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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 13 — Jun. 26, 2006
  • pp: 6201–6206
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Precise control of superluminal and slow light propagation by transverse phase modulation

I. Guedes, L. Misoguti, and S. C. Zilio  »View Author Affiliations


Optics Express, Vol. 14, Issue 13, pp. 6201-6206 (2006)
http://dx.doi.org/10.1364/OE.14.006201


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Abstract

We have used a heterodyne Z-scan technique to produce both superluminal and slow light propagation in media that present either thermal or Kerr nonlinearities. The sample position determines the magnitude and sign of the group velocity and this property was used to control it, with an experimental setup much simpler than those previously reported in similar investigations. The observed effect is attributed to the transverse phase modulation produced by a focused Gaussian beam, and is capable of producing both positive and negative group velocities in the range 1.5 m/s <|υg|<c.

© 2006 Optical Society of America

1. Introduction

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

10. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

]. 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

11. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200 (2003). [CrossRef] [PubMed]

]. 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

12. Q. Yang, J. T. Seo, B. Tabibi, and H. Wang, “Slow light and superluminality in Kerr media without a pump,” Phys. Rev. Lett. 95, 063902 (2005). [CrossRef] [PubMed]

] 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

13. P. Yeh, “2-wave mixing in nonlinear media,” IEEE J. Quantum Electron. QE-25, 484 (1989). [CrossRef]

], 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

14. M. Sheik-Bahae, A. A. Said, T. Wei, D. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760 (1990). [CrossRef]

].

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.

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

15. J. Penaforte, E. Gouveia, and S. C. Zilio, “Nondegenerate 2-wave mixing in GdAlO3-Cr3+,” Opt. Lett. 16, 452 (1991). [CrossRef] [PubMed]

]. 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

16. S. C. Zilio, J. C. Penaforte, E. A. Gouveia, and M. J. V. Bell, “Nearly degenerate 2-wave mixing in saturable absorbers,” Opt. Commun. 86, 81 (1991). [CrossRef]

]. 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

14. M. Sheik-Bahae, A. A. Said, T. Wei, D. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760 (1990). [CrossRef]

]:

T(z,ΔΦ0)=I(z,ΔΦ0)I(z,0)=14ΔΦ0(t)x(x2+1)(x2+9)
(1)

where x=z/z0, z0=kw02/2 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)Leff, 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

17. A. Yariv, Quantum Electronics, 3rd edition (John Wiiley and Sons, New York, 1989).

]: I(t)≈I0 [1+Γm sin(ωmt)], where Γ m is the modulation amplitude, ωm is the modulation frequency and I 0 is the on-axis irradiance at the focus. By solving Debyés equation one obtains:

Δn0(t)=n2I0{1+Γm1+δ2sin(ωmtγ)}
(2)

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

I(z,t)=I0[1A(x)B]+I0ΓmF(z,δ)sin(ωmt+φ)
(3)

with

φ=tg1[A(x)Bδ(1+δ2)A(x)B(2+δ2)]
(4)
F(z,δ)1A(x)B(2+δ21+δ2)
(5)
A(x)=4x(x2+1)(x2+9)
(6)

and B=kn2I0Leff. 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 Γm2 and n22.

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

18. L. C. Oliveira, T. Catunda, and S. C. Zilio, “Saturation effects in Z-scan measurements,” Jpn. J. Appl. Phys. 35, 2649 (1996). [CrossRef]

].

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.

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.

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 Δno=αP2πKdndT [19

19. J. G. Tian, C. Zhang, G. Zhang, and J. Li, “Position dispersion and optical limiting resulting from thermally-induced nonlinearities in chinese tea liquid,” Appl. Opt. 32, 6628 (1993). [CrossRef] [PubMed]

], 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

20. Handbook of Optical Materials, edited by M. J. Weber (CRC Press, Boca Raton, 2003).

], 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: Δt=L(1υg1υgair), where υg and υgair=c are the group velocities of the beam propagating through the material and in the air, respectively. Since Δt=φT/2π=φ/ωm, where T is the modulation period, we find the group velocity as υg=c(1+ωmL)1.

We observe that depending on the sign of φ, υg 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 υg. 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, |υg| 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).

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.

Acknowledgments

We acknowledge the support of the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). The authors thank L. De Boni for helpful discussions.

References and links

1.

R. W. Boyd and D. J. Gauthier, “Slow and Fast Light,” Prog. Opt. 43, 496, edited by E. Wolf (Elsevier, Amsterdam, 2002).

2.

A. Kasapiet al., “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. 74, 2447 (1995). [CrossRef] [PubMed]

3.

L. V. Hauet al., “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594 (1999). [CrossRef]

4.

M. M. Kashet al., “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229 (1999). [CrossRef]

5.

D. Budkeret al., “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767 (1999). [CrossRef]

6.

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A 46, R29 (1992). [CrossRef] [PubMed]

7.

S. P. Tewari and G. S. Agarwal, “Control of phase matching and nonlinear generation in dense media by resonant fields,” Phys. Rev. Lett. 56, 1811 (1986). [CrossRef] [PubMed]

8.

R. S. Benninkal., “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001). [CrossRef]

9.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature (London) 406, 277 (2000). [CrossRef]

10.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

11.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200 (2003). [CrossRef] [PubMed]

12.

Q. Yang, J. T. Seo, B. Tabibi, and H. Wang, “Slow light and superluminality in Kerr media without a pump,” Phys. Rev. Lett. 95, 063902 (2005). [CrossRef] [PubMed]

13.

P. Yeh, “2-wave mixing in nonlinear media,” IEEE J. Quantum Electron. QE-25, 484 (1989). [CrossRef]

14.

M. Sheik-Bahae, A. A. Said, T. Wei, D. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760 (1990). [CrossRef]

15.

J. Penaforte, E. Gouveia, and S. C. Zilio, “Nondegenerate 2-wave mixing in GdAlO3-Cr3+,” Opt. Lett. 16, 452 (1991). [CrossRef] [PubMed]

16.

S. C. Zilio, J. C. Penaforte, E. A. Gouveia, and M. J. V. Bell, “Nearly degenerate 2-wave mixing in saturable absorbers,” Opt. Commun. 86, 81 (1991). [CrossRef]

17.

A. Yariv, Quantum Electronics, 3rd edition (John Wiiley and Sons, New York, 1989).

18.

L. C. Oliveira, T. Catunda, and S. C. Zilio, “Saturation effects in Z-scan measurements,” Jpn. J. Appl. Phys. 35, 2649 (1996). [CrossRef]

19.

J. G. Tian, C. Zhang, G. Zhang, and J. Li, “Position dispersion and optical limiting resulting from thermally-induced nonlinearities in chinese tea liquid,” Appl. Opt. 32, 6628 (1993). [CrossRef] [PubMed]

20.

Handbook of Optical Materials, edited by M. J. Weber (CRC Press, Boca Raton, 2003).

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 3, 2006
Revised Manuscript: June 3, 2006
Manuscript Accepted: June 9, 2006
Published: June 26, 2006

Citation
I. Guedes, L. Misoguti, and S. C. Zilio, "Precise control of superluminal and slow light propagation by transverse phase modulation," Opt. Express 14, 6201-6206 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6201


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References

  1. R. W. Boyd and D. J. Gauthier, "Slow and Fast Light," Prog. Opt. 43, 496, edited by E. Wolf (Elsevier, Amsterdam, 2002).
  2. A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris "Electromagnetically induced transparency - propagation dynamics," Phys. Rev. Lett. 74, 2447 (1995). [CrossRef] [PubMed]
  3. L. V. Hau, Z. Dutton, C. H. Behroozi, and S. E. Harris, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999). [CrossRef]
  4. M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, "Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229 (1999). [CrossRef]
  5. D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation," Phys. Rev. Lett. 83, 1767 (1999). [CrossRef]
  6. S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29 (1992). [CrossRef] [PubMed]
  7. S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 56, 1811 (1986). [CrossRef] [PubMed]
  8. R. S. Benninkal., "Enhanced self-action effects by electromagnetically induced transparency in the two-level atom," Phys. Rev. A 63, 033804 (2001). [CrossRef]
  9. L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal light propagation," Nature (London) 406, 277 (2000). [CrossRef]
  10. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]
  11. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200 (2003). [CrossRef] [PubMed]
  12. Q. Yang, J. T. Seo, B. Tabibi, and H. Wang, "Slow light and superluminality in Kerr media without a pump," Phys. Rev. Lett. 95, 063902 (2005). [CrossRef] [PubMed]
  13. P. Yeh, "2-wave mixing in nonlinear media," IEEE J. Quantum Electron. QE-25, 484 (1989). [CrossRef]
  14. M. Sheik-Bahae, A. A. Said, T. Wei, D. Hagan and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. QE-26, 760 (1990). [CrossRef]
  15. J. Penaforte, E. Gouveia, and S. C. Zilio, "Nondegenerate 2-wave mixing in GdAlO3-Cr3+," Opt. Lett. 16, 452 (1991). [CrossRef] [PubMed]
  16. S. C. Zilio, J. C. Penaforte, E. A. Gouveia, and M. J. V. Bell, "Nearly degenerate 2-wave mixing in saturable absorbers," Opt. Commun. 86, 81 (1991). [CrossRef]
  17. A. Yariv, Quantum Electronics, 3rd edition (John Wiiley and Sons, New York, 1989).
  18. L. C. Oliveira, T. Catunda, and S. C. Zilio, "Saturation effects in Z-scan measurements," Jpn. J. Appl. Phys. 35, 2649 (1996). [CrossRef]
  19. J. G. Tian, C. Zhang, G. Zhang, and J. Li, "Position dispersion and optical limiting resulting from thermally-induced nonlinearities in chinese tea liquid,"Appl. Opt. 32, 6628 (1993). [CrossRef] [PubMed]
  20. Handbook of Optical Materials, M. J. Weber, ed. (CRC Press, Boca Raton, 2003).

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