Broadband High-Reflection Coating at 50 Degrees

[size=+3]Broadband High-Reflection Coating at 50 Degrees
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) s: j$ p7 u' D! Y# NIn 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.
) K( m7 c5 \& P$ bThe 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).
The 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%.
TFCalc can be used to design and refine this type of coating. The starting design is a 43-layer coating with the stack formula
4 X4 O- N5 E3 a1 hG 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
# b8 F% h2 ^2 ~# O( `8 c9 y. [a = 1.2000, b = 1.0000, c = 0.8000, group optimization finds
a = 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.
6 j& R8 B" J' ^; K$ n/ H$ l
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.
9 T) W  X: z7 o* o) e    H      97.87
    L     171.35
. C8 N6 U6 R+ h' Q; j8 N    H      80.56
( a  X! R6 d0 I1 f: a: t    L     152.15
& `! c6 p8 d9 c4 N1 f3 s    H      74.35
3 Q$ c, B3 {  g' t* D0 c2 B    L     154.04
    H      83.10
) r: I; ?' \# h" V- w7 F    L     178.45
( ~5 [2 c+ G$ J1 [    H      72.02
) Z2 J# b4 Z9 U& {, U7 @+ r    L     129.75
    H      85.36
    L     141.64
    H      82.96
    L     141.89
    H      71.17
    L     137.71
    H      69.96
    L     104.04
    H      55.42
9 R9 b9 ~( z2 U3 M( S) ~  x    L     113.89
    H      71.53
& |. H) v5 `8 ]    L     121.61
    H      63.21
- K8 T! S- I9 M+ {; |    L     111.11
    H      61.47
7 Z8 g- @9 |; G. ]+ E( Y, u    L     121.69
7 M) }3 @5 z2 g, @1 s0 k, G    H      63.51
/ S' i$ M* y2 r0 d, S8 K8 J    L     101.69
    H      56.86
    L     101.16
! Z& U9 l/ X+ y    H      52.26
3 s" }+ p; S- c* @    L      82.60
: O* ]4 @- T$ l    H      46.62
    L      94.22
    H      53.49
1 z! ^: [0 c5 i* ~( G; A3 w& V/ g    L      90.92
/ z4 B2 ~% Y5 a1 \" V  v( q    H      46.16
    L      86.62
( `% s/ X% c- K% P0 U    H      51.59
    L      91.80
5 ]+ ~) s- o9 Q7 o4 q1 B8 x" j8 |2 u( i    H      45.96
    L      80.64
    H      49.35