Design and plasma deposition of
dispersion-corrected multiband rugate filters
Daniel Poitras, Ste ´phane Larouche, and Ludvik Martinu
Inverse Fourier transform method has been commonly used for designing complex inhomogeneous optical
coatings. Since it assumes dispersion-free optical constants, introducing real optical materials induces
shifts in the position of reflectance bands in multiband inhomogeneous minus ?rugate? filters.
propose a simple method for considering optical dispersion in the synthesis of multiband rugate filter
designs.Model filters designed with this method were fabricated on glass and polycarbonate substrates
by plasma-enhanced chemical vapor deposition of silicon oxynitrides and SiO2?TiO2mixtures with
precisely controlled composition gradients. © 2002 Optical Society of America
310.1620, 080.2710, 080.2740, 260.2030, 310.1860.
Inhomogeneous coatings have found a niche among
the traditional optical coatings in specific applications
requiring the isolation of narrow wavelength bands;
these applications, in which rugate filters are fre-
quently used, include, for example, laser protective
eyewear, Raman and fluorescence spectroscopy, visor
displays, and others.The attractive features of inho-
mogeneous optical coatings are their improved optical
performance ?efficient suppression of harmonics1and
sidelobes2in the case of rugate filters and reduction of
scattering?, and enhanced mechanical properties3,4
?low stress and higher scratch and wear resistance?.
Most often, the design of rugate filters is based on
the inverse Fourier transform relation between a
spectral function Q??? and the refractive-index profile
n?z? ?z being the geometric thickness?.5–7
method has also been applied to the design of homo-
geneous multilayer filters.7–9
the method itself are problems, namely those related
However, inherent to
to the approximate representation of the Fourier re-
Only approximative Q functions have been
Neither absorption nor dispersion can be con-
sidered in the Fourier transform
Finite substrate and abrupt interfaces, which
can be introduced in the Fourier transform, cannot
easily be considered in the inverse transform, i.e.,
during the design of coatings.
One can address some of these issues by choosing a
more convenient Q??? function10,11or by using an
The problem of dispersion
can be addressed by use of iterations, or it can be
corrected when the inhomogeneous n?z? is trans-
formed into an equivalent two-material multilayer.14
In the present study, we use the inverse Fourier
transform approach and propose a simple way of de-
signing multiband rugate filters while considering the
dispersive optical constants of the materials used for
their fabrication. We then test this approach experi-
mentally by fabricating the designed filters on glass
and on polycarbonate substrates by plasma-enhanced
chemical vapor deposition ?PECVD; for a review, see
Ref. 15?.Two types of materials were used:
phous hydrogenated silicon oxynitride ?SiOxNy?16and
amorphous silicon dioxide?titanium dioxide ?SiO2?
TiO2? mixtures17with graded compositions.
The coatings and filters considered in this study were
fabricated in a capacitively coupled radiofrequency
Experimental Methodology:Plasma Deposition
When this research was performed, the authors were with the
Groupe de Recherche en Physique et Technologie des Couches
Minces ?GCM? and Department of Engineering Physics, E´cole
Polytechnique de Montre ´al, C. P. 6079, Station Centre-Ville, Mon-
tre ´al, Que ´bec H3C 3A7 Canada.
nrc.ca? is now with the Institute for Microstructural Sciences,
National Research Council of Canada, 1200 Montreal Road, Ot-
tawa, Ontario K1A 0R6, Canada.
Received 13 November 2001; revised manuscript received 13
© 2002 Optical Society of America
D. Poitras ?daniel.poitras@
L. Martinu’s e-mail address is
1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS5249
?rf, 13.56 MHz? PECVD system, described in more
detail in previous papers from our laboratory.16,17
mixture of silane ?SiH4?, nitrous oxide ?N2O?, and
ammonia ?NH3? was used for the deposition of SiOxNy
on both glass and polycarbonate substrates, and a
working pressure of 80 mTorr and an rf power of 100
W were used.The gas composition was adjusted by
use of computer-controlled flowmeters for N2O and
NH3, whereas the flow rate of SiH4was kept con-
stant.With basically the same plasma conditions as
above, a constant flow of O2, and computer-controlled
flows of TiCl4and SiCl4, a similar approach was ap-
plied for the deposition of TiO2?SiO2mixtures.
Ex situ transmission and spectroellipsometric
measurements were performed with a Lambda 19
spectrophotometer ?Perkin-Elmer, PerkinElmer In-
struments, Shelton, Connecticut? and a variable-angle
spectroscopic ellipsometer ?J. A. Woollam Company,
Lincoln, Nebraska?, respectively.
ties were determined with the WVASE32 software
?J. A. Woollam Company? and by use of the Cauchy
dispersion formula and the Urbach absorption tail.
To predict and control the refractive index and the
deposition rate rDat every moment of the film
growth, it was necessary to characterize precisely the
materials deposited in our PECVD system prior to
the fabrication of the inhomogeneous optical filters.
The calibration curves shown in Fig. 1 were obtained
from optical characterization of single-layer homoge-
neous films with different compositions.
of the refractive index at a wavelength of 550 nm
were ?1.46, 1.86? and ?1.46, 2.35? for SiOxNyand
curves were used for generating fabrication recipes
from optical designs represented by n?z? profiles, as
shown in Sec. 4.
We found that rDvaries with the gas mixture com-
position ?see Fig. 2? used to generate the results
shown in Fig. 1.This means that at every moment
of the film growth, the near-surface layer ?or sub-
layer? is deposited at a different instantaneous rD
value.Therefore, to prevent distortions in n?z?, we
applied two approaches:
time of individual sublayers to maintain a constant
thickness value for all sublayers or ?ii? maintain con-
stant index variations ??n? for all sublayers, and
change the gas flows at every ??n? step in the design
profile n?z? at time intervals determined by the rD?z?
The optical proper-
?i? adjust the deposition
3.Rugate Filter Design
The index profile of a single-band rugate filter with a
band of reflection RAat a wavelength ?A, obtained
from the inverse Fourier transform of an even func-
tion with a single-peak spectrum, is given by11
nA?x? ? nA?0?exp?
Design of Single-Band Rugate Filters
?xA?A?W?0???sin?2??Ax ? ?A?
where x is the double-centered optical thickness,
which is 0 at the center of the filter and varies as dx ?
2n?z?dz; ?A? ?A
? nmax??2 is the mean index; nminand nmaxare the
minimum and maximum index values for the mate-
2ln1 ? ?RA
?1is the wave number; nA?0? ? ?nmin
1 ? ?RA
is the truncated Q function18;
is the total optical thickness; ?Ais a phase factor;
w?x? is an apodization function; and W?0? is the con-
tinuous component of the Fourier transform of w?x?.
An effective apodization function is the Kaiser func-
wK?x? ?I0???1 ? 4x2?1?2?
where I0is the modified Bessel function of the first
kind and zero order; ? is a parameter controlling the
shape of the apodization envelope; and ??x? is a
Heaviside-type function, unity in the layer and null
of working gas composition during the plasma deposition of ?a?
SiOxNyand ?b? SiO2?TiO2.
Variation of the refractive-index dispersion as a function
5250 APPLIED OPTICS ? Vol. 41, No. 25 ? 1 September 2002
outside the layer.
Fourier transform of wK?x?, is given by
The continuous component of the
WK?0? ?sinh ?
Equation ?2?, which represents one of many possible
Q functions, has been shown to be accurate in the
case of quarter-wave stacks.18
n?z? profile given by Eq. ?1? is assumed to be disper-
sion free.The nonapplicability of this assumption
for real materials affects the efficiency of the harmon-
ics and sidelobes suppression, but it does not influ-
ence the position and the amplitude of the principal
reflectance band if n?z? is defined at its position ?A.
As stated above, the
Using dispersion-free materials would significantly
simplify the design of multiband rugate filters; a gen-
eral equation similar to Eq. ?1? can be easily devel-
oped for this purpose.11
m-band rugate filter with bands of intensities Riat
Design of Multiband Rugate Filters
The index profile of a
?x?i?i?Wi?0???sin?2??ix ? ?i?
The optimal design is obtained when all the individ-
ual band designs have the same optical thickness,
When using Eq. ?6?, it is assumed that n?x? is in-
dependent of ?; this independence restricts its use to
cases in which the materials are approximately dis-
persion free in the region of the rejection bands.
However, in practical situations for which dispersion
is a concern, designs based on Eq. ?6? result in dis-
tortions of the spectra. As illustrated in Fig. 3, the
dispersion changes the band positions, the band-
widths, and the Rivalues.
quirements only at the wavelength where the index
profile is defined.
To avoid the problems related to dispersion, we
propose a new approach illustrated in Fig. 4.
first step consists of isolating the individual reflec-
tance bands of the target and generating separate
design profiles ni?x?, ?i? for each of them by use of Eq.
?1?. The individual ni?x?i, ?i? are defined at the cen-
ter of the bands ?i, with ni?0? ? ?nmin??i? ?
nmax??i???2 and thickness values calculated from Eq.
The next step consists of combining the different
ni?x?i, ?i? profiles in a single design.
profiles are defined at different wavelengths, they can-
not be simply multiplied as in Eq. ?6?, and it is there-
fore necessary to redefine them at a common
wavelength ?0.To maintain the optical properties of
the individual profiles, we need to express the ni?x?i,
transformations ni?z, ?i? 3 ni?z, ?0? are then per-
formed with calibration data, such as those in Fig. 1.
For reducing the number of operations, we found it
convenient to choose ?0as the wavelength of one of the
bands.Our experience showed that the closer ?0is to
After being all redefined according to ?0, the pro-
ni?0? ? n?0?.
The filter meets the re-
Since the index
during the plasma deposition of ?a? SiOxNyand ?b? SiO2?TiO2.
Deposition rate as a function of working gas composition
cal filter designed with the inverse Fourier transform method, with
nondispersive ?solid line? and dispersive ?dotted line? optical con-
stants; the index profile is defined at 800 nm.
Calculated reflectance spectra of an inhomogeneous opti-
1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS5251
files are reexpressed as functions of x, ni?x?0, ?0?, and
multiplied to obtain the m-band rugate filter
n?x? ? K?
This equation is based on the
where K is a constant.
linearity of the Fourier transform relation between
the natural logarithm of the index profile and the
Q??? function, and it leads to
Since the Q??? functions of single-band rugate filters
are essentially null outside the rejection band,18one
The method described here can also be used to design
broadband reflectors, but its accuracy depends on the
validity of the Fourier transform approach, which
means that high-order internal reflections must be
To prevent the profile obtained from Eq. ?8? from
exceeding the available maximum index value, one
can apply a correction factor to n?x? to center the
profile onto the range of indices available,7with only
minor effect on the spectrum.19
method does not take full advantage of the entire
range of indices available.
tage by modifying the ?nmin, nmax? values prior to the
design or by performing on n?x? a transformation of
One can take full advan-
ncorr?x? ? a?n?x??b,(11)
designs are all expressed at a common wavelength ?0; 3, designs
are multiplied and normalized.
Schematic representation of the design n?x? of a two-band
1, separate designs are generated for every band; 2,
scribed in Fig. 4 applied to a triple-band rugate filter, with a
SiO2?TiO2mixture, with bands at 450, 600, and 750 nm:
refractive-index depth profile n?z? at ?0? 600 nm and ?b? refection
spectra ?solid line? compared with the results of Eq. 6 ?dotted line?.
Example of the application of the design approach de-
5252APPLIED OPTICS ? Vol. 41, No. 25 ? 1 September 2002
This operation centers and scales the profile to cover
the entire range of indices available.
b corresponds to a change of the Q values of the bands
and, accordingly, to a change of their peak values.11
In many cases, b ? 1, and this correction does not
significantly alter the spectrum.
ple correction factor can be introduced in the Q func-
tions to overcome such effect.
As a final step, the index profile is expressed as a
function of z. Figure 5 shows an example of a three-
band filter designed with this method and a compar-
ison with the results from Eq. ?6?.
can be added at the substrate–filter and filter–air
interfaces to reduce unwanted ripples due to inter-
ference of light reflected at those interfaces and at the
back of the substrate.
Otherwise, a sim-
Based on the calculated designs and taking into con-
sideration the calibration curves in Figs. 1 and 2, we
determined the time sequence of flows for the differ-
ent gases used for the plasma deposition of the filters.
Examples of double-band rugate filters fabricated on
glass and polycarbonate with SiOxNyare shown in
Fig. 6.The range of indices available for the design
was restricted ?Fig. 6?a?? to account for the fact that it
was impossible at that time to close the flowmeters
completely.One can see that the band positions
were not affected by dispersion.
between theoretical and experimental results for the
filter deposited on polycarbonate can be attributed to
the variation of rDduring the fabrication process.
triple-band rugate filter deposited on glass with
SiO2?TiO2is shown in Fig. 7.
The index gradient in our filters was due to
changes in gas flow rate ratios, which were con-
trolled by gas diffusion in the chamber.
fore believe that the inhomogeneous profile in the
fabricated coatings represents a real continuous
change of the material properties and not a suc-
Experimental Results and Discussion
design index profile, ?b? design variation of the gas flows during the fabrication, ?c? measured transmission spectra of the filter deposited
on glass ?solid line? compared with calculated spectra ?dotted line?, ?d? measured transmission spectra of the filter deposited on polycar-
Example of plasma-deposited, dispersion-corrected, double-band rugate filters with SiOxNywith bands at 450 and 633 nm:
1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS5253
shown, in comparison with multilayer systems, im-
proved mechanical properties ?better scratch and
wear resistance, lower stress, and better adhesion?
in inhomogeneous films owing to the absence of
of thin homogeneouslayers
Recent studies have
We have proposed a method for considering the
materials’ optical dispersion in the design of multi-
band rugate filters.This method is based on split-
ting an m-band filter to m single-band filters and
combining them with reference to a common wave-
length. We have demonstrated successful fabrica-
tion of such filters by PECVD with SiOxNyand
SiO2?TiO2. Further improvement of the inhomo-
geneous optical coatings performance is expected
during implementation of in situ real-time process
control, the research currently under way in our
The authors acknowledge the technical assistance
of Mr. Gilles Jalbert and Mr. Jir ˇı ´ Cˇerny ´.
was supported by the Natural Sciences and Engi-
neering Research Council ?NSERC? of Canada.
This research was first presented at the Optical
Society of America’s Eighth Topical Meeting on Op-
tical Interference Coatings held in Banff, Canada,
15–20 July 2001.21
1. C. H. Greenwalt, W. Brandt, and D. D. Friel, “Iridescent colors
of hummingbird feathers,” J. Opt. Soc. Am. 50, 1005–1013
2. W. H. Southwell, “Using apodization functions to reduce side-
lobes in rugate filters,” Appl. Opt. 28, 5091–5094 ?1989?.
3. D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut,
“Mechanical properties of plasma-deposited silicon-based in-
homogeneous optical coatings,” Surf. Coat. Technol. 111, 220–
4. M.-A. Raymond, S. Larouche, O. Zabeida, L. Martinu, and J. E.
Klemberg-Sapieha, “Tribological properties of PECVD optical
coatings,” in Proceedings of the Forty-Fourth Annual Technical
Conference of the Society of Vacuum Coaters ?Society of Vac-
uum Coaters, Albuquerque, N. Mex., 2001?, pp. 301–305.
5. L. Sossi, “A method for the synthesis of multilayer dielectric
interference coatings,” Izvestiya Akademii Nauk Estonskoi
SSR Fizika, Matematika 23, 229–237 ?1974?.
lation available from the Translation Services of the Canada
Institute for Scientific and Technical Information ?CISTI?, Na-
tional Research Council of Canada, 1200 Montreal Road,
Building M-55, Ottawa, Ontario, K1A 0R6 Canada.
6. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc.
Am. 57, 1529–1533 ?1967?.
7. J. A. Dobrowolski and D. Lowe, “Optical thin film synthesis
program based on the use of Fourier transforms,” Appl. Opt.
17, 3039–3050 ?1978?.
8. P. G. Verly, J. A. Dobrowolski, W. J. Wild, and R. L. Burton,
“Synthesis of high rejection filters with the Fourier transform
method,” Appl. Opt. 28, 2864–2875 ?1989?.
9. P. G. Verly, J. A. Dobrowolski, and R. R. Willey, “Fourier-
transform method for the design of wideband antireflection
coatings,” Appl. Opt. 31, 3836–3846 ?1992?.
10. B. G. Bovard, “Rugate filter design:
transform technique,” Appl. Opt. 29, 24–30 ?1990?.
11. B. G. Bovard, “Rugate filter theory:
32, 5427–5442 ?1993?.
12. P. G. Verly and J. A. Dobrowolski, “Iterative correction process
for optical thin film synthesis with the Fourier transform
method,” Appl. Opt. 29, 3672–3684 ?1990?.
13. H. Chang, S.-S. Lee, M.-R. Choi, and S. Lim, “Inhomogeneous
optical filter design with the use of a Riccati equation,” Micro-
wave Opt. Technol. Lett. 22, 140–144 ?1999?.
14. H. Fabricius, “Gradient-index filters:
index solution by taking into account dispersion,” Appl. Opt.
31, 5216–5220 ?1992?.
15. L. Martinu and D. Poitras, “Plasma deposition of optical films
and coatings: a review,” J. Vac. Sci. Technol. A 18, 2619–
16. D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi,
and L. Martinu, “Characterization of homogeneous and inho-
mogeneous Si-based optical coatings deposited in dual-
frequency plasma,” Opt. Eng. 35, 2690–2699 ?1996?.
17. S. Larouche, A. Amassian, S. C. Gujrathi, J. Klemberg-
Sapieha, and L. Martinu, “Multilayer and inhomogeneous
optical filters fabricated by PECVD using titanium dioxide
and silicon dioxide,” in Proceedings of the Forty-Fourth An-
the modified Fourier
an overview,” Appl. Opt.
conversion into a two-
band rugate filter fabricated on glass with SiO2?TiO2mixtures
with bands at 600, 800, and 1100 nm:
?b? measured transmission spectra ?solid line? compared with cal-
culated spectra ?dotted line?.
Example of plasma-deposited, dispersion-corrected, triple-
?a? design index profile and
5254APPLIED OPTICS ? Vol. 41, No. 25 ? 1 September 2002
nual Technical Conference of the Society of Vacuum Coaters
?Society of Vacuum Coaters, Albuquerque, N. Mex., 2001?,
18. B. G. Bovard, “Fourier transform technique applied to quar-
terwave optical coatings,” Appl. Opt. 27, 3062–3063 ?1988?.
19. A. Thelen, Design of Optical Interference Coatings ?McGraw-
Hill, New York, 1988?.
20. W. H. Southwell, “Gradient-index antireflection coatings,”
Opt. Lett. 8, 584–586 ?1983?.
21. D. Poitras, S. Larouche, and L. Martinu, “Design and plasma
deposition of dispersion-corrected multiband rugate filters,” in
Optical Interference Coatings, Vol. 63 of OSA Technical Digest
?Optical Society of America, Washington, D.C., 2001?, pp.
1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS5255