A novel technique for studying photon propagation in scattering media is proposed and demonstrated, as is believed, for the first time. Photons propagating through the medium, from a frequency-ramped single-mode diode laser, meet a reference beam from the same source, at a common detector, and beat frequencies corresponding to various temporal delays are observed by heterodyne techniques. Fourier transformation directly yields the temporal dispersion curve. Proof-of-principle experiments on polystyrene foam and a tissue phantom suggest, that the new method, when fully developed, may favorably compete with the more complex time-correlated single-photon counting (TCSPC) and the phase-shift methods, now much employed.
© 2009 Optical Society of America
Photon propagation in scattering media is a broad field with applications ranging from radiative transfer in astrophysics, atmospheric radiative balance in climatology, to light propagation in biological tissue. The latter field is much studied with regard to optical mammography, measurements of tissue oxygenation and bleeding due to vessel rupture, as well as for dosimetry in photodynamic therapy. An early application concerned brain monitoring ; in a subsequent study imaging of structures in living human tissue was demonstrated . Studies of photon propagation in tissue can be pursued in the spatial domain (CW lasers), but more frequently in the temporal domain using pulsed lasers (time-of-flight measurements) or modulated CW lasers (the phase-shift method, where also amplitude demodulation is monitored). An early overview of different techniques for studying light propagation in biological tissue is provided in . More recently, the general field of tissue optics has been much studied, where the aspects of wavelength dependence of the absorption and reduced scattering coefficients of tissue, µa and µs’, respectively, are important, yielding information on the concentration of tissue constituents. A number of discrete wavelengths generated in swiftly pulsed diode lasers can be used in connection with time-correlated single-photon counting (TCSPC) electronics . An example is given in , where the optical properties of human prostate tissue were determined. Alternatively, a short-pulse white-light source can be used in conjunction with a combination of a spectrometer and a time-resolving streak camera . Then, in principle the optical properties of the medium can be evaluated from data generated in a single laser pulse. Such time-resolved white-light spectroscopy has also been extended to the study of pharmaceutical preparations .
Time-resolved measurements are also common in the laser radar field  and in reflectometry in optical fiber networks , although the time scales then are much extended. As illustrated in recent measurements on scattering media at a few meters distance , there is a seam-less transition from the large to the small time scales.
In laser ranging and in reflectometry there is special emphasis on capturing the distinct echoes from interfaces. Likewise, in early medical studies much emphasis was on “gated viewing”, i.e. capturing the part of the pulse propagating through a medium without suffering scattering. Numerous techniques were developed for that purpose . In particular, a method was developed by Toida et al. , where a CW laser was used in a Mach-Zehnder interferometer configuration with one beam passing the scattering medium, and a frequency-shifted reference beam being joined on a common detector. The heterodyne signal between the two beams, detected at the difference frequency, is due to the ballistic (unscattered) signal only, since other components all have varying phase due to delays. In the present paper the same basic geometry is used; however, we now realize that also the scattered components can be analysed, and also sorted in time using the same approach, but now employing a CW laser with an added frequency ramp; a technique which is readily achieved with a diode laser with a linear current ramp superimposed on the driving current. We in this way demonstrate the recording of time-resolved photon propagation curves, such as those obtained in time-correlated single-photon counting (TCSPC), but using simpler equipment. The simplification is similar as the one obtained by using the phase-shift technique instead of the TCSPC method. The technique, to our knowledge here demonstrated for the first time, draws inspiration from heterodyne gated viewing , fiber optic reflectometry  and in particular from the frequency modulated continuous wave (FMCW) ranging technique , where single-scattering is assumed. We show that the extension to multiple scattering is straight forward and that only moderate modulation and detecting speeds are needed, but still a high spatial resolution and a good signal-to-noise ratio (SNR) are achieved within a short time of averaging.
The paper is organized in the following way. The measurement principle and the analysis method used in the new method are presented in the next section. Then proof-of-principle experiments on polystyrene foam and a tissue phantom are presented and the data retrieval procedure is explained. Finally, the new method is put in relation to previous approaches in view of complexity, temporal/spatial resolution and acquisition speed, and general conclusions are drawn.
2. Principle and analysis
Fig. 1 (a) and (b) show the experimental set-up used in the present experiments, and the basic idea of data retrieval, respectively. In the FMCW technique, the frequency of the laser source is linearly modulated with, e.g., a triangular waveform. The modulated light is then separated into two beams, a reference beam and a probe beam. The reference beam passes through a well-known distance while the probe beam passes through an unknown optical path length. Both beams arrive at the same detector and interfere with each other. A beat frequency (fb) proportional to the time delay (τ) between these two beams will be induced in the detected signal (see the right part of Fig. 1(b)). This specific frequency can be used to deduce the path length of the probe beam and hence realize the space ranging. In most reported applications of the FMCW technique, e.g. , the reflected photons are single-scattered, and the beat frequency is detected with high contrast and at a narrow bandwidth. However, in the case when the probe beam passes through a multiple scattering medium, the path lengths of the photons will show a distribution with a position of the maximum and a width, which are dependent on µ s’, µa and the thickness of the sample. This part of the beam is labeled as sig and is shown as a dashed curve in the left part of Fig. 1(b). If we denote the field of the reference beam at the detector as
the field of the signal light that arrives to the detector at the same time can be expressed as a superposition of different components of the beam with photons which have passed through different effective path lengths,
Here, S=2π·Δf/ΔT is the slope of the frequency modulation (see Fig. 1(b)), τi is the time delay of each component compared with the reference beam, and the square of the amplitude, |Aisig|2, is proportional to the number of photons of the corresponding component. The interference signal of Eref and Esig can be detected as
The first term is a DC component of the interference signal. The second term indicates the interference between Eref and each component of Esig. The third term corresponds to the interferences among different components of Esig. The intensity of the third term is much weaker than that of the second term containing the reference signal itself, and can thus be ignored. Obviously the second term consists of components with different frequencies (proportional to the delays τi), and different amplitudes (proportional to Aisig, and |Aisig|2 is proportional to the number of photons). Therefore, the distribution of the time delays (effective path lengths) of photons passing through a scattering sample can be obtained by simply analyzing the spectral power (|ArefAisig|) of the interference signal of Eq. (3) in the frequency domain.
Since the working principle is based on optical interference, the measuring range of τi is limited by the coherent length of the light source. A laser with a good coherence property is needed for the technique presented. Suitable laser sources include distributed feed-back (DFB) diode lasers (with coherence length of hundreds of meters), external cavity diode lasers (ECDLs; kilometers), and Nd:YAG ring lasers (tens of kilometers) . Good coherence properties ensure the heterodyne efficiency between the reference and probe beams. Similar as for the FMCW technique , the minimal resolvable optical path length (R) is determined by the frequency range of the modulation (Δf), i.e., R≈c/Δf (c is the speed of light in vacuum). Considering the µa and µs’ of a scattering medium, e.g., a tissue sample, being constants even if the frequency of the laser source is tuned across 500 GHz (corresponding to ~1 nm wavelength difference when the central wavelength is located at 750 nm), the resolution, R, of the technique presented is expected to be of the order of 0.5 mm, which, however, degrades due to nonlinearity of the wavelength ramping (which has been thoroughly discussed in publications, e.g. Ref. ), and Doppler broadening in the frequency domain. Motion in the medium under study, e.g. blood flow in human tissue, or even Brownian motion of scatters, as well as mechanical instabilities in the interferometric apparatus are sources of broadening. A more detailed discussion will follow in Section 4. One advantage originating from the heterodyne technique used is that the amplification is proportional to the amplitude of the reference beam (see the second term of Eq. (3)), allowing a high sensitivity of the system.
3. Experiments and results
For demonstration, an experimental system was built as shown in Fig. 1(a). A DFB diode laser (Toptica Photonics, LD-0760-0040-DFB-1) operating in a single longitudinal mode, was employed as the light source. The central wavelength is around 760 nm and the line-width is less than 3 MHz (corresponding to a coherence length of 100 m). By ramping the driving current with a triangular waveform, the tunable range of frequency of the diode is around 40 GHz (corresponding to an estimated resolution, R, of 0.75 cm, according to the expression R≈c/Δf). The collimated light is divided by the first beam splitter into two parts, with 4% working as reference beam and the remainder working as the probe beam. The probe beam experiences multiple scattering in the sample under test, and penetrates through as signal light, which is collected by an imaging lens. The second beam splitter is used to overlap the reference beam and the signal light at the detector, a PMT tube (Hamamatsu, R5070A). A delay line is inserted in the probe beam to set an offset delay between the signal light and the reference beam. The tunable attenuator is used to adjust the intensity of the reference beam, for optimizing the amplifying function while not saturating the PMT. The detecting direction of the PMT is specially set as 90o against the probing beam, for convenience to test both transmission and reflection versions of the technique, as shown in Fig. 1(a).
While the driving current is modulated by a 10 Hz triangular waveform, the signal from the PMT is amplified by a current-to-voltage amplifier (Femto, DLPCA-200), and recorded by a data-acquisition card (National Instruments, NI-6154) installed on a computer. In real time, the spectral power of the signal in the frequency domain is analyzed with a fast Fourier transform (FFT) program on the computer. The spectral curves are averaged 100 times within 10 s to achieve a better SNR. Here, averaging is required to extract the weak signal (due to multiple scattering of the samples) from the background noise of the PMT. Averaging time can be reduced by employing a more powerful light source.
Scattering samples of different materials were tested in the transmission mode. Fig. 2 shows the results corresponding to a piece of normal printing paper (white, 100 µm thick), a tissue phantom (a 10 mm thick sample of gelatin containing ink as absorber, µs’=6.9 cm-1 and µa=0.3 cm-1), and polystyrene foam (11 mm thick, µs’ is ~30 cm-1 and µa is ~0.001 cm-1, respectively). As expected from Eq. (3), the curves of (b) and (c) show extended tails in the frequency domain (or optical path length domain), due to massive multiple scattering in the tissue and the foam. The curve of (c) is broader than that of (b), since µs’ of the polystyrene foam is much higher than that of the tissue phantom in this experiment. The thin white paper exhibits a relatively narrow curve. However, the width of the curve (~3 cm) is broadened compared with the theoretical spatial resolution limit (0.75 cm for the 40 GHz sweep utilized). Due to lateral multiple scattering a broadening is expected and further, a Doppler broadening will occur due to the vibrations and instability of the paper sample and the interferometer structures during averaging in the measurements.
Pieces of polystyrene foam with different thickness (l) were also tested in the transmission mode, and the resulting curves are shown in Fig. 3. With increasing l, the curve moves towards longer path length and the tail extends longer, which matches the fact that photons experience longer effective path length in a thicker scattering sample. For a piece of polystyrene foam with 1 cm of thickness, the distribution of photon arrival times shows that even path lengths of half a meter occurs.
Similar results can be obtained when the scattering sample is tested in a reflection mode. A piece of polystyrene foam (with infinite thickness) is set as shown in the inset of Fig. 1(a). Instead of changing l, the gap (d) between the illuminating point and the observing point is varied. The result shown in Fig. 4 illustrates that with larger d, deeper penetrating photons can be detected, and longer effective path lengths can be seen. However, the SNR degrades as a penalty, since less photons survive to be observed at a position further away from the illuminating point.
Although only proof-of-principle experiments are carried out, we believe that the new technique presented can potentially compete with conventional time-resolved and phase-shift methods in view of degree of complexity, spatial resolution, and sensitivity (acquisition speed), and might even show advantages in some aspects.
In time-resolved systems, the spatial resolution in the measurements is limited by the pulse width of the light source. Initially, Q-switched solid-state lasers were frequently utilized in such systems. Later, more compact Q-switched fiber lasers were developed to replace complex and large-sized solid-state lasers. However, the output wavelengths of fiber lasers are very limited for spectroscopic applications, such as those considered, e.g., in  and . Theoretically, if a femto-second laser is employed, a highest resolution on the sub-micrometer level can be achieved. Again, this resolution is not practical in spectroscopic applications, due to the limitation that the Fourier relationship puts between the pulse width and the bandwidth of the laser source. Diode lasers are extremely compact, cost effective, tunable and available at many wavelengths. They can be pulsed at high repetition rates, but pulse energies are minute compared to those obtained from Q-switched lasers. Thus, transient digitizer techniques cannot be used for capturing time-resolved signals. Instead, the TCSPC technique is frequently employed to generate a histogram of individual photon arrival times at the detector, a PMT which is cooled down to reduce dark current counts. Fast electronics matching the pulse width of the laser source are required. The sampling time needed depends on the repetition rate and average output power of the laser source, and the sensitivity of the detector. An alternative technique is the phase-shift method, where the delay-induced phase shift and demodulation with regard to the source modulation depth are measured. Since both the laser and the detection system work in CW mode, the electronics required are much less complex compared with TCSPC methods. Modulation of the source and demodulation at the detector are achieved with readily available radio electronics. The spatial resolution of this technique is determined by the upper modulation (demodulation) frequency.
Both the TCSPC and the phase-shift methods operate at high frequencies since the techniques basically compete with the speed of light. In contrast, the new technique presented here transforms the fast phenomena down to audio frequencies by employing the heterodyne principle, where the slow beat frequency between two optical frequencies being different just because of the propagation delay is detected. The semiconductor laser source is repetitively ramped in frequency at a slow rate (of the order of 10 Hz) and the different beat frequencies corresponding to different delays are retrieved by Fourier transformation of the detected signal. The longer the ramp sweeps the laser optical frequency, the better the temporal resolution will be.
Using an external cavity diode laser with tuning range of several tens of nanometer, a resolution of tens of micrometer can be realized, which has been proven for the FMCW technique, e.g. in . However, in many spectroscopic applications, the tuning bandwidth of the laser source is more limited and the expected resolution degrades. One extremely attractive feature of the FMCW technique, being of heterodyne nature, is that signal amplification is achieved by making the reference beam suitably strong, and noisy signal amplification electronics are thus largely eliminated. These attractive features in combination make the technique here presented viable and competitive. We realize, however, that for heavy multiple scattering the light at the detector is distributed over a substantial area. The reference beam must thus likewise be expanded for spatial matching. We believe that by optimizing beam overlap the signal-to-noise ratio obtained in our preliminary work presented here could be substantially improved.
We note, that the new heterodyning technique by its nature is sensitive to Doppler shifts and Brownian motion. We recall, that tissue perfusion can be measured using the Doppler flowmetry technique (see, e.g. ), where also heterodyning between moving blood cells and fixed cellular structures is utilized to retrieve a distribution of Doppler-shifted frequencies. Actually, the situation has many similarities with the one presented in our paper. Not unexpectedly we noticed a signal broadening when measuring on an instable paper sample. Such phenomena can be expected to be seen in clinical applications, where tissue blood flow, heart beating, etc., can obviously superimpose a Doppler broadening on top of the beat frequencies induced by true photon propagation delays. While the perfusion induced broadening information can be useful in some contexts, its detrimental influence could be reduced by deconvoluting the Doppler broadening from the measured curve to yield a true propagation delay curve. This might be done by measuring the frequency-shift distribution when not ramping the laser frequency.
We note, that blood flow and instability-induced effects can be minimized by simply increasing the ramping rate of the diode laser, inducing larger propagation delay frequency shifts, while the Doppler broadening, due to physical motion, remains the same. For a typical superficial blood cell speed of 3 mm s-1 the Doppler broadening will be of the order of kHz, which is similar to the propagation delay shifts measured in our experiments. If the ramping rate is increased from 10 Hz to 1 kHz in the experiments, the propagation time delay frequency shifts will be 100 times larger, and a 1 kHz Doppler broadening will become insignificant. The only fundamental temporal resolution factor will now be the frequency excursion of the ramp. Clearly, the detection bandwidth needs to extended from ~10 kHz to ~1 MHz, which is, however, still reasonably modest.
In conclusion, we have demonstrated a new frequency-resolved technique to study photon migration in multiple-scattering media. Both transmission and reflection measurement modes were realized. Short averaging times, low ramping rates and low signal recovery frequencies are attractive features in first proof-of-principle applications of the technique presented. The new method is inherently technically less complex than customary time- or phase-resolving techniques, and it could be expected, that when fully refined it might favorably compete with such techniques, which have benefitted from decades of development. A detailed comparison with customary techniques will be the subject of forthcoming work. In particular, the influence of instabilities and Doppler effects due to blood flow (in clinical applications) on the technique presented needs to be further investigated.
This work was supported by a project grant from the Swedish Research Council and a Linnaeus grant to the Lund Laser Centre. Fruitful discussions with Tomas Svensson and helpful assistance from Erik Alerstam are gratefully acknowledged.
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