Optics and Photonics Journal, 2011, 1, 124-129
doi:10.4236/opj.2011.13021 Published Online September 2011 (http://www.SciRP.org/journal/opj)
Copyright © 2011 SciRes. OPJ
Synthesis of Chained Achromatic Layer Systems Forming
Controlled Low Transmittance Bands
Mohamed Medhat, Samy S. Farag
Physics Department, Faculty of Science, Ain Shams University, Cairo, Egypt
E-mail: mmedhat61@hotmail.com
Received June 2, 2011; revised July 4, 2011; accepted July 16, 2011
Abstract
The approach utilized in the design of achromatic periods formed of two layers a high-index and a low-index
is developed by combining the first two periods together. These two periods are therefore reduced to three
layers a central one, an upper layer and a lower layer. Both the upper and lower layers are matched to the
central at two different wavelengths. This forms the so called a combined achromatic period or the basic unit.
Multilayers showing low transmittance bands are then synthesized of this basic unit. Parameters used in the
synthesis of such multilayers are pointed out and their control role is investigated.
Keywords: Thin Films, Design, Light Propagation
1. Introduction
Recalling the procedure followed in forming two layer
combination [1-3] which are matched according to the
relation

HHH LLL
oN oN
hnG hnG
(1)
where nH and nL are the refractive indices for the high-
and low-layers, respectively, and the h are the physical
thicknesses. Multilayers systems of N such achromatic
periods were formed so that each period matched at spe-
cified wavelength λoN, where N stands for period number.
The different wavelengths, at which the different periods
are formed, are specified according to the quantity ξ de-
fined as


dd dd
oN HHoNLLoN
HH LL
hn hn
nh nh



(2)
In the present work equation (1) is modified to [4]

HHH LLL
oN oN
hnG hnG
MN
(3)
where and N are real numbers [5]. M
Another modification is introduced here in the multi-
layer synthesis process. In a previous work [2,3] multi-
layers of achromatic periods (Equ ation (1)) were built up
by calculating the first achromatic period then subse-
quent periods were automatically built up by the way of
(1) and (2). In the present work the first achromatic pe-
riod is calculated by choo sing an initial cen tral layer then
an upper layer is calculated by matching to this central at
the design starting wavelength, by way of (3), then a
lower layer is matched to the central at another wave-
length. The convention of lower and upper is considered
according to positions of substrate and surrounding me-
dium with respect to the first central. Once the above
three layers are calculated they are considered as the ba-
sic unit. The process of building this basic unit is the
same as combining the two middle layers in the first two
achromatic periods. Subsequent layers are calculated by
considering the lower layer in the basic unit a central for
the following layer, in a chained manner, and so on. By
this means each layer, except the first and the last layers
in whole system, serves as a central layer.
2. Design
Six quantities are first specified, namely the film materi-
als, starting wavelength, number of layers, design num-
ber
, and N. Table 1 shows design details for a
multilayer formed of Zinc Sulphide and Cryolite. These
materials are chosen here only for illustration without
having any particular significance in the theory i.e. any
two materials (high and low) will work. The first central
layer is chosen as a high index layer which is calculated
as a quarter of the starting wavelength 1060 nm. is
M
M
M. MEDHAT ET AL.125
chosen as 0.77, as 1.3 and ξ as 1.01. The upper layer,
to the air side, is then calculated by way of (3). By way
of (2) the second reference Wavelength, after the starting
wavelength, is then calculated and then the lower layer
by way of (3) is calculated but at the second reference. It
is worth mentioning here that the first central layer may
be chosen as a low index as well. This will not affect our
design approach. This last calculated layer is then con-
sidered the central for the subsequent layer, a new refer-
ence wavelength, by w a y of (2), and so on.
N
500
Design details are listed in Table 1 and the resulting
transmittance spectrum in Figure 1 shows high-trans-
mittance bands alternating with low transmittance bands
[6]. The transmittance simulations are performed by
means of a program based on an algorithm presented by
Liddell [7]. Throughout this paper only the region of
spectrum including the central second order low trans-
mittance band approx. 400 nm - 900 nm will be consid-
ered.
3. Illustrations
If in the above design all calculations are done by way of
(1) instead of (3) (M =1, N = 1) and ξ also taken as
unity the resultin g design is listed in Table 2 . Setting all
parameters to unity results in the suppression of the cen-
tral minimum and all even orders (Figure 2). Comparing
with thicknesses in Ta ble 1 it can be seen that increasing
the ratio hL/hH is one way for keeping transmittance
minima in place. Another way is assigning ξ values big-
ger than unity. This is shown in the design listed in Ta-
ble 3 and the resulting transmittance curve in Figure 3.
Notice the appearance of central minimum but in a de-
3004006007008009001000 1100 12001300
0
20
40
60
80
100
T%
Wavelength(nm)
Figure 1. Transmittance spectrum for design listed in Table
1. Increasing the ratio hL/hH is one way for manifesting the
second order central minimum in addition to enhancing
T% on the short wave side of this central minimum.
Table 1. Design details for initial parameters: Start wave:
1060, ξ: 1.01, M: 0.77, N: 1.3 no of layers: 25. Every
reference wave for every layer is the wavelength matching
this layer to the next layer in the table.
Layer
no.
Physical Thickness
(after multiplication by
or in nm)
M NMaterial Reference
waves λo
(nm)
0 Air
1 264 Cryolite 1060
2 91 ZnS 373
3 265 - 372
4 90.7 - 371
5 266 - 370
6 90.4 - 369
7 267 - 368
8 90.1 - 367
9 268 - 366
10 89.8 - 365
11 269 - 364.9
12 89.5 - 364
13 270 - 363
14 89.1 - 362
15 271 - 361
16 88.8 - 360
17 272 - 359
18 88.4 - 358
19 273 - 357
20 88.1 - 356
21 274 - 355
22 87.8 354
23 275 353
24 87.4 352
25 276
Substrate Glass
300 400 500 600 700 800 9001000110012001300
0
20
40
60
80
100
T%
Wavelength(nm)
Figure 2. Transmittance spectrum for design listed in Table
2. When ξ, and are taken as unity the second
order central minimum is suppressed.
MN
Copyright © 2011 SciRes. OPJ
M. MEDHAT ET AL.
126
300 400 500 600 700 800 9001000110012001300
0
20
40
60
80
100
T%
Wavelength(nm)
Figure 3. Transmittance spectrum for design listed in Table
3. Increasing ξ is another way for exhibiting the second
order central min imum.
Table 2. Design details for initial parameters: Start wave:
1060, ξ: 1, : 1, : 1 no of layers: 25. Reference wave-
length the same for all layers.
M N
Layer
no.
Physical Thickness
(after multiplication by
or in nm)
MNMaterial Reference
waves
λo (nm)
0 Air
1 203 Cryolite 1060
2 118 ZnS -
3 203 - -
4 118 - -
5 203 - -
6 118 - -
7 203 - -
8 118 - -
9 203 - -
10 118 - -
11 203 - -
12 118 - -
13 203 - -
14 118 - -
15 203 - -
16 118 - -
17 203 - -
18 118 - -
19 203 - -
20 118 - -
21 203 - -
22 118 -
23 203 -
24 118 -
25 203
Substrate Glass
Table 3. Design details for initial parameters: Start wave:
1060, ξ: 1.25, : 1, : 1 no of layers: 25. Every refer-
ence wave for every layer is the wavelength matching this
layer to the next layer in the table.
MN
Layer
no.
Physical Thickness
(after multiplication by
M or in nm) NMaterial Reference
waves
λo (nm)
0 Air
1 203 Cryolite 1060
2 118 ZnS 983
3 205 - 913
4 117 - 848
5 207 - 789
6 115 - 735
7 210 - 686
8 113 - 642
9 214 - 603
10 111 - 568
11 219 - 536
12 108 - 508
13 227 - 482
14 104 - 459
15 236 - 437
16 99 - 417
17 249 - 397
18 93 - 378
19 267 - 359
20 87 - 335
21 289 - 277
22 79 275
23 320 272
24 71 269
25 364
Substrate Glass
formed state. Here appears the refining effect of M
and .
N
Increasing the ratio hL/hH further by way of and
results in broadening the central minimum with re-
spect to the spectrum. This is shown in Figure 4.
M
N
Concerning the transmittance fluctuations on either
side [8-11] of the central minimum it is seen from Fig-
ures 1 and 3 that ripple on the short wavelength side is
less obvious than that on the long wavelength side. This
is another effect of increasing the ratio hL/hH.
If the opposite is done by increasing the ratio hH/hL,
ripple on the long wavelength side is greatly i mproved as
shown in Figure 5. Figure 6 is a further illustration for
this last design which is extended to 35 layers as an in-
Copyright © 2011 SciRes. OPJ
M. MEDHAT ET AL.127
300 400 500 600 700 800 9001000110012001300
0
20
40
60
80
100
T%
Wavelength(nm)
Figure 4. Transmittance spectrum for design listed in Table
4. Increasing the ratio hL/hH further results in broadening
the central minimum with respect to the spectrum.
Table 4. Design details for initial parameters: Start wave:
1060, ξ: 1.04, : 0.75, : 1.5 no of layers: 25. Every
reference wave for every layer is the wavelength matching
this layer to the next layer in the table.
M N
Layer
no.
Physical Thickness
(after multiplication by
or in nm)
MNMaterial Reference
waves
λo (nm)
0 Air
1 271 Cryolite 1060
2 78 ZnS 278
3 276 - 277.3
4 77 - 277.9
5 281 - 277.5
6 76 - 277.1
7 286 - 276.7
8 74 - 276.3
9 291 - 275
10 73 - 274.9
11 298 - 274.4
12 71 - 273.9
13 304 - 273.4
14 70 - 272.9
15 311 - 272.4
16 68 - 271.8
17 319 - 271.2
18 66 - 270.6
19 327 - 270
20 65 - 269
21 335 - 268.8
22 63 268.2
23 343 267
24 62 266
25 353
Substrate Glass
300 400 500 600 700 8009001000110012001300
0
20
40
60
80
100
T%
Wavelength(nm)
Figure 5. Transmittance spectrum for design listed in Table
5. Increasing the ratio hH/hL instead of hL/hH improves rip-
ple on the long wavelength side of the central minimum.
Table 5. Design details for initial parameters: Start wave:
1060, ξ: 1.12, : 1.4, : 0.7 no of layers: 25. Every ref-
erence wave for every layer is the wavelength matching this
layer to the next layer in the table.
M N
Layer
no.
Physical Thickness
(after multiplication by
or in nm)
MNMaterial Reference
waves
λo (nm)
0 Air
1 145 Cryolite 1060
2 169 ZnS 1019
3 146 - 980
4 168 - 943
5 146 - 908
6 167 - 874
7 147 - 841
8 166 - 810
9 148 - 780
10 165 - 752
11 149 - 725
12 164 - 700
13 150 - 676
14 162 - 653
15 152 - 632
16 161 - 611
17 154 - 592
18 159 - 574
19 156 - 557
20 156 - 541
21 158 - 526
22 154 512
23 161 498
24 151 485
25 164
Substrate Glass
Copyright © 2011 SciRes. OPJ
M. MEDHAT ET AL.
128
Table 6. Design details for initial parameters: Start wave:
1060, ξ: 1.12, : 1.4, : 0.7 no of layers: 35. Every ref-
erence wave for every layer is the wavelength matching this
layer to the next layer in the table.
MN
Layer
no.
Physical Thickness
(after multiplication by
or in nm)
MNMaterial Reference
waves
λo (nm)
0 Air
1 145 Cryolite 1060
2 169 ZnS 1019
3 146 - 980
4 168 - 943
5 146 - 908
6 167 - 874
7 147 - 841
8 166 - 810
9 148 - 780
10 165 - 752
11 149 - 725
12 164 - 700
13 150 - 676
14 162 - 653
15 152 - 632
16 161 - 611
17 154 - 592
18 159 - 574
19 156 - 557
20 156 - 541
21 158 - 526
22 154 512
23 161 498
24 151 485
25 164 472
26 148 460
27 167 449
28 145 438
29 172 427
30 141 417
31 177 407
32 137 397
33 182 387
34 132 378
35 189
Substrate Glass
300 400 500 600 700 8009001000110012001300
0
20
40
60
80
100
T%
Wavelength(nm)
Figure 6. Transmittance spectrum for design listed in Table
6. Increasing no. of layers in the previous design affects the
low transmittance bands rather than the high ones.
vestigation for the effect of increasing no. of layers. Al-
though the reflection bands are got more defined the
transmittance ones are seriously affected.
4. Conclusions
A new approach for synthesizing optical multilayer
structures is presented. The resulting transmittance spec-
trum is also studied. The essential modifications to the
basic theory introduced effective elements which re-
vealed valuable control on the transmittance spectral
characteristics of the variety of systems under study.
Those control actions also unveiled useful applications
for the systems presented.
5. References
[1] M. Medhat and S. S. Farag, “The Effect of the Group
Propagation of Waves on the Spectral Behavior of a Mul-
tilayer Ar Coating,” Optics & Laser Technology, Vol. 30,
No. 1, 1998, pp. 57-61.
[2] M. Medhat and S. S. Fa rag, “Achromatic Layers Applied
to Dielectric Filters,” Journal of Optics A: Pure and Ap-
plied Optics, Vol. 2, No. 6, 2000, pp. 529-533.
doi:10.1088/1464-4258/2/6/305
[3] M. Medhat and S. S. Farag, “Broad-Band Dielectric Mul-
tilayer Mirrors by Period Staggering,” Journal of Optics
A: Pure and Applied Optics, Vol. 3, No. 3, 2001, pp.
178-182. doi:10.1088/1464-4258/3/3/304
[4] M. Medhat, “Interferometric Comparison of Optical
Pathlengths: Rings of Equal Chromatic Order,” Optik,
Vol. 84, No. 3, 1990, pp. 77-82
[5] R. W. Ditchburn, “Light,” Academic Press Inc., London,
1976, p. 65.
Copyright © 2011 SciRes. OPJ
M. MEDHAT ET AL.
Copyright © 2011 SciRes. OPJ
129
[6] A. Thelen, “Multilayer Filters with Wide Transmittance
Bands,” Journal of the Optical Society of America, Vol.
53, No. 11, 1963, pp. 1266-1270.
doi:10.1364/JOSA.53.001266
[7] H. M. Liddell, “Computer-Aided Techniques for the De-
sign of Multilayer Filters,” Adam Hilger, Bristol, 1981.
[8] W. T. Welford, “Computations in Thin Film Optics,”
Vacuum, Vol. 4, No. 1, 1954, pp. 3-19.
[9] P. W. Baumeister, “Design of Multilayer Filters by Suc-
cessive Approximations,” Journal of the Optical Society
of America, Vol. 48, No. 12, 1958, pp. 955-958.
doi:10.1364/JOSA.48.000955
[10] R. Jacobsson, “Matching a Multilayer Stack to a
High-Refractive-Index Substrate by Means of an Inho-
mogeneous Layer,” Journal of the Optical Society of
America, Vol. 54, No. 3, 1964, pp. 422-423.
doi:10.1364/JOSA.54.0422_1
[11] L. Young and E. G. Cristal, “On a Dielectric Fiber by
Baumeister,” Applied Optics, Vol. 5, 1966, pp. 77-80.
doi:10.1364/AO.5.000077