Broadband Antireflection Graded-Index Coating

[size=+3]Broadband Antireflection Graded-Index Coating
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In an important 1962 paper,
: _, B4 Q! ^) L2 W+ QPeter H. Berning, "Use of Equivalent films in the design of infrared multilayer antireflection coatings", Journal of the Optical Society of America, Vol. 52(4), pp. 431-436 (April 1962).
5 C- i4 F9 |) none of the coatings discussed by the author is a step-graded-index antireflection coating for germanium. This coating consists of 20 quarter-wave layers whose indices vary linearly from 1.35 (the lowest practical index available) to 4 (the index of germanium), as illustrated below.
9 V" V2 K7 X% C8 D  `; y% t7 _! \
; d2 v! k4 N$ v! d5 k4 b$ w% b The coating has a very broad wavelength range over which the reflectance is low. The question arises: is the linear profile the best possible? We can use optimization to explore this problem. Here we use a feature unique to TFCalc -- the capability of optimizing the reflective index while keeping the optical thickness fixed. We use one continuous optimization target: R=0 for 3-20 microns. To make the reflectance as flat as possible, we set the power in the merit function to 16. Surprisingly, the resulting index profile looks like part of a sine curve:

2 r  d( [: W9 p3 v0 P5 i: j: z The performance of the optimized design is significantly better. The reflectances of the two designs are shown below.


# }9 _) x  i+ h# c7 E; @Here is the list of refractive indices in the optimized design, starting with the layer closest to the germanium substrate:
' [8 L2 m' M6 P6 O2 c$ m! q6 N# J    1   3.9356
    2   3.7482
    3   3.3986
    4   2.9121
    5   2.3790
    6   1.9044
) D/ {: \. i. g# D1 K0 ?2 q( |    7   1.5598
    8   1.3726
  P' b7 F0 X# v+ X& I( Y- r$ Z    9   1.3500
   10   1.5042
7 e) }3 v7 k3 q, W3 Y   11   1.8606
% H4 n+ N. I$ ]& ~5 s0 k" R7 c   12   2.4315
+ p2 ?4 B" m% S8 G# M   13   3.1456
- z. A8 z- D% `0 `# N   14   3.7738
" y+ V7 _3 w: b3 Z, M+ f   15   4.0000
- @7 x  F6 q" v$ v   16   3.6871
) x- d/ d/ R9 @, L   17   3.0181
3 A$ Y7 l6 K2 F" ?, u2 [. [   18   2.3016
& L! I0 _1 k2 ~1 g9 @0 T  w  x& p) W   19   1.7310
7 w) @. J8 J6 _* o! \* o# y7 k   20   1.3500
The reference wavelength for the quarter-wave optical thickness is 5.217 microns, which is the midpoint, in the frequency scale, between 3 and 20 microns.
2 w1 `7 a, k8 CNote: by using more layers, designs with more cycles of the sine curve can be found. However, the additional cycles do not improve the performance as dramatically as above.
4 O$ {7 b" }& yDownload DesignYou may download the Berning design and watch how TFCalc optimizes it. You will need a real copy (i.e., not a demo) of TFCalc. Click here for the Windows or Macintosh file. This design assumes that you have a substrate called G whose index is 4. The G substrate was installed with the TFCalc software.
# \1 Y% q! N2 z* V* Z' ]0 g5 RWindows users, when they download this file, will have to save it to disk. Be sure to save it with extension ".TFD".