Abstract

The reflection coefficient is calculated for radiation incident normally on a nonabsorbing, optically inhomogeneous, plane parallel layer bounded on either side by nonabsorbing semi-infinite media. The theory is applied to determine the reflectance of a layer in which the index of refraction varies nearly linearly with the thickness.

© 1958 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Mayer, Physik Dünner Schichten (Wissenschaftliche Verlagsgesellschaft M. B. H., Stuttgart, 1950), p. 193.
  2. V. A. Bailey, Phys. Rev. 96, 865 (1954).
    [Crossref]
  3. F. Scandone, J. phys. radium 11, 337 (1950).
    [Crossref]
  4. B. S. Blaisse, J. phys. radium 11, 315 (1950).
    [Crossref]
  5. F. Abelès, J. phys. radium 17, 190 (1956).
    [Crossref]
  6. J. F. Hall, J. Opt. Soc. Am. 47, 662 (1957).
    [Crossref]
  7. W. Kofink and E. Menzer, Ann. phys. 39, 388 (1941).
    [Crossref]
  8. J. Strong, J. phys. radium 11, 451 (1950).
    [Crossref]
  9. O. Heavens, Optical Properties of Thin Solid Films (Academic Press, Inc., New York, 1955), p. 63.
  10. See reference 1, p. 201.
  11. J. F. Hall and W. F. C. Ferguson, J. Opt. Soc. Am. 45, 74 (1955).
    [Crossref]
  12. J. F. Hall, J. Opt. Soc. Am. 46, 1013 (1956).
    [Crossref]
  13. J. F. Hall and W. F. C. Ferguson, J. Opt. Soc. Am. 45, 714 (1955).
    [Crossref]

1957 (1)

1956 (2)

F. Abelès, J. phys. radium 17, 190 (1956).
[Crossref]

J. F. Hall, J. Opt. Soc. Am. 46, 1013 (1956).
[Crossref]

1955 (2)

1954 (1)

V. A. Bailey, Phys. Rev. 96, 865 (1954).
[Crossref]

1950 (3)

F. Scandone, J. phys. radium 11, 337 (1950).
[Crossref]

B. S. Blaisse, J. phys. radium 11, 315 (1950).
[Crossref]

J. Strong, J. phys. radium 11, 451 (1950).
[Crossref]

1941 (1)

W. Kofink and E. Menzer, Ann. phys. 39, 388 (1941).
[Crossref]

Abelès, F.

F. Abelès, J. phys. radium 17, 190 (1956).
[Crossref]

Bailey, V. A.

V. A. Bailey, Phys. Rev. 96, 865 (1954).
[Crossref]

Blaisse, B. S.

B. S. Blaisse, J. phys. radium 11, 315 (1950).
[Crossref]

Ferguson, W. F. C.

Hall, J. F.

Heavens, O.

O. Heavens, Optical Properties of Thin Solid Films (Academic Press, Inc., New York, 1955), p. 63.

Kofink, W.

W. Kofink and E. Menzer, Ann. phys. 39, 388 (1941).
[Crossref]

Mayer, H.

H. Mayer, Physik Dünner Schichten (Wissenschaftliche Verlagsgesellschaft M. B. H., Stuttgart, 1950), p. 193.

Menzer, E.

W. Kofink and E. Menzer, Ann. phys. 39, 388 (1941).
[Crossref]

Scandone, F.

F. Scandone, J. phys. radium 11, 337 (1950).
[Crossref]

Strong, J.

J. Strong, J. phys. radium 11, 451 (1950).
[Crossref]

Ann. phys. (1)

W. Kofink and E. Menzer, Ann. phys. 39, 388 (1941).
[Crossref]

J. Opt. Soc. Am. (4)

J. phys. radium (4)

J. Strong, J. phys. radium 11, 451 (1950).
[Crossref]

F. Scandone, J. phys. radium 11, 337 (1950).
[Crossref]

B. S. Blaisse, J. phys. radium 11, 315 (1950).
[Crossref]

F. Abelès, J. phys. radium 17, 190 (1956).
[Crossref]

Phys. Rev. (1)

V. A. Bailey, Phys. Rev. 96, 865 (1954).
[Crossref]

Other (3)

H. Mayer, Physik Dünner Schichten (Wissenschaftliche Verlagsgesellschaft M. B. H., Stuttgart, 1950), p. 193.

O. Heavens, Optical Properties of Thin Solid Films (Academic Press, Inc., New York, 1955), p. 63.

See reference 1, p. 201.

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 (2)

Fig. 1
Fig. 1

The reflectance of an inhomogeneous layer is plotted as a function of p=21.4(d/λ). The reflectance determined by Eq. (8) is indicated by the dashed line, while the dotted line shows the reflectance calculated by the WKB method. The solid line is the reflectance of an inhomogeneous layer, Eq. (7), when Ra=Rs=0.

Fig. 2
Fig. 2

The calculated, Eq. (8), and measured reflectance of an inhomogeneous layer of magnesium fluoride and zinc sulfide on a quartz substrate. The reflectance determined by the WKB method is also shown.

Equations (27)

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

d r = limit Δ x 0 n ( x ) - n ( x + Δ x ) n ( x ) + n ( x + Δ x ) = - 1 2 n ( n x ) d x + 1 4 n [ 1 n ( n x ) 2 - 2 n x 2 ] ( d x ) 2 ,
d t = limit Δ x 0 2 [ n ( x ) n ( x + Δ x ) ] 1 2 n ( x ) + n ( x + Δ x ) = 1 - 1 8 n 2 ( n x ) 2 ( d x ) 2 + 1 8 n 2 ( n x ) [ 1 n ( n x ) 2 - ( 2 n x 2 ) ] ( d x ) 3 .
2 n x 2 < ˜ 1 n ( n x ) 2 .
T ( x ) = T ( x ) exp [ - j 0 x α ( x ) d x ] .
T 2 ( x ) d r = T 2 ( x ) exp [ - 2 j 0 x α ( x ) d x ] d r .
( x ) = 0 x exp [ - 2 j 0 x α ( x ) d x ] T 2 ( x ) d r ,
m ( x ) = - 0 x exp [ - 2 j 0 x α ( x ) d x ] × [ 1 - m - 1 ( x ) m - 1 * ( x ) ] 1 2 n ( n x ) d x ,
1 ( x ) = - 0 x exp [ - 2 j 0 x α ( x ) d x ] 1 2 n ( n x ) d x = 0 x d [ 1 ( x ) ] .
2 ( x ) = 0 x { 1 - R 2 e [ 1 ( x ) ] - I 2 m [ 1 ( x ) ] } × d { R e [ 1 ( x ) ] + j I m [ 1 ( x ) ] } 2 ( x ) = r - 1 3 r 3 - 0 x i 2 d r + j [ i - 1 3 i 3 - 0 x r 2 d i ] ,
m ( d ) = tanh r + j tanh i - 0 d { R e 2 [ m - 1 ( x ) ] + I 2 m [ m - 1 ( x ) ] } ( d r + j d i ) + 0 d tanh 2 r d r + j 0 d tanh 2 i d i ,
p ( d ) = 4 π λ 0 d n ( x ) d x 0 ,
1 ( d ) = - n 1 n 2 d n 2 n = - 1 2 ln n 2 n 1 = 1 * ( d ) .
m ( d ) = n 1 - n 2 n 1 + n 2 .
p ( x ) = 4 π n 1 λ 0 x d x 1 - δ x = - β ln ( 1 - δ x ) ,
β = 4 π n 1 n 2 n 2 - n 1 ( d λ ) .
1 ( d ) = j 2 β 0 d exp [ j β ln ( 1 - δ x ) ] j β δ d x 1 - δ x = j 2 β [ e j p ( d ) - 1 ] ,
m ( d ) m * ( d ) R 2 e [ 1 ( d ) ] + I 2 m [ 1 ( d ) ] = 1 2 β 2 ( 1 - cos p ) .
1 ( M d ) - π n n 1 d 0 M d exp ( - j 4 π n 1 x λ ) cos 2 π d x d x .
1 ( M d ) - n 4 n 1 ( 1 g + 1 g + 2 ) × [ sin 2 π M g - j ( 1 - cos 2 π M g ) ] ,
m 2 ( M d ) [ tanh π n M 2 n 1 ] 2 .
R 1 = R a + m ( d ) 1 + R a m ( d ) .
R = R 1 + R s e - i p 1 + R 1 R s e - i p = R a + m ( d ) + R s e - i p + R a m ( d ) R s e - i p 1 + R a R s e - i p + m ( d ) R s e - i p + R a m ( d ) ,
p = 4 π λ d n ( x ) d x
R R * { R a 2 + 2 R a R s cos p + R s 2 } + A { 1 + 2 R a R s cos p + R a 2 R s 2 } + B ,
A = R a [ m ( d ) + m * ( d ) ] + R s [ m ( d ) e i p + m * ( d ) e - i p ] + m ( d ) m * ( d )
B = R a [ m ( d ) + m * ( d ) ] + R s [ m ( d ) e - i p + m * ( d ) e i p ] .
A = ( R s - R a ) sin p β - 2 R s β sin p cos p + 1 - cos p 2 β 2             and             B = - ( R a + R s ) sin p β .