Open Access
Add to BibSonomyAbstract
We have developed a new experimental setup based on optical Kerr gating in order to isolate either the transmitted or the scattered light going through an optically thick medium. This selectivity can be obtained by finely tuning the focusing of the different laser beams in the Kerr medium. We have developed an experimental setup. A Monte Carlo simulation scheme generates an accurate model of scattering processes taking into account the time of flight, the geometry of the Kerr gating and the polarization. We show that our experimental setup is capable of analyzing the transmitted light with optical densities up to OD = 9.7, and scattered light beyond OD = 347 in poly-disperse silica spheres in water (distribution centered on ~0.9µm radius) at λ = 550 nm. Strongly positive correlations are obtained with simulations.
©2012 Optical Society of America
1. Introduction
The propagation of light in strongly scattering media such as clouds, paints or biological tissues has been receiving increasing attention. This research is mainly driven by a need for characterization of such media (particle sizing, chemical characterization, fine physical measurements, etc.). This often requires an instantaneous and accurate in situ measurement, using light/matter interaction. Depending on the application, the goal is to focus either on the transmitted light (i.e. the light which goes through the sample and does not interact with it) or on the scattered light. Transmitted light characterization is very useful for ballistic imaging [1], detection of objects hidden in turbid media [2], or for spray imaging [3]. It can also be used for spectral extinction measurement of optically thick media [4] such as combustion chambers and rocket motors. The goal is to accurately measure the transmitted fraction of energy while removing the scattered parasitic part, however this becomes difficult with increasing OD (Optical Density). The other approach consists of carefully analyzing the scattered part itself. A considerable variety of information on the microphysical structure of the system can be gained from the temporal profile analysis of the scattered light. Thanks to femtosecond lasers and techniques such as up conversion [5] or optical Kerr gating (OKG) [6], it is now possible to perform Time Of Flight (TOF) measurements and to characterize the temporal scattered intensity profile. A careful analysis of the forward scattered light gives access to the optical depth and/or the particle sizes, even if the ballistic contribution is negligible [7]. In a recent work, Barthélémy [8] has shown the great benefit of using femtosecond laser sources and OKG experiments for optical diagnosis: the high temporal resolution and increased power allow for spectral extinction measurements of thick media. In this present study, we will experimentally demonstrate that optical Kerr gating is a well suited technique to measure independently scattered or ballistic intensities of light travelling through optically thick media. This selectivity allows us to study, thanks to a unique set up, a great variety of scattering samples displaying a large range of optical densities. At small OD, spectral extinction measurements should give, after inversion, the particle size distribution of the sample [4]. At high OD, scattered light shows variable time delay depending on OD and particles sizes and could be used as a tool of characterization [7]. We validate our results using a Monte Carlo simulation scheme. In the first section, we describe the experimental set up we have developed and show how fine-tuning the focusing of the beam increases either the ballistic or the scattered fraction of the signal. In the second section, the numerical Monte Carlo scheme is introduced and used to evaluate the range of accessible OD and the efficiency of OKG filter. Finally, experimental measurements are carried out and compared with various numerical simulations in scattered and ballistic configurations. Correlations are quantified and limits of detection are discussed.
2. Optical Kerr gate experiment: two possible configurations
When a femtosecond laser pulse impinges on a dense scattering medium, the ballistic light (transmitted light) goes straight through the sample. It is linearly delayed only by the eventual refractive index changes through its pathway (Fig. 1 ). The scattered light undergoes a more complex pathway and is consequently more delayed.
Fig. 1 Interaction of an ultra short scattering pulse and a scattering medium, and resulting temporal intensity of the signal.
It is possible to temporally sample the forward scattered and/or ballistic light thanks to the optical Kerr gating method. The laser source generates two pulses (pump and probe (Fig. 2 )). The linearly polarized probe (using polarizer P1) goes through the sample. The pump is optically delayed and is used as a temporal gate as it induces an instantaneous birefringence [6] in a BK7 plate. A crossed polarizer P2 lets the probe pulse reach the detector only if the two pulses (pump and probe) perfectly overlap spatially and temporally as shown on Fig. 2. The pump is linearly polarized at 45° with respect to the probe for optimal efficiency. This is strictly equivalent to a virtual pinhole which stops all the probe light that does not overlap the pump light. When the delay between the two pulses is adjusted, it is possible to sample either the ballistic or the scattered light. The sampling is performed along a defined delay range in order to obtain the temporal profile of the scattered light. A typical gate of 100 fs should discriminate pathway differences of 30 µm. In comparison with up conversion, there is no need for phase matching which is convenient when the probe is tunable over a large spectrum as is the case in our setup (tunable OPA). The only limit is the transmission of the OKG plate (0.3 to 2 µm for BK7). Furthermore, this method allows the sampling of photons with different linear polarization states simply by adjusting the two polarizers (P1 and P2). Optical Kerr gating also provides optimized sampling either of the scattered or the ballistic light, obtained with a fine adjustment of the focusing of the pump and probe beams on the Kerr plate (lenses Lpp and Lpr). Hence, two experimental configurations arise:
Fig. 2 Schematic representation of the sampling set up. The probe beam crosses the scattering sample S, and the iris I1 constrains the area of interest. The iris I2 defines the angle of collection of the scattered light. A convergent lens Lpr focuses both the ballistic and scattered light. The ballistic light follows a Gaussian propagation and focuses at the focal point F’. The scattered light follows a geometric propagation and the sample is imaged at plane (A’). The OKG plate is placed either at F’ or A’. P1 and P2 are two crossed polarizers.
- 1) Ballistic photon measurement (BPM). The smallest possible virtual pinhole is set up to filter out as much scattered light as possible. Ideally, the Kerr plate should be placed at the focal plane of both lenses Lpp and Lpr. This necessitates a strong attenuation of the pump energy. Practically, the pump is slightly defocused to avoid any damage to the plate. This also makes the adjustment of the spatial overlap of both beams easier.
- 2) Scattered light measurement (SLM). The Kerr plate is placed at the image plane A’ of the sample. The pump is enlarged to overlap the majority of the sample image on the Kerr plate (i.e. the image of the iris I1). Here, the pump energy is raised to its maximum to maintain the same Kerr efficiency (7% at the ballistic peak for both configurations).
The beam size has been measured using a beam profiler (WincamD-UD23, Datary Inc., Boulder Creek, CA, USA). The different spot sizes ωpump and ωprobe on the OKG plate are reported in Table 1 (ω denotes the waist of the beam at 1/e). We have furthermore calculated the size of the scattered light spot Rspot on the OKG plate, induced by a single particle situated on the optical axis (red spot on Fig. 2). This diameter (2* Rspot) is equal to I2 at the lens position (O), and linearly decreases to almost 0 at the image plane (A’) (the Point Spread Function of our optical apparatus is less than 30 µm).
Table 1. Beam waist at 1/e (in µm) of pump (ωpump) and probe (ωprobe) and spot size Rspot on the OKG plate in the two sampling configurations
From these parameters, we evaluate the transmission of ballistic (Tbal) and scattered (Tscatt) photons through the OKG plate. Ballistic photons have a Gaussian transverse energy repartition Iprobe(r). The transmission at any point of the OKG plate is directly proportional to the pump energy Ipump. Integration of the transverse transmitted intensity profile ∫Iprobe(r)Ipump(r)dr gives, after a straightforward derivation and normalization, . The transmission of scattered light Tscatt is obtained using a similar derivation assuming that we have a homogeneous energy distribution all over the scattered disk. We then define and calculate the ratio η of these two transmissions:
where u = (Rspot/ωpump)2 . In the SLM configuration, η=0.75, i.e. transmissions of scattered and ballistic photons have the same order of magnitude. In the BPM configuration, η = 180 and the transmission of ballistic light is greatly enhanced. This analysis of efficiency has to be extended to every particle situated at a given distance r from the optical axis. In the BPM configuration, since Rspot>>ωpump, efficiency of OKG is homogeneous all over the sample and does not depend on r. In the SLM configuration, Kerr efficiency decreases with r. Indeed, it is directly governed by the pump profile intensity. One needs to project the Gaussian pump profile from the OKG plane, back to the sample plane, using Γ, the optical magnification of the probe line. The OKG efficiency is therefore equal to η(r) = η(0).exp(-r2/ωη2) where ωη = ωpump*Γ. Hence, the optimization of the scattered flux is a two step process. The first step consists of minimizing η by placing the OKG plate at A’. The second step consists of collecting the scattered flux coming from the largest part of the sample (i.e. by enlarging ωpump and consequently ωη ).3. Numerical simulations
In order to evaluate the relative efficiency of the temporal and spatial filtering, we have developed a temporal Monte Carlo scheme [9]. Using a computer simulation, we randomly send particles of light (that we call photons) through the sample. Their pathway through the sample is randomly built toward the detector, keeping track of their time of flight. We calculate the different density functions. The distance between two scattering events is governed by the extinction coefficients. The phase function determines the scattering angle. To model the time spent in the particle, one needs to introduce temporal phase functions [10] and Debye modes [11]. For basic geometries, a good agreement between our simulations and those found in Calba et al. [7] was obtained. Our model also takes into account the depolarization effect, and further results on this subject will be presented in a dedicated study. In order to carefully understand and predict the relative weight of the ballistic and scattered light, a very accurate description of the geometry of the system is required. We measured the size of the probe beam ωsample at the sample position and the pulse duration (FWHMprobe) was obtained from the laser specifications. Given these data, initialization of the photon is then done spatially and temporally on a Gaussian profile. After every initialization, the photon position is checked. If it remains within the iris I1, the corresponding ballistic contribution reaching the detector is calculated by multiplying by η.10-OD. Every scattering event is checked to be within the iris I1. We use a semi Monte Carlo detection by multiplying the corresponding intensity by the solid angle of collection Ω (defined by the iris I2). We also multiply the contribution by the radial Gaussian attenuation (ωη) due to the pump spatial profile. The photon is “killed” when it comes to the interface. We then sum the ballistic and scattered contributions and convolute the result with a temporal Gaussian profile (FWHMconvol) in order to reproduce the signal obtained experimentally for a pure ballistic signal. This last convolution merges the temporal size of the pump beam and other effects due to the non compensated dispersions in lenses, polarizer or the sample itself and the electronic response of the Kerr medium which is known to be faster than the 100 fs pulse duration [12]. The parameters used for the simulation are given in Table 2 .
Table 2. Geometrical parameters of the experimental setup
With this set of parameters, the maximum retrievable OD is evaluated in BPM configuration. The idea is to consider a given radius of the particle, and to increase the OD to a level where the parasitic collected scattered light within the iris I2 induces an error of 5% on the measured optical density, without OKG filtering. This maximum value for regular spatial filtering is reported in Table 3 for four different particle radii. The next step consists of evaluating the benefit of an OKG filtering. We integrate the simulated flux reaching the detector within 160fs around the zero delay. In this temporal window, the scattered light and the resulting error on OD are mainly due to the snake-like photons that undergo no major delay [13]. We then increase the OD until the snake-like parasitic photon induces a similar uncertainty (5%) on the OD measurement. The main result is that OKG allows increasing the maximum OD from 7.3 to 11.6 for small particles. The case of big particles is more complex as the weight of snake-like photons is more important since their scattering contribution is more peaked in the forward direction, and almost no delay can be observed. Yet, OKG filtering allows OD measurements up to 6.3 for 50µm particles.
Table 3. Maximum retrievable optical densities with ≤5% relative uncertainty for regular and OKG filtering. We consider a mono disperse distribution of silica particles in water illuminated by a λ = 550nm laser beam
4. Experimental study
4.1 Samples investigated and principle of measurement
We use silica particles in water suspensions, with a given radius of r = 0.8µm (value provided by the manufacturer). Samples are then prepared in order to predict the OD with the following formula OD = mp/MpLσext where mp is the mass concentration, Mp = 4πr3/3ρ is the particle mass using ρ = 2.2 (density of silica), L is the geometrical thickness travelled by the laser beam (here, 10−2 m) and σext is the extinction cross section calculated using Mie theory. The sample optical density is then checked through two distinct acquisitions: (i) the intensity of ballistic photons after temporal integration of their contribution through the sample (Isample) and (ii) the same contribution through calibrated densities (Iref). A straightforward derivation gives:
where ODref refers to a calibrated density. Given this direct protocol, we are able to measure OD up to 9.7 with the experimental conditions described in section 4.2. Samples with higher OD are characterized after dilution. For the highest OD of our sample set, we evaluated a global reproducibility of 7% RMS: twelve distinct samples were made and measured with the same experimental protocol, silica powder weighing, dilution with homogenization process and finally OKG measurement. The presently reported sample names refer to these OD measurements. Three samples are used for BPM experiments: OD8, OD8.6 and OD9 and four are used for SLM experiments: OD8.6, OD10.5, OD18 and OD42.7. A supplementary sample OD347 is used to determine the limit of detection of the system. When no more ballistic light can be detected (OD>9.7), the main characteristics of the signal are the delay momenta Δtn [14] of order n, expressed as:where t is the delay (in ps) and I(t) the signal intensity (a.u.). In order to compare experiments with simulations, the initial silica powder must be precisely characterized. Hence, a spectral inversion code is used to determine the particle size distribution. It is adapted from previous works [15], based on a numerical filtering [16] method and a positive constraint scheme [17]. Transmission spectra were recorded using a “Lambda 950” spectrometer (Perkin Elmer, Waltham, MA, USA) on suspensions of OD<1. Thanks to the introduction of a numerical noise in the inverse process, a PSD set centered on the same modal radius is obtained (Fig. 3b ). These PSD correspond to similar “normalized” absorption spectra (Fig. 3a). Furthermore, Monte Carlo simulations were run to evaluate the impact of the PSD shape on the temporal scattering profile (Fig. 3c); no visual influence can be noticed. The calculations of Δtn reveal a deviation ranging from 5% to 13% respectively for n = 1 and 3. A deepened study shows that lowering this relative standard deviation would imply a modification of the OD of ~0.1, which confirms that the PSD shape has a negligible influence on the studied objects at the current scale of measurement. In the following, PSD6 is used to conduct all the Monte Carlo simulations.Fig. 3 (b) different PSD evaluated by the spectral inversion code for the corresponding absorption spectra (a), giving similar temporal scattering profiles (c) in SLM configuration, for OD = 12.
4.2 Experimental setup
The laser source is a 3mJ - 1 kHz - 800 nm - 100 fs amplified Ti:Sapphire laser (Coherent “Libra HE”, Santa Clara, CA, USA). The laser beam is split in two before the compression stage: the two lines are compressed independently through identical compressor stages in order to avoid self phase modulation in the beam splitter. The pump beam (1 mJ/pulse) can be optically delayed using a motorized delay line kit (Newport, Irvine, CA, USA). The pulse energy is adjusted depending on the set configuration (1 mJ/pulse in SLM, few µJ/pulse for BPM) in order to have a similar laser fluence on the OKG plate. The probe beam (2 mJ/pulse) goes through a commercial optical parametric amplifier (TOPAS, Light Conversion Ltd., Vilnius, Lithuania) and a harmonic generator. This setup makes the 800 nm initial beam tunable continuously from 0.23 to 20 µm. This study presents results obtained for a single 550 nm SFI output (sum frequency between 800 nm and a produced Idler). The output energy is ~200 µJ/pulse. The probe beam is then routed through the sample, towards the OKG plate. As OKG sampling is based on the birefringence effects inside the Kerr medium, the extinction level of the crossed polarizer P2 should be sufficient to significantly reduce the amount of non-sampled probe beam, actually identified as background. This is particularly true in the SLM configuration when very low signals are acquired from very thick scattering media. After precise alignment of the probe line through lenses and plates and ensuring that the optics are birefringence free (i.e. constraint free), we find an extinction coefficient of ~10−6 which is close to the specification of a single polarizer (10−5). In other words, the background signal can be reduced to this value, meaning that P2 has a “leak” of one photon out of 106. The resulting ballistic and scattered sampled light are finally directed towards a spectrometer made of a monochromator (Newport, Oriel Instruments 74125, Stratford, MT, USA) and a PMT detector (Perkin Elmer MH973, Waltham, MA, USA). This additional spectral filter is imperative in order to eliminate the pump parasitic light and to spectrally filter the output of the TOPAS. Indeed, different wavelengths are generated in the TOPAS and the provided harmonic separators are not sufficient.
5. Results
Figure 4a shows the results obtained for OD8, OD8.6 and OD9 in the ballistic configuration. Experimental pump/probe scanning clearly reveals a ballistic peak of intensity, placed at the zero delay (Fig. 4b). Here after, the temporal lobe of scattered photons grows with OD and the maximum intensity is delayed: 0.8 ps (OD8), 0.9 ps (OD8.6), and 1 ps (OD9). Simulations fit the experimental data for all samples well, with computed OD of 8.1, 8.7 and 9.1 respectively for the three samples. For the sake of clarity, the curves are normalized to the experimental ballistic peak. Yet the great step forward of these experiments is the absolute agreement between experiments and simulations. Indeed, the latter are normalized to the same calibrated ODref as that used experimentally. We were able to reproduce the absolute level of both ballistic peak and scattered lobe with a slight adjustment of the critical parameters (η, Ω) within uncertainties measured on our setup. Only one set of parameters were sufficient to reproduce accurately all the different experiments.
Fig. 4 Experimental and simulated temporal scattering diagrams in BPM configuration, for samples OD8, OD8.6 and OD9. Intensities and zero time delay are normalized to the experimental ballistic light peak. Scattered light is represented in (a) with a zoom on ballistic photons in (b).
Figure 5 represents a compilation of experimental and simulated temporal scattering profiles in the SLM configuration. No ballistic light can be detected because of the high optical densities studied and the lack of efficiency of the SLM configuration for such detection. Whereas the OD8.6 sample clearly exhibits a ballistic peak in the BPM configuration, it is not visible here. Therefore, signals are normalized to the maximum peak intensity for all samples. We observe an increase of Δt1 with OD: 1.6ps (OD8.6), 2.4ps (OD10.5), 4.3ps (OD18) and 10.6ps (OD42.7). The simulations were computed with OD of 8.2, 11.8, 19.5 and 42.1 respectively. These adjustments are slightly higher than the 7% uncertainty over measured the OD (section 4.1) but could be explained by the sample instability (i.e. sedimentation, agglomeration). We have calculated the 3 first orders of Δtn, and deviations between the simulations and experiments are below 2%.
Fig. 5 Experimental and simulated temporal scattering diagrams in SLM configuration, for samples OD8.6, OD10.5, OD18 and OD42.7. Intensities are normalized to the maximum measured signal for each sample. Zero delay is the previously detected ballistic peak of OD8.6, not visible anymore in SLM.
Figure 6 represents the experimental result of the temporal scattering in the OD347 sample. The signal to background ratio is ~1. For this OD, it was not possible to perform the simulation. Three main reasons could be given. First, side interfaces of the sample modify the temporal signal even if the forward direction is considered. This point is under investigation and we have evidenced modification of temporal profiles for OD greater than 50 (for this geometry and PSD). Moreover, the charge rate of this last sample exceeds a few percent and for this regime, dependent scattering has to be considered. Finally, such an optical density requires enormous calculation resources. The estimated time needed for such a simulation is nearly a month using a 3GHz CPU, 16GB RAM personal computer. Meanwhile, a similar single in situ measurement is performed in five minutes.
4. Discussion
The BPM configuration allows for direct measurements of optical densities up to 9.7, which is comparable to Calba et al. [7]. The setup is tunable over a very large frequency range thanks to the use of TOPAS and the adapted efficient spectral filtering. The positioning of the OKG plate and convergent lenses is critical since it creates an optimal transmission ratio η in order to eliminate the maximum amount of scattered light, thus enhancing the detection of ballistic photons. Under routine operation, the pump can be defocused to favor the spatial overlap setting. This slight degradation decreases the OD upper limit of 9.7, but makes the set up more robust: less sensitivity to the pump/probe overlap quality and to sample instabilities (gas jets, liquid flows, living biological media). The SLM configuration provides an efficient detection of scattered light for optical densities above 300. In this case, beam waists (pump and probe) are optimized to enhance scattered light collection. For the thickest sample displayed in Fig. 6, the total intensity scattered in the forward direction is very weak. Only one incident photon out of 106 is scattered through the iris I2 and consequently reaches the detector if the OKG filtering is bypassed. Knowing that the Kerr effect efficiency (Keff) is adjusted to approximately 7% at the peak of ballistic light, this efficiency is spread over the total signal range (T~5 × 10−10 s) following the pump resolution (τ~10−13s) and becomes equal to Keffτ/T = 1.4 × 10−5 for the peak of scattered light shown on Fig. 6. 200 µJ incident probe pulse energy at 550 nm leads to a number of photons of about 104 on the detector at the scattered lobe peak. This is still far from the single photon detection limit, however explains the poor signal to noise ratio (the curve displayed is an average of 1000 pulses). A more important limiting factor is the contrast between the scattered signal and background (offset on Fig. 6) equal to 1. This order of magnitude can be readily evaluated as the ratio between the Kerr efficiency (~10−5) and the extinction of the probe line (10−6) taking into account the depolarization (effective signal divided by 2). This discussion shows that two challenges have to be simultaneously overcome to efficiently sample the forward scattered light. First, one needs to obtain the maximum contrast realizing an optimal alignment of the probe line, i.e. no remaining constraints, birefringence in lenses or Kerr plate. Then, the best link budget should be realized. Yet, the signal/background contrast appears to be the limiting factor. Indeed, despite an optimal extinction (10−6 compared to the extinction of a single polarizer specification of 10−5) and an improvable linked budget (more powerful laser, better Kerr medium, longer averaging and single photon detection) the signal background still has a stronger impact on Fig. 6. Hence, we have chosen an arbitrary ratio of detection in order to draw comparisons: 1) The scattered signal can be detected only if it is greater than one tenth of ballistic light (lower OD limit), and only if it is bigger than the leak background (upper OD limit) and 2) Ballistic light can be detected only if it is twice as big as the scattered light peak. These OD limits are reported in Fig. 7 , which summarizes the benefits of the two distinct configurations. The given values are specific to the present study, but the method is applicable to all types of samples. Indeed, these limits depend for example on absorption and particle size. A recent study shows that absorption by particles tends to reduce the mean time of flight of photons [18] and therefore the temporal filtering will be less efficient. If we consider particles smaller than these in the present work, the scattered light distribution will be more isotropic, and the forward scattered flux reduced. The upper limit of detection should therefore be shifted toward lower OD. As previously mentioned in this study, the precise PSD seems to have a limited impact on the temporal shape of the scattered light, as long as the corresponding extinction spectra are similar. This lack of correlation strengthens our results, as the precise knowledge of the PSD (existence of aggregates, non spherical particles…) apparently becomes trivial.
Fig. 7 Limits of detection of ballistic and/or scattered photons, relatively to the two different experimental configurations (BPM and SLM). The upper limits given for scattered light correspond to OD acquired with a signal / background ratio of ~1. The lower limit for scattered light corresponds to a scattered signal / ballistic signal ratio of 0.1. The upper limit for ballistic light corresponds to scattered signal / ballistic signal ratio of 2.
5. Conclusions
The present study reports a temporal analysis of scattered and ballistic light on a very large OD range [0, 347]. The extremely short acquisition time (compared to simulations) makes the present setup a reliable tool to compile a reference data bank of temporal signatures of scattered light emitted from calibrated very thick media. This opens a way for further possible optical diagnostics of very thick unknown samples (particle sizing, OD determination…). Furthermore, our study allows the validation of our Monte Carlo code. We are thus able to precisely model two different configurations, optimized either for ballistic or scattered light collection. The OKG filtering efficiency for spectral extinction measurements has been numerically evaluated and appears to work well for particles up to 50 µm radii. We are currently performing further experiments at different wavelengths and particle sizes. The temporal depolarization could provide additional information about the sample [19] and should be meticulously investigated.
Acknowledgments
This work is supported by the ONERA - The French Aerospace Lab Research Project “FEMTO”. The authors would like to thank Nicolas Rivière for his great technical and scientific support.
References and links
1. M. Paciaroni and M. Linne, “Single-shot, two-dimensional ballistic imaging through scattering media,” Appl. Opt. 43(26), 5100–5109 (2004). [CrossRef] [PubMed]
2. L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical kerr gate,” Science 253(5021), 769–771 (1991). [CrossRef] [PubMed]
3. C. Schultz, J. Gronki, and S. Andersson, “Multi-species laser-based imaging measurements in a diesel spray,” SAE Transactions 113(4), 1032–1042 (2004).
4. M. Barthélémy, N. Rivière, L. Hespel, and T. Dartigalongue, “Pump probe experiment for high scattering media diagnostics,” in Reflection, scattering, and diffraction from surfaces, Z.-H. Gu and L.M. Hanssen, eds., Proc SPIE 7065, 70650Z (2008).
5. A. P. Vandevender and P. G. Kwiat, “High efficiency single photon detection via frequency up conversion,” J. Mod. Opt. 51(9–10), 1433–1445 (2009).
6. P. P. Ho, N. L. Yang, T. Jimbo, Q. Z. Wang, and R. R. Alfano, “Ultrafast resonant optical Kerr effect in 4-butoxycarbonylmethylurethane polydiacetylene,” J. Opt. Soc. Am. B 4(6), 1025–1029 (1987). [CrossRef]
7. C. Calba, L. Méès, C. Rozé, and T. Girasole, “Ultrashort pulse propagation through a strongly scattering medium: simulation and experiments,” J. Opt. Soc. Am. A 25(7), 1541–1550 (2008). [CrossRef] [PubMed]
8. M. Barthélémy, Apport d’une source laser femtoseconde amplifiée pour la mesure de spectre d’extinction d’un milieu diffusant optiquement épais. PhD thesis, Université de Toulouse, 2009.
9. N. Rivière, M. Barthélémy, T. Dartigalongue, and L. Hespel, “Modeling of femtosecond pulse propagation through dense scattering media,” in Reflection, scattering, and diffraction from surfaces, Z.-H. Gu and L.M. Hanssen, eds., Proc SPIE 7065, 70650X (2008).
10. L. Méès, G. Gréhan, and G. Gouesbet, “Time-resolved scattering diagram for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194(1-3), 59–65 (2001). [CrossRef]
11. A. E. Hovenac and J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9(5), 781–795 (1992). [CrossRef]
12. R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear susceptibilities of glasses,” Phys. Rev. B 11(2), 964–967 (1975). [CrossRef]
13. N. Pfeiffer and G. H. Chapman, “Monte Carlo simulations of the growth and decay of quasi-ballistic photon fractions with depth in an isotropic medium,” in Optical Interactions with Tissue and Cells XVI, S.L. Jacques and W.P. Roach, eds. Proc. SPIE 5695, 136–147 (2005).
14. W. F. Long and D. H. Burns, “Particle sizing and optical constant measurement in granular samples using statistical descriptors of photon time-of-flight distributions,” Anal. Chim. Acta 434(1), 113–123 (2001). [CrossRef]
15. L. Hespel and A. Delfour, “Mie light-scattering granulometer with adaptive numerical filtering. I. Theory,” Appl. Opt. 39(36), 6897–6917 (2000). [CrossRef] [PubMed]
16. G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurement using analytic eigenfunction theory,” J. Atmos. Sci. 49(22), 2074–2081 (1992). [CrossRef]
17. J. P. Butler, J. A. Reeds, and S. V. Dawson, “Estimating solutions of first kind integral equation with nonnegative constraints and optimal smoothing,” SIAM J. Numer. Anal. 18(3), 381–397 (1981). [CrossRef]
18. M. Kervella, F.-X. d’Abzac, F. Hache, L. Hespel, and T. Dartigalongue, “Picosecond time scale modification of forward scattered light induced by absorption inside particles,” Opt. Express 20(1), 32–41 (2012). [CrossRef] [PubMed]
19. S. G. Demos and R. R. Alfano, “Temporal gating in highly scattering media by the degree of optical polarization,” Opt. Lett. 21(2), 161–163 (1996). [CrossRef] [PubMed]
