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Abstract
The scattering of an ultrashort laser pulse by an air bubble in water is investigated by means of the Lorenz-Mie theory and the Debye expansion. A 70 fs, 800 nm pulse is considered as a plane wave with a Gaussian temporal envelope. The transient response is treated with the theory derived from Gouesbet and Gréhan [Part. Part. Syst. Charact. 17, 213–224 (2000)], taking now into account chromatic dispersion and absorption of water. It is observed that contrary to the case of water droplet in air, the Debye modes p ≥ 1 start their transient scattering at the same time and the same angle (≈90°) and for a large size parameter, they differentiate as time elapses. A parametric study on the size parameter and the spatial extension of the pulse is performed to identify regimes where the different Debye mode are distinguishable in time. Dependence on the scattering angle is also treated. Finally, by considering pulse chirp, it is shown that the laser/bubble distance has an influence on the separability of modes p = 0 and p = 1.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of ultrashort laser pulses, below the pico-second, allows to generate light wave packets of a few oscillations and to treat light matter interaction as transient. In the field of the Mie scattering, referred to as Lorenz-Mie Theory (LMT) herein, the transient illumination by ultrashort laser pulses was pioneered by Chowdhury et al. [1] and Shifrin & Zolotoz [2]. Later, Gouesbet and Gréhan [3] showed that the two aforementioned approaches were equivalent. They showed that a short pulse scattered by a liquid droplet in air allows to separate in time the different scattering modes, thus avoiding their interferences and providing a smoother scattered intensity with respect to the direction of radiation [4]. Further application of their theory to water droplets [5] allowed to develop simulation methods to predict the global scattering of a laser pulse by a cloud of droplets [6,7]. More recently, Lock & Laven [8,9] investigated the two first Debye modes of a short pulse scattered by a droplet. Their representation of the intensity in a map versus the time and the scattered angle allowed to separate in time different phenomena, thus providing better measurements for the velocity of the surface wave, and showing that glory originates from the interferences of individual surface waves. They also showed the slow transition of the zeroth Debye mode from diffraction to surface wave radiation when the scattering angle increases. Świrniak & Mroczka [10] showed for a homogeneous cylinder and an optical fiber that time diagrams provide more tangible insights on the scattering process and the particle’s morphology compared to the Discrete Fourier Transform.
An important parameter in the LMT for continuous illumination is the size parameter $x=\pi d_p n / \lambda _0$ that compares the diameter $d_p$ of the scattering particle to the carrier wavelength $\lambda _0$ of the incident ray, $n$ being the refractive index of the propagation medium. With a transient illumination due to an ultrashort pulse, in addition to $x$, it was shown in [5] that the size of the particle must be compared to the spatial extent $\Delta s = c_0 \Delta t / n$ of the pulse, in the propagation direction. The terms $c_0$ and $\Delta t$ are respectively the speed of light in vacuum and the pulse duration defined at "Full With at Half Maximum" (FWHM). The parameter $\delta = d_p / \Delta s$ links both scales: $x= \delta \Delta t \omega _0/2$ where $\omega _0$ is the carrier angular frequency of the pulse. To our knowledge, the case of an ultrashort laser pulse scattered by air bubbles in water has not been investigated theoretically, numerically, or experimentally. It differs from the case of water droplets in air, the refractive index is now lower in the bubble and due to dispersive nature of water, beam chirping will affect propagation in water. Another important difference between the scattering of droplets and bubbles, is the presence of the critical scattering region for bubbles. For an incident angle greater than the critical angle $\theta _C$ (e.g. $\theta _C\approx 48.59^{\circ }$ in the case of air bubbles in water), total reflection occurs and triggers several remarkable but extremely complex phenomena. These are sharp intensity gradients, strong interferences between high-order rays, tunneling effect and the Goos-Hänchen effect. Please refer to [11] for an extensive collection of references on theses effects. Although these phenomena are complex in nature and difficult to model in Geometrical Optics Approximation, they are inherently taken into account in the LMT, and hence they will not be investigated in this study. Also, the laser pulse intensity is assumed below the threshold for nonlinear effects, such as Kerr effect, to be significant.
This manuscript is a first step in the direction of the transient of spherical air bubble scattering; it presents simulations of an ultrashort ($\Delta t = 70$ fs, $\lambda _0=800$ nm) laser plane pulse propagating in water and scattered by a single air bubble. Beam chirping and absorption are also treated. The size parameter is varied between 50 and 5000; $x=5000$ is approximately the upper bound for bubbles to remain spherical [12]. We followed the Generalized Lorenz-Mie Theory (GLMT) by Gouesbet and Gréhan [3], which describes the transient of Lorenz-Mie scattering when the incident wave is not necessarily planar but of arbitrary shape. Even though the present study considers an illumination by a planar wave, we selected the GLMT because of its clarity and because the illumination by a Gaussian beam will be investigated in the next steps.
The body of the paper is organized as follows. The theory by Gouesbet and Gréhan is briefly recalled and adapted to dispersive medium in Section 2. The comparison of a water droplet with an air bubble is shown in Section 3, and the influence of the size parameter is tackled in Section 4. The chirping effect and the influence of the propagation distance in water are investigated in Section 5.
2. Methodology
The results of the LMT and Debye expansion for a continuous illumination were computed according to the methods presented in [13] and [14], respectively. The scattering of a continuous plane wave was validated for a droplet in air and a bubble in water on the results of [15] for LMT. The Debye series were validated for droplet in air with real and complex refraction indices on the results from [14]. Please note that even though the Debye expansion fails to accurately predict the scattering of bubbles in the immediate vicinity of the critical scattering angle, it is correct outside the critical region. Hence, we will show the five first terms of the Debye expansion in the rest of the paper for illustrative purpose. The methodology to calculate the scattering of an ultrashort pulse by a spherical particle is presented following [3].
Let $\boldsymbol{\psi }$ be the incident electric (E) or magnetic (H) field for a pulsed plane wave:
where the $\boldsymbol{\psi }_0$ is the amplitude of the incident wave, $\omega _0$ is the angular frequency of the carrier and $g(\tau )$ is the temporal envelope of the pulse. The variable $\tau$ is the argument of the propagating wave $\tau = t - z/c$ where $t$, $z$ and $c$ are the time, the propagation direction and the phase velocity in the medium. To compute the time-dependent scattering, the theory of Gouesbet and Gréhan [3] consists in convoluting in the frequency domain the scattered wave by an incident monochromatic continuous wave ($\boldsymbol{\psi }^{cw}$) with the Fourier Transform of the pulse envelop. It is briefly recalled in the following for a non-dispersive medium. The scattered field is given by:In a dispersive medium, the phase velocity depends on the frequency, $c(\omega )$, which means that the argument of the exponential that models the propagation $\tau (\omega ) = t - z/(c(\omega ))$ depends on the frequency $\omega$, and thus, cannot represents the pulse in the temporal domain. Hence the notation $g(t,z)$ is preferred to $g(\tau )$. Because $c(\omega )$ has a priori no analytical form, the envelop of the pulse cannot be expressed analytically in the temporal form and must be expressed with the inverse Fourier Transform of the frequency envelop. The incident plane wave (Eq. (1)) thus becomes:
To facilitate the numerical integration of Eq. (8), it is expressed as in [3]:
3. Droplet/bubble comparison
In this part we compare the scattering of the same pulse by a water droplet in air with an air bubble in water of the same size parameter. One of the fundamental difference between these two cases is that the laser pulse propagates more slowly in the particle than in the surrounding medium for the droplet case. To the contrary for the bubble in water, the light pulse is faster in the particle. This leads to major differences in the temporal regime as shown in the following. We neglect the chirping effect in water and we consider a wave where the electric field is in the incident plane (TM wave). The operating conditions are given in Table 1, where the value of the wavelength is expressed in vaccum. As we decided to keep the size parameter $x$ constant, it was necessary to set the droplet and bubble diameters to 95 and 127 µm, respectively. Also, as the pulse travels in the propagating medium at different velocities, it is necessary to express time and space in reduced coordinates. We define $z^{*} = z/r_p$ the axial coordinate normalized by the particle radius. For instance, $z^{*}=-1, 0$ and $1$ correspond to the "back", the center and the "front" of the particle surface, respectively. Concerning the time, we define $t^{*} = t c_0/(n_{\textrm {pm}} r_p)$ the reduced time normalized by the time needed for the light to travel a distance equal to the particle radius. Hence $t^{*}$ can be regarded as the position of the pulse in the reference frame of the particle. For instance, $t^{*}=-1, 0$ and 1 correspond to the peak of the pulse at the "back" particle surface, at the particle center and the "front" particle surface, respectively. Thanks to this representation, the peak of the pulses is always located at $t^{*}=z^{*}$, and thus is at the same position in both cases. The simulation is run from $t^{*}=-1-4/\delta$ when the pulse is completely in front of the particle, to $t^{*}=\max (12,1+4/\delta )$ when the pulse is completely outside the particle and most of the internal reflections are completed.
The evolution is shown in Figs. 2 and 3. The top part depicts the location of the pulse with respect to the particle (droplet or bubble) where the vertical lines represent the boundaries of the particle, while the polar plot on the bottom part shows the instantaneous scattered intensity normalized by the incident intensity ($I_1/I_0$) for a droplet in air (top hemisphere) and a bubble in water (bottom hemisphere).
When the pulse reaches the back of the particle ($t^{*}=-1$) the wave is reflected (mode 0 in Debye expansion) on the surface of the particle and propagate backwards ($\theta \approx 180^{\circ }$). As the pulse moves forward ($t^{*}= -0.37$), only the mode $p=0$ (reflection plus diffraction) is visible for the droplet because as the light is slower in water, the $p\ge 1$ modes are still propagating inside the droplet. Note the singularity at $\theta =74^{\circ }$ which corresponds to the Brewster’s angle $\theta _B\approx 53^{\circ }$ for air/water. For the bubble, as light travels faster in air, the refraction/internal reflection modes ($p\ge$1) are already scattered at $\theta \approx 70^{\circ }$, thus leading to interferences predicted by the LMT. Note that at this position ($t^{*}= -0.37$) the maximum of the scattered intensity is in the vicinity of the critical scattering angle, and hence the Debye modes are not rigorously accurate for the case of the bubble. The effect of the Brewster’s angle for water/air ($\theta _B\approx 37^{\circ }$) is visible at $\theta =106^{\circ }$ where the reflected intensity is zero. At $t^{*}= 0.27$ the droplet scatters only the mode $p=0$ in the sector 20-60$^{\circ }$ whereas the bubble already scatters $p=1$ in the full forward mode ($\theta =0^{\circ }$) although mode $p=0$ has its maximum at only $\theta =45^{\circ }$. This highlights that the faster photons in the full forward scattering mode are the ones passing by the center of the bubble.
Fig. 2. Time response of a $x=500$ and $\delta =6.07$ particle. Top: position of the pulse with respect to the particle. Bottom: scattered intensity from LMT (black plain line) and Debye expansion (colored dashed line) for a droplet in air (top hemisphere) and bubble in water (bottom hemisphere). See Visualization 1.
Table 1. Operating parameters for the droplet/bubble comparison.
Figure 3 focuses on times when the pulse has passed the particle. At $t^{*}=1$ the peak of the pulse reaches the virtual detector on the front surface of the particle ($\theta =0^{\circ }$). The intensity is maximum in both cases. For larger $t^{*}$, the modes $p\ge 1$ are quickly "swept" along the meridians of the bubble whereas they are much stretched in time for the droplet. At $t^{*}=2$ the $p=2$ mode of the bubble reaches $\theta = 180^{\circ }$ whereas mode $p=1$ of the droplet only starts its sweep at $\theta = 0^{\circ }$ towards larger $\theta$. At $t^{*}=3$, mode $p=3$ starts to sweep forwards while mode p=1 for the droplet is halfway to the backward scattering ($\theta = 90^{\circ }$). For $t^{*}>3$ no modes are visible for the bubble whereas the droplet will scatter the modes $p\ge 2$.
To analyze the temporal response of all scattering angles versus the time, the maps of the total intensity $I_t$ versus $t^{*}$ and $\theta$ are shown in Fig. 4 in a grey scale arbitrary limited between $10^{-6}$ and $10^{2}$. For a randomly polarized electromagnetic wave, the total intensity is the arithmetic mean of the intensities of the TM and TE waves: $I_t=(I_1 + I_2)/2$. Arbitrary iso-values of Debye modes are superimposed to identify the peaks. These are $5 \ 10^{-6}$ and $0.5$ of the maximum value for modes $p=0$ and $p=1$, respectively. For larger modes it is $0.01$ of the maximum value. The horizontal strips at roughly constant time correspond to a Debye mode sweeping along the meridians of the particle. The general picture is that modes are well separated in time and in angles for the liquid droplet, whereas they are concentrated in the same $(t^{*},\theta )$ region for the air bubble. In full forward scattering ($\theta =0^{\circ }$), the largest peak is attributed to the reflection/diffraction mode ($p=0$). It arrives first for the droplet and last for the bubble. For the bubble, all the modes $p\ge 1$ arrive almost together, and by zooming in the region where modes are large, one could see that there exists a location $(t^{*},\theta )$ where the mode $p=1$ could be discernible from $p=0$. Also after $t^{*}>5$, all the scattering modes of the bubble have already radiated.
Fig. 4. Map of intensity versus the scattering angle and the time for the water droplet (left) and the air bubble (rigth) for $x=500$ and $\delta =6.07$.
In conclusion, due to the fact that the light travel faster in the bubble, all refraction/internal reflection modes are scattered almost together, whereas they are clearly separated in time in the case of a droplet. It is to be highlighted that for the bubble case, the photons traveling through the bubble without being deflected (mode $p=1$) are faster than the photons travelling in the propagation medium (i.e. water) only. This is opposite to the scattering by a droplet, where the photons crossing the particle are slower than the ones traveling in air only.
4. Influence of bubble diameter
In this part we focus on a bubble only and we investigate the influence of the size parameter $x$ with four different values: 5, 50, 500 and 5000; the corresponding values for $\delta$ are 0.061, 0.61, 6.1 and 61, respectively. Other parameters are the same as in Section 3, especially beam chirping is neglected. First, the mean scattered total intensity is calculated as $\langle I_t(\theta ) \rangle = \int _{t_{\textrm {ini}}}^{t_{\textrm {end}}} I_t(t,\theta ) \, \mathrm d t / \Delta t_{cw}$ where $\Delta t_{cw}$ is the duration of a square pulse of the same energy and the same maximum power as the Gaussian pulse, such as $\Delta t_{cw} \cdot \max (g^{2}(t,z)) = \int _{t_{\textrm {ini}}}^{t_{\textrm {end}}} g^{2}(t,z) \, \mathrm d t$, see Fig. 5.
Fig. 5. Illustration of $\Delta t$ and $\Delta t_{cw}$ for a pulse of $\Delta t=70$ fs. Note that the square of the pulse is displayed.
In this case, $\langle I_t(\theta ) \rangle$ can be compared to the intensity scattered by the illumination with a Continuous Monochromatic Wave $I_{\textrm {CMW}}(\theta )$. They are depicted in Fig. 6 (top hemisphere) versus their CMW counterpart (bottom hemisphere) for all investigated $x$.
Fig. 6. Scattered intensity for bubbles of different $x$. For each plot, the top hemisphere is the time-average of the response to the pulse, and the bottom hemisphere is the scattered intensity for a continuous monochromatic wave (CMW) at $\nu _0$.
For $x=5$, $\langle I_t(\theta ) \rangle$ is almost identical to $I_{\textrm {CMW}}(\theta )$, which partially validates the methodology and suggests that when $x \le 5$ the temporal effects of an ultrashort illumination are negligible on the scattering. At $x=50$, the interference fringes in the back-scattering region ($\theta \approx 180^{\circ }$) are reduced for the total scattering (LMT) only. At $x=500$, these fringes in the back-scattering region are reduced for all modes, as also found in the case of a water droplet [4]. The fringes in the forward scattering are also somewhat less intense for the overall signal (LMT), and the regular lobes for $p=2$ and $p=4$ have disappeared. For $x=5000$, the effects mentioned for $x=500$ are stronger, with almost no fringes anymore for any mode.
The temporal evolution of the scattered intensity $I_1$ is shown for $x=50$ and 5000 at different snapshots $t^{*}$ in Fig. 7; the pulse envelope is also depicted versus $z^{*}.$ Note that the pulse is the same in both cases, but as $z^{*}$ is normalized by the bubble diameter, the pulse for $x=5000$ looks like a Dirac delta function (large parameter $\delta$). For $x=50$, the wavelength $\lambda _0$ is only one order of magnitude lower than the bubble diameter and hence the pulse (that is several time of magnitude wider than $\lambda _0$) is significantly wider than the bubble. This results in a very diffuse illumination of the bubble, thus impacting all the angle almost simultaneously, leading to the typical interference fringes for each mode individually, and the overall scattering response as predicted by the LMT. Note that the lobes from the LMT between $\theta =0$ and $45^{\circ }$ are visible as in Fig. 6, which corroborates the fact that temporal effects of a short pulse are negligible for $x \le 50$.
Fig. 7. Time response of a bubble for $(x,\delta ) = (50,0.61)$ (top hemisphere) and $(5000,61)$ (bottom hemisphere). See Visualization 2.
For $x=5000$, the pulse acts as a scan of the different angles of incidence. For instance at $t^{*}=-1$, it triggers the reflective mode around $\theta =180^{\circ }$ while at $t^{*}=-0.5$, it impacts the bubble at $\theta \approx 120^{\circ }$. Hence, as the different incidence angles are separated in time, there are less interferences. It is well illustrated here that the angle of incidence of the max of the pulse $\theta _{\max }$ is related to $t^{*}$ with $t^{*} = \cos \theta _{\max }$. Linearization for $\delta \gg 1$ shows that the bandwidth $\Delta \theta$ on which angles are scanned varies from $2/\sqrt \delta$ (at $\theta \approx 180^{\circ }$) to $2/\delta$ (at $\theta \approx 90^{\circ }$), e.g. from 14$^{\circ }$ to 1.9$^{\circ }$ for $\delta =60.67$.
The maps of the total intensity $I_t$ versus $(t^{*},\theta )$ are shown in Fig. 8 for $x=500$ and 5000. Only the modes $p=0$ to 2 are displayed at the iso-value of 1 for the sake of clarity. The meridian-sweeping of the modes are visible with different "strips" across the $\theta$ axis. For $x=5000$, as $\delta \gg 1$, the scanning effect of the pulse is visible with very well defined strips. The differentiation of the Debye modes in three groups in near-forward scattering is well visible when the first strip coming from $180^{\circ }$ splits in three distincts "sub-strips" corresponding to the modes $p=1$, $p\ge 2$ and $p=0$. The mode $p=0$ can be distinguished for $\theta \lessapprox 30^{\circ }$ and the mode $p=1$ can be distinguished from the modes $p\ge 2$ for $\theta <10^{\circ }$. Light starts coming out of the bubbles for $t^{*}\ge 0.5$ with peak of activity between $t^{*} = 0.5-1$.
Fig. 8. Map of intensity versus the scattering angle and the time for $(x,\delta ) = (500,6.1)$ (left) and $(5000,61)$ (right).
To conclude this part, it was seen that for $x\le 50$ and $\delta <1$, the temporal effects are almost not visible on the scattering properties of the bubble. For $x\ge 500$ and $\delta >1$, the modes $p=0$ and $p=1$ can be separated both in time and intensity. One could argue that the time response of the scattering process could be driven not only by $x$ but also by $\delta =d_p / \Delta s$, i.e. the ratio of the bubble diameter to the pulse width. It is investigated in the next part.
5. Pulse chirp - propagation in a dispersive medium
The pulse chirp due to chromatic dispersion is taken into account. The refractive index is treated as a complex function of the wavelength $n(\lambda ) = n_R(\lambda ) - i \, n_I(\lambda )$. The real and imaginary parts correspond to dispersion and extinction, respectively. The real part $n_R$ is modeled by expressing the molar refraction of the Lorentz-Lorenz function $LL = (n_R^{2}-1)/[(n_R^{2}+2)\rho ]$ with a rational function depending on temperature and on $1/\lambda ^{2}$ [16,17]. Originally, the LL function is used to relate the refractive index to the polarizability of a medium. It was derived by Lorenz in 1869 in the field of optics and by Lorentz in 1878 for electromagnetism [18]. The square of the index in the LL function comes originally from expressing the speed of light in a medium with the angular frequency and the magnitude of the wave vector as $c = \omega / \| \textbf{k} \|$ and substituting it in the wave equation [19]. Note that, even if the LL function can be used to describe a complex refraction index, the correlations from [16,17] focus on the real part. Hence, the imaginary part $n_I$ is calculated by a polynomial of the third order in $\lambda$ proposed by Mesenbrink [20]. The refractive index versus $\lambda$ ranging from 400 to 1200 nm is shown in Fig. 9.
Fig. 9. Refraction index of water. Real part (left) from [17] at a temperature of $20^{\circ }$C, and imaginary part (right) from [20]
The envelop of a pulse of $\Delta t=70$ fs and $\lambda _0=800$ nm is simulated after a propagation over a distance $L_0$ of 1, 10 and 100 mm in three types of medium: (i) a non-dispersive and non-attenuating medium ($n=n_R(\lambda _0$)), (ii) a dispersive and non-attenuating medium ($n=n_R(\lambda$)) and (iii) a dispersive and attenuating medium ($n=n_R(\lambda ) - i \, n_I(\lambda$)). The results are shown in Fig. 10, where the pulse length $\Delta s$ is defined as the total pulse length at half the maximum, in the same manner as the FWHM. For a $L_0$ of 1 and 10 mm, the signal is weakly deformed with a $\Delta s$ almost constant whereas at 100 mm the pulse is strongly stretched in space. The delay between the propagation in dispersive and non-dispersive media comes classically from the expression of the group velocity $v_g$:
Fig. 10. Dispersion of a pulse of $\Delta t=70$ fs and $\lambda _0=800$ nm in water at $20^{\circ }$ at 1, 10, and 100 mm from the source. In the legend, dispersion and extinction are abbreviated to "disp." and "ext.".
The scattering of a 70 fs laser pulse is simulated for an air bubble in water of size parameter $x=500$ at different distances $L_0$ of 1, 10 and 100 mm, leading to a parameter $\delta$ of 6.92, 6.60, 2.10, respectively. Since $x$ is kept unchanged, the different scattering regimes in this case are solely due to the pulse width and not to the wavelength. The time response of the scattering is given in polar plot in Fig. 11 for different $t^{*}$ and $L_0=1$ and 100 mm (top and bottom hemisphere, respectively). The effect of the dispersion of the pulse width is clearly visible. For $L_0=1$ mm, the pulse is thinner than the bubble and hence it acts a scan "in time" of the incidence angles. Note that in this case, the pulse is not as sharp as the one for $x=5000$ in Fig. 7. For $L_0=100$ mm, the pulse is wider than the bubble and most of the incidence angles scatter at the same time, leading to more interferences than for $L_0=1$ mm. This suggests that $\delta$ and the beam chirping have an influence on the time separation of the different modes.
Fig. 11. Time response of a bubble located at 1 mm (top hemisphere) and 100 mm (bottom hemisphere) from the source. See Visualization 3.
The time signals of $I_t$ at $\theta = 10^{\circ }$ versus $t^{*}$ is given in Fig. 12 for the three distances. Figure 12 corresponds to the signal over a vertical line in the map of intensity versus $t^{*}$ and $\theta$ (e.g. Fig. 8). The angle of $10^{\circ }$ was selected because the modes $p=0$ and $p=1$ are more likely to be distinguished in near-forward scattering. As the chirping effect stretches the incident wave in space and time, the scattered intensity follows the same trends, leading to wider peaks in the time diagram. This stretching in time and space leads to a decrease of the energy concentration in space, and hence the peaks have a lower intensity when increasing $L_0$, thus reducing contrasts on a detector. Also, as the illumination is longer for larger $L_0$, it produces more interferences that can be misinterpreted as troughs and would lead to false peak detection. For instance, the trough between the two peaks at $t^{*}\approx 1$ for $L_0=100$ mm is due to interference because the modes $p=0$ and $p=1$ are larger than the total signal predicted by the LMT. This trough leads to two peaks which could be taken as the peaks of modes $p=0$ and $p=1$. However, it is not the case because the peak of mode $p=0$ is located within the first lobe from the LMT, and this could lead to erroneous interpretation. Another general effect is that, contrary to the influence of $x$, the time delay between the peaks is kept unchanged. The main conclusion from Fig. 12 is that a bubble which is sufficiently large to separate modes at a small distance to the laser, could become too small when distance increases.
Finally, the time-averages of the scattered intensities for the three distance are identical and hence they are not reported here. This confirms that the $\delta$ parameter, in these investigated conditions, does not influence of the time-averaged scattering regimes but only affects the temporal evolution.
6. Conclusion
We applied the Lorenz-Mie theory and Debye expansion to investigate the scattering of an ultrashort laser pulse (70 fs, 800 nm) by a spherical air bubble in water. In near-forward scattering, it was found that contrary to the case of a water droplet in the air where the modes are clearly separated in time, all internal modes ($p \ge 1$) mostly overlap in time for a size parameter $x\le 50$. When $x$ increases, the modes can be distinguished in time. They are separated in three groups of mode $p=1$, and $p\ge 2$ and $p=0$, in order of appearance in the time signal. The most favorable angle to distinguish these groups is around $10^{\circ }$. For large $x$ and $\delta$, the short laser pulse acts as a scan of the incident angle starting at the back of the particle ($180^{\circ }$) and sweeping towards its front, thus reducing the occurrence of interferences. Pulse chirp stretches the scattered intensity in time, which reduces the separation between the modes $p=0$ and $p=1$. However, it does not affect the time-averaged scattering cross section. This is a first study and for a given experimental configuration, the spatial stretch $\Delta s$ need to be optimized as a function of the pulse length, $\Delta t$, and the carrier frequency, $\omega _0$. Furthermore, an extensive parametric study on the effect of $x$ and $\delta$ is necessary to specify their role on temporal and time-averaged scattering.
Funding
Office of Naval Research (N000141712616); U.S. Department of Energy (DE-NE0008747).
Acknowledgments
The authors acknowledge the support of the US Office of Naval Research (N000141712616) under the supervision of Drs. Thomas Fu and Woei-Min Lin, and the US Department of Energy (DE-NE0008747).
Disclosures
The authors declare no conflicts of interest.
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