Gain Equalization (or Flattening) Filters

The terms "gain equalization" and "gain flattening" come from the fiberoptic telecommunications field, where erbium-doped fibers are used to amplify the light traveling through the fiber. These amplifiers have a wavelength-dependent gain; i.e., some wavelengths are amplified more than others. A gain-flattening filter restores all wavelengths to approximately the same intensity. Erbium-doped fiber amplifiers (EDFA) operate in the 1530-1565 nm wavelength range.
7 b$ _0 l. x. A' SThis type of filter has many applications. For example, for visible wavelengths 380-780 nm, a filter could be constructed to convert any continuous-spectrum illuminant into white light; e.g., light from an ordinary light bulb could be filtered to produce almost pure white light.
* f) i# w$ H: j) iAn introduction of the design of this filter can be found in the paper
) W. C& ~. J* l3 `. BM. Tilsch, C.A. Hulse, K.D. Hendrix, R.B. Sargent, "Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs", presented at the 1999 NFOEC conference.
The plot below shows the relative intensity of the EDFA output. If this were plotted on the logarithmic dB scale, the output would have a range of 8 dB.
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0 |0 p# I1 I7 z. C* S5 j$ k' l We will use TFCalc's optimization capabilities to design a gain-flattening reflection filter which will flatten (or equalize) the output of the EDFA so that it is flat with a tolerance of ±0.5 dB. In general, the tolerance is tighter for real filters.
TFCalc has the capability of optimizing the product of the illuminant (EDFA) and the reflectance (or transmittance) of a coating. We want this product, which is the actual output of the filter, to be a flat as possible. As can be seen from the plot shown above, even if the filter reflects (or transmits) a 100% at the endpoints (1527.3 and 1566.3 nm), the output would only be about 20%. Hence, for the entire wavelength range, the flattened output of the filtered EDFA can be no more than 20%. The design becomes somewhat easier if the target output is less than 20%.
: m2 m% }+ s0 p6 k. S0 |, j: t$ oReflection Filter
Tilsch et al studied the reflection filter in their paper. To design such a filter, we started with the initial 100-layer design (HL)^50, where H represents 1.2 QWOT of index 2.25 and L represents 1.2 QWOT of index 1.45. The reference wavelength is 1530 nm. The substrate has index 1.44. We design this for normal incidence; in reality, the filter will be positioned at a small angle. For the optimization target we use
EDFA * Reflectance = 17.5% for wavelengths from 1527.3 to 1566.3
A total of 201 equally-spaced targets are used. To make the output as flat as possible, we use power=16 during optimization. The result is a 91-layer design. In white light, the performance of the filter is shown below.
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! w( Y. _; M" y7 F$ x6 B The plot below shows the output of the filtered EDFA. If this is plotted on a dB scale, it becomes apparent that the design, if it could be produced without errors, meets the ±0.5 dB requirement.
6 d5 @: `) B3 ?: ~

Transmission Filter
8 Z3 V7 V2 X. n1 E1 UTo discover whether there is any advantage in using a transmission filter, we decided to design one. Most of the details are the same as for the reflection filter. However, this time we started with the 101-layer design (HL)^50 H. The target is
EDFA * Transmittance = 17.5% for wavelengths from 1527.3 to 1566.3
We used one continuous target. Again, we use power=16. By constraining the layer thicknesses to be greater than 0.5 QWOT during optimization, we discover a 101-layer design having a very flat (less than ±0.12 dB) output, as displayed below on a dB scale.

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0 a2 T; ?/ q# EManufacturability
1 n9 r& `# a% [# p9 X) w) n1 RBy randomly varying the thickness of layers, TFCalc can determine the manufacturability of a design. For the designs mentioned in the Tilsch et al paper, a maximum error of 0.05% is required for production. Note that, using this error limit, if a layer is 100 nm thick, then the thickness must be monitored to within ±0.5 angstrom! The plot below shows the range of the performance of the reflection filter for 1000 random designs when layers are allowed to vary uniformly by ±0.05%.

Even 0.05% seems barely acceptable. Using TFCalc's sensitivity optimization capability, we can try to find a design that is less sensitive to manufacturing errors. The procedure is rather slow, but the results may be worth the wait. We chose to optimize the sensitivity of 1000 random designs. On a 1 GHz Pentium, this requires several hours. Below is the sensitivity analysis of the improved design. Note that the optimizer has found a design that is significantly less sensitive to thickness errors.

2 V. b9 V, O- O% b% e& H+ x Although the sensitivity optimization seems to work by making the performance of the filter flatter when the intensity of the EDFA is high, the optimizer actually does more. Numerical experiments show that, in general, merely designing a filter that is flatter where the EDFA intensity is high will still be very sensitive to thickness variations. The sensitivity optimization actual finds a less sensitive design, which also happens to have a flatter output at wavelengths where the EDFA has higher intensity.
# _4 u; T) Y6 q+ B4 UThis work is not nearly as exhaustive as in the Tilsch et al paper. However, it does show how optimizing the sensitivity can improve the performance of a design.
% V! s9 c5 R3 j6 u+ hHere are the two designs; Design 1 is the original reflection design; Design 2 is the reflection design optimized for sensitivity. The first layer is closest to the substrate and thickness is given in nm:
1 n6 _8 Z0 z  v* B: [  }Material  Design1   Design2
; q  v1 b) F( J! m7 o( qN225      173.22    183.23
- @- N, Z& H  f! G+ m; |N145      266.11    283.45
. D+ E8 r) F3 ^N225      170.37    187.68
N145      281.70    310.91
, I: B1 |! p* ]) P& J1 i% A2 NN225      241.67    232.70
2 U' M% M  l9 ?7 q# CN145      499.80    486.55
, v/ X9 b( W3 `* [5 eN225      149.42    123.93
; J+ a$ n2 Y) q  gN145      216.86    239.55
N225      170.23    182.06
# R, Y0 ]6 O4 h% @* ^/ [N145      276.29    300.51
+ x+ l0 s/ t9 G8 h" GN225      162.25    185.72
N145      260.42    264.95
8 ?$ U( I$ m6 O; {! B' P+ ?# u+ Y1 ~, ZN225      176.10    151.72
$ e8 T4 y. N3 N$ R) V2 qN145      249.71    215.61
$ Z: b% `2 k5 k( j. YN225      389.76    404.62
N145      244.24    250.29
7 E" K7 K/ m; X! W! f# BN225      176.51    176.04
N145      269.84    258.26
' |, W* s8 A: _2 ^N225      172.68    161.16
2 v# l% U  f# W& A/ qN145      351.00    358.60
N225      318.17    316.63
N145      285.93    279.08
; g- p/ }3 F5 x/ Q) ]% F, HN225      158.51    181.70
N145      246.70    282.89
N225      177.51    169.19
8 i* V) `; _4 d' [N145      290.04    253.96
N225      166.26    165.89
/ j: G( G0 Q+ k4 H1 v/ P: R2 [N145      265.69    263.71
/ @6 s' V4 ^( W# |+ F9 AN225      187.39    176.44
4 s6 L7 E; P* X: |% _2 K. d+ m; {N145      311.12    322.47
N225      266.31    283.07
N145      353.44    322.81
N225      183.75    179.50
N145      268.79    273.76
N225      177.71    179.47
N145      267.34    275.20
N225      180.82    186.31
N145      265.61    285.18
1 H8 J% ?: T9 I; G+ l) }$ r$ u& ?N225      179.56    191.58
/ h. d  r' j, C: H3 }/ a) zN145      310.69    307.51
8 i% m; Y' A1 C3 tN225      201.44    191.85
N145      337.80    335.97
: I( e7 O% N+ g- eN225      546.46    539.71
N145      419.51    418.70
N225      180.85    182.18
, e/ V9 C) B! F+ c! ]2 p" NN145      261.04    270.87
N225      173.18    179.08
/ E1 E. X, x) @5 W# n, D% y3 p# iN145      264.37    276.37
N225      172.93    185.79
) t, |1 k2 A$ A$ p' AN145      264.15    256.11
( C2 x% L8 G3 o' x9 n( IN225      175.67    168.30
5 w# P% c9 y+ c: Y3 u! D. `: ~N145      265.07    260.12
N225      178.47    199.99
N145      251.01    244.43
N225      182.79    199.31
N145      596.29    582.84
5 o+ ~0 f8 B# `N225      115.52    114.77
N145      244.92    217.38
N225      173.55    177.19
N145      263.13    271.27
N225      168.53    174.83
N145      265.62    276.17
N225      172.31    179.17
1 ]% ]- \) T3 oN145      264.86    274.15
N225      175.52    180.52
" v! Z) X, ?, D9 R/ r2 m* n% IN145      280.45    288.85
( |* ]! z. s0 `+ F( KN225      176.08    183.83
N145      243.23    239.70
2 G6 z+ O. r: NN225      245.25    215.08
4 _3 C+ x" h$ H( G6 |- t2 R5 tN145      177.49    231.26
N225      358.93    358.44
N145      262.76    255.62
: S7 O3 v6 M7 u) oN225      176.24    171.92
8 U! W" Q/ A+ a7 y( C( q: uN145      245.65    223.47
N225      166.69    160.65
N145      255.30    259.85
N225      165.34    169.67
N145      256.16    275.05
5 T. ^/ u1 L2 P/ d4 q2 a3 q6 {% V& }N225      412.72    403.73
N145      239.67    218.93
N225      175.49    175.58
N145      273.85    297.29
6 j6 g7 i+ G1 X. \N225      140.82    169.58
N145      297.28    309.52
N225      219.98    193.75
N145      493.44    484.74
% k8 ?) `7 B1 D9 a" s5 f* l1 XN225      168.57    157.29
* f8 K* o* S+ @5 Z( ^% @: |N145      282.43    319.13
N225      175.48    186.11
4 b- V% D- f% I8 v+ U8 X/ L+ vN145      274.12    303.92
N225      173.02    181.17