Abstract

We obtain the optical response of a multilayered system in which the interfaces have a common periodicity but arbitrary shape by extending a published method previously restricted to interfaces with identical profiles [ J. Opt. Soc. Am. 72, 839 ( 1982)]. Numerical comparisons with other modeling results for grating systems are presented that show the effectiveness of the new method.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. T. K. Gaylord, M. C. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  3. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  4. M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  6. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
    [CrossRef]
  7. 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]
  8. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
    [CrossRef]
  9. 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]
  10. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  11. L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
    [CrossRef]
  12. D. T. Nguyen, M. L. Rustgi, “Diffraction of plane waves from a multilayered film with a grating surface,” J. Opt. Soc. Am. B 9, 1850–1856 (1992).
    [CrossRef]
  13. G. M. Gallatin, “Properties and applications of layered grating resonances,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.158–167 (1987).
    [CrossRef]
  14. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profile and thickness,” J. Opt. Soc. Am. 8, 755–762 (1991).
    [CrossRef]
  15. E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).
  16. J. B. Harris, T. W. Preist, J. R. Sambles, “A differential formalism for multilayer gratings made with uniaxial materials,” J. Opt. Soc. Am. A (to be published).
  17. E. Popov, M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
    [CrossRef]

1995 (2)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

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]

1994 (3)

1992 (1)

1991 (2)

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

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profile and thickness,” J. Opt. Soc. Am. 8, 755–762 (1991).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

1985 (1)

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

1982 (1)

1981 (1)

1978 (1)

1973 (1)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Awada, K. A.

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profile and thickness,” J. Opt. Soc. Am. 8, 755–762 (1991).
[CrossRef]

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

Bryan-Brown, G. P.

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

Cadilhac, M.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Chandezon, J.

Cornet, G.

Cotter, N. P. K.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

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]

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, 6393–6400 (1991).
[CrossRef]

Gallatin, G. M.

G. M. Gallatin, “Properties and applications of layered grating resonances,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.158–167 (1987).
[CrossRef]

Gaylord, T. K.

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

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

Harris, J. B.

J. B. Harris, T. W. Preist, J. R. Sambles, “A differential formalism for multilayer gratings made with uniaxial materials,” J. Opt. Soc. Am. A (to be published).

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

Li, L.

Mashev, L.

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

Maystre, D.

Moharam, M. C.

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

Moharam, M. G.

Nash, D. J.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Nguyen, D. T.

Pai, D. M.

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profile and thickness,” J. Opt. Soc. Am. 8, 755–762 (1991).
[CrossRef]

Petit, R.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
[CrossRef]

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

Preist, T. W.

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]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, “A differential formalism for multilayer gratings made with uniaxial materials,” J. Opt. Soc. Am. A (to be published).

Rustgi, M. L.

Sambles, J. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

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]

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

J. B. Harris, T. W. Preist, J. R. Sambles, “A differential formalism for multilayer gratings made with uniaxial materials,” J. Opt. Soc. Am. A (to be published).

J. Opt. Commun. (1)

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Phys. Rev. B (2)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps generated by surface plasmons on gratings,” Phys. Rev. B 51, 11,164–11,167 (1995).
[CrossRef]

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

Proc. IEEE (1)

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

Other (3)

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

J. B. Harris, T. W. Preist, J. R. Sambles, “A differential formalism for multilayer gratings made with uniaxial materials,” J. Opt. Soc. Am. A (to be published).

G. M. Gallatin, “Properties and applications of layered grating resonances,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.158–167 (1987).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Definition of the index labels for the interfaces and media in the multilayer system.

Fig. 2
Fig. 2

Orientation of the field components at the (j + 1)st interface.

Fig. 3
Fig. 3

Comparisons between the theory of Nguyen and Rustgi12 and that presented in this paper. The figures show p-polarized reflectivity for a dielectric–silver system with different dielectric overlayers on the silver. The curves indicate the scattering matrix theory, and the dots are the theory from the paper of Nguyen and Rustgi. The silver has a dielectric constant of (ω) = (0.04 + i7.5)2 and is 50 nm thick, and the surrounding medium has a dielectric constant of 3.6. The grating has a pitch of 400 nm, and light of wavelength 1059.7 nm (photon energy of 1.17 eV) is incident upon it. (a) Results for a 50-nm-thick dielectric overlayer with a dielectric constant of 3.0, (b) results for a 100-nm-thick overlayer with a dielectric constant of 3.13.

Fig. 4
Fig. 4

Comparisons between the scattering matrix theory (curves) and the predictions of Gallatin’s perturbative approach (dots) for three systems of different thicknesses.13 The structure consists of a dielectric layer with a flat upper (incident) surface and a rectangular periodic profile on the lower surface, surrounded by air. The dielectric has a permittivity of 5.29 and is 200 nm thick, and the grating has a pitch of 370 nm and light of wavelength 632.8 nm is used. (a), (b), (c) Results for groove depths of 5, 10, and 20 nm, respectively.

Fig. 5
Fig. 5

p Transmissivity at normal incidence of two gratings of equal pitch separated by an air-filled gap as a function of relative phase. The gratings (considered to be semi-infinite with a dielectric constant of 2.5) have a pitch of 800 nm and an amplitude of 100 nm, and light of wavelength 632.8 nm is incident in the top layer. The gap is 1.58 μm thick.

Fig. 6
Fig. 6

Dependence of the p reflectivity for a grating-coupled waveguide on the relative phase between the upper and lower periodic surfaces of the waveguide. The dielectric waveguide has a permittivity of 3.0, is 1 μm thick, and is surrounded by semi-infinite vacuum. Both surfaces have a pitch of 800 nm and an amplitude of 40 nm, and light of wavelength 632.8 nm is used. Variation with relative phases of (a) 0, (b) 90, and (c) 180 deg.

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

y j = d j + a j ( x ) ,
d Q + 1 = 0 ,
d j = - ( e Q + e Q - 1 + e j ) ,             j = 1 , , Q .
v = x ,             u = y - a j ( x ) ,             w = z .
δ r = e 1 δ v + e 2 δ u + e 3 δ w ,
e 1 = i + a j ,             e 2 = j ,             e 3 = k ,
a = d d v [ a j ( v ) ]             ( in the j th layer ) .
= e 1 v + e 2 u + e 3 w ,
e 1 = i ,             e 2 = j = a i ,             e 3 = k .
e i · e j = δ i j ,             e i × e j = e i j k e k ,             e i × e j = e i j k e k .
A = e 1 A v + e 2 A u + e 3 A w
A = e 1 A 1 + e 2 A 2 + e 3 A 3 .
× A = ( A 3 u - A 2 w ) e 1 + ( A 1 w - A 3 v ) e 2 + ( A 2 v - A 1 u ) e 3 .
e 1 · A 1 + a 2 = A 1 1 + a 2 ,             e 3 · A = A 3 ,
e 2 · A 1 + a 2 = A u 1 + a 2 .
A 2 = C A u + D A 1 ,
C = 1 1 + a 2 ,             D = a 1 + a 2 ,
A v = C A 1 = D A u ,             A w = A 3 .
× A = i γ Q ,
A 3 u - w ( C A u + D A 1 ) = i γ Q v = i γ ( C Q 1 - D Q u ) ,
A 1 w - A 3 v = i γ Q u ,
v ( C A u + D A 1 ) - A 1 u = i γ Q w = i γ Q 3 .
× E = i γ 1 H ,             × H = - i γ 2 E ,
γ 1 = ω Z 0 c μ r ,             γ 2 = ω Z 0 c r ,
F = E ,             G = γ 1 H
F = Z 0 H ,             G = - Z 0 γ 2 E ,
× F = i G ,             × G = - i γ 1 γ 2 F ,
F u = i C G + D F v ,
G u = v ( D G ) + v ( i C F v ) + i γ 1 γ 2 F .
F ( u , v ) = m F m ( u ) exp ( i α m v ) ,
G ( u , v ) = m G m ( u ) exp ( i α m v ) ,
α m = k x + m K ,
ζ p ( u ) = q M p q ϕ q b q ,
ζ ( u ) ( F ^ - N , , F ^ n ; G ^ - N , , G ^ N ) ,
ζ p ( u ) = q M p q exp [ i r q ( u - d j ) ] b q .
cos Θ = 1 + a j a j + 1 ( 1 + a j 2 ) 1 / 2 ( 1 + a j + 1 2 ) 1 / 2 , sin Θ = a j + 1 - a j ( 1 + a j 2 ) 1 / 2 ( 1 + a j + 1 2 ) 1 / 2 .
G ^ j + 1 ( 1 + a j + 1 2 ) 1 / 2 = G ^ j cos Θ + G ^ u j sin Θ ( 1 + a j 2 ) 1 / 2 ,
G ^ j + 1 = G ^ j + Δ j ( C j G ^ u j + D j G ^ j ) ,
Δ j = d d v [ a j + 1 ( v ) - a j ( v ) ]
F j + 1 = F j .
G ^ j + 1 = G ^ j + Δ j ( D j G ^ j + η j C j F ^ j v ) ,
η j = 1 / μ r j             for TE , = 1 / r j             for TM .
u j = d j + 1 + ( a j + 1 - a j ) d j + 1 + Δ j .
m F m j + 1 ( u j + 1 ) exp ( i α m v ) = m F m j ( u j ) exp ( i α m v )
m q M ( + ) m q j + 1 b q j + 1 exp ( i α m v ) = m q M ( + ) m q j ϕ q j ( e j ) b q j exp [ i r q j Δ j ( v ) ] exp ( i α m v ) ,
q M ( + ) n q j + 1 b q j + 1 = m q M ( + ) m q j ϕ q j ( e j ) b q j L n m q j ,
L n m q j 1 D 0 λ g exp { - i [ Δ j ( v ) ( - r q j ) + ( n - m ) K v ] } d v ,
Δ j exp [ i r q j Δ j ( v ) ] exp ( i α m v ) = 1 r q j s ( s - m ) K L s m q j × exp ( i α s v )
F v ( u , v ) = m F m ( u ) ( i α m ) exp ( i α m v ) .
m q M ( - ) m q j + 1 b q j + 1 exp ( i α m v ) = m q M ( - ) m q j ϕ q j ( e j ) b q j exp [ i r q j Δ j ( v ) ] exp ( i α m v ) + p D p j exp ( i p K v ) m q M ( - ) m q j ϕ q j ( e j ) b q j Δ j × exp [ i r q j Δ j ( v ) ] exp ( i α m v ) - η j p C p j exp ( i p K v ) × m q M ( + ) m q j ϕ q j ( e j ) b q j α m Δ j exp [ i r q j Δ j ( v ) ] exp ( i α m v ) = m q M ( - ) m q j ϕ q j ( e j ) b q j exp [ i r q j Δ j ( v ) ] exp ( i α m v ) + p D p j exp ( i p K v ) m q M ( - ) m q j ϕ q j ( e j ) b q j 1 r q j × s ( s - m ) K L s m q j exp ( i α s v ) - η j p C p j exp ( i p K v ) × m q M ( + ) m q j ϕ q j ( e j ) b q j α m r q j × s ( s - m ) K L s m q j exp ( i α s v ) .
q M ( - ) m q j + 1 b q j + 1 = m q M ( - ) m q j ϕ q j ( e j ) b q j L n m q j + p m q M ( - ) m q j ϕ q j ( e j ) b q j × ( n - p - m ) K r q j D p L ( n - p ) m q j - η j p m q M ( + ) m q j ϕ q j ( e j ) b q j × α m r q j ( n - p - m ) K C p L ( n - p ) m q j .
q M n q j + 1 b q j + 1 = q X n q j ϕ q j ( e j ) b q j ,
X ( + ) n q j = m M ( + ) m q j L n m q j ,
X ( - ) 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 D p j - η j α m M ( + ) m q j C p j ] L ( n - p ) m q j .
b j = I ( j + 1 ) b j + 1 ,
I ( j + 1 ) = [ ϕ j ( e j ) ] - 1 [ X j ] - 1 M j + 1 ,
[ ϕ j ( e j ) ] p q = ϕ q j ( e j ) δ p q .
y ( x ) = y 0 cos [ ( 2 π / D ) x ] , y ( x ) = y 0 cos [ ( 2 π / D ) x ] + y 1 cos [ 2 ( 2 π / D ) x ] , y ( x ) = y 0 cos [ ( 2 π / D ) x ] + y 1 cos [ 2 ( 2 π / D ) x ] + y 2 cos [ 3 ( 2 π / D ) x ] ,
exp [ i r q j Δ j ( v ) ] exp ( i α m v ) = s L s m q j exp ( i α s v ) .
i [ r q j Δ j ( v ) + α m ] exp [ i r q j Δ j ( v ) ] exp ( i α m v ) = i [ r q j Δ j ( v ) + α m ] s L s m q j exp ( i α s v ) = s ( i α s ) L s m q j exp ( i α s v ) .
Δ j ( v ) exp [ i r q j Δ j ( v ) ] exp ( i α m v ) = ( 1 / r q j ) s ( s - m ) K L s m q j × exp ( i α s v ) ,

Metrics