Abstract

A pedagogical presentation of the propagation of electromagnetic waves in stratified media is given. The usual 2 × 2 matrix analysis is simplified and generalized. A Bloch factor eigenvalue equation is obtained, valid for all periodic stratifications. The implications of the band structure of an infinite periodic structure for reflection by a finite structure are demonstrated. Some features of the reflectivity are shown to be universal. In the long-wave limit, a periodic stratification with an arbitrary dielectric function profile (z) within a unit cell is shown to be equivalent to a homogeneous anisotropic medium, with ordinary and extraordinary dielectric constants given by o = 〈〉 and e−1 = 〈−1〉.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, N. F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Clarendon, Oxford, 1936; reprinted by Dover, New York, 1958).
  2. See L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, New York, 1946; reprinted by Dover, New York, 1953), Chap. I, for a brief historical review.
  3. J. W. S. Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. 24, 145–159 (1887).
  4. J. W. S. Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
    [CrossRef]
  5. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dan les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).
  6. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Chap. 12.
    [CrossRef]
  7. J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
    [CrossRef]
  8. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,”J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  9. A. Yariv, P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,”J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Chap. 14.
  11. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  12. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 354–355.
  13. J. Lekner, “Reflection and refraction by uniaxial crystals,”J. Phys. Condens. Matter 3, 6121–6133 (1991), Sec. 5.1.
    [CrossRef]
  14. J. P. van der Ziel, M. Ilegems, R. M. Mikulyak, “Optical birefringence of thin GaAs–AlAs multilayer films,” Appl. Phys. Lett. 28, 735–737 (1976).
    [CrossRef]
  15. M. Kitagawa, M. Tateda, “Form birefringence of SiO2/Ta2O5periodic multilayers,” Appl. Opt. 24, 3359–3363 (1985).
    [CrossRef] [PubMed]

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,”J. Phys. Condens. Matter 3, 6121–6133 (1991), Sec. 5.1.
[CrossRef]

1990 (1)

J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
[CrossRef]

1985 (1)

1977 (2)

1976 (1)

J. P. van der Ziel, M. Ilegems, R. M. Mikulyak, “Optical birefringence of thin GaAs–AlAs multilayer films,” Appl. Phys. Lett. 28, 735–737 (1976).
[CrossRef]

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dan les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

1917 (1)

J. W. S. Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

1887 (1)

J. W. S. Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. 24, 145–159 (1887).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dan les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 354–355.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 354–355.

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Chap. 14.

Brillouin, L.

See L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, New York, 1946; reprinted by Dover, New York, 1953), Chap. I, for a brief historical review.

Hong, C. S.

Ilegems, M.

J. P. van der Ziel, M. Ilegems, R. M. Mikulyak, “Optical birefringence of thin GaAs–AlAs multilayer films,” Appl. Phys. Lett. 28, 735–737 (1976).
[CrossRef]

Jones, H.

See, for example, N. F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Clarendon, Oxford, 1936; reprinted by Dover, New York, 1958).

Kitagawa, M.

Lekner, J.

J. Lekner, “Reflection and refraction by uniaxial crystals,”J. Phys. Condens. Matter 3, 6121–6133 (1991), Sec. 5.1.
[CrossRef]

J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
[CrossRef]

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Chap. 12.
[CrossRef]

Mikulyak, R. M.

J. P. van der Ziel, M. Ilegems, R. M. Mikulyak, “Optical birefringence of thin GaAs–AlAs multilayer films,” Appl. Phys. Lett. 28, 735–737 (1976).
[CrossRef]

Mott, N. F.

See, for example, N. F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Clarendon, Oxford, 1936; reprinted by Dover, New York, 1958).

Rayleigh, J. W. S.

J. W. S. Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

J. W. S. Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. 24, 145–159 (1887).

Tateda, M.

van der Ziel, J. P.

J. P. van der Ziel, M. Ilegems, R. M. Mikulyak, “Optical birefringence of thin GaAs–AlAs multilayer films,” Appl. Phys. Lett. 28, 735–737 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Chap. 14.

Yariv, A.

Yeh, P.

Am. J. Phys. (1)

J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
[CrossRef]

Ann. Phys. (Paris) (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dan les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. P. van der Ziel, M. Ilegems, R. M. Mikulyak, “Optical birefringence of thin GaAs–AlAs multilayer films,” Appl. Phys. Lett. 28, 735–737 (1976).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. Condens. Matter (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,”J. Phys. Condens. Matter 3, 6121–6133 (1991), Sec. 5.1.
[CrossRef]

Phil. Mag. (1)

J. W. S. Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. 24, 145–159 (1887).

Proc. R. Soc. London Ser. A (1)

J. W. S. Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

Other (6)

See, for example, N. F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Clarendon, Oxford, 1936; reprinted by Dover, New York, 1958).

See L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, New York, 1946; reprinted by Dover, New York, 1953), Chap. I, for a brief historical review.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Chap. 14.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 354–355.

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Chap. 12.
[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 (5)

Fig. 1
Fig. 1

Dielectric function profile for a (high–low)4 dielectric mirror, drawn to scale with n1 = 1, nh = 2.35 (ZnS), nl = 1.38 (MgF2), and n2 = 1.5 (glass). For maximum reflectivity at normal incidence and wavelength λ the layer thicknesses are dh = λ/4nh and dl = λ/4nl (a quarter-wave stack).

Fig. 2
Fig. 2

Real and imaginary parts of ϕs and ϕp for the ZnS − MgF2 high–low structure as a function of the angle of incidence. The upper plot is for the design frequency ω0 for high reflectivity at normal incidence, at which dh = λh/4 and dl = λl/4 (the λ/4 stack) and thus δh = π/2 = δl. The lower plot is drawn for ω = 1.3ω0.

Fig. 3
Fig. 3

Band structure parameter ϕ = arcos(½ tr M) for a λ/4 stack as a function of the frequency, drawn for normal incidence onto the high–low stack of Fig. 1. The stop band is centered at the design frequency ω0, with half-width given by Eq. (44).

Fig. 4
Fig. 4

Frequency dependence of the normal-incidence reflectivity of a dielectric multilayer, drawn for 20 ZnS–MgF2 bilayers on glass. The multilayer is tuned for high reflectivity at ω = ω0 (it is a quarter-wave stack at the design frequency).

Fig. 5
Fig. 5

Reflectivities of a 20-bilayer high–low stack as a function of the angle of incidence at (top) ω = ω0 and (bottom) ω = 1.3ω0. The parameters are the same as those in Fig. 2. The stop-band edges are at 53.10 for the p waves when ω = ω0 and at 41.4° for the s wave and 59.7° for the p wave when ω = 1.3ω0.

Equations (69)

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

[ m 11 m 12 m 21 m 22 ] N = [ m 11 S N - S N - 1 m 12 S N m 21 S N m 22 S N - S N - 1 ] ,
S N = sin ( N ϕ ) sin ϕ ,             cos ϕ = ½ ( m 11 + m 22 ) .
2 ( cos ϕ ) S N - S N - 1 = S N + 1 .
E y ( z , x , t ) = exp [ i ( K x - ω t ) ] E ( z ) ,
B y ( z , x , t ) = exp [ i ( K x - ω t ) ] B ( z ) ,
d 2 E d z 2 + q 2 E = 0 ,             d d z ( 1 d B d z ) + q 2 B = 0 ,
q 2 ( z ) = ( z ) ω 2 / c 2 - K 2 .
K = n 1 ( ω / c ) sin θ 1 = n 1 ( ω / c ) sin θ 2 ,
q 1 = n 1 ( ω / c ) cos θ 1 ,             q 2 = n 2 ( ω / c ) cos θ 2 .
E ( z ) = f F ( z ) + g G ( z ) ,
( E b E b ) = [ m 11 m 12 m 21 m 22 ] ( E a E a ) ,
( E a E a ) = [ F a G a F a G a ] ( f g ) A ( f g ) ,
( E b E b ) = [ F b G b F b G b ] ( f g ) B ( f g ) .
M = B A - 1 = W - 1 [ - ( F , G ) ( F , G ) - ( F , G ) ( F , G ) ] .
W = F G - F G ,             W = 0 ,
( F , G ) F a G b - G a F b , ( F , G ) F a G b - G a F b , ( F , G ) F a G b - G a F b , ( F , G ) F a G b - G a F b .
det M = W - 2 [ ( F , G ) ( F , G ) - ( F , G ) ( F , G ) ] = 1.
( F , G ) = sin δ , ( F , G ) = q cos δ , ( F , G ) = - q cos δ , ( F , G ) = q 2 sin δ ,
M = [ cos δ q - 1 sin δ - q sin δ cos δ ]
( t s exp ( i q 2 b ) i q 2 t s exp ( i q 2 b ) ) = [ m 11 m 12 m 21 m 22 ] × ( exp ( i q 1 a ) + r s exp ( - i q 1 a ) i q 1 [ exp ( i q 1 a ) - r s exp ( - i q 1 a ) ] )
r s = exp ( 2 i q 1 a ) q 1 q 2 m 12 + m 21 + i q 1 m 22 - i q 2 m 11 q 1 q 2 m 12 - m 21 + i q 1 m 22 - i q 2 m 11 ,
t s = exp [ i ( q 1 a - q 2 b ) ] × 2 i q 1 q 1 q 2 m 12 - m 21 + i q 1 m 22 - i q 2 m 11 .
( z ) ω 2 / c 2 2 m 2 [ E - V ( z ) ] .
( B b B ˜ b ) = [ m 11 m 12 m 21 m 22 ] ( B a B ˜ a ) .
M = U - 1 [ - ( C ˜ , D ) ( C , D ) - ( C ˜ , D ˜ ) ( C , D ˜ ) ] ,
U = C D ˜ - C ˜ D ,             U = 0
det M = U - 2 [ ( C , D ) ( C ˜ , D ˜ ) - ( C , D ˜ ) ( C ˜ , D ) ] = 1.
M = [ cos δ Q - 1 sin δ - Q sin δ cos δ ] .
exp ( i q 1 z ) - r p exp ( - i q 1 z ) B ( z ) n 2 n 1 t p exp ( i q 2 z ) .
( n 2 n 1 t p exp ( i q 2 b ) i Q 2 n 2 n 1 t p exp ( i q 2 b ) ) = [ m 11 m 12 m 21 m 22 ] × ( exp ( i q 1 a ) - r p exp ( - i q 1 a ) i Q 1 [ exp ( i q 1 a ) + r p exp ( - i q 1 a ) ] ) ,
- r p = exp ( 2 i q 1 a ) × Q 1 Q 2 m 12 + m 21 + i Q 1 m 22 - i Q 2 m 11 Q 1 Q 2 m 12 - m 21 + i Q 1 m 22 + i Q 2 m 11 ,
n 2 n 1 t p = exp [ i ( q 1 a - q 2 b ) ] × 2 i Q 1 Q 1 Q 2 m 12 - m 21 + i Q 1 m 22 + i Q 2 m 11 .
M = M N M N - 1 M 2 M 1 .
R s + T s = 1 ,             R p + T p = 1 ,
( ψ n + 1 ψ n + 1 ) = M ( ψ n ψ n ) ,
( ψ n + 1 ψ n + 1 ) = β ( ψ n ψ n ) .
( M - β I ) ( ψ n ψ n ) = 0 ,
β 2 = 2 β cos ϕ + 1 = 0 ,
β ± = cos ϕ ± ( cos 2 ϕ - 1 ) 1 / 2 = exp ( ± i ϕ ) .
e ξ = cos ϕ + ( cos 2 ϕ - 1 ) 1 / 2 .
[ c l q l - 1 s l - q l s l c l ] [ c h q h - 1 s h - q h s h c h ] = [ c l c h - q l - 1 q h s l s h q h - 1 c l s h + q l - 1 s l c h - q l s l c h - q h c l s h c l c h - q l q h - 1 s l s h ] ,
cos ϕ s = c l c h - ½ s l s h ( q l - 1 q h + q h - 1 q l ) = cos ( δ l + δ h ) - s l s h [ ( q l / q n ) 1 / 2 - ( q h / q l ) 1 / 2 ] 2 .
cos ϕ p = c l c h - ½ s l s h ( Q l - 1 Q h + Q h - 1 Q l ) = cos ( δ l + δ h ) - s l s h [ ( Q l / Q h ) 1 / 2 - ( Q h / Q l ) 1 / 2 ] 2 .
Δ ω ω 0 = 2 π arcsin ( n h - n l n h + n l ) .
σ N = S N - 1 S N = sin [ ( N - 1 ) ϕ ] sin ( N ϕ ) = cos ϕ - sin ϕ cot ( N ϕ ) ,
r s = q 1 q 2 m 12 + m 21 + i q 1 ( m 22 - σ N ) - i q 2 ( m 11 - σ N ) q 1 q 2 m 12 - m 21 + i q 1 ( m 22 - σ N ) + i q 2 ( m 11 - σ N ) ,
t s = 2 i q 1 S N - 1 q 1 q 2 m 12 - m 21 + i q 1 ( m 22 - σ N ) + i q 2 ( m 11 - σ N ) .
R s = r s 2 = 1 - 4 q 1 q 2 S N - 2 ( q 1 q 2 m 12 ) 2 + m 21 2 + q 1 2 ( m 22 - σ N ) 2 + q 2 2 ( m 11 - σ N ) 2 + 2 q 1 q 2 S N - 2 ,
T s = q 2 q 1 t s 2 = 1 - R s .
- r p = Q 1 Q 2 m 12 + m 21 + i Q 1 ( m 22 - σ N ) - i Q 2 ( m 11 - σ N ) Q 1 Q 2 m 12 - m 21 + i Q 1 ( m 22 - σ N ) + i Q 2 ( m 11 - σ N ) ,
n 2 n 1 t p = 2 i Q 1 S N - 1 Q 1 Q 2 m 12 - m 21 + i Q 1 ( m 22 - σ N ) + Q 2 ( m 11 - σ N ) .
T p = q 2 q 1 t p 2 = 4 Q 1 Q 2 S N - 2 ( Q 1 Q 2 m 12 ) 2 + m 21 2 + Q 1 2 ( m 22 - σ N ) 2 + Q 2 2 ( m 11 - σ N ) 2 + 2 Q 1 Q 2 S N - 2 .
σ N 2 - 2 σ N cos ϕ + 1 = [ ( sin ϕ ) / sin ( N ϕ ) ] 2 = S N - 2 .
r s q 1 - q 2 q 1 + q 2 ,             - r p Q 1 - Q 2 Q 1 + Q 2 .
Im ϕ = log [ cos ϕ + ( cos 2 ϕ - 1 ) 1 / 2 ] .
S N 2 = [ sinh ( N Im f ) sinh ( Im f ) ] 2 ,
q o 2 = o ω 2 / c 2 - K 2 ,             q e 2 = o ω 2 / c 2 - ( o / e ) K 2 ,
E o ~ ( 0 , 1 , 0 ) ,             E e ~ [ q e , 0 , - ( 0 / e ) K ] .
ϕ s = q s ( d h + d l ) = q s d ,             q s 2 s ω 2 / c 2 - K 2 ,
ϕ p = q p ( d h + d l ) = q p d ,             q p 2 s ω 2 / c 2 - ( s / p ) K 2
s = f h h + f l l ,             p = h l f h l + f l h ,
f h = d h d h + d l ,             f l = d l d h + d l
p - s = - ( h - l ) 2 f h f l f h l + f l h .
M s = [ 1 - a b d z q 2 ( z ) ( b - z ) b - a - a b d z q 2 ( z ) 1 - a b d z q 2 ( z ) ( z - a ) ]
cos ϕ s = ½ tr M s = 1 - ½ ( b - a ) a b d z q 2 ( z ) + .
s = 1 b - a a b d z ( z ) .
M p = [ 1 - a b d z [ q 2 ( z ) / ( z ) ] z b d ζ ( ζ ) a b d z ( z ) - a b d z q 2 ( z ) / ( z ) 1 - a b d z ( z ) z b d ζ q 2 ( ζ ) / ( ζ ) ] .
s p ( b - a ) 2 = a b [ d z / ( z ) ] z b d ζ ( ζ ) + a b d z ( z ) z b d ζ / ( ζ ) ,
1 p = 1 1 b - a a b d z ( z ) .

Metrics