Abstract

It is shown that a multilayer with repeated symmetric unit structure is equivalent to a homogeneous and optically biaxial medium for obliquely incident light, the three principal axes being normal to the plane of incidence, normal to the film plane, and normal to the other two. Unit thickness dependence and incident angle dependence of the dielectric tensor and the effective thickness are calculated using exact formulas. It is found that, at small unit structure thickness, the equivalent medium is almost uniaxial, the optical axis being normal to the film, and the effective thickness is very close to the mechanical one. Absorbing films are also envisaged and the domain of validity (in terms of unit structure thickness) of the uniaxial equivalent medium is defined by numerical examples.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Herpin, “Calcul du pouvoir réflecteur d’un systéme stratifié quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).
  2. F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 637 (1950).
  3. H. A. Macleod, “A New Approach to the Design of Metal-Dielectric Thin Film Optical Coatings,” Opt. Acta 25, 93–106 (1978).
    [CrossRef]
  4. O. Hunderi, “Effective Medium Theory and Nonlocal Effects for Superlattices,” J. Wave-Material Interaction 2, 29–39 (1987).
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 67.
  6. Ref. 2, p. 777.
  7. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 82–93.
  8. Ref. 5, pp. 705–708.
  9. L. I. Epstein, “The Design of Optical Filters,” J. Opt. Soc. Am. 42, 806–810 (1952).
    [CrossRef]

1987

O. Hunderi, “Effective Medium Theory and Nonlocal Effects for Superlattices,” J. Wave-Material Interaction 2, 29–39 (1987).

1978

H. A. Macleod, “A New Approach to the Design of Metal-Dielectric Thin Film Optical Coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

1952

1950

F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 637 (1950).

1947

A. Herpin, “Calcul du pouvoir réflecteur d’un systéme stratifié quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).

Abelés, F.

F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 637 (1950).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 67.

Epstein, L. I.

Herpin, A.

A. Herpin, “Calcul du pouvoir réflecteur d’un systéme stratifié quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).

Hunderi, O.

O. Hunderi, “Effective Medium Theory and Nonlocal Effects for Superlattices,” J. Wave-Material Interaction 2, 29–39 (1987).

Macleod, H. A.

H. A. Macleod, “A New Approach to the Design of Metal-Dielectric Thin Film Optical Coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 67.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 82–93.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 82–93.

Ann. Phys. Paris

F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 637 (1950).

C. R. Acad. Sci.

A. Herpin, “Calcul du pouvoir réflecteur d’un systéme stratifié quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).

J. Opt. Soc. Am.

J. Wave-Material Interaction

O. Hunderi, “Effective Medium Theory and Nonlocal Effects for Superlattices,” J. Wave-Material Interaction 2, 29–39 (1987).

Opt. Acta

H. A. Macleod, “A New Approach to the Design of Metal-Dielectric Thin Film Optical Coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 67.

Ref. 2, p. 777.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 82–93.

Ref. 5, pp. 705–708.

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

Unit cells of the periodic multilayer systems simulated in this paper: A[a/2|b|a/2]; B[b/2|a|b/2] with ɛa = 6 and ɛb = 2.

Fig. 2
Fig. 2

Unit thickness dependence of the effective dielectric functions ɛx, ɛy, and ɛz (a) and effective thickness and de (b) of a symmetric multilayer A or B (see Fig. 1).

Fig. 3
Fig. 3

Incident angle dependence of the effective dielectric functions ɛx, ɛy, and ɛz (a) and effective thickness de (b) of a symmetric multilayer A or B (see Fig. 1).

Fig. 4
Fig. 4

Dependence of the effective characteristics of a periodic multilayer system (B type) on an absorption in layer b ( ɛ b is the variable): (a) and (b) real and imaginary parts of the effective dielectric functions (ɛx,ɛy,ɛz); (c) and (d) real and imaginary parts of the effective thickness de.

Fig. 5
Fig. 5

Dependence of the imaginary parts of the effective characteristics of periodic multilayer systems (A and B) on unit thickness du of the period for an absorbing a layer ( ɛ b = 4): (a) imaginary part of effective dielectric functions ɛx, ɛy, ɛz; (b) imaginary part of effective unit thickness de.

Equations (21)

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

b / 2 a b a b a b / 2 ,
M j = | cos α j ( i / p j ) sin α j ( i p j ) sin α j cos α j | ,
M unit = M ( b / 2 ) * M ( a ) * M ( b / 2 ) = | m 11 m 12 m 21 m 22 | ,
m 11 = m 22 = cos α a cos α b - sin α a sin α b ( p a / p b + p b / p a ) / 2 , m 12 = ( i / p a ) [ sin α a cos α b + cos α a sin α b ( p a / p b + p b / p a ) / 2 + sin α b ( p a / p b - p b / p a ) / 2 ] , m 21 = ( i p a ) [ sin α a cos α b + cos α a sin α b ( p a / p b + p b / p a ) / 2 - sin α b ( p a / p b - p b / p a ) / 2 ] .
m 11 = m 22
m 11 m 22 - m 12 m 21 = 1
M unit N = | M 11 M 22 M 21 M 22 | = | m 11 U N - 1 ( v ) - U N - 2 ( v ) m 12 U N - 1 ( v ) m 21 U N - 1 ( v ) m 22 U N - 1 ( v ) - U N - 2 ( v ) | ,
v = ( m 11 + m 22 ) / 2 = m 11 ,
U N ( v ) = sin [ ( N + 1 ) arccos ( v ) ] / ( 1 - v 2 ) 1 / 2
U N ( w ) = sin [ ( N + 1 ) w ] / sin w .
| M 11 M 12 M 21 M 22 | = | cos β ( i / P ) sin β ( i P ) sin β cos β |
P 2 = - M 21 / M 12 = m 21 / m 12 , sin 2 β = M 12 M 21 = m 12 m 21 U N - 1 ( V ) .
P = ( m 21 / m 12 ) 1 / 2 β = N arccos [ ( m 11 + m 22 ) / 2 ] = N arccos ( m 11 ) . }
P s = ( ɛ x - S 2 ) 1 / 2 , β s = [ 2 π D e / λ ] · ( ɛ x - S 2 ) 1 / 2 ; }
P p = [ ɛ y / ( 1 - S 2 / ɛ z ) ] 1 / 2 β p = [ 2 π D e / λ ] · [ ɛ y · ( 1 - S 2 / ɛ z ) ] 1 / 2 . }
d e = [ λ / 2 π ] · arccos ( m 11 ) s / ( m 21 / m 12 ) s 1 / 2 , ɛ x = S 2 + ( m 21 / m 12 ) s , ɛ y = [ λ / 2 π d e ] ( m 21 / m 12 ) p 1 / 2 arccos ( m 11 ) p , ɛ z = S 2 / [ 1 - ɛ y ( m 12 / m 21 ) p ] . }
P s = ( ɛ x - S 2 ) 1 / 2 , β s = [ 2 π D s / λ ] ( ɛ x - S 2 ) 1 / 2 ; }
P p = [ ɛ x / ( 1 - S 2 / ɛ z ) ] 1 / 2 , β p = [ 2 π D p / λ ] [ ɛ x · ( 1 - S 2 / ɛ z ) ] 1 / 2 . }
d s = [ λ / 2 π ] arccos ( m 11 ) s / ( m 21 / m 12 ) s 1 / 2 , ɛ x = S 2 + ( m 21 / m 12 ) s , d p = [ λ / 2 π ] arccos ( m 11 ) p / ( m 21 / m 12 ) p 1 / 2 ɛ x , ɛ z = S 2 / [ 1 - ɛ x ( m 12 / m 21 ) p ] . }
ɛ o = f a ɛ a + f b ɛ b ( = ɛ x = ɛ y ) ,             1 / ɛ e = f a / ɛ a + f b / ɛ b ( = 1 / ɛ z ) ,
f a = d a / ( d a + d b ) ;             f b = d b / ( d a + d b ) .

Metrics