Abstract

Results of the theoretical computations of reflectance and transmittance values of periodic dielectric multi-film stacks with symmetrical periods, each consisting of films of three different refractive indices, are reported. The analysis yields good estimates of the bandwidths of the low reflectance and high reflectance zones for different compositions of the periodic structures, and one of the structures offers the possibility of use as a heat reflecting mirror. This particular structure when combined with a previously published design gives a heat reflecting mirror design with good spectral characteristics and seems to be relatively easy to fabricate, while the earlier design, which exhibits somewhat superior spectral characteristics, appears to be extremely difficult to fabricate.

© 1988 Optical Society of America

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References

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  1. H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
  2. A. Thelen, “Multilayer Filters with Wide Transmittance Bands,” J. Opt. Soc. Am. 53, 1266 (1963).
    [CrossRef]
  3. B. S. Verma, R. Bhattacharyya, V. V. Shah, “Optical Admittance of an All-Dielectric Unsymmetrical Multilayer near the Monitoring Wavelength,” Appl. Opt. 25, 315 (1986).
    [CrossRef] [PubMed]
  4. A. F. Turner, P. W. Baumeister, “Multilayer Mirrors with High Reflectance over an Extended Spectral Region,” Appl. Opt. 5, 69 (1966).
    [CrossRef] [PubMed]

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1966 (1)

1963 (1)

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Figures (3)

Fig. 1
Fig. 1

(a), (b) Reflectance vs wavelength plots for periodic symmetrical multifilm stacks involving three different indices. The geometry of the stack and other parameters are given in each plot.

Fig. 2
Fig. 2

Reflectance vs wavelength plot for a heat reflecting mirror of our design: 1.0 | { 0.93 ( A B 2 C B A ) } 6 { A 2 ( A 2 B C B A 2 ) 6 A 2 } | 1.52 ,where nA = nL = 1.30, nB = nI = 1.73, nC = nH = 2.30, λref = 475 nm.

Fig. 3
Fig. 3

Reflectance vs wavelength plot for a heat reflecting mirror of the design proposed by Thelen2: 1.0 | [ 1.1 ( A 2 C A 2 ) ] ( A 2 C A 2 ) 5 [ 1.25 ( A 2 C A 2 ) ] [ 0.57 ( A B C D A ) ] 8 × [ 0.642 ( A B 2 C B A ) ] 8 A 2 | 1.50 ,where nA = 1.38, nB = 1.781, nC = 2.30, nD = 1.90, λref = 860 nm.

Equations (41)

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air ( n 0 ) | ( A 2 B C B A 2 ) p | ( n s ) substrate ,
M A 2 = ( cos δ / 2 i / n A sin δ / 2 i n A sin δ / 2 cos δ / 2 ) , M B = ( cos δ i / n B sin δ i n B sin δ cos δ ) , M C = ( cos δ i / n c sin δ i n c sin δ cos δ ) ,
δ = 2 π D λ = π 2 · λ 0 λ ,
M 1 = M B M C M B = ( cos γ i E sin γ i E sin γ cos γ ) ,
cos γ = cos δ [ 1 - ( 2 + n B n C + n C n B ) sin 2 δ ] , E = n B ( 2 + n B n C + n C n B ) cos 2 δ - n B n C ( 2 + n B n C + n C n B ) cos 2 δ - n C n B ,
M = M A 2 M 1 M A 2 = ( cos Γ i N sin Γ i N sin Γ cos Γ ) ,
cos Γ = cos γ cos δ - 1 2 ( n A E + E n A ) sin γ sin δ , N = n A sin δ cos γ + 1 2 ( n A E + E n A ) cos δ sin γ - 1 2 ( n A E - E n A ) sin γ sin δ cos γ + 1 2 ( n A E + E n A ) cos δ sin γ + 1 2 ( n A E - E n A ) sin γ .
M p = ( cos p Γ i / N sin p Γ i N sin p Γ cos p Γ ) ;
( P Q ) = M p ( 1 n s ) , P = cos p Γ + i n s N sin p Γ , Q = n s cos p Γ + i N sin p Γ ,
Y = Q P = n s cos p Γ + i N sin p Γ cos p Γ + i n s / N sin p Γ .
R = | n 0 - Y n 0 + Y | 2 = ( n 0 - n s ) 2 cos 2 p T + ( n 0 n s N - N ) 2 sin 2 p Γ ( n 0 + n s ) 2 cos 2 p Γ + ( n 0 n s N + N ) 2 sin 2 p Γ .
δ ( λ ) = δ ( λ 0 ) + ( λ - λ 0 ) δ λ | λ = λ 0 + = π 2 - π 2 ( λ - λ 0 λ 0 ) + π 2 - θ ,
θ = π 2 ( λ - λ 0 λ 0 ) .
cos δ sin θ , sin δ 1.
cos γ = - ( 1 + n B n C + n C n B ) sin θ ,
sin γ ± ( 1 - ½ cos 2 γ ) as γ 1 if θ 1 = - 1 + 1 2 ( 1 + n B n C + n C n B ) 2 sin 2 θ ,
E n B 2 n C [ 1 + 1 2 ( 2 + n B n C + n C n B ) ( n B n C + n C n B ) sin 2 θ ] .
cos Γ C 1 - C 2 sin 2 θ ,
C 1 = 1 2 ( n A n C n B 2 + n B 2 n A n C ) , C 2 = 1 + ( n B n C + n C n B ) + ( n A n B + n B n A ) + 1 2 ( n C n A + n A n C ) + ½ C 1 .
1 2 ( x + 1 x ) 1.
C 1 = 1 for n B 2 = n A n C > 1 otherwise .
sin θ = C 1 + 1 C 2 ,
Δ λ λ 0 = 4 π sin - 1 C 1 + 1 C 2 ,
cos Γ = 1 ,
sin θ = C 1 - 1 C 2
Δ λ λ 0 = 4 π sin - 1 C 1 - 1 C 2 ,
δ ( λ ) = δ ( 2 λ 0 ) + ( λ - 2 λ 0 ) δ λ | λ = 2 λ 0 + π 4 - π 2 · λ - 2 λ 0 4 λ 0 = π 4 - ψ ,
ψ = π 8 ( λ - 2 λ 0 λ 0 ) .
cos δ cos ( π / 4 - ψ ) = 1 2 ( 1 + sin ψ ) , sin δ sin ( π / 4 - ψ ) = 1 2 ( 1 - sin ψ ) .
cos Γ - A 1 + A 2 sin 2 ψ ,
sin ψ = A 1 - 1 A 2 or Δ λ 2 λ 0 = 8 π sin - 1 A 1 - 1 A 2 ,
1.0 | ( A 2 B C B A 2 ) 6 | 1.52             ( P = 6 )
1.0 | ( L 2 I H I L 2 ) p | 1.52             ( p = integer 6 )
1.0 ( A B 2 C B A ) 10 1.52 ,
1400 12 467 4 nm ;
1.0 | { 0.93 ( A B 2 C B A ) } 6 { A 2 ( A 2 B C B A 2 ) 6 A 2 } | 1.52 ,
1.0 | { 0.93 ( A B 2 C B A ) } 6 { A 2 ( A 2 B C B A 2 ) 6 A 2 } | 1.52 ,
1.0 | [ 1.1 ( A 2 C A 2 ) ] ( A 2 C A 2 ) 5 [ 1.25 ( A 2 C A 2 ) ] [ 0.57 ( A B C D A ) ] 8 × [ 0.642 ( A B 2 C B A ) ] 8 A 2 | 1.50 ,
a 1 = 1 2 2 ( n B n C + n C n B ) , a 2 = 2 + a 1 , a 3 = 1 - a 1 2 , a 4 = a 1 a 2 / a 3 , a 5 = n B 1 - 1 2 ( n B n C - n C n B ) 1 + 1 2 ( n B n C - n C n B ) , a 6 = n B ( 2 + n B n C + n C n B ) ( n B n C - n C n B ) 2 { 1 - 1 2 ( n B n C - n C n B ) } 1 / 2 { 1 + 1 2 ( n B n C - n C n B ) } 1 / 2 .
A 1 = 1 2 [ a 1 - a 3 2 ( a 5 n A + n A a 5 ) ] , A 2 = a 2 2 + ( 1 - a 4 ) 2 2 ( a 6 n A - n A a 6 a 5 2 ) + a 4 2 2 ( a 5 n A + n A a 5 ) .
a 1 1 2 ,             a 3 1 2 ,

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