A novel method is proposed to generate slow and fast lights at arbitrary signal wavelength in benzocyclobutene (BCB) polymer, which eliminates the requirement on the optical nonlinearity or the resonant effect at the signal wavelength with the help of the thermo-optic nonlinear effect induced by a control beam at a different but fixed wavelength. The signal group velocity can be precisely tuned simply by scanning the position of the BCB sample along the light propagation direction. Another advantage is the ability to control chromatically the group velocity of the signal beam by adjusting the control beam in the BCB sample, which, in essence, is to control the refractive index change experienced by the signal beam. This method provides an active chromatic control on the group velocity of light at arbitrary signal wavelength and therefore may have important potential applications in optical communication network and optical information processing.
©2009 Optical Society of America
Intensive studies on manipulating the group velocity of lights have been carried out because of many important potential applications associated with slow and fast lights . In 1999, by use of the electromagnetically induced transparency (EIT) effect , Hau et al.  and Kash et al.  decelerated the group velocity of light pulses to 17 m/s in an ultra-cold gas of sodium atoms at 450 nK and 90 m/s in an optically dense hot rubidium gas at about 360 K, respectively. Later Wang et al. used gain-assisted linear anomalous dispersion to demonstrate superluminal light propagation with a negative group velocity of vg=-c/310 in an atomic caesium gas . In the following years, a flurry of studies on slow and fast lights were carried out in various dispersive systems by use of different physical mechanisms, and great progresses have been made. M. Fleischhauer and M. D. Lukin proposed to store light pulses by mapping the quantum state of light pulses onto the EIT media via dark-state polaritons . Liu et al. brought the laser pulses to a complete stop by using the EIT effect in a magnetically trapped cold cloud of sodium atoms, demonstrating experimentally effective storage and retrieval of light pulses . Later the pulse storage was also demonstrated in a Pr3+:Y2SiO5 crystal at 5 K using the EIT effect by Turukhin et al. . This pulse storage technique has great potential applications in quantum information processing and optical communication. Besides the EIT effect, other mechanisms such as the coherent population oscillation in Cr3+-doped ruby and alexandrite crystals [9, 10], the stimulated Brillouin scatterings [11, 12] and stimulated Raman scatterings  in optical fibers, the dispersive wave mixing in nonlinear materials [14, 15], and the artificial resonance in photonic structures [16, 17, 18, 19] were also employed to generate slow and fast lights. Note that among the various techniques to generate slow and fast lights, optical nonlinearity or resonance at the signal wavelength are in general required. Therefore, either specific signal wavelength (for example, by use of EIT effect ) or complicated fabrication processes (for example, by use of the artificial resonance in photonic structures [16, 17, 18, 19]) are necessary. On the other hand, the resonant effects are usually accompanied by significant signal attenuation or amplification and strong high-order dispersion effect during the generation of slow and fast lights. Such requirements put serious limitations on the practical applications of slow and fast lights.
In this paper, we proposed a new method to generate slow and fast lights which eliminates the requirement on the optical nonlinearity and the optical resonance at the signal wavelength. Therefore, it is possible to control the group velocity of light at arbitrary signal wavelength. This technique also offers simple ways to precisely tune the group velocity of lights. Thus it may be of a wide range potential applications in optical delay lines and buffers.
2. Scheme for active chromatic control on group velocity of light at arbitrary wavelength
Heterodyne Z-scan technique has been used to get slow and fast lights previously by use of transverse phase modulation produced by a focused Gaussian beam, and precise control on the group velocity of slow and fast lights was achieved [20, 21]. As mentioned above, optical nonlinearity such as kerr or thermo-optic nonlinearity at the signal wavelength is required in these experiments.
Here we proposed a novel method to actively control the group velocity of light at arbitrary wavelength based on heterodyne Z-scan technique. The scheme for such an active control on slow and fast lights is shown in Fig. 1. Two beams of different wavelengthes were employed in this scheme, one was named as the signal beam, and the other was the control beam. These two beams were made to propagate collinearly through a beam splitter BS, and were then modulated sinusoidally in intensity with a small modulation depth of 5%~10% using an electro-optic modulator EO. The modulated beams were split into the reflected and transmitted output ports of another beam splitter. The component at the signal wavelength in the reflected output port was selected out by use of a narrow bandwidth bandpass spectral filter (filter1) and was detected by a photo-detector D1, this modulated beam at the signal wavelength was served as a reference beam during the measurement of the group velocity of slow and fast lights. The combined collinear beams at the transmitted output port were focused into an optical material by a lens L1 with a focal length of approximately 12 cm. Note that the signal and the control beams were at different wavelengthes, therefore the focal length of the lens L1 for these two beams was slightly different. We assume that the material is optically insensitive to the signal wavelength, whereas it is of optical nonlinearity at the control wavelength. In our following experiments we used a benzocyclobutene (BCB) polymer sample and its optical properties will be characterized in section 3. The component at the signal wavelength in the transmitted beams after the BCB sample was again selected out by another narrow bandwidth bandpass filter (filter2) and its on-axis temporal intensity evolution was monitored by another photo-detector D2. The output from the photo-detector D2 was compared in an oscilloscope with that of the photo-detector D1 in the reference arm. In this way the time delay Δt experienced by the signal beam, therefore, the group velocity of the signal beam, can be measured.
We note that, although the signal beam cannot induce any optical nonlinearity itself, it will experience the refractive index change induced by the control beam as the signal beam and the control beam propagate collinearly in the material. Therefore the nonlinearity induced by the control beam will also have an effect on the wavefront of the signal beam during its propagation in the material, and induce a transverse phase modulation of the signal beam, just as the effect acting on the control beam itself. It is evident that this active chromatic control on the group velocity of light eliminates the requirements on the optical nonlinearity or resonance at the signal wavelength, while it is more easy to have very strong optical nonlinearity at a fixed specific control wavelength. In the following sections, we will give a detailed theoretical analysis and experimental verification on this novel method based on a thermo-optic nonlinearity of the BCB polymer at the control wavelength.
3. Material characterization
The BCB polymer is commercially available from Dow Chemical Company. A bulk BCB polymer with a dimension of 25×20×2.1 mm3 was used in the experiments. Figure 2 shows the transmittance spectrum of the BCB sample measured when light propagates along the direction with a thickness of 2.1 mm. It is seen that the BCB polymer is absorptive in the visible. We confirmed experimentally that the BCB polymer is of thermo-optic nonlinearity in the visible. For convenience, we set the control wavelength at 532 nm. The absorption coefficient α1 and the linear refractive index n 0 at 532 nm were measured to be 1.1 cm-1 and 1.56, respectively. The thermal conductivity K was ~0.29 Wm-1K-1, and the thermo-optic nonlinear coefficient dn/dT at 532 nm was measured to be -3.1×10-5 K-1 by using a typical Z-scan method [22, 23]. Note that the sample is transparent with negligible absorption for near infrared lights with wavelength longer than 700 nm. No noticeable thermo-optic nonlinear effect was observed at moderate light intensities at the near-infrared wavelength. Therefore we set the signal wavelength at 785 nm.
4. Theoretical analysis
In the case when the electro-optic modulator EO is absent in Fig. 1, and assuming that the control and signal beams are of TEM00 mode and propagate along the +z direction with intensity profiles I 1(r, z 1) and I 2(r, z 2) as
respectively. Where r is the transverse coordinate with the beam center as the coordinate originate, z 1 (or z 2) is the on-axis position displacement with respect to the beam waist position of the control beam (or the signal beam) after the lens L1, and w 10 and w 20 are the beam waist radii of the focused control and signal beams, respectively. The beam radii at position displacement z 1,2 are w 1,2=w 10,20(1+x 2 1,2)1/2 with x 1,2=z 1,2/z 10,20 being the dimensionless position displacement and z 10,20=k 1,2 w 2 10,20/2, k 1,2=2π/λ1,2 are the wave vectors with λ1,2 being the control and signal wavelengthes. I10,20 are the on-axis irradiance intensity at focus of lens L1 (i.e., at z 1,2=0). The focused control beam will induce a transverse temperature gradient in the BCB sample with its rear surface at the position displacement z 1,2, which will result in a nonlinear refraction term Δn 1(r, z 1, z′) through thermo-optic nonlinearity on the approximation of thin sample 
Where z′ is the propagation depth in the sample, L is the thickness of the BCB sample, α 1 is the absorption coefficient at the control wavelength, τ=w 2 1 ρC/4K is the relaxation time constant of the thermo-optic nonlinearity, which is dependent on the radius of the control beam w1 in the sample, K, ρ and C are the thermal conductivity, the density and the specific heat of the BCB polymer sample, respectively. For the steady state (t→∞), the nonlinear refractive index term is expressed as
Since the signal beam propagates collinearly with the control beam in the sample, it will also feel the refractive index change induced by the control beam, and therefore its propagation dynamics will be influenced by the nonlinearity induced by the control beam. Thus the transverse distribution of the refractive index change experienced by the signal beam can be written as
where η is a dispersion parameter characterizing the refractive index dispersion of the BCB polymer, which can be approximately set to be unit due to the weak linear refractive index dispersion of the BCB polymer between the control wavelength (532 nm) and the signal wavelength (785 nm). So the transverse distribution of the refractive index change Δn 2(r, z 1, z′) can be expressed as
Therefore one can obtain the phase shift of the signal beam Δϕ 2(r, z 1, z′) within the sample through the following equation
here , is the incident power of the control beam on the sample, and Le f f=[1-exp(-α 1 L)]/α 1, respectively. On the approximation of w 2(z 2)≈w 1(z 1) at the exit surface of the sample, the normalized far-field on-axis z-scan transmittance of the signal beam with a small diameter pinhole as shown in Fig. 1 can then be obtained [22, 23, 24, 25]
In the case when the control and signal beams are modulated sinusoidally in intensity in the form of
with a modulation angular frequency ωm and a small modulation depth Γ1,2, respectively. The dynamics of the nonlinear refractive index term Δn 1(r, z 1, z′, t) induced by the control beam can be evaluated by the Debye relaxation equation of the form: , where in our case. The solution of the Debye relaxation equation can be expressed as
where γ=arctanδ, and δ=ωmτ. Thus the temporal evolution of the transverse distribution of the refractive index change Δn 2(r, z 1, z′, t) experienced by the signal beam can be written as
The on-axis irradiance of the signal beam after the pinhole can be obtained 
where I′20 is the averaged intensity of the far-field on-axis irradiance of the signal beam without thermo-optic nonlinearity, and
and the phase delay φ experienced by the signal beam is
which results in a time delay Δt of the signal beam after the BCB sample as
where c is the light speed in vacuum. Therefore the group velocity vg of the signal beam can be obtained as
It is evident that, with the help of the thermo-optic nonlinearity at the control wavelength, slow and fast lights can be generated without any requirement on the optical nonlinearity at the signal wavelength, achieving an active chromatic control on the group velocity of lights while eliminating the requirement on the nonlinearity or the resonance at the signal wavelength.
5. Experimental results and discussions
In this section, we will give an experimental demonstration of the above proposed active chromatic control on slow and fast lights without the optical nonlinearity at the signal wavelength. The experimental setup is shown in Fig. 1, where the modulation depth of the signal and the control beams were set to be in the range of 5%~10%. Note that the signal and the control beams were at different wavelength, therefore the focal length and the beam waist of these two beams were also different. It was measured that the difference of the focal length between the signal beam and the control beam was 2.1 mm. The beam waist radii of the focused control and signal beams were measured to be w 10=35 µm and w 20=47 µm, respectively. We see that the waist radius of the control beam is slightly smaller than that of the signal beam, this will have an effect on the group velocity of the signal beam but it should be small because only the on-axis temporal intensity variation of the signal beam was detected by the photo-detector D2 in the measurement.
Figures 3 and 4 show typical temporal traces of the on-axis intensity variations of the signal beam when the BCB polymer sample was placed at z 2=-6.4 mm and z 2=13.6 mm, respectively. Where the incident powers of the control and the signal beams were set to be 17.1 mWand 0.71 mW, respectively. The modulation frequency ωm/2π for both beams was 10 Hz, and the modulation depth for the control beam was Γ1=6% and that for the signal beam was Γ2=9%, respectively. For comparison, the corresponding temporal traces of the intensity variation of the reference beam detected by the photo-detector D1 are also plotted in Figs. 3 and 4. The time delay Δt was measured to be 1.3 ms at z 2=-6.4 mm in Fig. 3 and -5.4 ms at z 2=13.6 mm in Fig. 4, corresponding to a subluminal light propagation with a group velocity of 1.62 m/s and a superluminal light propagation with a group velocity of -0.39 m/s, respectively.
From Eqs. (15) and (18), one obtains the effective refractive index neff (δ) experienced by the modulated component of the signal beam which is given by neff (δ)=n 20+cφ/Lω 2, where n 20 and ω 2 are the linear refractive index of the BCB polymer at the signal wavelength and the angular frequency of the signal beam, respectively, and the transmittance of the modulated component of the signal beam is characterized by F(x 1,x 2,δ) (see Eq. (17)). Figure 5 shows the dispersion curves of neff and F(x 1,x 2,δ) at z 2=-6.4 mm and 13.6 mm, corresponding to the cases of Figs. 3 and 4, respectively. It is seen that both neff and F(x 1,x 2,δ) are highly dispersive, and a gain peak in F(x 1,x 2,δ) corresponds to a positive slope of the dispersion curve of neff and a slow light propagation, while a transmittance dip in F(x 1,x 2,δ) corresponds to a negative slope of the dispersion curve of neff and a fast light propagation. The results clearly indicate that the time delay and advance measured in Figs. 3 and 4 are due to the slow and fast light propagation. It is evident that slow and fast lights can be generated although no optical nonlinearity is induced by the signal beam itself.
More interestingly, it is possible to tune precisely the time delay Δt experienced by the signal beam simply by scanning the sample along the light propagation direction, as predicted by Eqs. (18) and (19) and demonstrated experimentally in Fig. 6. Here the experimental conditions such as the incident power, the modulation frequency and depth of the signal and the control beams were set to be the same as those in Figs. 3 and 4, and only the sample position z 2 was scanned along the light propagation direction. In Fig. 6, the solid squares are the measured time delay Δt, and the red curve is a theoretical fit to the measured data using Eqs. (18) and (19). Note that the relaxation time constant τ of the thermo-optic nonlinearity is dependent on the control beam radius w 1 in the sample  which is a function of the position displacement z 1 or z 2. The relaxation time constant τ was measured to be 13 ms at the position displacement z2=13.6 mm, the values of τ at other positions can therefore be determined. Quantitative agreement is achieved between the theoretical prediction and the experimental data, and even transition between the subluminal and the superluminal light propagation can be achieved easily.
Due to the limited relaxation time constant of the thermo-optic nonlinearity of the BCB polymer, the time delay Δt experienced by the signal beam is dependent on the modulation frequency ωm/2π. Figure 7 shows the dependence of the time delay Δt on the modulation frequency ωm/2π with the BCB polymer at the position z 2=13.6 mm, i.e., with a relaxation time constant of τ=13 ms. The incident powers and the modulation depth of the control and the signal beams were set to be the same as those of Fig. 4. The time delay Δt is larger at a lower modulation frequency, and it decreases monotonously with increasing modulation frequency and tends to be negligibly small with modulation frequencies higher than 50 Hz.
As can be seen from Eqs. (13) and (14), the temporal dynamics of the transverse distribution of the refractive index change experienced by the signal beam is mainly determined by the temporal dynamics of the thermo-optic nonlinearity induced by the control beam. Therefore, it is possible to control the group velocity of the signal beam through the control beam by means of adjusting the incident power of the control beam, the relative position and the overlap between the signal and the control beams. As an example, figure 8 shows the tunability of the time delay Δt by adjusting the relative waist position between the control beam and the signal beam with a fixed sample position z 2=19.3 mm. In this experiment, the sample position was fixed with respect to the waist of the focused signal beam, while the spot size and the focal waist position of the control beam were tuned by scanning the lens L1 along the beam propagation direction of the control beam. Here we modified the experimental setup by replacing the lens L1 in the dashed rectangle in Fig. 1 with the optical element set in the inset of Fig. 8, in which filter3 and filter4 were narrow bandwidth bandpass filters with central wavelengthes at 532 nm and 785 nm, respectively. The focal length of the lens L2 was the same as that of the lens L1.
The incident powers of the control and signal beams were 8.4 mW and 0.4 mW, respectively. The modulation frequency was set to be 10 Hz for both beams, and the modulation depth were Γ1=4.5% and Γ2=5.2%, respectively. It is evident that it is able to chromatically tune the group velocity of the signal beam simply by adjusting the control beam in the sample but without the requirement on the optical nonlinearity at the signal wavelength.
For practical applications such as optical delay lines and buffer memories, issues such as the maximum time delay and the delay-bandwidth product have to be considered for a specific slow and fast light scheme. The maximum time delay/advance depends not only on the group velocity of light pulses, but also on parameters such as the material thickness, the absorption coefficients at the control and the signal wavelengthes, and the requirements of specific applications [26, 27]. Under the experimental conditions of Fig. 4, a time advance of the order of ~25 ms is achievable for a BCB polymer of 1-cm thickness, which corresponds to a delay-bandwidth product of ~0.25. Moreover, trade-off between the maximum time delay/advance and the bandwidth of the signal pulses has to be considered. In general, temporal shorter signal pulses with broader bandwidth experience smaller time delay/advance, as indicated by the results shown in Fig. 7. To avoid profile distortion of the slow or fast light pulses, the bandwidth of the signal pulses should be within the slow or fast light window which is determined by the dispersion curve of neff shown in Fig. 5. Therefore, high intensity light pulses and materials with fast response rate and large optical nonlinearity at the control wavelength are preferred in order to achieve large delay-bandwidth product. Fortunately, one advantage of our scheme is that it is possible to achieve strong optical nonlinearity for a certain material by choosing an appropriate control wavelength. Also, under the constrain of the Kramers-Kronig relation shown in Fig. 5, a steeper slope of the dispersion curve of neff results in a larger modification on the group velocity of light, therefore, a larger time delay/advance. Certainly, practical issues on real-life applications of slow and fast lights are more complicated than proof-of-principle demonstration of slow and fast lights in this paper. A detailed discussion on these issues surely deserves a full-length paper and therefore is out of the scope of this paper.
In conclusion, we have proposed a novel technique to actively and chromatically control the group velocity of lights at arbitrary wavelength without any requirement on the nonlinearity or the resonant effect at the signal wavelength. The group velocity of the signal light can be tuned precisely by adjusting the nonlinearity induced by the control beam at a fixed wavelength other than the signal wavelength. One advantage of this technique is that it is relatively easy to achieve strong nonlinearity at a specific control wavelength, which results in a very effective tunability on the group velocity of the signal light. Both slow and fast lights, and even transition between subluminal and superluminal light propagation, can be achieved simply by scanning the sample position or by adjusting the control beam in the sample. This technique may have potential applications in optical information processing and optical communication network.
This work is financially supported by the MOE Cultivation Fund of the Key Scientific and Technical Innovation Project (708022), the NSFC (60678021, 10804054), the 973 program (2007CB307002), the CNKBRSF (2006CB921703), the 111 project (B07013), and the RFDP (200800551034).
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