + ɛ ( cos 2 Θ sin 2 Φ + cos 2 Φ ) ,
ɛ n m = x i q n x j q m ɛ i j ,             ɛ m n = q n x i x j q m ɛ j i , ɛ n m = q n x i q m x j ɛ i j ,             x i ( x , y , z ) ,             q n ( v , u , w ) , x i q n = [ 1 s 0 0 1 0 0 0 1 ] ,             q n x i = [ 1 0 0 - s 1 0 0 0 1 ] ,
ɛ n m = [ x i q n ] [ a b c b d e c e f ] [ x i q m ] = [ a + 2 s b + s 2 d b + s d c + s e b + s d d e c + s e e f ] ,
ɛ m n = [ q n x i ] [ a b c b d e c e f ] [ x i q m ] = [ a + s b b c ( 1 - s 2 ) b + s ( d - a ) d - s b e - s c c + s e e f ] ,
ɛ n m = [ q n x i ] [ a b c b d e c e f ] [ q m x i ] = [ a b - s a c b - s a - 2 s b + s 2 a + d e - s c c e - s c f ] .
E 1 u = v ( X 1 E 1 + X 2 E 3 + γ X 3 H 1 + i X 3 H 3 v ) - i H 3 ,
E 3 u = i γ X 4 E 1 + D E 3 v + i γ X 2 E 3 + i X 5 H 1 - γ X 3 H 3 v ,
H 1 u = v ( γ C E 1 ) + i X 6 E 1 + v ( i C E 3 v ) + i X 7 E 3 + v ( D H 1 ) - i γ X 8 H 1 + X 8 H 3 v ,
H 3 u = i X 9 E 1 - γ C E 3 v + i X 10 E 3 - i γ X 4 H 1 + X 1 H 3 v ,
X 1 = - ɛ 12 ɛ 22 ,             X 2 = - ɛ 3 2 ɛ 22 ,             X 3 = - 1 ɛ 22 , X 4 = - C ɛ 1 2 ɛ 22 , X 5 = C - γ 2 ɛ 22 ,             X 6 = C ( ɛ 31 - ɛ 1 2 ɛ 3 2 ɛ 22 ) , X 7 = ɛ 33 - ( ɛ 3 2 ) 2 ɛ 22 , X 8 = ɛ 3 2 ɛ 22 ,             X 9 = C { γ 2 + C [ ( ɛ 1 2 ) 2 ɛ 22 - ɛ 11 ] } , X 10 = C ( ɛ 1 2 ɛ 3 2 ɛ 22 - ɛ 13 ) .
E 1 u = v ( D E 1 - γ C ɛ H 1 - i C H 3 v ) - i H 3 ,
E 3 u = D E 3 v + i C ( 1 - γ 2 ɛ ) H 1 + γ C ɛ H 3 v ,
H 1 u = v ( D H 1 + γ C E 1 + i C E 3 v ) + i ɛ E 3 ,
H 3 u = D H 3 v + i C ( γ 2 - ɛ ) E 1 - γ C E 3 v .
Ψ ( v , u ) = m Ψ m ( u ) exp i α m v [ α m = ( ɛ Q + 1 ) 1 / 2 sin θ cos ϕ + m K ] ,
X μ ( v ) = p X p μ exp i p K v ,             μ = 1 - 10 ,
C ( v ) = p C p exp i p K v ,             D ( v ) = p D p exp i p K v .
- i E 1 ( m ) u = n [ α m X m - n 1 E 1 ( n ) + α m X m - n 2 E 3 ( n ) + γ α m X m - n 3 H 1 ( n ) - ( α m α n X m - n 3 + δ m n ) H 3 ( n ) ] ,
- i E 3 ( m ) u = n [ γ X m - n 4 E 1 ( n ) + ( α n D m - n + γ X m - n 2 ) E 3 ( n ) + X m - n 5 H 1 ( n ) - γ α n X m - n 3 H 3 ( n ) ] ,
- i H 1 ( m ) u = n [ ( X m - n 6 + γ α m C m - n ) E 1 ( n ) + ( X m - n 7 - α m α n C m - n ) E 3 ( n ) + ( α m D m - n - γ X m - n 8 ) H 1 ( n ) + α n X m - n 8 H 3 ( n ) ] ,
- i H 3 ( m ) u = n [ X m - n 9 E 1 ( n ) + ( X m - n 10 - γ α n C m - n ) E 3 ( n ) - γ X m - n 4 H 1 ( n ) + α n X m - n 1 H 3 ( n ) ] .
- i u ( E 1 ( m ) E 3 ( m ) H 1 ( m ) H 3 ( m ) ) = [ α m X m - n 1 α m X m - n 2 γ α m X m - n 3 - α m α n X m - n 3 - δ m n γ X m - n 4 α n D m - n + γ X m - n 2 X m - n 5 - γ α n X m - n 3 X m - n 6 + γ α m C m - n X m - n 7 - α m α n C m - n α m D m - n - γ X m - n 8 α m X m - n 8 X m - n 9 X m - n 10 - γ α n C m - n - γ X m - n 4 α n X m - n 1 ] ( E 1 ( m ) E 3 ( m ) H 1 ( m ) H 3 ( m ) ) .
- i ξ ( u ) u = T ξ ( u ) ,
ξ ( u ) = q = 1 8 M + 4 b q V q exp i r q u ,
ξ j ( u ) = M j ϕ j ( u ) b j ,
E 3 j + 1 = E 3 j ,
E 1 j + 1 = E 1 j + Δ j E 2 j ,             Δ j = d d v [ s j + 1 ( v ) - s j ( v ) ] ,
M n q j + 1 b q j + 1 = Ω n q j ϕ q j ( e j ) b q j ,
Ω ( 1 ) n q j = m M ( 1 ) m q j L n m q j + p m [ M ( 1 ) m q j X p 1 + M ( 2 ) m q j X p 2 + γ M ( 3 ) m q j X p 3 - M ( 4 ) m q j α m X p 3 ] × ( n - p - m ) K r q j L ( n - p ) m q j ,
Ω ( 2 ) n q j = m M ( 2 ) m q j L n m q j ,
Ω ( 3 ) n q j = m M ( 3 ) m q j L n m q j + p m [ γ M ( 1 ) m q j C p - M ( 2 ) m q j α m C p + M ( 3 ) m q j D p ] × ( n - p - m ) K r q j L ( n - p ) m q j ,
Ω ( 4 ) n q j = m M ( 4 ) m q j L n m q j ,
L n m q j = 1 λ g 0 λ g exp - i [ ( n - m ) K v - Δ j r q j ] d v .
I j + 1 = [ ϕ j ( e j ) ] - 1 [ Ω j ] - 1 M j + 1 ,
b + Q + 1 = S 12 b - Q + 1 ,             b - 0 = S 22 b - Q + 1 , S = [ S 11 S 12 S 21 S 22 ] ,
Ψ s ( u , v , w ) = m Ψ m s ( u ) exp i ( α m v + β s u + γ w ) ,
( E 3 ) m s = E 3 s L m - s ( - β s ) ,
( H 3 ) m s = H 3 s L m - s ( - β s ) ,
( E 1 ) m s = 1 f { [ β s - α s β s ( m - s ) K ] H 3 s + γ α m E 3 s } × L m - s ( - β s ) ,
( H 1 ) m s = - ɛ f { [ β s - α s β s ( m - s ) K ] E 3 s - γ ɛ α m H 3 s } × L m - s ( - β s ) ,
incident field :             β s = - ( ɛ Q + 1 - α s 2 - γ 2 ) 1 / 2 ,             s = 0 , reflected field :             β s = + ( ɛ Q + 1 - α s 2 - γ 2 ) 1 / 2 ,             s = ± 0 , 1 , 2 , , transmitted field :             β s = - ( ɛ 0 - α s 2 - γ 2 ) 1 / 2 ,             s = ± 0 , 1 , 2 , .
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 ] ,
T ^ = Q S 22 ( M 21 S 12 + M 22 ) - 1 ( M ^ 21 R ^ + L ) ,
[ M ^ 11 0 M ^ 12 0 M ^ 21 0 M ^ 22 0 ] - 1 [ M 11 M 12 M 21 M 22 ] = [ I 0 0 Q ] .

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