ANALYTICAL SCIENCES DECEMBER 2007, VOL. 23
2007 © The Japan Society for Analytical Chemistry

NotesAnalysis of a Quasi - Elastic Laser Scattering Spectrum
Using the Maximum Entropy MethodIsao TSUYUMOTO
Department of Environmental Chemistry, Kanazawa Institute of Technology,
7-1 Ohgigaoka, Nonoichi, Ishikawa 921– 8501, Japan

We have applied the maximum entropy method (MEM) to the analysis of quasi-elastic laser scattering (QELS) spectraand have established a technique for determining capillary wave frequencies with a higher time resolution than that of theconventional procedure. Although the QELS method has an advantage in time resolution over mechanical methods, it
requires the averaging of at least 20 – 100 power spectra for determining capillary wave frequencies. We find that theMEM analysis markedly improves the S/N ratio of the power spectra, and that averaging the spectra is not necessary for
determining the capillary wave frequency, i.e., it can be estimated from one power spectrum. The time resolution of theQELS attains the theoretical limit by using MEM analysis.
(Received August 20, 2007; Accepted October 15, 2007; Published December 10, 2007)

Introduction
Surface tension measurements are widely used for quality
control in various industrial processes, such as washing, coating,penetration, dispersion and emulsification, because they provide
useful information on surfactant dynamics at liquid surfaces orliquid/liquid interfaces. The measurements are also helpful for
fundamental research on interfacial thermodynamics andproperties. A number of techniques, such as the Wilhelmy
method, the Du Nouy method, and the spinning drop method,
have been developed for surface tension measurements.1 Thesemechanical techniques, however, are not suitable for timeresolved and remote-sensing measurements of high-temperatureliquids, such as the fused silicon, the inside of a globe, and thesurface of a planet.
The quasi-elastic laser scattering (QELS) method is used tomonitor the spectrum of QELS by capillary waves (ripplons),
which are spontaneously generated at liquid surfaces. The
surface tension is calculated from the capillary wave frequency
observed in the fast Fourier transformed power spectrum. Many
researchers have reported that the QELS method is appropriatefor surface tension measurements, indicating that the surface
tensions calculated from the capillary wave frequencies agreed
with the other methods.2–8 We fabricated a conventional laser
heterodyne QELS measurement system using a diffraction
grating as a local oscillator, and have reported on the dynamics
of the mass transfer of surfactants at a liquid/liquid interface.9–13
Furthermore, we devised a new principle of the QELS method
using a monoluminous flux detection technique in a beam
divergence, and have achieved 50 – 100 times higher S/N ratios
than the conventional heterodyne system.14 Although the S/Nratio was substantially improved by a new technique, averagingof 20 – 100 power spectra was necessary to estimate thecapillary wave frequency.
The maximum entropy method (MEM) was first reported as aprospecting technique by Burg in 1967.15 The method has been

applied to time-series analysis in various fields, such as
meteorology, geophysics, economy, and biorhythm.16 In the
instrumental analysis area, there are also many reports17–19 on theapplications of MEM, such as to calculate the electron density
distribution from the X-ray diffraction pattern20 and to analyze
the interferogram of IR spectroscopy in place of the fast Fouriertransform (FFT).21 The MEM provides a much higher resolutionof the spectrum than the FFT. In the present study, weinvestigated the applicability of the MEM to the QELS method,and found that the resolution of the spectrum was substantiallyimproved. The capillary wave frequencies were successfully
estimated by the MEM without averaging the spectra.

Principle
The QELS method
The principle of the new monoluminous flux QELS methods
using a beam divergence has already been reported in detailpreviously.14 In short, the QELS method measures the capillary
wave frequency of a selected wavelength, and the surface
tension is calculated from the relationship between thefrequency and the wavelength. A laser beam with a continuous
intensity distribution in the divergence is incident normal to thesurface (Fig. 1). The light intensity has a maximum at the centerof the irradiated spot and decreases monotonously with thedistance from the center. Because of the divergence, the angleof incidence varies as a function of the distance from the center.
The strongest part of the beam is incident exactly normal to thesurface, and its transmitted light is used as a local light (F Hz).
The surrounding parts of the beam are incident obliquely andare scattered almost nondirectionally by capillary waves ofvarious wavelengths. Some parts of the light are scatteredexactly normal to the surface with a scattering angle of θ andare mixed with the local light to generate an optical beat. By
locating a photodiode at the strongest part of the transmitted
light, the optical beat between the local light and the scatteredANALYTICAL SCIENCES DECEMBER 2007, VOL. 23
functions, C(k), of the time-series. In the MEM analysis, it isessential to find autocorrelation functions that maximize theentropy, H. We can obtain the autocorrelation functions bysolving the Yule–Walker equation,
C(k) = a1C(k − 1) + a2C(k − 2) + … + amC(k − m).

(4)

Equation (4) has the same structure as the auto-regressive modelof the time-series,
xk = a1xk−1 + a2xk−2 + ... + amxk−m,(5)

and thus we can find ai as the least-square solutions of Eq. (5).
The power spectrum is calculated from the solutions using theWiener–Khintchine relation,

Fig. 1 Measurement principle of the QELS method of the newmonoluminous flux technique utilizing beam divergence, which wehave previously reported. The surrounding parts of the beam (F Hz)
incident obliquely to the surface are scattered by a capillary wave
(f Hz) with a Doppler shift (f Hz). An optical beat (f Hz) is generated
by mixing the local light (F Hz) and the scattered light (F ± f Hz).
The beat frequency (f Hz) corresponds to the capillary wave
frequency (f Hz).

light is observed in the power spectrum. Because of thecontinuous spatial distribution of the incident beam, thescattered beam generating the optical beat consists of multiple
scattering angles. The scattering angle that generates thegreatest signal intensity can be determined from the angle of thebeam divergence, and it gives the wavelength of the observed
capillary wave as follows.
The electric field intensity generated by all the scattered lightat a distance of r from the center is approximately expressed by
the following equation:
E(r) ∝ 2πr exp (−r2/a2).

(1)

By differentiating Eq. (1), we find the maximum at r = a / 2.
This corresponds to a scattering angle of θ0 / 2, where θ0 is thebeam divergence of the light source. In the present study, thebeam divergence of the light source is 1 mrad, and thus the
wavelength of the observed capillary wave is calculated to be
0.752 mm.
The surface tension, γ is calculated from the relationshipbetween the capillary wave frequency, f, and the capillary wave
wavenumber, k, using Lamb’s equation,1f=

31 γ 2 2k ,2π  ρ (2)where ρ is the density of the liquid.

1log(det Cm),
2PM$t
1m

3 akexp(2Pifk$t)

2(6)

k 1

where PM and M are the prediction error and the order of theMEM, respectively. We calculated the power spectra using aself-made program according to an algorithm proposed by Burg.

Experimental
The experimental setup is the same as what we reported
previously.14 The beam from a diode-pumped YAG laser(CrystaLaser, model GCL-025S, 532 nm, 25 mW) is incidentnormal to the liquid surface after passing through the bottom of
the sample cell. An aperture (1 mm diameter) is located at thestrongest part of the transmitted light, and the selected light isdetected by a photodiode (Hamamatsu Photonics S1290). Data
were collected by an A-D converter after amplification. The
sampling frequency was 10 kHz (t = 0.1 ms), and one data record
consisted of 4096 data points. The collected data weretransformed into the frequency region (0 – 5 kHz) by using theMEM or the FFT. Window functions were not used in thepresent study. The MEM spectrum consisted of 4096 datapoints, whereas the FFT spectrum had 2048 data points (bothtransformed from 4096 data points in the time region). The FFTspectrum was only averaged 100 times for the reduction ofnoise. Moreover, the averaged spectrum was converted to thetime region again by the inverse FFT, and MEM was performedon the time region data. This procedure was performed tocompare the averaged FFT spectrum with the MEM.21
We used a surfactant aqueous solution (10 mM,cetyltrimethylammonium bromide, Kanto Chemical Co., Inc.)
for measurement samples. In order to monitor the surface
tension change, a piece of concrete specimen containing thesurfactant was immersed in distilled water.

Results and Discussion

The MEM
The MEM is based on the idea to find the spectrum thatmaximizes the information entropy.15 Since the theoreticaldetails of the MEM have been reviewed in various literature, webriefly give here an outline of the MEM. When discrete timeseries x1, x2, x3, x4, ... , xm are observed every Δt, the entropy, H,
is expressed byH=P( f ) (3)where Cm is the Toeplitz matrix, consisting of an autocorrelation

The FFT power spectrum of quasi-elastic laser scattering fromsurfactant aqueous solution surface with averaging of 100 times
is shown in Fig. 2a, and the corresponding MEM spectrum (theorder M = 15) is shown in Fig. 2b. The broad peaks appearing
at 0.6 – 0.7 kHz are the capillary wave frequencies with awavelength of 0.752 mm. The MEM spectrum was obtainedwith a much higher S/N ratio than the FFT. The FFT spectrum
was fitted with a Gaussian function, and the peak frequency wasestimated at 0.684 kHz. As for the MEM spectrum, the peakfrequency was estimated at 0.699 kHz from second

ANALYTICAL SCIENCES DECEMBER 2007, VOL. 23

Fig. 2 (a) FFT power spectrum from a surfactant aqueous solution
surface with averaging of 100 times. (b) MEM power spectrum
calculated from time-series data obtained by the inverse FFT of (a).
The curves are deviated for clarity.

Fig. 3 (a) FFT power spectrum from a surfactant aqueous solution
surface without averaging. A broad peak due to the capillary wave isindicated by an arrow. (b) MEM power spectrum calculated fromraw time-series data of 4096 sampling points without averaging. The
curves are deviated for clarity.

differentiation. Substituting the frequency, the wavelength, andthe density into Eq. (2), the surface tensions of surfactant
aqueous solutions were calculated at 31.7 and 33.1 mN/m fromthe FFT and the MEM spectra, respectively. The values of thepeak frequency and the surface tension agreed well with eachother. We consider that a small discrepancy between the FFT
and the MEM was within the experimental error. Both the FFT
and the MEM power spectra were successfully used for surface
tension measurements, provided they were sufficiently averaged(20 – 100 times).
Figure 3 a shows the FFT power spectrum without averaging,and Fig. 3b the corresponding MEM spectrum (the order M =27). The peak was not definitely observed in the FFT spectrum(depicted by an arrow), and the peak frequency could not bedetermined. The signals from the capillary wave seemed to beburied in noise. On the other hand, the MEM spectrum showeda definite curve, and we could calculate the peak frequency fromits second differentiation. The calculated peak frequency was0.690 kHz (surface tension 32.2 mN/m), which corresponds tothe values from Fig. 2. The peak frequency of the capillary
waves was successfully estimated by the MEM without
averaging, whereas the peak was buried in noise in the FFT.
Our result indicates that the time resolution of the QELS wasimproved to the theoretical limit of the present setup, 0.4096 s(4096 data points with sampling frequency of 10 kHz). We also
confirmed that the surface tension change of surfactant aqueous
solution can be monitored by the MEM spectra without
averaging, and that the values agreed with those from the FFTspectra with averaging 100 times.
The MEM was found to afford a higher S/N spectra of thequasi-elastic laser scattering method, and it successfullyenhanced the time resolution of the QELS by reducing theaveraging times. We should note the following points inpractical applications of the MEM. In the MEM analysis, theorder M, which is the counterpart of the maximum lag of theautocorrelation function in Eq. (4) or of the order in the autoregressive model in Eq. (5), is the only free parameter.
Inappropriate M values give no peaks or disproportionate curvesin the spectra, which can be regarded as being incorrect in every
respect, and thus it is of great significance to find an appropriate
M value. Although some procedures have been reported to findthe optimum order M, such as the final prediction error (FPE)and Akaike’s information criterion (AIC),22 such determined Mvalues brought about disproportionate spectral curves. Wefound empirically that the appropriate M value for the QELS

method was in the range of 6 to 30, which was dependent on thesampling conditions. We used M = 15 for the time-series data
without averaging (Fig. 3), and M = 27 for the data from theaveraged spectra (Fig. 2). The data above were all obtained by
the new monoluminous flux QELS method, but additionally, theMEM can also analyze the data of the conventional heterodyne
QELS method using the diffraction grating similarly (M = 7,1024 data points with sampling frequency 100 kHz). On theother hand, Sasaki et al. have reported that the adequate M lies
within the range of N/3 < M < N/2, as for the interferograms ofIR data, where N is the number of the sample points.21 We cansay that the appropriate order M is dependent on the type ofpeaks that we want to be enhanced. For example, two small
sharp peaks appearing at around 4 kHz in the FFT spectrum(Fig. 2a) disappeared in the MEM spectrum (Fig. 2b). This isdue to the choice of the appropriate order, M = 15. If thespectrum is analyzed by MEM of order, M = 1500, the broad
capillary wave peak becomes discrete, and the two small sharp
peaks are enhanced. Once the appropriate order M wasdetermined, we could use the same M value as long as theexperimental conditions, such as the data points, sampling
frequency, and observable peaks were the same. MEM analysis
takes a rather longer computation time than the FFT, but owing
to the recent rapid progress in information processing
technology, we can carry out MEM analysis in real-timemeasurements. The procedures of MEM and FFT werecompleted within 1 and 0.1 s using a standard personal
computer, respectively.
The theoretical limit of the time resolution was 0.4 s becauseof the sampling frequency of 10 kHz and the sample point
number, 4096. This is because the peak frequency of thecapillary wave (λ = 0.752 mm) was 0.6 – 0.7 kHz. We canincrease the sampling frequency, i. e., the observable capillary
wave frequency by decreasing the wavelength of the waves.
This can be easily done by increasing the divergence angle ofthe incident light. It is possible to achieve a time resolution of 1ms by increasing the sampling frequency to 1 MHz.
In conclusion, we applied MEM to calculations of the power
spectra for the quasi-elastic laser scattering from liquid surfaces
in place of FFT, and found that the S/N ratio of the spectra weregreatly improved. Although the averaging of 20 – 100 spectra
was necessary to estimate the peak frequency in the FFT power
spectra, the MEM spectra provided the correct peak frequency
without averaging.
We also found empirically that theappropriate order M for the MEM analysis of the QELS spectraANALYTICAL SCIENCES DECEMBER 2007, VOL. 23

was in the range of 6 to 30. MEM has a great advantage intime-resolution, and it will be possible to achieve a timeresolution of 1 ms by increasing the sampling frequency to 1
MHz. In addition, there is also a possibility of further
improvements of FFT as well, like for MEM, by combining
with the other data processing methods, such as nonlinear
optimization methods, such as the auto-regressive model.

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