November 2024
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Reverse saturation of absorption is a relatively rare phenomenon in light–matter interaction, as it requires a few conditions to be fulfilled. We observe that ruby exhibits a very strong reverse saturation of absorption at 473 nm. Furthermore, we measure the group velocity of a modulated laser beam in ruby and observe that the peaks of the pulses appear more than a hundred microseconds earlier than the reference signal. A theoretical model based on coherent population oscillation would suggest a fast-light effect with an extremely large and negative group index of −(1.7 ± 0.1) × 10⁶ in consistency with the observed temporal advancement. We propose that this pulse advancement can also be described by time-dependent absorption of ruby. Our study helps to understand the nature of the fast- and slow-light effects in transition-metal-doped crystals such as ruby and alexandrite with potential applications in optical memories and delay lines.
![Quantitative analysis of the shadow contrast. (A) Direct image of the illuminating light transmitted through the ruby for a 15 W object laser beam. The vertical line of lower brightness is the shadow of the green laser beam. (B) For the 1 mm wide ${y}$ y -region in (A) indicated by the orange lines, the experimental (dots, error bars are the standard error from 21 trials) and theoretical (solid blue line and shaded error bands from the model in the Theoretical Simulations) relative transmittance $T$ T through the ruby cube of the blue illuminating light is plotted for six object-laser optical-powers. For clarity, the six datasets are separated vertically by 0.2 alongside their respective power $P$ P of the green object laser and contrast ${\cal C}$ C [(Eq. (1)]. For each, the horizontal black solid line at the left marks a transmittance of $T = 1$ T = 1 relative to the transmittance when the object laser is absent. The solid red line is the Gaussian fit of the measured laser beam spatial profile, which shows that the shadow shape is the same as the object laser spatial profile. (C) Peak contrast (experiment: orange circles with plotted but not visible error bars in both ${\cal C}$ C and $P$ P ; theory: green triangles with error bars) for the six power values in (B) along with a linear fit with zero-intercept.](https://www.researchgate.net/publication/384533623/figure/fig3/AS:11431281290462675@1731611270354/Quantitative-analysis-of-the-shadow-contrast-ADirect-image-of-the-illuminating-light_Q320.jpg)
![520 mW continuous-wave laser beam at 473 nm is focused using a 100 mm focal length plano–convex lens ${{\rm L}_1}$ L 1 to a spot size of 20 µm onto the input face of rotating ruby rod. The rod spins around its axis driven by a stepper motor. The laser beam at the output of the crystal is imaged onto a CCD camera with unity magnification using a 4-f system consisting of two lenses ${{\rm L}_2}$ L 2 and ${{\rm L}_3}$ L 3 of focal length $f = 150\;{\rm mm} $ f = 150 m m . The CCD camera captures the beam, with a frame rate of 1000 fps, as the stepper motor is rotated at various speeds. An ND filter is placed between the dielectric mirror and lens 2, ${{\rm L}_2}$ L 2 , for nonlinear measurements, and between ${{\rm L}_1}$ L 1 and the ruby for linear measurements. The CCD camera images at different $z$ z positions using a translation stage. Measurements are taken at $z = 0$ z = 0 , ${\rm z} = {0.762}$ z = 0.762 , and $z = 1.524\;{\rm cm} $ z = 1.524 c m to measure the transverse shift, as well as the output angle of the beam as it exits the crystal. The fluorescence filter $FF$ F F (high transmission near 473 nm) is used to minimize fluorescence. The dielectric mirror $DM$ D M is used as a neutral density filter with low absorption to limit the beam intensity for high-power tests, while also minimizing image distortions due to aberrations induced by thermal nonlinearities in a standard neutral density filter. Input beam power was controlled by a half-wave plate and a polarizing beam splitter before the ruby crystal. M, mirror; HWP, half-wave plate; PBS, polarizing beam splitter; BD, beam dump; ${{\rm L}_1}$ L 1 , plano–convex lens [ ${\rm f} = {100}\;{\rm mm}$ f = 100 m m ]; ${{\rm L}_2}$ L 2 , plano–convex lens [ ${\rm f} = {150}\;{\rm mm}$ f = 150 m m ]; ${{\rm L}_3}$ L 3 , plano-convex lens [ ${\rm f} = {150}\;{\rm mm}$ f = 150 m m ]; FF, fluorescence filter; DM, dielectric mirror; ND, neutral density filter [O.D. 1]; CCD, charge-coupled device.](https://www.researchgate.net/publication/369774624/figure/fig2/AS:11431281253270155@1718841281677/520mW-continuous-wave-laser-beam-at-473nm-is-focused-using-a-100mm-focal-length_Q320.jpg)
![(a) Experimentally measured and (b) simulated COI trajectories in the nonlinear regime (input laser power of 100 mW). COI trajectories of o- and e- beams [schematically shown in Fig. 1(d)] are plotted for an input laser power of 100 mW, considered as the nonlinear regime for different rotational speeds ( $\Omega$ Ω ) in units of deg/s. At low speeds, o- and e-beams couple to each other, causing significant variation in the traces of the COI upon rotation. Therefore, these patterns are different from the figure-eight-like patterns seen in the linear regime. At high speeds, the deviations from a figure-eight pattern start to average out. All that remains in the high-speed limit is that the figure-eight patterns are shifted from one another for positive and negative rotation speeds as a result of nonlinear deflection.](https://www.researchgate.net/publication/369774624/figure/fig4/AS:11431281253190456@1718841282266/aExperimentally-measured-and-bsimulated-COI-trajectories-in-the-nonlinear-regime_Q320.jpg)
![FIG. 1. (a) Schematic drawing of a subwavelength atomic cloud composed of N atoms at random positions for different realizations. N R is the number of realizations. The diameter D of the atomic cloud is smaller than the wavelength. (b) The induced electric and magnetic multipole moments of each realization obtained from multipole expansion of the induced current [using Eqs. (B1) in the Appendix]. (c) Coherent electric and magnetic polarizabilities as a function of frequency detuning obtained from the induced multipole moments shown in panel (b). (d) Coherent and incoherent scattering cross sections and contribution of each multipole moment obtained from Eq. (4). The ensemble-averages are obtained from 10 000 realizations of the atomic cloud with radius R = 0.2λ a composed of N = 25 atoms.](https://www.researchgate.net/publication/352598108/figure/fig1/AS:1037356958236673@1624336580175/a-Schematic-drawing-of-a-subwavelength-atomic-cloud-composed-of-N-atoms-at-random_Q320.jpg)
![FIG. 2. Coherent and incoherent electric and magnetic polarizabilities obtained from the induced multipole moments and defined as α j i = α j i + δα j i , where i ∈ {ED, MD}, j ∈ {D, OD}. ED (MD) denotes the electric (magnetic) dipole and D (OD) represents diagonal (off-diagonal) components of the polarizability tensor [see Eq. (5)]. (a, b) Electric dipole polarizabilities as a function of frequency detuning. Note that α D ED = 0, whereas α OD ED = 0. The thickness of the shaded lines show the fluctuating components of the polarizabilities. (c) Relation between coherent and incoherent polarizabilities obtained from the conservation of energy, see the left and right sides of Eq. (6). [(d)-(f)] Same as panels [(a)-(c)] for the magnetic dipole polarizability of the atomic cloud.](https://www.researchgate.net/publication/352598108/figure/fig2/AS:1037356958228480@1624336580215/Coherent-and-incoherent-electric-and-magnetic-polarizabilities-obtained-from-the-induced_Q320.jpg)
![FIG. 3. Selective excitation of ED and MQ moments using four TE polarized plane waves. (a) Schematic drawing of a subwavelength atomic cloud when illuminated by four plane waves. φ is the relative phase between the plane waves. (b) Normalized ensemble-averaged scattering cross section of the atomic cloud as a function of frequency detuning and the relative phase φ. The red and green dashed lines depict selective excitation of pure ED and MQ moments, respectively. [(c)-(e)] Normalized ensemble-averaged total scattering cross sections as a function of frequency detuning for different phase (c) φ = 2mπ , (d) φ = (2m + 1)π , and (e) φ = (2m + 1)π/2. The ensemble-averages are obtained from 10 000 realizations of the atomic cloud.](https://www.researchgate.net/publication/352598108/figure/fig3/AS:1037356958228481@1624336580243/Selective-excitation-of-ED-and-MQ-moments-using-four-TE-polarized-plane-waves-a_Q320.jpg)
![FIG. 4. Selective excitation of MD and EQ moments using four TM polarized plane waves. (a) Schematic drawing of a subwavelength atomic cloud when illuminated by four plane waves. φ is the relative phase between the plane waves. (b) Normalized ensemble-averaged scattering cross section of the atomic cloud as a function of frequency detuning and the relative phase between input plane waves. The orange and purple dashed lines depict selective excitation of pure MD and EQ moments, respectively. [(c)-(e)] Normalized ensemble-averaged scattering cross sections as a function of frequency detuning for different phase (c) φ = 2mπ , (d) φ = (2m + 1)π , and (e) φ = (2m + 1)π/2. The ensemble-averages are obtained from 10 000 realizations of the atomic cloud.](https://www.researchgate.net/publication/352598108/figure/fig4/AS:1037356958224385@1624336580288/Selective-excitation-of-MD-and-EQ-moments-using-four-TM-polarized-plane-waves-a_Q320.jpg)
