Broadband Antireflection Graded-Index Coating

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In an important 1962 paper,
    Peter 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).

one 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.

Plot of AR design

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:

Plot of index profile

The performance of the optimized design is significantly better. The reflectances of the two designs are shown below.

Plot of AR design

Here is the list of refractive indices in the optimized design, starting with the layer closest to the germanium substrate: 

    1   3.9356
    2   3.7482
    3   3.3986
    4   2.9121
    5   2.3790
    6   1.9044
    7   1.5598
    8   1.3726
    9   1.3500
   10   1.5042
   11   1.8606
   12   2.4315
   13   3.1456
   14   3.7738
   15   4.0000
   16   3.6871
   17   3.0181
   18   2.3016
   19   1.7310
   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.

Note: 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.

Download Design

You 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.

Windows users, when they download this file, will have to save it to disk. Be sure to save it with extension ".TFD".

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