Broadband High-Reflection Coating at 50 Degrees

[size=+3]Broadband High-Reflection Coating at 50 Degrees
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% {8 U. t5 s! ?, ^3 rIn 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.
the 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.
/ L2 j7 {6 l  a+ B( eThe 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).
' i! Y; l' l; Z1 W' XThe 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%.
. |3 }0 u" w& S  r1 h$ LTFCalc can be used to design and refine this type of coating. The starting design is a 43-layer coating with the stack formula
G 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
a = 1.2000, b = 1.0000, c = 0.8000, group optimization finds
  P3 v/ K# ^2 B" m4 ~9 W( ja = 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.
0 t1 r" q2 R0 M+ v. O$ G( G0 s
+ T- N3 v4 V% z& `! c) z; K 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.
% l  @, A! L. f0 P5 L6 U
- y7 b" t" u# t, o7 Y- S Here is the last design, with the first layer closest to the substrate and thickness given in nm.
    H      97.87
    L     171.35
    H      80.56
    L     152.15
1 q8 j/ i0 c7 n  f  X, p$ f    H      74.35
    L     154.04
; S; A+ g: x  Z; @- [$ G    H      83.10
/ J+ l7 s) ?1 H! o: u    L     178.45
# ^+ S& ~* g9 V' s9 U0 m4 R4 D1 z    H      72.02
- U( u; a  o: L; F6 z: W    L     129.75
. [$ j6 v3 j" e! H    H      85.36
    L     141.64
    H      82.96
    L     141.89
& g3 k8 A. J  Q! W    H      71.17
    L     137.71
  x) ]; [$ W& V- I0 a% }2 g    H      69.96
: {6 j" J3 X) M    L     104.04
    H      55.42
$ a7 P1 r+ h9 F    L     113.89
    H      71.53
* b* J8 r5 T8 W$ ~    L     121.61
5 r" M6 v- S: g0 a    H      63.21
, R  G( U5 b' z% b6 q0 f% H    L     111.11
    H      61.47
    L     121.69
) `: @0 g# R  u    H      63.51
    L     101.69
8 S. @4 p2 J+ W! {    H      56.86
4 E4 ~; F4 h3 a, b  p/ g. ^    L     101.16
    H      52.26
    L      82.60
/ ~6 G2 Y, \$ U; |    H      46.62
    L      94.22
    H      53.49
    L      90.92
    H      46.16
. O3 u& r( x$ P/ x1 `    L      86.62
/ X. E* _! h0 A) x; ]  w! n    H      51.59
& Q2 A  N: h0 Q8 F5 p& l1 P    L      91.80
    H      45.96
    L      80.64
    H      49.35