## Abstract

We measure the sub-picosecond optical impulse response of a system consisting of a varying 1D diffusive medium and a stationary hidden object. It is shown that by averaging the temporal optical *field* response of a diffusive medium (as opposed to the optical intensity response) the signal-to-noise ratio of the object’s reflection can be improved considerably. The Spectral-Ballistic-Imaging technique is used to reconstruct the optical-field impulse response with a 200fs temporal resolution.

©2006 Optical Society of America

Optically turbid media such as biological tissue, ‘milky’ glass or milk in a glass, scatter illuminating light, hiding optical information embedded within or behind the medium. Various techniques have been studied to overcome this problem. For example, averaging is a well known method to improve the signal to noise ratio (SNR) of the image of a stationary object embedded in a dynamic noisy environment [1–4]. Another well known technique is to employ some type of gating (e.g., “time gating”[5–9], “spatial gating” [10–11], “polarization gating” [12–15], and “angle gating” [16]). In these techniques most of the noise is prevented from penetrating the measurement gate. For example, in the time-gating technique the first-arriving photons are separated from the rest (diffusive photons) by a fast shutter, giving a ballistic-image of the medium.

In this work, we describe a method for improving the SNR of the image of an object embedded in a dynamic scattering medium by combining temporal gating *and* optical-field averaging. As can be seen from the list of references, the two techniques (time gating and field averaging) are well known in the literature; however, in this letter we show that the Spectral Ballistic Imaging (SPEBI) technique is a very good candidate to implement the two techniques simultaneously with a very high temporal resolution. With this technique it is possible to demonstrate the noise reduction in the entire impulse response. When a photon is scattered from a moving scatterer, not only is its path modified but so is its frequency [17]. In fact, it is clear that the number of scattered photons that do not undergo a frequency shift is negligible. On the other hand, all the photons that were not scattered cannot experience this frequency (i.e., energy) shift. Thus, a spectrally narrow filter can separate the ballistic (non-scattered) photons from the scattered ones (in many respects this reasoning is similar to that of ref. [4]). This is exactly temporal averaging.

It should be pointed out that this reasoning is valid even in the adiabatic case; so long as the measurements are taken at finite intervals. This can be illustrated for the simple 1D case shown in Fig. 1. In this figure the scatterer is represented as a local change in the index of refraction. The transmission coefficient of the scatterer is independent of its location (*x*
_{1} or *x*
_{2} in the figure), but the reflection coefficient is not, gaining a certain phase which is proportional to its location δ=2(*x*
_{2}-*x*
_{1})*k*, where *k*=2*π*/λ is the wavenumber of the photons (λ is the wavelength). Therefore, when many measurements of the reflection and transmission coefficients are acquired at different scattering locations and averaged, the reflection coefficients average to zero, while the transmission coefficients are unaffected by the different locations.

Mathematically, this can be written

where the angular brackets and the subscript *x* denote averaging over the scatterer’s location, φ_{t} and φ_{r} are the phases of the transmission *t* and reflection *r* coefficient respectively.

Therefore, this distinction between reflection and transmission is independent of the velocity of the scatterer so long as the averaging time is large enough. Since the transmission coefficient corresponds to photons which did not experience scattering, it is unaffected by averaging (as Eq. (1) suggests), while the reflection coefficient corresponds to the scattered photons and therefore averages to zero. It should be emphasized that in Eqs. (1) and (2) the averaging is done over the optical fields coefficients (〈*A*exp(*i*φ)〉) and not separately on the amplitudes (〈*A*〉 or 〈*A*
^{2}〉) and phases (〈φ〉)[2–3].

We can generalize this reasoning to any system with multiple scatterers, without regard for the number of scatterers involved. If a scatterer is in a dynamic state during averaging over the fields, then it can be regarded as if its reflection scattering coefficients are zero. In the 1D case it means that the scatterer can be replaced with an equivalent object that has the *same transmission coefficient but a zero reflection coefficient*. Therefore, after averaging only the stationary scatterers are effectively contributing. Note, that unlike ref. 4, the scatterers’ motion can be random and not driven by an externally-controlled source.

Clearly, this averaging method can be used to image a turbid or a diffusive medium. Suppose we wish to image an object in a glass of milk. In principle, we can send a beam (or a train of pulses) of light into the medium and sample the reflection or transmission coefficient (depending on whether we desire reflection or transmission imaging) at a certain rate, with the time between successive measurements long enough for the medium to vary. Averaging over the coefficients will lead to the ballistic (non diffusive) image of the medium. The problem is, of course, that this averaging should take place over the electromagnetic *field*, while optical square-law detectors measure the beam’s power. To average over the field its amplitude *and phase* must be measured simultaneously. While the amplitude measurement is quite straightforward the phase measurement is more complicated, and in general requires interferometric measurements, which are relatively complicated and inherently noisy. Most time-gating experiments use ultra fast detectors (e.g., streak cameras); however they cannot measure the pulse’s phase. A technique that allows measuring both amplitude and phase simultaneously is the Frequency Resolved Optical Gating (FROG) method [18–19]. However, this technique utilizes both interferometric equipment and nonlinear dynamics, which means that very intense beams are required, and therefore it cannot be implemented for diffusive medium imaging.

To overcome this problem we use the SPEBI technique [20]. In this method, both the phase and amplitude of the electromagnetic field are measured almost directly without the need for complicated interferometric measurements, and it does not require nonlinear processes. Similar methods were used in the past for fiber dispersion measurements [21–24], however, in this paper, as in previous work [25–26] we use this technique to reconstruct the impulse response of a diffusive medium.

The idea to employ temporal averaging of the dynamic scatter to improve imaging is not new (see, for example Refs. [2–4]), however we apply this method in conjunction with the SPEBI time-gating technique by averaging directly over the full optical fields to reconstruct the quasi-ballistic impulse response of the medium with a 0.2ps temporal resolution.

The SPEBI technique [20,25,26] operates in the frequency domain. Instead of measuring the impulse response, the spectral response of the medium *H*(ω) is measured. That is, for every frequency ω in the spectral domain of the pulse the amplitude *A*(ω) and phase φ(ω) of *H*(ω) are evaluated. However, instead of measuring the phase φ directly, the derivative τ_{D}≡*d*φ/*d*ω, which can be interpreted as the time delay, is measured directly in a much more robust way. Since the phase difference Δφ is measured directly for every ω, the phase φ can easily be resolved by simple numerical integration.

In an experiment we implement the SPEBI technique to reconstruct a 200fs FWHM impulse response of a varying 1D diffusive medium. The sample in this experiment (the diffusive medium) consists of twelve 0.15mm glass plates with random orientation (see Fig. 2). Therefore, a lateral motion of the medium is equivalent to a random change in the plate’s locations.

To reconstruct the impulse response, we used a tunable laser, and scanned the entire optical telecommunication C-band (1535nm–1565nm) spectral range. Such a spectral range corresponds to a sinc-shape pulse whose FWHM is ~0.2ps. To measure the phase difference Δφ the laser beam is modulated with an external Mach-Zehnder modulator, controlled by a network analyzer. The spectral width of the laser (~200kHz) is considerably narrower than the modulation frequency Ω/2π=1GHz, therefore, the modulated beam (with the carrier frequency ω) that enters the etalon can be written *I _{in}*(

*t*)

*I*

_{0}[1+cos(Ω

*t*)].

If the transmitted beam (the beam that passed through the etalon) is attenuated and phase shifted, it can be written approximately as [20] *I _{out}*(

*t*)=η(Ω,ω)

*I*

_{0}{1+cos[Ω

*t*+ΔΦ(Ω,ω)]}. After measuring

*I*(

_{out}*t*) with the electronic detector, the network analyzer compares the input and output signals to evaluate both η(Ω,ω) and ΔΦ(Ω,ω).

There is a simple relation between the phase shift ΔΦ of the *intensity* wave to the phase φ of the EM *field* [20]: ΔΦ(Ω,ω) is equal to the phase difference between the two side bands of the modulated beam, which means that ΔΦ/Ω is an approximation of *d*φ/*d*ω, or, more accurately, ${\tau}_{D}=d\phi \u2044d\omega ={\displaystyle \underset{\Omega \to 0}{lim}}\left[\Delta \Phi (\Omega ,\omega )\u2044\Omega \right]$. Therefore, the frequency response *H*(ω) of the system is simply (up to an arbitrary global phase)

In practice the modulation frequency is Ω/2π=1GHz and not Ω→0, which would result in extremely small phase differences and low signal-to-noise ratio. We then scan the entire spectral range (from 1535nm to 1565nm) with spectral steps of δλ=0.01nm (which corresponds to frequency steps of δω~1.25GHz), and measure both η and ΔΦ for the entire spectral range. Under these conditions the spectral response of the medium can be approximated at the discrete frequencies ω_{N}≡ω_{0}+*N*δω for every integer *N*, by

where *N*≡(ω_{N}-ω_{0})/δω, and ω_{0} is the lower boundary of the spectral range. After evaluating the spectral response *H*(ω) of the medium we reconstruct the impulse response by a simple fast-Fourier-transform. We repeat the experiment at ten different locations on the sample. By doing so, we simulate a varying medium. Ten impulse responses of the medium are acquired in this fashion.

In Fig. 3 all of the ten impulse responses are superimposed with different colors. The reflection response from the mirror is marked by an ellipse. All the other peaks are due to reflections from the glasses. Clearly, we can use this impulse response to separate between the mirror reflection and the surrounding noise. This method is equivalent to the well known time-gating technique, although here the measurements were taken in the *frequency* domain.

However, since in this experiment we measure all the parameters of the electromagnetic field (both amplitude and phase) we can do more. In fig. 4 the *field* of the impulse responses is plotted. As can be seen, in the region of the impulse response that corresponds to the reflection from the mirror (the stationary component) – all the pulses are in-phase, and therefore do not cancel each other by averaging their fields. On the other hand, the reflections from the other non-stationary components (we focus on one of them in this figure) are out-of-phase and therefore cancel-out by averaging.

In Fig. 5 we show how such averaging can improve the SNR. Not only do the phases of the peaks vary from one pulse to the next, but their temporal locations vary as well. Therefore, even intensity averaging reduces the noise peaks more than it reduces the reflection from the mirror (see the lower panel of Fig. 5). However, this kind of averaging cannot eliminate the noise, it only smears it. On the other hand, if we average over the *fields* (see the upper panel of Fig. 5) the noise, which correspond to the reflections from the varying parts of the system is reduced considerably. In this figure the SNR is increased by a factor of 3 (which concurs with the square root of the number of measured impulse responses).

In this work we simulated a diffusive medium by a relatively simple structure; however, this is a proof of principle, which can easily be generalized to any varying diffusive medium.

To summarize, in this paper the SPEBI technique was used to measure the impulse-response of a diffusive medium with a 200fs temporal resolution. Since the SPEBI technique measures the pulses’ amplitude and phase simultaneously, it is possible to apply averaging over intensities of the response as well as over the fields. It is shown that when field-averaging is executed over a varying diffusive medium, the reflections from the moving scatterers are reduced considerably with respect to the reflections from the stationary objects. This property of field averaging increases the SNR of the measurement and can be used to image objects in dynamic diffusive media.

This research was supported by the Israel Science Foundation (grant no. 144/03-11.6).

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