Broadband High-Reflection Coating at 50 Degrees

[size=+3]Broadband High-Reflection Coating at 50 Degrees
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6 |* a! W& I* U+ ~, n. k0 zIn the paper,
Konstantin V. Popov, J.A. Dobrowolski, Alexander V. Tikhonravov, and Brian T. Sullivan, "Broadband high-reflection multilayer coatings at oblique angles of incidence," Applied Optics, Vol. 36, No. 10, 1 April 1997, pp. 2139-2151.
# O+ I3 @4 q3 _$ K5 U% k) othe authors examine the problem of creating broadband dielectric reflectors using contiguous quarter-wave stacks. It is an interesting paper -- well worth reading if you need to design this type of coating. The following example is taken from the paper.
5 @$ o; T5 A" g, `& f. E5 NThe goal is to design a high reflector which operates in the wavelength range 400-800 nm at a 50 degree angle of incidence. Because this coating is at 50 degrees, the reflectance of S polarization will be much higher than P. Hence, we can concentrate on controlling the reflectance for P polarization. The coating, composed of layers of L (index 1.45) and H (index 2.35), is to be deposited on glass G (index 1.52). The incident medium is air (index 1.0).
! v8 M; t& l5 ~7 M2 y+ M. J: [& BThe formulas in the paper lead to a 3-stack (7 periods/stack) design whose stacks are centered at wavelengths 705, 555, and 437 nm. To increase the reflectance, the authors add one additional H layer next to the substrate. The result is a 43-layer coating whose average P reflectance is 98%.
, z9 T. H- V) B' [1 eTFCalc can be used to design and refine this type of coating. The starting design is a 43-layer coating with the stack formula
6 ?. J* [# w$ nG a(HL)^7 b(H(LH)^7) c(LH)^7 air where H and L represent 1 QWOT (quarter-wave optical thickness) at a reference wavelength of 550 nm and incident angle of 50 degrees, and the factors a, b,and c must be determined using "group" optimization. (This is called group optimization because we vary the thickness of a group of layers; all layers in the same group keep the same relative thickness to the other layer in the group.) Using one continuous target (Rp=100% for 400-800 nm at 50 degrees) and starting values of
  j$ T( z* c4 S7 I5 ra = 1.2000, b = 1.0000, c = 0.8000, group optimization finds
/ {9 R: t8 _6 ~5 ga = 1.3260, b = 1.0171, c = 0.7855. These values correspond to stacks centered at 729, 559, and 432 nm. The average P reflectance is about 98%. The performance for P polarization is displayed below.

If all layers are optimized to improve the reflectance, the average P reflectance stays approximately the same; however, the minimum reflectance is substantially higher, as shown below.

Here is the last design, with the first layer closest to the substrate and thickness given in nm.
6 F, K8 ^# t  s1 X( f$ v    H      97.87
/ _* g. C0 O6 ]8 e    L     171.35
6 ^) }; ], I' C! t1 t7 u6 g* Z    H      80.56
    L     152.15
# m7 l' n6 `. D    H      74.35
    L     154.04
) U8 x& C4 ]) k7 s" J7 i) M  P    H      83.10
! t' r" |8 L  C1 y9 T    L     178.45
    H      72.02
$ _5 k& |% ?1 }7 Z6 [    L     129.75
3 w5 x) @& K. t7 Y, f2 `    H      85.36
    L     141.64
    H      82.96
    L     141.89
    H      71.17
    L     137.71
+ U" M3 P( J% j7 ?9 V    H      69.96
1 h1 w& R- U7 H( |2 ?* V  V9 d. {    L     104.04
% j* o: r$ v# S: G4 x  ?    H      55.42
% U5 [) k% ]4 a- W/ Q4 K# A% n    L     113.89
    H      71.53
    L     121.61
* ^; K7 q" h: M$ V/ ]5 o    H      63.21
+ W/ r; S$ H6 j% d    L     111.11
    H      61.47
    L     121.69
    H      63.51
    L     101.69
    H      56.86
+ s; k* N. U9 |6 @, r    L     101.16
    H      52.26
    L      82.60
    H      46.62
    L      94.22
    H      53.49
) U* Y" Z, y. H% `$ Z, U: i- _    L      90.92
    H      46.16
    L      86.62
    H      51.59
! g, |9 o5 v  d; i* H" Z5 \    L      91.80
    H      45.96
    L      80.64
    H      49.35