Available online at www.sciencedirect.com

www.elsevier.com/locate/optcom

Optics Communications 281 (2008) 494–498

High angle phase modulated low coherence interferometryfor path length resolved Doppler measurements

of multiply scattered light

Babu Varghese a,*, Vinayakrishnan Rajan a, Ton G. Van Leeuwen a,b,Wiendelt Steenbergen a

a Biomedical Technology Institute, Biophysical Engineering Group, University of Twente, P.O. Box 217, NL-7500AE Enschede, The Netherlandsb Academic Medical Center, Laser Center, Meibergdreef 9, 1105 AZ, Amsterdam, The Netherlands

Received 28 June 2007; received in revised form 26 September 2007; accepted 26 September 2007

Abstract

We describe an improved method for coherence domain path length resolved measurements of multiply scattered photons in turbidmedia. An electro-optic phase modulator sinusoidally modulates the phase in the reference arm of a low coherence fiber optic Mach–Zehnder interferometer, at a high phase modulation angle. For dynamic turbid media this results in Doppler broadened phase modu-lation interference peaks at the modulation frequency and its multiples. The signal to noise ratio is increased by almost one order ormagnitude for large modulation angles and the shape of the spectral peaks resulting from the interference of Doppler shifted samplewaves and reference light is not changed. The path length dependent Doppler broadening is compared with the theoretical predictionsin the single scattered and diffusive regimes. The experimentally measured optical path lengths are validated with the Monte Carlotechnique.� 2007 Elsevier B.V. All rights reserved.

OCIS codes: 170.3340; 120.3180; 170.4580; 170.3890

Path length resolved photon intensity in scatteringmedia, and the associated Doppler spectra, can be mea-sured using coherence gated [1–5] and wavelength modu-lated [6] interferometric systems, in which the limitedtemporal coherence acts as a band pass filter in selectingthe photons that have traveled a specific path length.Michelson [1,2] and Mach–Zehnder based interferometricsetups [3,4] for measurements on completely dynamicmedia have been published. In a preliminary study, weshowed that path length distributions in almost static andmixed static-dynamic media can be measured by modulat-ing the optical path length in the reference arm [7]. This willenable path length resolved measurements in mixed mediasuch as tissue perfused with blood.

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.09.050

* Corresponding author. Tel.: +31 534891080; fax: +31 534891105.E-mail address: b.varghese@utwente.nl (B. Varghese).

In phase modulated low coherence interferometry at lowmodulation angles, the power spectrum shows an interfer-ence peak at the modulation frequency only, with a Dopp-ler broadening depending on the dynamic properties of themedium. This technique has been explored in single scatter-ing spectroscopy for analyzing the characteristics of extre-mely dense colloidal suspensions [8] and in industrialmetrology for measuring target displacements [9]. For highpeak phase modulation angles, frequency sidebands areobserved at multiples of the modulation frequency. Usu-ally, either the phase modulation angle is kept sufficientlysmall to avoid frequency sidebands [8], or the unwantedhigh-order harmonics are removed by low-pass filtering [9].

While usually the effect of higher order spectral bands isavoided, in this manuscript we deliberately generate anduse them to enhance the signal. We show that the pathlength distributions and path length resolved Doppler

B. Varghese et al. / Optics Communications 281 (2008) 494–498 495

spectra of multiply scattered light can be obtained from thefirst order peak in the power spectrum corresponding to thephase modulation frequency plus higher order peaks at twoor three times the modulation frequency. Optical pathlength distributions are measured from the area of multipleinterference peaks in the power spectrum, in a bandwidthof 2 kHz around these peaks. The average Doppler shiftby diffusion broadening is measured from the FWHM ofthe Doppler broadened phase modulation interferencepeaks at the modulation frequency and at higher harmon-ics, appearing in the photodetector signal power spectrum.Furthermore, we compare the path length dependentDoppler broadening measured for short and large opticalpath lengths with the theoretical predictions for single scat-tering and diffusive regimes respectively.

We use a fiber optic Mach–Zehnder interferometer(Fig. 1) with a superluminescent diode (Inject LM2-850,k = 832 nm, DkFWHM = 17 nm, coherence lengthLC = 18 lm) that yields 2 mW of power from the single-mode pigtail fiber as the light source. A single-mode fiberoptic coupler with a splitting ratio of 90:10 is used to createa reference arm (10%) and a sample arm (90%). Single-mode fibers (mode field diameter = 5.3 lm, NA = 0.14)are used for illumination, while multimode graded-indexfibers (core diameter = 100 lm, NA = 0.29) are used fordetection, providing a large detection window. The coher-ence length of the light source, and the intermodal disper-sion in the detection fiber, define the path length resolutionof the measurement. The centre-to-centre fiber distance is300 lm. The path length of the reference arm is varied byreflection of the light in a translatable retroreflector andthe position of the retroreflector is adjusted to yield anoptical path length equal to the optical path length of a cer-

Fig. 1. Schematic of the fiber optic Mach–Zehnder interferometer. Single-mode and gradient index multimode fibers are shown by thin and thicklines, respectively. SLD denotes superluminescent diode, PD is thephotodetector, LP is a linear polarizer, PM is the electro-optic phasemodulator, 90:10 and 50:50 are single-mode and multimode fiber couplers,respectively.

tain part of the photons in the sample arm. The referencebeam is polarized using a linear polarizer and its phase issinusoidally modulated at 7 kHz using an electro-opticbroadband phase modulator (New Focus Model 4002).The peak optical phase shift (D/) applied to the modulatorwas increased from 0.51 to 2.04 rad so that for the latterphase modulation amplitude frequency sidebands are pres-ent in the spectra.

Water suspensions of polystyrene microspheres (Poly-sciences Inc) of B0.77 lm (g = 0.85) are used to make ascattering phantom with a reduced scattering coefficientðl0sÞ of 2 mm�1. At the detector (New Focus Model 2001photo receiver) the light that is scattered in the sampleand the light from the reference arm after passing througha 50:50 fiber coupler are mixed. The AC photocurrent ismeasured with a 12 bit analogue to digital converter(National Instruments), sampling at 50 kHz, the signalwas sampled for 52 s to get an average of 1000 spectraand is Fourier transformed. Squaring this Fourier trans-form yields the power spectrum of the signal. To getsmooth curves we use an average of 1000 spectra. The posi-tion of the retroreflector that corresponds to the zero opti-cal path length in the medium was determined by replacingthe sample by a mirror on a distance of �13 mm to the tipsof the two fibers. From the position of the retroreflector forwhich a heterodyne signal was obtained, the position for azero path length of light probing the scattering samplescould be obtained. We measured spectra for a range ofpositions of the retroreflector in the reference arm. TheAC power of the heterodyne spectra is normalized by theDC2 component to render the results independent of thepower of the light source. The variation of the DC valueof the reference beam for the whole path length is within4%.

Monte Carlo simulations were performed [10] to vali-date the optical path length distributions. In the simula-tion, 100000 photons are injected into a simulatedmedium with scattering properties equal to those of theexperimental medium, using Mie scattering functions,and with an absorption coefficient of 0.001 mm�1. Eachphoton returning to the detection fiber (fiber separa-tion = 300 lm, NA = 0.29) is assumed to be detected,and its optical path length is recorded.

According to diffusive wave spectroscopy (DWS), in thecase of diffusive scattering, the power spectrum of diffusivelight that is heterodyne detected is Lorentzian and the line-width, pf0 = k2DBL(1 � g)/l depends on the self-diffusioncoefficient, (DB = KBT/3pga) of the particles in Brownianmotion [1]. Here k is the wave number in the scatteringmedium, l is the photon mean free scattering path,g = hcoshi is the scattering anisotropy of the medium, L

is the geometrical photon path length (optical pathlength/refractive index of water), KB is the Boltzmann con-stant, T is the temperature (293 K), g is the viscosity of thesuspending liquid [g = 1.0 cps for water] and a is the hydro-dynamic diameter (B0.77 lm) of the scattering particles[2,11]. For calculating the Doppler shift corresponding to

496 B. Varghese et al. / Optics Communications 281 (2008) 494–498

single scattering, we used the expression f0 = q2DB, withphoton momentum transfer q = 2ksinh/2, a function ofthe scattering angle h.

Fig. 2 shows power spectra measured for the water sus-pension of polystyrene microspheres for D/ = 0.51 and2.04 rad. For the lowest modulation angle, the power spec-trum is composed of an interference peak at the phasemodulation frequency, and a low frequency component.This low frequency component is composed of interferenceof the scattered light from the sample and the componentof the unmodulated reference light, and mutual interfer-ence of light backscattered by the sample. When the phasemodulation amplitude is increased, more light is conveyedfrom the unmodulated to the modulated part of the refer-ence beam. As a consequence the level of the power spec-trum at low frequencies (0–5 kHz) is reduced, as can beobserved from Fig. 2. Apart from an increase of the spec-tral peak at the modulation frequency, sidebands at higherharmonics 2f and 3f (14 kHz and 21 kHz) are formed. Theshape of the spectral peaks is the same and can be charac-terized by their amplitudes and line widths. When the sig-nal to noise ratio is expressed as the amplitude of thesignal minus the noise in the spectrum composed of fre-quency sidebands at the multiples of the modulation fre-quency divided by the amplitude of the signal whenphase modulation is switched off, for the retroreflectorposition where maximum interference is measured, itappears that the signal to noise ratio is about 4.7 (D/= 0.51 rad) at the modulation frequency. For large phasemodulation angles, the amplitudes of the interference sig-nals at the modulation frequency and higher harmonicsare increased and thus, when the information from theseusually unwanted and often-removed high-order harmon-ics are utilized by deliberately modulating at large phasemodulation angles (D/ = 2.04 rad), the signal to noiseratio is increased to 44 (by a factor of 9), as compared to

Fig. 2. Power spectra measured for water suspension of polystyrenemicrospheres for two different peak optical phase shifts (D/ = 0.51 and2.04 rad), with the position of the retroreflector corresponding to anoptical path length difference of 1.3 mm.

a situation where the peak phase modulation angle is keptlower (D/ = 0.51).

The optical path length distribution obtained by addingthe areas of all interference peaks (after subtraction of thebackground noise, and within a bandwidth of ±2 kHzaround all centre frequencies) as measured for the largemodulation angle is compared with the area of the peakat the modulation frequency for the lower modulationangle in Fig. 3. The signal to noise ratio enhancementdue to the modulation angle increase is clear from Fig. 3,in which the amplitude of the path length distribution mea-sured for large modulation angle is 9 times higher than thelow modulation angle case. The distributions are similarwhen normalized to the maximum value of the path lengthdistribution at the largest phase modulation angle, andsmoother curves are obtained for high phase modulationangles, as shown in the inset of Fig. 3. For sinusoidal phasemodulation, the phase modulated field amplitude at the kthsideband is given by Jk(D/), where Jk is the Bessel functionof order k and D/ is the peak optical phase shift [12]. Thetheoretically expected increase of the levels of the opticalpath length distributions is indicated by the solid trianglesin Fig. 3, where the peak amplitudes are indicated relativeto the peak amplitude for the highest phase modulationangle. Theoretically, the maximum gain in signal to noiseratio that could be obtained by further increase of peakoptical phase shift is 9.4 (D/ = 2.3 rad). In Fig. 3, theresults of Monte Carlo simulations are normalized withthe maximum value for comparison with experimentalresults. There is a good agreement between the experimen-tal data and the simulation results.

The FWHM of the Lorentzian fit to the phase modula-tion peak at the modulation frequency and higher harmon-

Fig. 3. Optical path length distributions measured for a scatteringmedium (g = 0.85, l0s ¼ 2:0mm�1, la = 0.001 mm�1) for different peakoptical phase shifts (D/ = 0.51, 1.02, 1.53, 2.04 rad) and the theoreticallyexpected values corresponding to different peak optical phase shifts (solidtriangles). Inset: Optical path length distributions normalized with theirrespective maximum values for D/ = 0.51 and 2.04 rad (logarithmic scale).

B. Varghese et al. / Optics Communications 281 (2008) 494–498 497

ics for different peak phase modulation angles is comparedwith the predicted linewidth based on diffusive wave spec-troscopy (Fig. 4). The FWHM of the interference signalis about 50 Hz in a static medium. Thus, the bandwidthbroadening of the interference signal for the dynamic med-ium results from the Doppler shift imparted to the interfer-ing photons by the Brownian motion of particles andincreases with optical path length due to multiple scatter-ing. The shapes of the interference peaks at the modulationfrequency and higher harmonics are similar when normal-ized to the same maximum values and the FWHM, whichfor Lorentzian spectra represents the average Doppler shiftcan be extracted from these interference peaks. For opticalpath lengths greater than 3.25 mm, the amplitude of theinterference signal is low at the higher harmonics and anestimation of width based on the Lorentzian fit to the spec-tra results in large errors. The experimental results are ingood agreement with the predictions of DWS within 10%for optical path lengths from 1.2 mm. In Fig. 4, theoreticalpredictions are given for the linewidth broadening for sin-gle scattering as a function of the optical path length. Inour dual fiber geometry, the value of sin (h/2) for singlyscattered light increases with the optical path length, caus-ing the single scattering Doppler broadening to increasewith the optical path length [13]. For single scattering thevalue of h will be slightly varying with the position in thetissue, the consequence of which is indicated by the errorbars in the single scattering data in Fig. 4. In previousreports on path length resolved DLS spectroscopy, singlyscattered light has been observed for short optical pathlengths [1,2,13] and they showed the dependence of detec-tion of multiply scattered light on the geometry of thedetection optics and on the anisotropy of the scattering

Fig. 4. The FWHM of the spectrum of interference peaks at themodulation frequency and higher harmonics as a function of the pathlength through the scattering medium (g = 0.85, l0s ¼ 2:0mm�1,la = 0.001 mm�1) for different peak optical phase shifts (D/ = 0.51,1.02, 1.53, 2.04 rad) is compared with the Doppler broadening in the singlescattering and diffusive scattering regime.

[1] and a theoretical model was developed to predict thistransition regime across the full range of path lengths fromsingle scattering through diffusive transport [2]. We observethat the measured linewidth for short optical path lengthsis above the calculated linewidth for singly scattered light(35 Hz). This is expected, since the photon mean free scat-tering path of 76 lm indicates that light is scattered multi-ple times before being detected, even for the shortestoptical path length of 0.5 mm.

To summarize, in this manuscript we present animproved low coherence interferometry method for mea-suring path length distributions and path length dependentDoppler broadening of multiple scattered light from turbidmedia. The improvement is based on sinusoidal phasemodulation of the optical path length in the referencearm at high phase angles, leading to interference peaks atboth the phase modulation frequency and higher harmon-ics. Rather than avoiding or neglecting these peaks, pathlength distributions and path length resolved Dopplerinformation is obtained from all interference peaks. Weshowed an enhancement of the signal to noise ratio by afactor of 9 in measured path length distributions and statis-tical averaging can be done on measured Doppler shiftsfrom the FWHM of spectrum of the sidebands. Goodagreement between the experimental path length distribu-tion and Monte Carlo simulations was found. The pathlength dependent Doppler broadening measured for largepath lengths is shown to agree with DWS within 10%.The modulation frequency can be shifted to higher fre-quencies, where often the noise level is lower and its spec-trum is more flat than for low frequencies. Hence, furtherincrease of the SNR might be obtained by using largerphase modulation frequencies.

We aim to apply this method on living tissue, forinstance to perform path length sensitive measurementsof tissue perfusion and to measure path length distributionswhich might reveal information about the tissue structureand function.

Acknowledgements

This work was sponsored by the Netherlands Technol-ogy Foundation STW (Grant TTF 5840) and the Biomed-ical Technology Institute of the University of Twente.

References

[1] K.K. Bizheva, A.M. Siegel, D.A. Boas, Phys. Rev. E 58 (1998) 7664.[2] A. Wax, C. Yang, R.R. Dasari, M.S. Feld, Appl. Opt. 40 (2001) 4222.[3] A.L. Petoukhova, W. Steenbergen, F.F.M. de Mul, Opt. Lett. 26

(2001) 1492.[4] A.L. Petoukhova, W. Steenbergen, T.G. van Leeuwen, F.F.M. de

Mul, App. Phys. Lett 81 (2002) 595.[5] D. Huang, E.A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W.

Chang, M.R. Hee, T. Flotte, K. Gregory, C.A. Puliafito, J.G.Fujimoto, Science 254 (1991) 1178.

[6] J.M. Tualle, E. Tinet, S. Avrillier, Opt. Commun. 189 (2001) 211.[7] B. Varghese, V. Rajan, T.G. Van Leeuwen, W. Steenbergen, J.

Biomed. Opt. 12 (2007) 024020.

498 B. Varghese et al. / Optics Communications 281 (2008) 494–498

[8] K. Ishii, R. Yoshida, T. Iwai, Opt. Lett. 30 (2005) 555.[9] U. Minoni, E. Sardini, E. Gelmini, F. Docchio, D. Marioli, Rev. Sci.

Instrum. 62 (1991) 2579.[10] F.F.M. de Mul, M.H. Koelink, M.L. Kok, P.J. Harmsma, J. Greve,

R. Graaff, J.G. Aarnoudse, Appl. Opt. 34 (1995) 6595.

[11] A.G. Yodh, P.D. Kaplan, D.J. Pine, Phys. Rev. B 42 (1990) 4744.[12] A. Yariv, Quantum Electronics, second ed., Wiley, New York, 1975,

p. 341.[13] B. Varghese, V. Rajan, T.G. Van Leeuwen, W. Steenbergen, Opt.

Express (2007) 9157.