Abstract

An exact expression of reflection coefficient from inhomogeneous planar structures is derived in a simple and very explicit form by using the analytical transfer matrix method. It is revealed that the reflection coefficient is directly dependent on a single phase integral over the structure. Different from the WKBJ and the integral expansion methods, the presented phase integral is accumulated not only by the main waves, but also by the total subwaves, which are propagated in the structure with continuously varying index profile.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. R. Jacobsson, "Light reflection from films of continuously varying refractive index," in Progress in Optics, Vol. 5, E.Wolf, ed. (Elsevier, North-Holland, 1966), pp. 247-286.
    [CrossRef]
  2. F. Abeles, "Optical properties of inhomogeneous films," Natl. Bur. Stand. (U.S) Misc. Publ. 256, 41-58 (1964).
  3. P. Yeh and S. Sari, "Optical properties of stratified media with exponentially graded refractive index," Appl. Opt. 22, 4142-4145 (1983).
    [CrossRef] [PubMed]
  4. M. Kildemo, O. Hunderi, and B. Drevillon, "Approximation of the reflection coefficient for rapid real time calculation of inhomogeneous films," J. Opt. Soc. Am. A 14, 931-939 (1997).
    [CrossRef]
  5. J. F. Hall, "Reflection coefficient of optically inhomogeneous layers," J. Opt. Soc. Am. 48, 654-657 (1958).
    [CrossRef]
  6. M. Kildemo, "Real-time monitoring and growth control of Si-gradient-index structures by multiwavelength ellipsometry," Appl. Opt. 37, 113-124 (1998).
    [CrossRef]
  7. J. M. Vigoureux, "Polynomial formulation of reflection and transmission by stratified planar structures," J. Opt. Soc. Am. A 8, 1697-1701 (1991).
    [CrossRef]
  8. J. M. Vigoureux, "Use of Einstein's addition law in studies of reflection by stratified planar structures," J. Opt. Soc. Am. A 9, 1313-1319 (1992).
    [CrossRef]
  9. J. M. Vigoureux, "The reflection of light by planar stratified media: the grupoid of amplitudes and a phase 'Thomas precession', " J. Phys. A 26, 385-393 (1993).
    [CrossRef]
  10. Z. Cao, Y. Jiang, Q. Shen, X. Dou, and Y. Chen, "Exact analytical method for planar optical waveguides with arbitrary index profile," J. Opt. Soc. Am. A 16, 2209-2212 (1999).
    [CrossRef]
  11. Z. Cao, Q. Liu, Y. Jiang, Q. Shen, and X. Dou, "Phase shift at a turning point in a planar optical waveguide," J. Opt. Soc. Am. A 18, 2161-2163 (2001).
    [CrossRef]
  12. Y. C. Ou, Z. Cao, and Q. Shen, "Formally exact quantization condition for nonrelativistic quantum systems," J. Chem. Phys. 121, 8175-8178 (2004).
    [CrossRef] [PubMed]
  13. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965).
  14. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  15. J. Chilwell and I. Hodgkinson, "Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides," J. Opt. Soc. Am. A 1, 742-753 (1984).
    [CrossRef]

2004 (1)

Y. C. Ou, Z. Cao, and Q. Shen, "Formally exact quantization condition for nonrelativistic quantum systems," J. Chem. Phys. 121, 8175-8178 (2004).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

1998 (1)

1997 (1)

1993 (1)

J. M. Vigoureux, "The reflection of light by planar stratified media: the grupoid of amplitudes and a phase 'Thomas precession', " J. Phys. A 26, 385-393 (1993).
[CrossRef]

1992 (1)

1991 (1)

1984 (1)

1983 (1)

1964 (1)

F. Abeles, "Optical properties of inhomogeneous films," Natl. Bur. Stand. (U.S) Misc. Publ. 256, 41-58 (1964).

1958 (1)

Appl. Opt. (2)

J. Chem. Phys. (1)

Y. C. Ou, Z. Cao, and Q. Shen, "Formally exact quantization condition for nonrelativistic quantum systems," J. Chem. Phys. 121, 8175-8178 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Phys. A (1)

J. M. Vigoureux, "The reflection of light by planar stratified media: the grupoid of amplitudes and a phase 'Thomas precession', " J. Phys. A 26, 385-393 (1993).
[CrossRef]

Natl. Bur. Stand. (U.S) Misc. Publ. (1)

F. Abeles, "Optical properties of inhomogeneous films," Natl. Bur. Stand. (U.S) Misc. Publ. 256, 41-58 (1964).

Other (3)

R. Jacobsson, "Light reflection from films of continuously varying refractive index," in Progress in Optics, Vol. 5, E.Wolf, ed. (Elsevier, North-Holland, 1966), pp. 247-286.
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

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 (4)

Fig. 1
Fig. 1

Planar structure.

Fig. 2
Fig. 2

One-layer planar structure.

Fig. 3
Fig. 3

Two-layer planar structure.

Fig. 4
Fig. 4

Comparison of ATM and WKBJ.

Equations (42)

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

M ( h j ) = [ cos ( κ j h j ) 1 κ j sin ( κ j h j ) κ j sin ( κ j h j ) cos ( κ j h j ) ] ( j = 1 , 2 , , m ) ,
κ j = k 0 n j cos θ j ,
κ j = ( k 0 2 n j 2 β 2 ) 1 2 .
E y ( x ) = { A 0 exp ( i κ 0 x ) + B 0 exp ( i κ 0 x ) ( x < 0 ) A s exp ( i κ s x ) ( x > d ) ,
κ 0 = k 0 n 0 cos θ 0 ,
κ s = k 0 n s cos θ s ,
[ E y ( 0 ) E y ( 0 ) ] = j = 1 m M ( h j ) [ E y ( d ) E y ( d ) ] .
[ E y ( 0 ) E y ( 0 ) 1 ] j = 1 m M ( h j ) [ 1 E y ( d ) E y ( d ) ] = 0 .
E y ( x j ) E y ( x j ) = p j ( j = 1 , 2 , , m ) ,
p s = i κ s .
[ i κ 0 A 0 B 0 A 0 + B 0 1 ] j = 1 m M ( h j ) [ 1 p s ] = 0 .
i κ 0 A 0 B 0 A 0 + B 0 = p 1 ,
p j = κ j tan [ tan 1 ( p j + 1 κ j ) κ j h j ] ( j = 1 , 2 , , m )
p m + 1 = p s .
ϕ j = tan 1 ( p j κ j ) ,
ϕ j = l π + tan 1 ( p j + 1 κ j ) κ j h j = l π + tan 1 ( κ j + 1 κ j tan ϕ j + 1 ) κ j h j ( l = 0 , 1 , 2 , , j = 1 , 2 , , m 1 ) .
κ j h j + [ ϕ j + 1 tan 1 ( κ j + 1 κ j tan ϕ j + 1 ) ] = l π + ( ϕ j + 1 ϕ j ) .
κ m h m = l π + tan 1 ( p s κ m ) ϕ m .
j = 1 m κ j h j + j = 1 m 1 [ ϕ j + 1 tan 1 ( κ j + 1 κ j tan ϕ j + 1 ) ] = l π + tan 1 ( p s κ m ) ϕ 1 ,
exp ( i 2 ϕ 1 ) = exp { i 2 [ j = 1 m κ j h j + j = 1 m 1 ( ϕ j + 1 tan 1 ( κ j + 1 κ j tan ϕ j + 1 ) ) tan 1 ( p s κ m ) ] } .
tanh 1 u = 1 2 ln ( 1 + u 1 u ) .
exp [ i 2 tan 1 ( p s κ m ) ] = exp [ 2 tanh 1 ( κ s κ m ) ] = κ m κ s κ m + κ s = r m s .
j = 1 m κ j h j = 0 d κ ( x ) d x ,
κ ( x ) = ( k 0 2 n 2 ( x ) β 2 ) 1 2 ,
j = 1 m 1 [ ϕ j + 1 tan 1 ( κ j + 1 κ j tan ϕ j + 1 ) ] = 0 d p p 2 + κ 2 d κ d x d x .
K ( x ) = κ ( x ) + p p 2 + κ 2 d κ d x ,
exp ( i 2 ϕ 1 ) = r m s exp [ i 2 0 d K ( x ) d x ] .
κ 0 κ 1 A 0 B 0 A 0 + B 0 = i p 1 κ 1 ,
r = B 0 A 0 = r 01 + exp ( i 2 ϕ 1 ) 1 + r 01 exp ( 12 ϕ 1 ) .
r = r 01 + r m s exp [ i 2 0 d K ( x ) d x ] 1 + r 01 r m s exp [ i 2 0 d K ( x ) d x ] ,
r 01 = κ 0 κ 1 κ 0 + κ 1 ,
exp ( i 2 ϕ 1 ) = r 12 exp ( i 2 κ 1 h 1 ) ,
r 12 = κ 1 κ 2 κ 1 + κ 2 .
r = r 01 + r 12 exp ( i 2 κ 1 h 1 ) 1 + r 01 r 12 exp ( i 2 κ 1 h 1 ) .
exp ( i 2 ϕ 1 ) = r 23 exp { i 2 [ ( κ 1 h 1 + κ 2 h 2 ) + ( ϕ 2 tan 1 ( κ 2 κ 1 tan ϕ 2 ) ) ] } .
exp ( i 2 ϕ 2 ) = r 23 1 exp ( i 2 κ 2 h 2 ) ,
exp [ i 2 tan 1 ( κ 2 κ 1 tan ϕ 2 ) ] = exp [ i 2 tan 1 ( κ 2 κ 1 κ 3 κ 2 tan ( κ 2 h 2 ) 1 + p 3 κ 2 tan ( κ 2 h 2 ) ) ] = r 12 + r 23 exp ( i 2 κ 2 h 2 ) 1 + r 12 r 23 exp ( i 2 κ 2 h 2 ) ,
r 23 = κ 2 κ 3 κ 2 + κ 3 .
exp ( i 2 ϕ 1 ) = r 12 + r 23 exp ( i 2 κ 2 h 2 ) 1 + r 12 r 23 exp ( i 2 κ 2 h 2 ) exp ( i 2 κ 1 h 1 ) ,
r 123 = r 12 + r 23 exp ( i 2 κ 2 h 2 ) 1 + r 12 r 23 exp ( i 2 κ 2 h 2 ) .
r = r 01 + r 123 exp ( i 2 κ 1 h 1 ) 1 + r 01 r 123 exp ( i 2 κ 1 h 1 ) .
r = r 01 + r m s exp [ i 2 0 d κ ( x ) d x ] 1 + r 01 r m s exp [ i 2 0 d κ ( x ) d x ] .

Metrics