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Modeling beam propagation in a moving nonlinear medium
Abstract
Fully describing light propagation in a rotating, anisotropic medium with thermal nonlinearity requires modeling the interplay between nonlinear refraction, birefringence, and the nonlinear group index. Incorporating these factors into a generalized coupled nonlinear Schrödinger equation and fitting them to recent experimental results reveals two key relationships: the photon drag effect can have a nonlinear component that is dependent on the motion of the medium, and the temporal dynamics of the moving birefringent nonlinear medium create distorted figure-eight-like transverse trajectories at the output. The beam trajectory can be accurately modeled with a full understanding of the propagation effects. Efficiently modeling these effects and accurately predicting the beam's output position has implications for optimizing applications in velocimetry and beam steering. Understanding the roles of competitive nonlinearities gives insight into the creation or suppression of nonlinear phenomena like self-action effects.
... However, the RSA spectral region is expected to span tens of nanometers [27]. In addition, the strong RSA in ruby allows one to study negative photon drag and other phenomena in a moving nonlinear medium [26,28], as well as to observe the shadow of a laser beam [29]. ...
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.
Light propagating in a moving medium is subject to light drag. While the light drag effect due to the linear refractive index is often negligibly small, the light drag can be enhanced in materials with a large group index. Here we show that the nonlinear refractive index can also play a crucial role in the propagation of light in moving media and results in a beam deflection. We perform an experiment with a rotating ruby crystal that exhibits a very large negative group index and a positive nonlinear refractive index. The negative group index drags the light opposite to the motion of the medium. However, the positive nonlinear refractive index deflects the beam along with the motion of the medium and hinders the observation of the negative drag effect. Hence, we show that it is necessary to measure not only the transverse shift of the beam but also its output angle to discriminate the light drag effect from beam deflection. Our work provides insight into applications for all-optical control of light trajectories, particularly for beam steering, mode sorting, and velocimetry.
A moving dielectric medium can displace the optical path of light passing through it, a phenomenon known as the Fresnel-Fizeau optical drag effect. The resulting displacement is proportional to the medium's velocity. In this paper, we report on the observation of an anomalous optical drag effect, where the displacement is still proportional to the medium's speed but along the direction opposite to the medium's movement. We conduct an optical drag experiment under conditions of electromagnetically induced transparency and observe the transition from normal, to null, to anomalous optical drag by modification of the two-photon detuning.
Dispersive shock waves are fascinating phenomena occurring when nonlinearity overwhelms linear effects, such as dispersion and diffraction. Many features of shock waves are still under investigation, as the interplay with noninstantaneity in temporal pulses transmission and nonlocality in spatial beams propagation. Despite the rich and vast literature on nonlinear waves in optical Kerr media, spatial dispersive shock waves in nonlocal materials deserve further attention for their unconventional properties. Indeed, they have been investigated in colloidal matter, chemical physics and biophotonics, for sensing and control of extreme phenomena. Here we review the last developed theoretical models and recent optical experiments on spatial dispersive shock waves in nonlocal media. Moreover, we discuss observations in novel versatile materials relevant for soft matter and biology.
When a pulsed light beam propagates through ruby, it is delayed by a slow-light mechanism. This mechanism has been the subject of debate (Wisniewski-Barker et al 2013 New J. Phys. 15 083020; Kozlov et al 2014 New J. Phys. 16 038001; Wisniewski-Barker et al 2014 New J. Phys. 16 038002). To distinguish between the two main proposed mechanisms, we investigate the trailing edge of a square-wave pulsed laser beam propagating through ruby. Our observation of a pro-nounced tail on the trailing edge of the transmitted pulse cannot be explained solely by the effects of a time-varying absorber acting upon the incident pulse. Therefore, our observation of the creation of a tail at the trailing edge of the pulse provides evidence for a complicated model of slow light in ruby that requires more than pulse reshaping. The different delays of individual Fourier components of the pulse signal explain the pulse distortion that occurs upon transmission through the ruby and must be accounted for by any model that attempts to describe the effects of slow light in ruby.
Measurements of the Fresnel drag in moving glass media were performed using a passive ring-resonator method. The effective drag coefficient was measured for various glass samples of different indices of refraction and dispersions. When we averaged 15 separate measurements, we obtained overall agreement with theory that was well within our ±2.8 × 10⁻⁴ measurement uncertainty, thus providing, to our knowledge, the most accurate verification of Fresnel drag thus far.
Several groups have recently reported the observation of slow light propagation speeds based on the process of coherent population oscillations (CPO) under conditions such that the laser line width exceeds the width of the CPO transparency window. We explain this apparently paradoxical result by noting that the CPO effect is sensitive only to intensity variations of the incident laser field that occur on time scales comparable to or shorter than the population relaxation time of the material medium. Laser linewidths are associated primarily with phase fluctuations or with intensity fluctuations on much shorter time scales.
A light beam normally incident upon an uniformly moving dielectric medium is, in general, subject to bendings due to a transverse Fresnel-Fizeau light drag effect. In most familiar dielectrics, the magnitude of this bending effect is very small and hard to detect. Yet, the effect can be dramatically enhanced in strongly dispersive media where slow group velocities in the m/s range have been recently observed taking advantage of the electromagnetically induced transparency effect. In addition to the usual downstream drag that takes place for positive group velocities, we discuss a significant anomalous upstream drag which is expected to occur for negative group velocities. Furthermore, for sufficiently fast speeds of the medium, higher-order dispersion terms are found to play an important role and to be responsible for light propagation along curved paths or the restoration of the time and space coherence of an incident noisy beam. The physics underlying this class of slow-light effects is thoroughly discussed.
We describe a simple, reliable, and inexpensive experiment for teaching the basic features of slow light propagation using a solid saturable absorber at room temperature. The experiment is accessible to undergraduate and graduate students. We provide an explanation and analysis based on the coherent population oscillations model, which describes the propagation of light across a ruby crystal. © 2008 American Association of Physics Teachers.
First results from a study on tunable frequency-selective surfaces (FSSs) used for beam-steering applications are reported. In particular, the scattering response of a multilayer passband FSS based on varactor-loaded rectangular slots has been investigated. Simulated and measured results show that tunable FSSs can be employed to locally control the phase of the transmitted field with limited amplitude losses.
Transmission through a spinning window slightly rotates the polarization of the light, typically by a microradian. It has
been predicted that the same mechanism should also rotate an image. Because this rotary photon drag has a contribution that
is inversely proportional to the group velocity, the image rotation is expected to increase in a slow-light medium. Using
a ruby window under conditions for coherent population oscillations, we induced an effective group index of about 1 million.
The resulting rotation angle was large enough to be observed by the eye. This result shows that rotary photon drag applies
to images as well as polarization. The possibility of switching between different rotation states may offer new opportunities
for controlled image coding.
The real part n2′ of the complex nonlinear refractive-index coefficients of ruby crystal (Cr³⁺:Al2O3) are measured by the Mach–Zehnder interferometric method. The n2′ values measured at 514.5 nm for the light polarization perpendicular (σ) and parallel (π) to the crystal c axis are (2.81 ± 0.36) × 10⁻⁸ and (1.33 ± 0.17) × 10⁻⁸ cm²/W, respectively. From the results obtained the ratio of the polarizability difference between the excited and ground states of Cr³⁺ in ruby for σ polarization to that for π polarization is found to be 1.05 ± 0.30.
The ratio of the imaginary part to the real part of the complex nonlinear refractive index has been measured by transient two-wave mixing in acridine-orange-doped fluorophosphate glass, fluorescein-doped boric-acid glass, and ruby at argon-ion wavelengths. The method enables the individual signs of the components of the nonlinear index to be obtained. A simple theory based on a single homogeneously broadened absorption from the ground state agrees with measurements of the phase of the nonlinear index in acridine but not in fluorescein, where our data suggest that excited-state absorption plays an important role.
We have observed slow light propagation with a group velocity as low as 57.5+/-0.5 m/s at room temperature in a ruby crystal. A quantum coherence effect, coherent population oscillations, produces a very narrow spectral "hole" in the homogeneously broadened absorption profile of ruby. The resulting rapid spectral variation of the refractive index leads to a large value of the group index. We observe slow light propagation both for Gaussian-shaped light pulses and for amplitude modulated optical beams in a system that is much simpler than those previously used for generating slow light.
We have observed both superluminal and ultraslow light propagation in an alexandrite crystal at room temperature. Group velocities as slow as 91 meters per second to as fast as -800 meters per second were measured and attributed to the influence of coherent population oscillations involving chromium ions in either mirror or inversion sites within the crystal lattice. Namely, ions in mirror sites are inversely saturable and cause superluminal light propagation, whereas ions in inversion sites experience conventional saturable absorption and produce slow light. This technique for producing large group indices is considerably easier than the existing methods to implement and is therefore suitable for diverse applications.
A new method of polarized light analysis is described in which a highly sensitive electronic detector specific for birefringence is used to identify the crystalline axes of an object and then measure its phase retardation due to birefringence. The microscopic system employed in the method consists of an electronic birefringence detection system (BDS), a microscope with strain-free lenses, and a driven stage for passing the specimen at appropriate velocities across the image of an aperture placed at the field stop and imaged in the specimen plane by the condenser. The detector registers retardations directly as voltage at a constant deflection sensitivity of ca. 1.1 v per angstrom unit over a range of 120 angstrom units. The basal rms noise level is 0.002 A for a spot 36 micro in diameter formed by a 95 x, N. A. 1.25 objective pair, and increases in proportion to the reciprocal of the diameter of the scanning spot. The increase in noise with high resolution scanning can be offset by increasing the instrumental time constant, which is adjustable in decades between 0.004 and 0.4 seconds. A number of difficult problems in high extinction polarization microscopy are avoided by the use of modulated light and a rapid electronic detector. For example: (a) The measured distribution of birefringence is unaffected by the usual diffraction anomaly; therefore polarization rectifiers are not required. (b) The detector is selective for birefringence, so that there is no problem in separating contrast due to different optical properties (e.g. dichroism, light scattering). (c) The speed and sensitivity are both increased by between one and two orders of magnitude over that attainable by visual or photographic methods, thereby rendering a vast number of weakly birefringent, light-scattering, and motile objects readily analyzable for the first time with polarized light.
A simple method for optically measuring the thermal diffusivity of solids is demonstrated. The thermal displacement created on a substrate by a focused laser beam is determined from the divergence that it induces in a weak probe beam. The dynamics of the surface lens and the amplitude of the probe beam’s divergence are then used to determine the thermal diffusivity of the substrate. Several materials that span a wide range of thermal properties are studied.
A differential interferometric method for direct measurements of the real part of the nonlinear index of refraction n2 has been developed. With this method we were able to measure n2 as low as 10⁻¹⁰ cm²/W. This method was used to measure n2 in both ruby and GdAlO3:Cr⁺³ for the first time. Our results show that the real part of n2 in ruby is an order of magnitude larger than the imaginary part. This result clarifies the controversy about the dispersive or absorptive origin of the phase conjugation in ruby- and cromium-doped crystals.
The change in the speed of light as it propagates through a moving material has been a subject of study for almost two centuries. This phenomenon, known as the Fresnel light-drag effect, is quite small and usually requires a large interaction path length and/or a large velocity of the moving medium to be observed. Here, we show experimentally that the observed drag effect can be enhanced by over 2 orders of magnitude when the light beam propagates through a moving slow-light medium. Our results are in good agreement with the theoretical prediction, which indicates that, in the limit of large group indices, the strength of the light-drag effect is proportional to the group index of the moving medium.
Epitaxial ruby thin film is deposited via pulsed laser deposition technique. The charateristic R-lines in PL spectra confirmed ruby phase in the film. Raman spectra showed (C) -axis growth of the film. The nonlinear absorption coefficient and nonlinear refractive index of film were found to be (beta sim 5.2 times 10^{mathrm {mathbf {-5}}}) m/W and (n_{mathrm {mathbf {2}}}sim 3.1 times 10^{mathrm {mathbf {-5}}}) m (^{mathrm {mathbf {2}}}) /W, respectively. Further, the sub as well as superluminal propagation of 130-ns Gaussian pulse in 3.5- (mu ) m-thick PLD ruby thin film is reported in this paper. A delay of 17 ns is observed at film orientation of +45°, whereas pulse advancement of 12 ns is observed at −45°. Pulse broadening and compression are also observed with film rotation.
Subluminal and superluminal propagation of a light pulse in Kerr materials has been investigated. Group velocities as slow as much less than 1 mm per second to as fast as negative several thousands meters per second can easily be obtained in the Kerr medium, which possesses a large nonlinear refractive index and long relaxation time, such as Cr3+-doped alexandrite, ruby, and GdAlO3. The physical mechanism is the strong highly dispersive coupling between different frequency components of the pulse.