Optical spatial shock waves in nonlocal nonlinear media

Figures & data

Figure 1. Phase chirp $u(x)$ (a,b), and amplitude $\sqrt{\rho (x,z)}$ (c,d) for transverse dimensionality $D=1$ and different values of $z$, as indicated. (a,c) are obtained by Eqs. (18) with $ϵ={10}^{-3}$. (b,d) are simulations of the result of the system (17) with $D=1,\alpha =0,\chi =-1,{\sigma }^2=5$.

Figure reprinted with permission from [29]. Copyright 2007 by the American Physical Society.

Figure 2. Pictorial representation of an energy landscape. When the system is in the proximity of a local maximum it obeys the RHO Hamiltonian, in figure ${\hat{H}}_{RO}$. In the proximity of the minimum the system obeys the Hamiltonian of a harmonic oscillator, in figure ${\hat{H}}_{HO}$. The two Hamiltonians are explicitly written in the two the corresponding text boxes, with the related dynamical systems and the discrete eingenvalues. Insets show the transverse profiles of the respective eigenfunctions, bounded on right-hand side for the harmonic oscillator, unbounded on the left-hand side for the RHO.

Reprinted by permission from Macmillan Publishers Ltd. from [38]. Copyright 2015.

Figure 3. (a) $||{\mathfrak{f}}_n^{-}(x)|{|}^2$ in Eq. (23) for increasing even order $n$; (b) corresponding phase chirps ${\mathrm{\partial }}_x\mathrm{A}\mathrm{r}\mathrm{g}\left[{\mathfrak{f}}_n^{-}(x)\right]$; (c) weights $p_n(0)$ [Eq. (27)] of the GV expansion of a Gaussian wave packet.

Figure reprinted with permission from [37]. Copyright 2015 by the American Physical Society.

Figure 4. (a) Numerical solution of Eq. (19) with $p={10}^4$ and ${\sigma }^2=10$; (b) projection on GVs for increasing order $n$ for $\alpha =0.3$ and $\gamma =8$; continuous lines are from Eq. (19), dots are from Eq. (27); (c) as in panel (b) for $\gamma =24$.

Figure reprinted with permission from [37]. Copyright 2015 by the American Physical Society.

Figure 5. Experimental transverse intensity profiles of an initial Gaussian beam propagating in a thermal medium. Measurements are performed for varying input power $P=\pi W_0^2{I}_0$. Insets show the 2D output patterns.

Figure reprinted with permission from [29]. Copyright 2007 by the American Physical Society.

Figure 6. (a) Experimental setup. Authors of [38,40] collected the transmitted and fluorescence images of the laser beam propagating in RhB samples. Two types of launching lenses L1 were used: a cylindrical and a spherical, for the 1D and 2D experiments, respectively. The top fluorescence image of the propagating beam was collected by a microscope placed above the RhB samples. The second lens is spherical and was used to collect the transverse output profile. (b, c) Top-view intensity distribution as obtained from 2D experiment (b) and numerical simulations (c). Respectively experimental (d) and numerical (e) sections of the images (b) and (c) taken at $z=0.2$ (red), 0.6 (green) and 0.9 mm (blue).

Reprinted by permission from Macmillan Publishers Ltd. from [38]. Copyright 2015.

Figure 7. (a) Observed intensity decay at different laser powers, obtained by slicing along $X\simeq 0.1$mm the top-view intensity distribution the propagation direction (see the yellow line in Figure 6b). (b) Numerically calculated decays in the conditions of panel A. (c) Peak region of the experimental curve at $P=450$mW. The superposition of the first two exponential decays unveils the presence of two GVs, the fundamental state, $n=0$ (slowly decaying) and the first excited state, $n=2$ (fastly decaying). (d) Decay rates vs $P$ for the fundamental state, ${\mathrm{\Gamma }}_0$ (filled circles) and the excited state, ${\mathrm{\Gamma }}_2$, (triangles).

Reprinted by permission from Macmillan Publishers Ltd. from [38]. Copyright 2015.

Figure 8. (a, b) CCD image of the light beam at laser powers $P=2$W and $4$W, respectively; the bottom panels show the normalized intensity profile at the maximum waist along $Y=0$. (c) Analytical solution obtained by Eq. (37) changing Gaussianly the power $P$ in the $y$ direction. (d) As in (c) but for higher powers; the bottom panels show the slice of panel (c) and (d) at $y=0$, i.e., Eq. (37) square modulus for $W=1.5$ and $\gamma \simeq 12$ and $\gamma \simeq 40$, respectively. (e) Log-scale normalized intensity as a function of power, as obtained by slicing along $Y$ a region in panel (b). The slopes of the straight lines give the GV decay rates (${\gamma }_1=-8\pm 0.4$ and ${\gamma }_2=-1.6\pm 0.1$). Their quantized ratio is $5.0\pm 0.4$ as expected from theory [37]. (f) Intensity oscillations for different power values. (g) Measured oscillations period $T$ as a function of power; continuous line is the fit function $T\propto \sqrt[4]{P}$, as expected by the theory; the inset shows the same curve of (g) with $P^{1/4}$ as abscissa axis.

Reprinted with permission from [40]. Copyright 2016 Optical Society of America.

Figure 9. (a, b) Trajectories $x(z)$ and (b, d) phase space $(x,v)$, respectively, with disorder strength $\eta =0$ and $\eta =0.1$. $z$ varies from $z=0$ to $z=3$.

Figure reprinted with permission from [32]. Copyright 2012 by the American Physical Society.

Figure 10. Transverse intensity patterns for different input power $P$ and silica spheres concentration $c$: (a) $P=5mW$, $c=0$w$/$w, (b)$P=400mW$, $c=0$w$/$w, (c) $P=5mW$, $c=0.017$w$/$w, (d) $P=400mW$, $c=0.017$w$/$w, (e) $P=5mW$, $c=0.030$w$/$w, (f) $P=400mW$, $c=0.030$w$/$w. White 1D curves show the measured section of the intensity profiles vs $X$.

Figure reprinted with permission from [32]. Copyright 2012 by the American Physical Society.

Figure 11. Far-field intensity profiles at the output of the silica aerogel for $P_{in}$ ranging from $1$mW to $1$W, and input beam waist $w_0$ ranging from $43\mu$m to 1.4 mm. Images in the second and third rows correspond to the same incident laser power and beam size, but different positions of the incident laser beam.

Reprinted with permission from [35]. Copyright 2014 Optical Society of America.

Figure 12. Output beam intensity transverse profiles, coming out from a $2$mm long M-Cresol/Nylon solution. Input power varies: (a) $P_{in}=2\mu$W, (b) $P_{in}=5$mW, (c) $P_{in}=10$mW, (d) $P_{in}=20$mW.

Reprinted with permission from [36]. Copyright 2014 Optical Society of America.

Figure 13. Output beam waist for varying hemoglobin concentration and input power. (a) Detected beam diameter as function of input power through the hemoglobin solutions for four different concentrations (Hb1-Hb4): $2.4$, $5.1$, $8.6$, and $15.0$ million cells per mL. Nonlinear self-focusing of the beam occurs around $100$mW for high concentrations of hemoglobin, but it subsequently expands into thermal defocusing rings at high powers. (b-e) Output beam transverse intensity profiles for (b) self-trapped beam at high concentration and low power, (c) DSW at high concentration and high power, (d) self-trapped beam at low concentration and low power, (e) DSW at low concentration and high power.

Reprinted by permission from Macmillan Publishers Ltd. from [43]. Copyright 2019.

Figure 14. Experimental setup. A CW laser beam, $\lambda =532$nm, is sent through a 4-f telescope. A ground-glass plate G, placed in the midst of the telescope (on the focus of the first lens), generates a speckle pattern. The incoherent beam impinges the samples, a cylindrical tube filled with a solution of methanol and graphene nanoscale flakes, with waist $W_0=2.3$mm, while the initial coherence length ${\lambda }_c^0$ is controlled by changing the beam size on G. A CDD camera detects the output.

Reprinted by permission from Macmillan Publishers Ltd. from [39]. Copyright 2015.

Figure 15. Experimental observation at short-range regime. (a, b) Experimental beam profiles of the output intensity recorded at power a $P=0.05$W, (b) $P=2.50$W. (c, d) Zooms on details of (b) that evidence the development of shocklets. (e, f) Numerical simulations of NLSE equation, and (g, h) corresponding zooms.

Reprinted by permission from Macmillan Publishers Ltd. from [39]. Copyright 2015.

Figure 16. Experimental observation at long-range regime. (a-c) One-dimensional intensity transverse profiles along $x$ at $y=0$. (d-f) Two-dimensional intensity profiles. The asymmetry in the lower part of the beam is due to convection within the sample. (g-i) Numerical simulations of NLSE. (j-l) Experimental and (m-o) numerical spectrograms: the Z-shaped distortion reveals a dramatic coherence degradation on the annular boundaries of the beam (the coherence length decreases at the shock front), while a significant coherence enhancement occurs in the internal region of the beam. Input beam power: (a, d, g, j, m) $P=0.05$W, (b, e, h, k, n) $P=1.25$W, (c, f, i, l, o) $P=2.50$W.

Reprinted by permission from Macmillan Publishers Ltd. from [39]. Copyright 2015.

Ghofraniha N, Conti C, Ruocco G, et al. Shocks in nonlocal media. Phys Rev Lett. 2007;99:043903.

 PubMed Web of Science ®Google Scholar

Gentilini S, Braidotti MC, Marcucci G, et al. Physical realization of the Glauber quantum oscillator. Sci Rep. 2015;5:15816.

 PubMed Web of Science ®Google Scholar

Gentilini S, Braidotti MC, Marcucci G, et al. Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media. Phys Rev A. 2015;92:023801.

 Web of Science ®Google Scholar

Braidotti MC, Gentilini S, Conti C. Gamow vectors explain the shock profile. Opt Express. 2016;24:21963.

 PubMed Web of Science ®Google Scholar

Ghofraniha N, Gentilini S, Folli V, et al. Shock waves in disordered media. Phys Rev Lett. 2012;109:243902.

 PubMed Web of Science ®Google Scholar

Gentilini S, Ghajeri F, Ghofraniha N, et al. Optical shock waves in silica aerogel. Opt Express. 2014;22:1667.

 PubMed Web of Science ®Google Scholar

Smith V, Cala P, Man W, et al. Dark soliton attraction and optical spatial shock waves observed in m-cresol/nylon solutions. In: CLEO: 2014. Optical Society of America; 2014. p. FW3D.1. Washington, D.C. (USA): OSA Publishing.

 Google Scholar

Gautam R, Xiang Y, Lamstein J, et al. Optical force-induced nonlinearity and self-guiding of light in human red blood cell suspensions. Light: Sc Appl. 2019;8:31.

 PubMed Web of Science ®Google Scholar

Xu G, Vocke D, Faccio D, et al. From coherent shocklets to giant collective incoherent shock waves in nonlocal turbulent flows. Nat Commun. 2015;6:8131.

 PubMed Web of Science ®Google Scholar