1. Introduction

The search for new methods that can allow the precise control of the group velocity υ_g of a light wave has largely increased in the past few years, motivated not only by the understanding of basic concepts, but also by promising technological applications that include controllable delay lines and quantum information devices [1–5]. It is known that when a light pulse propagates through an optical medium in a condition where its frequency is near a strong and narrow resonance, it experiences a rapid frequency change in the refractive index that can significantly affect its group velocity [1]. This effect has already been used to produce slow light propagation [2–5], but in order to circumvent the strong absorption associated to the refractive index dispersion by Kramers-Kronig relations, electromagnetically induced transparency (EIT) has to be employed. Such a quantum interference effect allows the propagation of light through an otherwise opaque medium, while still keeping the necessary strong normal dispersion for the generation of slow light [6–8]. In addition, gain-assisted process or electromagnetically induced absorption produce the necessary anomalous dispersion that can be used for superluminal light propagation [9]. However, these techniques require sophisticated experimental setups that are not suitable for practical applications and simpler methods of producing superluminal and slow light propagation are sought. Furthermore, it would be of great interest that these methods could also allow the precise control of the group velocity.

Another quantum coherence technique, based on the creation of a spectral hole due to population oscillations, was used by Bigelow et al. for the observation of slow light propagation in a ruby crystal at room-temperature [10]. Later, they used an alexandrite crystal and showed to be possible to switch between slow and superluminal light propagation by changing the laser wavelength to access chromium ions in inversion or mirror sites [11]. Although this technique still leaves a few drawbacks such as the low transmittance (typically 10^-2–10^-3) at the wavelength used, the relatively narrow spectral bandwidth (~100 nm) for devices operation and the inadequacy of continuously changing the group velocity, it certainly simplifies previously introduced experimental apparatus. More recently, Yang et al. [12] investigated theoretically slow and fast propagation of a light pulse in Kerr materials. Basically, they used a nearly degenerate two-wave mixing geometry to calculate the energy transfer between the pump and probe beams, which is a highly dispersive process. The nonlinear coupling between these beams is the physical mechanism responsible for either slow or fast light propagation. Although this sensitive method was often used in the past for measuring optical nonlinearities [13], it has a few drawbacks that include the difficulty of precisely overlapping the beams and the susceptibility to mechanical vibrations because of its interferometric nature. Therefore, it was almost completely abandoned in favor of a single beam technique, named Z-scan, which offers simplicity as well as very high sensitivity [14].

In this work we show that the heterodyne Z-scan technique can be used to produce both superluminal and slow light propagation in materials whose index of refraction is intensity or fluence-dependent, besides having a finite response time. Moreover, we demonstrate the possibility of continuously tuning the group velocity simply by changing the sample position and measuring the changes in the far field intensity pattern due to the nonlinear transverse phase induced in the sample. Owing to the finite response time of the optical nonlinearity, the lens induced by the transverse phase profile will present a time lag when a small sinusoidal modulation is superposed to a continuous wave (cw) laser beam. By measuring the phase of the signal detected at the far field position one can determine the value of the group velocity of the transmitted beam.

2. Method

2.1 Experimental setup

In our experimental setup, shown in Fig. 1, we use the second-harmonic (λ=532 nm) of a diode pumped Nd:YVO₄ cw laser as the exciting source. The beam can be attenuated by means of neutral density filters, if necessary, and then passes through an electro-optic modulator (EOM) driven by a function generator that places a few percent (5–10%) of sinusoidal amplitude modulation ranging from 10 to 300 Hz. The modulated beam is focused onto the sample with a 10-cm-focal length lens, resulting in a beam waist at the focus of w ₀=17 µm. The energy transmitted through a small aperture placed at the far field is measured with a Si detector plugged into a dual-phase lock-in amplifier triggered by the modulation signal. The lock-in measures both the amplitude and phase of the transmitted beam, but only the last is enough to provide information whether slow or fast light was produced. We also used another Si detector to measure the signal coming from a glass slide placed after the EOM and used this signal as the reference to the lock-in, as usually done in similar experiments. However, we found that the phase was the same as that coming directly from the function generator, which was then used for simplicity.

 

Fig. 1. Experimental setup used to observe superluminal and slow light. L:lens, S:sample, D:detector, A:aperture.

Download Full Size | PDF

2.2 Samples

We used samples presenting the two different classes of optical nonlinearities mentioned earlier. First, we used a well-characterized 1.4 mm-thick ruby sample (linear absorption, α=2.3 cm^-1), which is known to present an electronic Kerr nonlinearity (n₂≈18×10^-6 cm²/kW @ 514 nm, T₁=3.4 ms) and negligible thermal effects [15]. In this kind of chromium-doped material, the electronic nonlinearity has its origin in the susceptibility difference between the ground (⁴A₂) and the excited metastable (²E) states, and also in their populations [16]. Second, we used a sample where the optical nonlinearity has a thermal origin, namely, the azo-compound Disperse Orange 3 (DO3) dissolved in chloroform, placed in a 2 mm-thick quartz cuvette. In this case, the azo group presents a radiationless photo-isomerization process that generates an appreciable amount of heat and induces a thermal lens. We adjusted the DO3 concentration such that the linear absorption coefficient was around 1.7 cm-1.

2.3 Theoretical considerations

To develop an understanding of our results, we consider a cubic nonlinearity and a small nonlinear phase change. When the aperture at the far field is small, the measured normalized Z-scan transmittance is given by [14]:

where x=z/z₀, z₀=k${\mathrm{w}}_0^2$/2 is the diffraction length of the beam, k=2π/λ is the wave vector, λ is the wavelength, and ΔΦ₀ is the on-axis phase shift at the focus, given as: ΔΦ₀=kΔn₀(t)L_eff, with L=(1-e^-αL)/α eff being the sample’s effective length. The dynamic refractive index change, Δn₀(t), can be evaluated assuming a Debye relaxation equation of the form: τ(dΔn₀/dt)+Δn₀=n₂I(t), where τ is the relaxation time and n ₂ is the nonlinear refractive index. The modulated laser beam intensity is approximately given by [17]: I(t)≈I₀ [1+Γ_m sin(ω_mt)], where Γ _m is the modulation amplitude, ω_m is the modulation frequency and I ₀ is the on-axis irradiance at the focus. By solving Debyés equation one obtains:

where δ=ω_mτ and γ=tg^-1δ. The on-axis irradiance measured by the detector is obtained by substituting Eq. (2) in Eq. (1), resulting in:

with

and B=kn₂I₀L_eff. The DC term on the right-hand side of Eq. (3) corresponds to the usual Z-scan signature, while the second term is the out-of-phase modulated signal to be measured by the lock-in. The phase given by Eq. (4) can be positive or negative, such that the modulation is delayed or advanced with respect to the reference, resulting in slow or fast light. Moreover, the phase can be adjusted by changing the value of x. The above set of equations was obtained by neglecting terms in ${\mathrm{\Gamma }}_{\mathrm{m}}^2$ and ${\mathrm{n}}_2^2$.

3. Results and discussion

The dual phase lock-in measures directly the amplitude and phase of the modulation. By scanning the sample along the z-direction, the amplitude can be normalized to the one obtained for |z|≫z₀ such that the values of Γ _m and I₀ are eliminated, leaving only F(z,δ). Figure 2 shows the results obtained for the ruby sample at δ=1.5 and I₀=1.7 kW/cm², as well as the theoretical plot of Eqs. (4) and (5), with the parameters n₂=1.33×10^-6 cm²/kW and T₁=3.4 ms, which agree with those reported in the literature, provided that saturations effects are considered [18].

 

Fig. 2. Normalized amplitude (a) and phase (b) of the modulation for the ruby sample. The dots correspond to the experimental points while solid lines are plots of Eqs. (4) and (5), with the parameters given in the text.

Download Full Size | PDF

Although the normalized amplitude of the modulation signal is quite noisy due to poor surface quality of our sample, the phase measurement is not sensitive to such linear noise and its measurement can become an important tool for the characterization of n ₂ and τ of saturable absorbers. The same behavior with respect to the noise applies to the DO3/chloroform solution, as seen in Fig. 3 for a power of 0.33 mW. However, the amplitude in this case does not follow exactly the theory because we assumed a cubic nonlinearity and for a thermal nonlinearity there is heat diffusion, which broadens the Z-scan signature.

 

Fig. 3. Normalized amplitude (a) and phase (b) of the modulation for the DO3 sample at P=0.33 mW. The dots correspond to the experimental points while solid lines are theoretical curves using Eqs. (4) and (5), with the parameters given in the text.

Download Full Size | PDF

We regard the nonuniform heating caused by the nearly cw modulated beam as a steady state, such that the on-axis nonlinear index change at focus can be determined in terms of the thermo-optic coefficient dn/dT as $〈\Delta n_o〉=\frac{\mathrm{\alpha P}}{2\mathrm{\pi K}}\frac{\mathrm{dn}}{\mathrm{dT}}$ [19], where K is the thermal conductivity and P is laser power. Using the tabulated values of dn/dT (-5.98×10^-4 K^-1) and K (0.117 W/m.K) for chloroform [20], Eqs. (4) and (5) provide the solid curves shown in Fig. 3, which are in good agreement with the experimental data despite the fact that these equations are not adequate to describe a thermal nonlinearity.

An important feature of our results is that the phase can be set as positive or negative, depending on the sample position. This is very important to allow a precise control of the group velocity of the light wave, which can be calculated by writing the time delay induced by the transverse phase modulation as: $\mathrm{\Delta t}=L\left(\frac{1}{{\upsilon }_g}-\frac{1}{{\upsilon }_g^{\mathrm{air}}}\right)$, where υ_g and ${\mathrm{\upsilon }}_{\mathrm{g}}^{\text{air}}$=c are the group velocities of the beam propagating through the material and in the air, respectively. Since Δt=φT/2π=φ/ω_m, where T is the modulation period, we find the group velocity as ${\upsilon }_g=c{\left(1+\frac{\mathrm{c\phi }}{{\omega }_{m}L}\right)}^{-1}$.

We observe that depending on the sign of φ, υ_g can be smaller (φ>0) or greater (φ<0) than c, corresponding to slow and fast light. Figure 4 depicts the group velocity corresponding to the data of Fig. 3 together with a plot with the expression for υ_g. In the vicinity of z=0, the phase is very small and can be positive (z>0) or negative (z<0), leading to a dispersive-like behavior with an abrupt change at z=0, ranging from -c to +c. At this region, the group velocity can be precisely controlled only if one has a good resolution translation stage. On the other hand, for |z|>z₀, the phase changes slowly with the sample position, making easier the group velocity control. In the vicinity of ±z₀, |υ_g| can be as small as 1.5 m/s.

 

Fig. 4. Group velocity for the DO3 sample as a function of its position (squares) and theoretical fit (solid line).

Download Full Size | PDF

4. Conclusions

In summary, we have demonstrated that both ultra-slow and superluminal light propagation can be achieved with the same sample just by reversing its position with respect to the focal plane. Besides its inherent simplicity and sensitiveness, the technique works even with thermal nonlinearities, such as that of Chinese tea. Indeed, we carried out measurements in this material and found that the signal is also easily observed, which makes the method attractive for teaching purposes if a low-cost 5 mW green microchip laser pointer is used. Finally, and most important, the present method has the special attractive of continuously changing the group velocity, being enough for this purpose to translate the sample along the beam propagation direction.