Icarus 190 (2007) 274–279
www.elsevier.com/locate/icarus
Infrared reflectance spectroscopy on thin films: Interference effects
B.D. Teolis, M.J. Loeffler, U. Raut, M. Famá, R.A. Baragiola ∗
University of Virginia, Laboratory for Atomic & Surface Physics, Engineering Physics, Charlottesville, VA 22904, USA
Received 4 August 2006; revised 7 February 2007
Available online 18 April 2007
Abstract
Laboratory simulations of processes on astronomical surfaces that use infrared reflectance spectroscopy of thin films to analyze their composition and structure often ignore important optical interference effects which often lead to erroneous measurements of absorption band strengths
and give an apparent dependence of this quantity on film thickness, index of refraction and wavelength. We demonstrate these interference effects
experimentally and show that the optical depths of several absorption bands of thin water ice films on a gold mirror are not proportional to film
thickness. We describe the method to calculate accurately band strengths from measured absorbance spectra using the Fresnel equations for two
different experimental cases, and propose a way to remove interference effects by performing measurements with P -polarized light incident at
Brewster’s angle.
© 2007 Elsevier Inc. All rights reserved.
Keywords: Ices, IR spectroscopy; Experimental techniques
1. Introduction
Most experiments that simulate the effects of temperature
and radiation on astronomical ices are performed on thin films.
In a typical arrangement, gas is condensed onto a cold substrate in vacuum until a film of desired thickness is attained.
The film is then analyzed by infrared absorption spectroscopy,
which is highly sensitive to composition because of the specificity of vibrational frequencies to molecular bonding. For this
reason, infrared absorption spectroscopy has been employed
extensively in the simulation of processes that occur on icy surfaces in space, such as gas adsorption and trapping (for recent
work see, e.g., Chaabouni et al., 2000a; Manca et al., 2000;
Borget et al., 2001; Horimoto et al., 2002; Hudson et al., 2002;
Collings et al., 2003; Takaoka et al., 2004; Loeffler et al.,
2006a) and the synthesis of new species by radiolysis in a variety of pure and mixed ices (e.g., Baratta et al., 1994; Gerakines
et al., 1996; Chaabouni et al., 2000b; Hudson and Moore, 2001;
* Corresponding author. Address for correspondence: University of Virginia,
Department of Materials Science & Engineering, 140 Chemistry Drive, Charlottesville, VA 22904, USA.
E-mail address: [email protected] (R.A. Baragiola).
0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.icarus.2007.03.023
Caro and Schutte, 2003; Gomis et al., 2004; Loeffler et al.,
2005, 2006b, 2006c; Zheng et al., 2006).
Infrared absorption spectroscopy can be done in transmission or reflectance geometries (Tolstoy et al., 2003). In transmission, films are deposited onto an infrared transparent substrate positioned between the IR source and detector. In reflectance, the substrate is highly reflective and the detector
is positioned to measure the intensity of the specularly reflected light. In both geometries, the intensity of the reflected
and transmitted light is determined by the absorbance of the
film and the reflectance and transmittance of the film and substrate surfaces. Optical interference effects are typically minor
in transmission, making data analysis simpler. Nevertheless,
in laboratory simulations of radiation effects, the nonmetallic substrates used in transmission are particularly susceptible
to effects such as electrostatic charging or changes in optical properties by damage or chemical alterations (Clark and
Crawford, 1973). These problems, together with physical limitations on experimental geometry, often require the use of reflectance. One advantage of reflectance is the ability to use
a quartz-crystal microbalance as a substrate to measure the
mass of the films (e.g., Westley et al., 1998; Loeffler et al.,
2006a). However, Fresnel reflection from the film surface complicates the analysis of thin film reflectance, since interfer-
Infrared reflectance of thin films: Interference effects
275
ence between light reflected from the surfaces of the film and
the substrate must be taken into account (Jacob et al., 2000;
Milton and Leung, 2002), as noted in our recent study of radiation synthesis of hydrogen peroxide from water ice (Loeffler
et al., 2006c). Unfortunately, interference effects are frequently
overlooked.
2. Band strengths in reflectance
In many previous infrared studies on thin films, the column density η of an absorbing species (molecules/cm2 ) has
been calculated from absorption bands in transmission infrared absorption (e.g., Jiang et al., 1975; Hudgins et al., 1993;
Gerakines et al., 2005; Gomis et al., 2004; Bernstein et al.,
2006) from the expression
BT cos ϕf
(1)
,
A
where the angle of refraction φf of light in the film is usually
calculated from the angle of incidence φ0 and the real part of
the refraction index of the film, n, using φf ≈ sin−1 (sin φ0 /n),
which is the low absorbance limit of Snell’s law:
sin ϕ0
−1 sin ϕ0
−1
ϕf = sin
(2)
= sin
,
n
n − iλα/4π
η=
with n = n − iλα/4π the complex refraction index of the film
and α the absorbance. BT , the integrated band area, is given in
terms of the optical depth, − ln(T /T 0 ) and the wavelength λ of
the light by
BT = − ln(T /T 0 ) d(1/λ),
(3)
where T /T 0 is the ratio of the transmittance of the film to that
of the bare substrate. The band strength A is determined by
measuring BT and applying Eq. (1) when η is known from an
independent method. This calculation assumes that I decays
exponentially with αz/ cos φf as predicted by the Beer’s law,
where z is the film thickness. The cosine factor in Eq. (1) accounts for the dependence of the optical path length on φf .
Several authors have used the same approach for reflectance
measurements (e.g., Hudson and Moore, 2001; Bennett et al.,
2004; Hudson et al., 2005; Zheng et al., 2006; Wada et al., 2006)
and derived what we call an apparent absorption strength A
from
BR cos ϕf
A =
(4)
2η
where the factor 2 appears because the light goes through the
film twice. Here, the band area in reflectance is
BR = − ln(R/R 0 ) d(1/λ),
(5)
with R/R 0 the ratio of the reflectance of the film to that of the
bare substrate.
To test the validity of this approach in an infrared reflectance
experiment, we measured the reflectance spectra (in randomly
polarized light) of (amorphous) water ice films of different column densities, vapor deposited onto a gold-coated quartz crystal microbalance (Fig. 1). We determined the film thickness
Fig. 1. Infrared reflectance spectra of different column densities of water ice
films deposited onto a gold substrate at 110 K (incidence angle = 35◦ , randomly polarized light source): ice thicknesses are (a) 2, (b) 4 and (c) 6 in
units of 1018 H2 O/cm2 . The table gives the column densities (in units of 1018
H2 O/cm2 ) of the films of the figure measured by the microbalance (ηM ) and
calculated from the absorption band areas (ηr ) using band strengths measured
by Hudgins et al. (1993) in transmission, a procedure that assumes the validity of Beer’s law. Apparent band strengths A , calculated by using Eq. (4) with
ηM , are also shown, in units of 10−18 cm/molecule. Note the discrepancies
between ηr and ηM , as well as the dependence of A on column density.
from the oscillation of R/R 0 with thickness measured at visible and ultraviolet wavelengths (e.g., Westley et al., 1998). The
column densities were derived from the microbalance reading
(ηM ) and found to be proportional to the film thickness, as expected. In contrast, the optical depths and integrated band areas
are not proportional to z but oscillate due to optical interference (Fig. 2). As a result, calculations of the column density or
band strength from the band areas (Fig. 1, inset) using Eqs. (4)
and (5) give inconsistent results and differ drastically from ηM
and A, respectively (up to a factor of 8 in Fig. 1).
Optical interference depends strongly on the change in
phase δ of the waves reflected once from the substrate on traversing the film, given by
z
δ = 2πn cos ϕf .
(6)
λ
The effect is easiest to see when the optical depth within
an absorption band is plotted versus z. While the optical depth
trends higher with increasing film thickness, it also oscillates as
the real part of δ rotates through cycles of 2π . This behavior
is shown in Fig. 2a, where the optical depth measured at the
maximum of the 1640 cm−1 band of a water ice film is plotted
versus ηM and z. The optical depth deviates greatly from the
linear increase obtained by assuming the validity of Beer’s law
in reflectance.
For a given ϕ0 , the optical depth can be well fit with the natural logarithm of the expression for the reflectance ratio R/R 0
of a film:
R(z, n, α, λ)
RS + RP
,
= S
P |2
R 0 (λ)
|rs0 |2 + |rs0
(7)
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B.D. Teolis et al. / Icarus 190 (2007) 274–279
Fig. 2. (a) Optical depth vs film thickness and column density measured in reflectance at ∼1640 cm−1 (incidence angle = 35◦ , randomly polarized light source) for
a water ice film on a gold substrate at 110 K (magenta open circles). Also shown is a fit of Eq. (7) to the data using n = 1.32 and α = 0.14 µm−1 (blue solid line),
a calculation (Heavens, 1991) of the optical depth in transmission times two (to compare with reflectance, since most transmitted light traverses the film thickness
only once) of the same film on a cesium iodide substrate using n = 1.32, α = 0.14 µm−1 (orange solid circles), and a Beer’s law dependence (dashed line). (b) Fits
of (a) and Beer’s law estimation plotted to 40 µm. |Êf | and |Ês | are the moduli of the electric field of the light or the amplitudes of the waves reflected from the film
surface and once from the gold substrate, respectively.
where, for random polarization, the reflectances for light polarized perpendicular or parallel to the incidence plane are averaged (Born and Wolf, 1999; also see Appendix A for the
expressions for R S and R P and the Fresnel coefficients r). The
fit to the data of Fig. 2 yields n = 1.32 and α = 0.14 µm−1 , in
good agreement with previously reported values (e.g., Hudgins
et al., 1993). Substituting Eq. (7) into Eqs. (4) and (5), we see
that band areas and apparent band strengths depend on n, ϕ0
and thickness z:
cos ϕf
R(z, n, α, ϕ0 , λ)
ln
A = −
(8)
d(1/λ).
2η
R 0 (ϕ0 , λ)
Interference oscillations are greatest when the amplitude of the
wave reflected from the substrate is sufficiently attenuated by
absorption inside the film that its magnitude is similar to that
of the wave reflected from the surface, as shown in Fig. 2b,
where the fit of Fig. 2a is extrapolated to greater thickness. Interference diminishes at large thicknesses where the reflected
intensity is dominated by thickness independent Fresnel reflection from the film–vacuum interface.
We point out that optical interference has a much smaller
effect on a typical transmission experiment because the reflectance at both the film–vacuum and the film–substrate interfaces is small and reflected light must reflect from at least
two interfaces to be transmitted through the film. The small
contribution of reflected light to the total transmitted intensity
also decays with film thickness faster than directly transmitted light, since multiply reflected light travels a greater distance
through the absorbing medium than directly transmitted light.
As a result, interference between multiply reflected and directly
transmitted light diminishes with increasing film thickness, and
Fig. 3. Flow chart showing the steps involved in the iterative procedure used in
Section 3.
the optical depth converges to Beer’s law (Fig. 2), with the total
transmitted intensity dominated by light paths without reflection. We note that the optical depth in reflectance and transmittance, after correcting for the factor of 2 increase in optical path
length, closely agree near particular thicknesses (as in Fig. 2).
These coincidences might explain the occasional agreement in
absorption band strength measured in transmission and the apparent band strength measured in reflectance using Eq. (4).
The strong influence of optical interference on absorption
band strengths places several constraints on the interpretation of
Infrared reflectance of thin films: Interference effects
277
Fig. 4. The optical depth calculated from the natural logarithm of Eq. (7) for water ice films on a gold substrate at different wavelengths versus film thickness, for
incidence at (a) 0 and (b) 87 ◦ . The optical depth is divided by 2αz/ cos ϕf (with ϕf ≈ sin−1 (sin ϕ0 /n)) to compare with Beer’s law, which predicts that this ratio
is unity and independent of thickness. Wavelengths plotted are (in µm): 3 (black: 1.2, 2.5), 6 (red: 1.3, 0.15), 12 (green: 1.27, 0.44), 23.6 (blue: 1.4, 0.016), 47 (light
blue: 1.55, 0.19), 65 (magenta: 1.75, 0.1) and 100 (orange: 1.8, 0.018), where the values inside the brackets are n and α (in µm−1 ). Note that the deviation from
Beer’s law is more drastic at high incidence angle, where Fresnel reflection from the film surface is more intense.
reflectance spectra. For example, (i) an apparent band strength
derived at one film thickness cannot be used with Beer’s law to
interpret band areas measured in films of different thickness, (ii)
band strengths measured in transmission cannot be used with
Beer’s law in reflectance experiments to calculate the concentration of an absorbing species and (iii) the ratios of areas of
different bands at a given film thickness are generally different from those that would be measured at a different thickness
or in transmission (i.e., Fig. 1). The effects are generally worse
at high angles of incidence due to the more intense Fresnel reflection from the film surface. This is demonstrated in Fig. 4,
where the optical depth calculated from the natural logarithm
of Eq. (7) is plotted versus film thickness in different absorption bands of water ice (using n and α from Warren, 1984) and
compared for low and high φ0 (0 and 87 ◦ , respectively, see
Figs. 4a and 4b). One can see that, although in both cases there
are strong deviations from Beer’s law, at 87 ◦ , where reflection
from the film surface is large, the effects are much stronger.
These examples underscore the need for an accurate method for
the interpretation of absorption bands measured in reflectance.
In the following sections we discuss three potential solutions to
this problem.
3. Direct calculation of α(λ)—Fixed film thickness
In this approach the procedure used by several authors in
transmission (e.g., Hagen et al., 1981; Hudgins et al., 1993;
Baratta and Palumbo, 1998) is adapted for the determination of
the n(λ) and α(λ) functions from an infrared reflectance spectrum, which requires numerically adjusting n and α at all λ to
match the expression for thin film reflectance given in Eq. (7) to
the measured spectrum. This procedure also requires measuring
the thickness z, which can be done, e.g., by the usual method of
laser interferometry. Equation (7) contains two unknown vari-
ables (n and α) that are related via the Kramers–Kronig dispersion relation (Born and Wolf, 1999):
λ2
n(λ) = 1 +
2π 2
∞
0
α(λ )
dλ .
λ 2 − λ2
(9)
If the absorption in the experimental spectrum (bound between λ0 and λmax ) is well separated from absorption in other
parts of the spectrum (e.g., separation between vibrational and
electronic transitions in an insulator), n(λ) can be approximated
as
λ2
n(λ) ≈ n0 +
2π 2
λ
max
λ0
α(λ )
dλ ,
λ 2 − λ2
(10)
where n0 is the refraction index in the region of negligible
absorption at wavelengths below λ0 , and can be measured
by interferometry at a fixed wavelength during film growth,
which also gives the thickness z (e.g., Hudgins et al., 1993;
Westley et al., 1998; Moore and Hudson, 2000; Loeffler et al.,
2005). With only two unknown variables remaining (n and α),
the system of Eqs. (7) and (10) can be computed using the iterative procedure illustrated in Fig. 3 to yield solutions for n(λ)
and α(λ). The procedure requires input values R/R 0 , z and
n0 , as well as a reasonable initial guess for n(λ) and α(λ)
(e.g., α(λ) = 0 and n(λ) = 1.3). Once α(λ) is found, the band
strength A (not the apparent band strength A ) can be obtained
via the expression
z
α(λ) d(1/λ),
A=
(11)
η
given the column density η of the absorbent. The integral,
evaluated over the absorption band, is known as integrated absorbance of the band (Smith et al., 1985). Equation (11) is valid
278
B.D. Teolis et al. / Icarus 190 (2007) 274–279
for a single component system. In the case where the absorbent
is in a matrix with absorbance α0 , one needs to do a baseline
subtraction of α0 from α before using Eq. (11). When, as in radiation experiments, one needs to quantify the presence of an
absorbent of known A, one can use the procedure discussed
above to derive α and obtain η from Eq. (11), rather than using
Beer’s law.
4. Measurement of α(λ) during film deposition
Invocation of the Kramers–Kronig condition is unnecessary
when the reflectance spectrum is measured at different film
thicknesses, as can be accomplished while the film is condensed. If the spectra are measured for N different thicknesses
at constant thickness intervals z, then one obtains a system
of N different Eq. (7). Provided N is sufficiently large to resolve the oscillations due to interference in the reflectance (as in
Fig. 2), the resulting system of equations can be solved numerically at consecutive wavelengths across the spectrum resulting
in solutions for n(λ), α(λ), and z. When α(λ) is determined,
absorption band strengths can be computed via Eq. (11). The
accuracy of the solutions for n(λ) and α(λ) can be determined
by verifying that the solution for z is wavelength independent.
We note that the procedure also yields the final film thickness,
Nz.
5. Brewster’s angle technique
As an alternative approach, one can remove the effect of interference experimentally by minimizing the reflection at the
film surface. This can be achieved by performing the reflectance
measurements with incident P -polarized light at Brewster’s angle φb :
ϕb = tan−1 n,
(12)
which, for α = 0, reduces the reflectance of the film surface to
zero, thereby eliminating optical interference effects. If α > 0,
as in an absorption band, φb is the approximate angle needed
to minimize the surface reflection. The precise angle for reflectance minimization deviates from φb for α > 0 (Kim and
Vedam, 1986), but the deviation is not significant for weak absorption, as in Eq. (14) below. Optical interference is negligible
if the absorbance is sufficiently weak that the amplitude |rfP | of
the wave reflected from the surface is much less than that of the
wave reflected from the substrate |rsP | exp(−αz/ cos φf ):
P P −αz/ cos ϕ
f,
r r e
(13)
s
f
where φf is approximately sin−1 (sin φ0 /n) = sin−1 (sin φb /n)
for low absorbance (see Eq. (2)). Taking |rsP | to be unity (i.e.,
a highly reflective substrate) and using the expression for rfP
(Eq. (18d)), we can express the above condition as
− cos ϕf n cos ϕf − cos ϕb ,
ln
α
(14)
z
n cos ϕf + cos ϕb where we have approximated nf as n, which assumes that
α 4πn/λ. Therefore, provided that the above condition is
satisfied, optical interference is negligible at φ0 = φb and the
film reflectance obeys Beer’s law, which enables us to approximate α(λ) as
− cos ϕf
R
α(λ) ≈
(15)
ln
,
2z
R0
and the strength of an absorption band as
− cos ϕf
ln(R/R0 ) d(1/λ).
A≈
2η
(16)
Moreover, φb is fairly insensitive to n; e.g., as n increases
from 1 to 2, φb changes from 45 to 60 ◦ , thus requiring only
small adjustments of the measurement angle (or none at all) for
different parts of the film spectrum with different n. While this
approach may be more challenging experimentally due to limits on the system geometry, the simplification of data analysis
constitutes a major advantage.
Acknowledgment
This work was supported by NASA’s Planetary Geology and
Geophysics program.
Appendix A
The expressions for R S and R P used in Eq. (7) and given by
Heavens (1991) are as follows:
S
rf + rsS e−2iδ 2
,
R S (z, n, α, ϕ0 , λ) = (17a)
1 + rfS rsS e−2iδ P
rf + rsP e−2iδ 2
P
,
R (z, n, α, ϕ0 , λ) = (17b)
1 + rfP rsP e−2iδ where the Fresnel reflection coefficients rf of the film surface
and those of the substrate surface rs and rs0 with and without a
film are:
cos ϕf − n cos ϕ0
,
rfS =
(18a)
cos ϕf + n cos ϕ0
n cos ϕs − ns cos ϕf
,
rsS =
(18b)
n cos ϕs + ns cos ϕf
cos ϕs − ns cos ϕ0
S
rs0
(18c)
=
,
cos ϕs + ns cos ϕ0
n cos ϕf − cos ϕ0
,
rfP =
(18d)
n cos ϕf + cos ϕ0
ns cos ϕs − n cos ϕf
,
rsP =
(18e)
ns cos ϕs + n cos ϕf
ns cos ϕs − cos ϕ0
P
rs0
(18f)
=
.
ns cos ϕs + cos ϕ0
Here, ns is the complex refraction index of the substrate [which
can be measured (e.g., Westley et al., 1998) or obtained from
published tables (e.g., Palik and Ghosh, 1997)] and φs , the angle of refraction of light in the substrate, is given by
sin ϕ0
.
ϕs = sin−1
(19)
ns
Infrared reflectance of thin films: Interference effects
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