Page 1

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

OCIS codes:

310.1620, 080.2710, 080.2740, 260.2030, 310.1860.

We

1.

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

Introduction

This

However, inherent to

to the approximate representation of the Fourier re-

lation:10,11

•

Only approximative Q functions have been

found

•

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

iterative approach.12,13

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.

amor-

2.

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.

lmartinu@mail.polymtl.ca.

Received 13 November 2001; revised manuscript received 13

March 2002.

0003-6935?02?255249-07$15.00?0

© 2002 Optical Society of America

D. Poitras ?daniel.poitras@

L. Martinu’s e-mail address is

1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS 5249

Page 2

?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

SiO2?TiO2mixtures, respectively.

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?

value.

A

The optical proper-

The ranges

The calibration

?i? adjust the deposition

3. Rugate Filter Design

A.

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?

? sin??A??w?x??,

Design of Single-Band Rugate Filters

Q?A

?xA?A?W?0???sin?2??Ax ? ?A?

(1)

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-

rial used;

2ln1 ? ?RA

?1is the wave number; nA?0? ? ?nmin

Q?A?1

1 ? ?RA

(2)

is the truncated Q function18;

xA?

2Q?A

??Aln?nmax?nmin??W?0??

(3)

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-

tion

wK?x? ?I0???1 ? 4x2?1?2?

I0???

??x?,(4)

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

Fig. 1.

of working gas composition during the plasma deposition of ?a?

SiOxNyand ?b? SiO2?TiO2.

Variation of the refractive-index dispersion as a function

5250APPLIED OPTICS ? Vol. 41, No. 25 ? 1 September 2002

Page 3

outside the layer.

Fourier transform of wK?x?, is given by

The continuous component of the

WK?0? ?sinh ?

?I0???. (5)

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

B.

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

wavelengths ?iis

ni?0?exp?

? sin??i??wi?x??.

Design of Multiband Rugate Filters

The index profile of a

n?x? ??

i?1

m

Q?i

?x?i?i?Wi?0???sin?2??ix ? ?i?

(6)

The optimal design is obtained when all the individ-

ual band designs have the same optical thickness,

x?i? x0?

2

? ln?nmax?nmin??

i?1

m

Q?i

?i?Wi?0??, (7)

and ?i?1

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.

?7?.

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,

?i?intermsoftheirgeometricthicknessvaluesz.

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

abandwavelength,thebetteristhefilterperformance.

After being all redefined according to ?0, the pro-

m

ni?0? ? n?0?.

The filter meets the re-

The

Since the index

The

Fig. 2.

during the plasma deposition of ?a? SiOxNyand ?b? SiO2?TiO2.

Deposition rate as a function of working gas composition

Fig. 3.

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

Page 4

files are reexpressed as functions of x, ni?x?0, ?0?, and

multiplied to obtain the m-band rugate filter

n?x? ? K?

i?1

This equation is based on the

m

ni?x?0, ?0?, (8)

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

Q???exp?i????? ??

i?1

m

Qi???exp?i?i????. (9)

Since the Q??? functions of single-band rugate filters

are essentially null outside the rejection band,18one

can write

R??? ??

i?1

m

Ri???. (10)

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

negligible.

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

the form

However, this

One can take full advan-

ncorr?x? ? a?n?x??b, (11)

Fig. 4.

rugate filter:

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,

Fig. 5.

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-

?a?

5252 APPLIED OPTICS ? Vol. 41, No. 25 ? 1 September 2002

Page 5

where

b ?

ln?nmax?nmin?

ln?max?n?x???min?n?x???, (12)

a ?

?nmax?b

max?n?x??. (13)

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.

The exponent

Otherwise, a sim-

Quintic layers20

4.

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

Some discrepancy

A

We there-

Fig. 6.

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-

bonate.

Example of plasma-deposited, dispersion-corrected, double-band rugate filters with SiOxNywith bands at 450 and 633 nm:

?a?

1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS5253

Page 6

cession

inhomogeneous coatings?.

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

interfaces.3,4

ofthinhomogeneouslayers

?quasi-

Recent studies have

5.

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

laboratory.

The authors acknowledge the technical assistance

of Mr. Gilles Jalbert and Mr. Jir ˇı ´ Cˇerny ´.

Conclusion

This study

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

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Fig. 7.

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1 September 2002 ? Vol. 41, No. 25 ? APPLIED OPTICS 5255

Thin film optics