1. Introduction

In a recent paper [1], sub-exponential attenuation of optical pulses propagating through deionized water was reported. While the reduced loss was not striking, combined with other observations, it was concluded that the pulses were optical Brillouin precursors [2, 3]. The possibility of reduced absorption is of great practical importance, making confirmation of these results of interest. Specific applications benefitting from this would be free-space communications in water and air, and medical imaging. For communications purposes small improvements in the received signal can improve the bit-error rate by orders of magnitude. Medical imaging has a similar situation where, e.g. in breast cancer imaging, each centimeter of propagation through the tissue reduces the signal one order of magnitude. Even though the absorption coefficient is 3–5 times smaller than the scattering coefficient it still contributes, for conventional light pulses, exponentially to the overall reduction of the signal amplitude.

Sub-exponential, or non-Beer’s law behaviour, has been attributed to at least three different phenomena; i) zero-degree pulses [4], ii) self-induced transparency [5], and iii) initial, Sommerfeld and Brillouin precursors [3]. All three have been discussed and compared by Barrett [6]. In contrast to this work there are publications by Roberts [7, 8], which assert that mathematically, non-exponential decay is not possible for pulses without dc-content.

For the purposes of this communication we focus on non-exponential decay by excitation of precursors. Although the work by Varoquaux [9] makes a strong argument that zero-degree pulses may be a particular manifestation of precursors, it is more straightforward to follow the analysis using precursors as a starting point. The known methods for launching zero-degree pulses require specific phase-shaped pulses with respect to a particular two-level resonance, conditions we know were not met for the experiments in [1]. The intensity was limited to obtain only linear interactions, and so we do not include self-induced transparency in our discussion.

The three types of precursors represent the transient response of the medium to an incoming electromagnetic pulse [9]. The initial precursor is due to the inertia of the electrons in the medium, the first photons in the propagating pulse go through the medium at the speed of light in vacuum, as if the medium were not present. These photons have never been observed and are most likely too few in number ever to be detected. The second precursor (Sommerfeld) consists primarily of the original pulse’s high-frequency photons and propagates through the medium at almost the speed of light (in vacuum) and with relatively minor interactions with the medium. This precursor also carries little energy. The third precursor (Brillouin) can contain significant amounts of energy, and comes from the low-frequency photons present in a broadband pulse. It can arrive either before or after the main pulse, depending on the wavelength-dependent properties of the medium. The Brillouin precursor is due to a strong (but linear) interaction between the medium and the pulse, and is the precursor believed to have been observed in [1]. The detailed theoretical analysis of precursors is complicated and somewhat controversial; for a good introduction, see [10, 11]. In this paper, we present the relevant aspects of the theory, explore the implications of exponential absorption for a broad-spectrum pulse, and examine the experimental results [1] in this context. This analysis makes it clear that the absorption measured in [1] deviates strongly from Beer’s law.

2. Theoretical Background

Mathematically, a pulse propagating through a linear medium obeys the wave equation;

where f is the electromagnetic pulse and v is the speed of light in the medium.

Following Sommerfeld [2], neglecting any transverse spatial variations and assuming harmonically varying pulses, we can then write down the exact solution of the wave equation, as a function of distance z and time t, using an inverse Fourier-Laplace transform

where F̃(0,ω) is the spectrum of the original pulse launched into the medium and k(ω) is the frequency dependent wavenumber. The assumption that transverse spatial variations of the pulse can be ignored is readily satisfied experimentally by carefully collimating the beam at a size large enough to avoid thermal blooming and other optical nonlinearities [1].

Solving this integral analytically is challenging. However, an asymptotic analysis using the method of steepest descents can be used, yielding for the Brillouin precursor amplitude,

where $\phi \left(\omega \right)=i\left(k\left(\omega \right)-\omega \frac{t}{z}\right)$ is the phase function and ω_s is a frequency for which φ(ω_s ) represents an extremum or saddle point (i.e. φ′(ω_s )=0).

The difficulty comes in trying to assess the specific z-dependence of Eq. (3), as φ(ω) is a function of time, space and frequency. In order to suppress the exponential dependence of $e^{iz\phi \left({\omega }_s\right)}$, we have the constraint that φ(ω_s )≡0. In Oughstun’s analysis [3] we therefore need to know k(ω) in order to determine the effect of this constraint. In addition, there is an explicit dependence on z in the denominator, [φ″(ω_s ) z]^1/2 which is linked to the second derivative of φ(ω). If, for example, we are at an extremum for which, φ″(ω_s ) ≠0, then the denominator varies as above. If we look at solutions in the vicinity of a saddle point, where φ″(ω_s ) =0 and φ‴(ω_s ) ≠0, then a Taylor expansion yields that the denominator varies as [φ‴(ω_s ) z]^1/3. In addition, the particular saddle-point frequency we are examining (ω_s ) is a function of z and t so it changes as we move through the medium, making φ‴(ω_s ) a slowly varying function of z. The complete solution is a sum of terms with different ω_s , complicating the z-dependence still further.

A different approach, independent of the specific form of k(ω), taken by Roberts [7, 8] postulates that the solution to (2) for a bandwidth limited pulse is bounded by;

where k(ω)=k_r +ik_i and $e^{-k_i^{\mathit{min}}z}$ corresponds to exponential attenuation according to the smallest absorption coefficient within the frequency region [ω_min ,ω_max ] of the pulse. According to Roberts the critical issue is whether or not the pulse has any dc-content. In his formulation, if (and only if) there is dc content, Oughstun’s analysis is valid and we can have less than exponential attenuation.

Our interpretation of the two treatments presented above is that Robert’s analysis is correct for the steady-state solutions (main pulse) but not for the transients (representing the precursors), independent of the original pulse’s dc-content. The exact form of the pulse decay with distance is not readily determined, other than that the losses will be sub-exponential.

3. Modeling

A key issue in looking for Brillouin precursors is therefore determination of whether there is sub-exponential absorption; we present a detailed analysis, and apply it to available data. In [1], the absorption of broadband optical pulses propagating through different distances of deionized water up to a total length of 4.7 meters was measured with the setup shown in Fig. 1. Briefly, the experiments were performed as follows: 100fs pulses from a MaiTai laser were spectrally broadened by sending them through a short piece of hollow core fiber, and the resultant 500nm wide pulse spectrum was bandlimited by passage through a 80nm (FWHM) bandpass filter. The shaped pulses sent through the water were 500 fs long, with a Gaussian spectral profile.

 

Fig. 1. Experimental set-up. Where, OI-optical isolator, OF-optical fiber, BE-beam expander, W-water tube, and PMT-photomultiplier tube.

Download Full Size | PDF

In [1], the measured transmission was compared to exponential decay of the somewhat arbitrarily chosen wavelength, $e^{-{\alpha }_{640}z}$, which tacitly assumed that all the energy of the pulse was at l=640nm.

 

Fig. 2. Pulse spectrum (dashed line) and the absorption coefficient for water from [12] over the wavelength range of interest.

Download Full Size | PDF

As is clearly observed fromFig. 2, the absorption coefficienta(λ) [12] is strongly dependent on wavelength over the spectrum (F_λ (0,λ)) of the original pulse, revealing the analysis in [1] to be a conservative estimate, but oversimplified.

Because of the spectral variation of α, the pulse will decay according to

where $F_{\lambda }(0,\lambda )=A\exp \left[-\frac{1}{2}{\left(\frac{\lambda -{\lambda }_c}{\Delta \lambda }\right)}^2\right]$, λ_c is the carrier wavelength, Δλ is the spectral width of the gaussian, A is a normalization constant and α(λ) is the wavelength-dependent absorption coefficient taken from the data in Segelstein’s thesis [12]. The evolution of a Gaussian input pulse under exponential decay of each wavelength is shown in Fig. 3, by plotting the calculated spectrum for a series of propagation distances. The bulk of the energy is lost from the long wavelengths before the short wavelengths are attenuated significantly.

 

Fig. 3. Computed spectral evolution of the pulse for a series of propagation distances.

Download Full Size | PDF

A more realistic test of exponential decay of a broad-spectrum pulse is thus a plot comparing the measured data to the integral of Eq. (5) over wavelength, where we sum the spectral energy to obtain the integrated transmission as a function of z:

In Fig. 4, we show the results of this calculation for the center wavelength and FWHM used in the experiments, and compare this to the data of [1]. Numerical integration of Eq. (6) yields the solid red curve, with the somewhat non-intuitive result that exponential decay does not plot as a straight line on a log scale. However, it reveals that the experimental attenuation observed is much less than would be expected for properly weighted, but purely exponential decay. The deviation from exponential decay is now evident over distances of less than one meter.

In Fig. 4 we have also introduced the reference exponential used in [1] to illustrate that for large z, the slope on the log plot approaches -$k_i^{\mathit{\text{min}}}$ (from eqn (4)). The longer wavelengths are absorbed quickly and the curve gradually breaks over to a slope based on the smallest α values. However, since the spectral weight at these wavelengths is small, the asymptotic slope is approached slowly, and most of the pulse energy has been absorbed before the limiting slope is reached. The theoretically predicted energy vs. distance for purely exponential, but spectrally weighted, absorption is thus substantially lower than for a single-valued exponential representative of the low loss region. Although the slope will eventually reach -$k_i^{\mathit{\text{min}}}$ , the energy never recovers from the losses at short z.

 

Fig. 4. Transmitted energy as a function of distance for experimental measurements and two models.

Download Full Size | PDF

With this more detailed description of exponential attenuation of a broadband pulse, the excess transmission of the measured data of [1] is quite remarkable.

4. Conclusion

Comparison of the available models for the propagation of Brillouin precursors yields a wide array of possible z-dependences for the absorption of light, particularly when a broadband pulse is being examined. The one clear prediction is less than exponential absorption. For a rapidly varying absorption coefficient, the test for whether or not a broadband pulse has sub-exponential absorption must include the effects of the initial intensity at each wavelength. We show that the measured data [1] is unequivocally indicative of substantially reduced absorption, and that the deviation from Beer’s law begins at shorter distances than previously thought. The most likely explanation for this behavior, in conjunction with observations of pulse breakup [1], is that precursors are being generated in the water by these short pulses.