Abstract

Reflectance and transmittance of electromagnetic radiation from an exponentially graded-index multilayer dielectric structure are investigated theoretically. It is shown that such a structure maximizes the optical transmission in the case in which this transmission is approximated by summing intensities. Transmittance and reflectance of coherent radiation are also calculated for both thick and thin layers and are compared with the ones obtained by incoherent summation.

© 1983 Optical Society of America

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References

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  1. See, for example, R. Jacobsson, “Light Reflection From Films of Continuously Varying Refractive Index,” in Progress in Optics, Vol. 5, E. Wolf, Ed. (Wiley, New York, 1966), p. 249.
    [CrossRef]
  2. R. E. Thun et al., Automat. Control. 14, 26 (1961).
  3. C. M. Horwitz, Appl. Phys. Lett. 36, 727 (1980).
    [CrossRef]
  4. W. H. Lowdermilk, D. Milam, Appl. Phys. Lett. 36, 891 (1980).
    [CrossRef]
  5. W. H. Lowdermilk, D. Milham, Laser Focus64 (1980).
  6. See, for example, V. S. Ban, D. A. Kramer, J. Mater. Sci. 5, 978 (1970).
    [CrossRef]
  7. See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 51.
  8. P. Yeh, A. Yariv, C. Hong, J. Opt. Soc. Am. 62, 423 (1977);67, 438 (1977).
    [CrossRef]
  9. Z. Knittle, Optics of Thin Films (Wiley, New York, 1976), p. 484.
  10. G. Bauer, Ann. Phys. Leipzig 19, 434 (1934).
    [CrossRef]
  11. B. Sheldon, J. S. Haggerty, A. G. Emslie, J. Opt. Soc. Am. 72, 1049 (1982).
    [CrossRef]

1982 (1)

1980 (3)

C. M. Horwitz, Appl. Phys. Lett. 36, 727 (1980).
[CrossRef]

W. H. Lowdermilk, D. Milam, Appl. Phys. Lett. 36, 891 (1980).
[CrossRef]

W. H. Lowdermilk, D. Milham, Laser Focus64 (1980).

1977 (1)

1970 (1)

See, for example, V. S. Ban, D. A. Kramer, J. Mater. Sci. 5, 978 (1970).
[CrossRef]

1961 (1)

R. E. Thun et al., Automat. Control. 14, 26 (1961).

1934 (1)

G. Bauer, Ann. Phys. Leipzig 19, 434 (1934).
[CrossRef]

Ban, V. S.

See, for example, V. S. Ban, D. A. Kramer, J. Mater. Sci. 5, 978 (1970).
[CrossRef]

Bauer, G.

G. Bauer, Ann. Phys. Leipzig 19, 434 (1934).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 51.

Emslie, A. G.

Haggerty, J. S.

Hong, C.

Horwitz, C. M.

C. M. Horwitz, Appl. Phys. Lett. 36, 727 (1980).
[CrossRef]

Jacobsson, R.

See, for example, R. Jacobsson, “Light Reflection From Films of Continuously Varying Refractive Index,” in Progress in Optics, Vol. 5, E. Wolf, Ed. (Wiley, New York, 1966), p. 249.
[CrossRef]

Knittle, Z.

Z. Knittle, Optics of Thin Films (Wiley, New York, 1976), p. 484.

Kramer, D. A.

See, for example, V. S. Ban, D. A. Kramer, J. Mater. Sci. 5, 978 (1970).
[CrossRef]

Lowdermilk, W. H.

W. H. Lowdermilk, D. Milham, Laser Focus64 (1980).

W. H. Lowdermilk, D. Milam, Appl. Phys. Lett. 36, 891 (1980).
[CrossRef]

Milam, D.

W. H. Lowdermilk, D. Milam, Appl. Phys. Lett. 36, 891 (1980).
[CrossRef]

Milham, D.

W. H. Lowdermilk, D. Milham, Laser Focus64 (1980).

Sheldon, B.

Thun, R. E.

R. E. Thun et al., Automat. Control. 14, 26 (1961).

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 51.

Yariv, A.

Yeh, P.

Ann. Phys. Leipzig (1)

G. Bauer, Ann. Phys. Leipzig 19, 434 (1934).
[CrossRef]

Appl. Phys. Lett. (2)

C. M. Horwitz, Appl. Phys. Lett. 36, 727 (1980).
[CrossRef]

W. H. Lowdermilk, D. Milam, Appl. Phys. Lett. 36, 891 (1980).
[CrossRef]

Automat. Control. (1)

R. E. Thun et al., Automat. Control. 14, 26 (1961).

J. Mater. Sci. (1)

See, for example, V. S. Ban, D. A. Kramer, J. Mater. Sci. 5, 978 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

Laser Focus (1)

W. H. Lowdermilk, D. Milham, Laser Focus64 (1980).

Other (3)

See, for example, R. Jacobsson, “Light Reflection From Films of Continuously Varying Refractive Index,” in Progress in Optics, Vol. 5, E. Wolf, Ed. (Wiley, New York, 1966), p. 249.
[CrossRef]

Z. Knittle, Optics of Thin Films (Wiley, New York, 1976), p. 484.

See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 51.

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Figures (4)

Fig. 1
Fig. 1

Schematic drawing of a multilayer dielectric structure.

Fig. 2
Fig. 2

Reflectivity spectrum of an exponentially graded multilayer structure with N = 5, n0 = 2.8, ns = 3.5. Rmax = 0.0125. The flat horizontal line has been calculated using Eq. (5).

Fig. 3
Fig. 3

Reflectivity spectrum of a graded transition layer with a hyperbolic profile, n0 = 2.8, ns = 3.5.

Fig. 4
Fig. 4

Reflectivity spectrum of a graded transition layer with an exponential profile, n0 = 2.8, ns = 3.5.

Equations (32)

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n ( z ) = n 0 z < z 0 , n 1 z 0 < z < z 1 , n 2 z 1 < z < z 2 , n N z N 1 < z < z N , n s z N < z ,
1 T = N + i = 0 N 1 T i ,
( T n i ) = 0 and ( 2 T n i 2 ) < 0 .
n i = n 0 ( n s n 0 ) i / N + 1 i = 1 , 2 , , N
T = 1 1 + ( N + 1 ) sinh 2 [ 1 2 ( N + 1 ) log ( n s n 0 ) ] .
T 1 1 4 ( N + 1 ) log 2 ( n s n 0 ) .
2 π λ n i t i = ϕ i = 1 , 2 , , N ,
r = C sin ( N + 1 ) y A sin ( N + 1 ) y sin Ny ,
C = 1 x 2 x exp ( i ϕ ) ,
A = 1 + x 2 x exp ( i ϕ ) ,
y = cos 1 ( 1 + x 2 x cos ϕ ) ,
x = ( n s n 0 ) 1 / N + 1 .
R = | r | 2 = 1 1 + sin 2 y | C | 2 sin 2 ( N + 1 ) y .
R e = 1 1 + sin 2 y / | C | 2 .
n ( z ) = n 0 1 ( n s n 0 n 0 ) z L ,
Φ = lim it N ϕ N = 2 π λ 0 L n ( z ) dz = 2 π n 0 n s L ( λ ( n s n 0 ) log ( n s n 0 ) .
| C | 1 2 N log ( n s n 0 ) ,
y 1 N [ Φ 2 ( ½ log n s n 0 ) 2 ] 1 / 2 .
R = 1 1 + Φ 2 ¼ log 2 ( n s / n 0 ) ¼ log 2 ( n s / n 0 ) sin 2 Φ 2 ¼ log 2 ( n s / n 0 ) .
sin 2 Φ 2 ¼ log 2 ( n s n 0 ) = 1 .
R e = 1 1 + 4 Φ 2 log 2 ( n s / n 0 ) .
n ( z ) = n 0 exp [ ( z / L ) log ( n s / n 0 ) ] .
r = [ J 0 ( y 0 ) i J 1 ( y 0 ) ] [ Y 0 ( y s ) i Y 1 ( y s ) ] [ Y 0 ( y 0 ) i Y 1 ( y 0 ) ] [ J 0 ( y s ) i J 1 ( y s ) ] [ J 0 ( y 0 ) + i J 1 ( y 0 ) ] [ Y 0 ( y s ) i Y 1 ( y s ) ] [ Y 0 ( y 0 ) + i Y 1 ( y 0 ) ] [ J 0 ( y s ) i J 1 ( y s ) ] ,
y 0 = 2 π n 0 L λ log ( n s / n 0 ) y s = 2 π n s L λ log ( n s / n 0 ) .
R = ( n s n 0 ) 2 16 Φ 2 n 0 2 n s 2 ( n s 2 + n 0 2 2 n 0 n s cos 2 Φ ) ( L / λ ) 1 ,
R = ( n 0 n s n 0 + n s ) 2 [ 1 α ( 4 π L λ ) 2 ] ( L / λ ) 1 ,
Φ = 2 π λ 0 L n ( z ) dz = 2 π L ( n s n 0 ) λ log ( n s / n 0 ) ,
α = n 0 n s 4 log 2 ( n s / n 0 ) [ 1 4 n 0 2 n s 2 ( n s 2 n 0 2 ) 2 log 2 ( n s n 0 ) ] .
R = ¼ log 2 ( n s n 0 ) sin 2 Φ Φ 2 ( L / λ ) 1 ,
R = | C | 2 sin 2 ( N + 1 ) y sin 2 y ,
R = ( N + 1 ) | C | 2 .
R = ( N + 1 ) sinh 2 [ 1 2 ( N + 1 ) log ( n s n 0 ) ] .

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