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OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Controlling pulse delay by light and low magnetic fields: slow light in emerald induced by transient spectral hole-burning Rajitha Papukutty Rajan,1 Hans Riesen,1,* and Aleksander Rebane2,3 1
School of Physical, Environmental, and Mathematical Sciences, The University of New South Wales, UNSW Canberra, Canberra, ACT 2600, Australia 2 Physics Department, Montana State University, Bozeman, Montana 59715, USA 3
National Institute of Chemical Physics and Biophysics, Tallinn, Estonia *Corresponding author: [email protected]
Received July 4, 2013; revised September 5, 2013; accepted October 2, 2013; posted October 3, 2013 (Doc. ID 193112); published November 5, 2013 Slow light based on transient spectral hole-burning is reported for emerald, Be3 Al2 Si6 O18 ∶Cr3 . Experiments were conducted in π polarization on the R1 3∕2 line (2 E←4 A2 ) at 2.2 K in zero field and low magnetic fields B‖c. The hole width was strongly dependent on B‖c, and this allowed us to smoothly tune the pulse delay from 40 to 154 ns between zero field and B‖c 15.2 mT. The latter corresponds to a group velocity of 16 km∕s. Slow light in conjunction with a linear filter theory can be used as a powerful and accurate technique in time-resolved spectroscopy, e.g., to determine spectral hole-widths as a function of time. © 2013 Optical Society of America OCIS codes: (300.0300) Spectroscopy; (300.6240) Spectroscopy, coherent transient; (300.6320) Spectroscopy, high-resolution. http://dx.doi.org/10.1364/OL.38.004546
In recent years, the ability to produce slow light has generated considerable interest because of its potential in optical data storage, signal and quantum-data processing, optical switching, and pulse regeneration [1–3]. The propagation of an optical pulse through a medium is determined by its group velocity, vg . The group velocity is related to the group refractive index, ng , as, vg
c ng
1
with ng given by ng n ω
dn : dω
(2)
The first term in Eq. (2) is the refractive index and the second term is determined by dispersion. Under normal conditions, the dispersion dn∕dω is very small and ng ≈ n. Rapid variation in dn∕dω can produce extremely large, small, and negative group velocities. An increase in ng by increasing dn∕dω gives slow (subluminal) light, while fast (superluminal) light [4] is characterized by a group velocity beyond the speed of light in vacuum, c, or if ng is negative, even a negative group velocity [5]. A number of techniques have been used over recent years to slow down light. First, electromagnetically induced transparency [6,7] has been used widely in slow light experiments. In 1995, a group velocity of c∕165 was observed in a 10 cm Pb vapor cell by Kasapi et al. [8]. Experiments were carried out more rigorously since 1999 when Kash et al. [9], Hau et al. [10], and Budker et al. [11] measured group velocities of 90, 17, and 8 m∕s, respectively, by applying different methodologies. Slow light based on coherent population oscillation (CPO) has also been reported. Schwarz and Tan [12] were the first to predict CPO-based spectral holes and 0146-9592/13/224546-04$15.00/0
in 1983, Hillman et al. [13] observed this phenomenon for the first time via employing ruby. We note here that related coherent population trapping was also reported for ruby at room temperature [14]. Bigelow et al. [15] used the CPO-based spectral hole in ruby to generate slow light and observed a group velocity of 57 m∕s. We point out that drawing a distinction between slow light and a distortion of the pulse shape due to scattering remains rather subtle [16–18] as the whole notion of welldefined group velocity does not account for the detection of photons, which of course remains an essentially quantum-mechanical process. Ku et al. [19] reported a group velocity of 9600 m∕s in semiconductor quantumwell structures induced by CPO. Slow light based on (conventional) spectral hole-burning [20] was first proposed by Shakhmuratov et al. [21]. They also proposed a waveguide geometry to slow down light in an optical dense medium and presented calculations based on a linear filter theory [22]. Applying persistent hole-burning, Hahn and Ham [23] observed a group velocity of 75 m∕s in a Pr3 -doped Y2 SiO5 crystal. Subsequently, Lauro et al. obtained a group velocity reduction of 105 in Tm:YAG [24]. More recently, slow light induced by transient spectral hole-burning (THB) in ruby was observed and a group velocity of c∕1400 was reported [25]. THB involves the selective excitation of a subset of optical centers from the ground state to an excited state, which results in the depletion of the ground state at the laser frequency within the inhomogeneously broadened transition. The population storage in the excited state leads to a narrow spectral hole in the absorption spectrum, which can be read out by rapidly scanning the laser frequency. The spectral hole leads to low absorption and high dispersion at the laser frequency in the sample. There are a vast number of rare earth or transition metal ion-doped systems that display THB; moreover, THB allows the shaping of spectral holes, and © 2013 Optical Society of America
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
hence, there are numerous possibilities and opportunities for slow and fast light applications based on THB. Therefore, it is important to explore this potential. The present Letter reports our study of slow light in a laboratory grown pale-green emerald (Be3 Al2 Si6 O18 ∶Cr3 , 0.04 wt. %) by THB and its dependence on a low external magnetic field. The emerald crystal was cut parallel to the crystal c axis and polished to a path length of 2.5 mm. The optical properties of emerald have been studied for many years [26,27 and references therein]. The green color of emerald is due to the substitution of a small percentage of Al3 by Cr3 ions and the concentration of Cr3 determines the strength of the green color. The spin forbidden electronic transitions (ΔS ≠ 0) of Cr3 from the 4 A2 ground state to the lowest excited state 2 E (4 A2 → 2 E) are the most interesting features of emerald. At low temperatures, the radiative decay 4 A2 ←2 E results in two (relatively) sharp zero phonon lines, called R-lines, at ∼679 nm and ∼682 nm. The 4 A2 → 2 E transitions in emerald are strongly temperature dependent, subject to inhomogeneous broadening, and, naturally, display the Zeeman effect [27,28]. The crystal was cooled to 2.2 K by mounting the sample on the cold finger of a Janis/Sumitomo SHI-4.5 closed cycle refrigerator (CCR) using crycon grease. For polarized transmission spectra, the light from a 50 W tungstenhalogen light bulb was passed through a Spex 1402 monochromator (1200 grooves∕mm grating), a colored glass filter (FGL435), and a chopper blade (Thorlabs MC1000) before being focused onto the sample. A polarizer was also placed in front of the sample. The transmitted light was detected by a photomultiplier (Hamamatsu R928) and the signal was processed by a pre-amplifier (Femto DLPCA-200) and a lock-in amplifier (Stanford Research Systems SR810). Figure 1 shows the polarized absorption spectra for the R-lines (2 E←4 A2 transitions) for π and σ polarization. The inset shows the corresponding energy level diagram for the 4 A2 ground and 2 E excited states and indicates the π-polarized 2 E←4 A2 transitions used in the slow light
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experiments. The diagram also illustrates the linear Zeeman splitting upon the application of an external magnetic field B‖c for the 2 E and 4 A2 levels. We note that the apparent value of absorbance of 1.67 for the R1 line in π polarization was affected to a lesser extent by luminescence and to a greater extent by depolarization. Depolarization was clearly seen in the σ-polarized spectra since the R1 3∕2 line should be fully π polarized as unequivocally follows from spectra measured in α polarization [27]. The same experiment was conducted in nominally chrome-free emerald (0.0017 wt. % Cr3 ) to obtain the ratio between the R-lines; using this ratio an absorbance of 2.5 was calculated for the emerald sample at 2.2 K. We note here that the inhomogeneously broadened linewidth of the R1 3∕2) line was approximately 30 GHz. Slow light experiments by THB were conducted in the R1 3∕2 line in a zero field and for various low magnetic fields B‖c [25]. The 682.30341 nm laser light from an external cavity diode laser, ECDL (Toptica DL 100), was focused onto the emerald crystal by a 150 mm lens in π polarization. To avoid back reflection, an isolator was placed just after the ECDL. A pair of 250 mm diameter Helmholtz coils generated the external magnetic field B‖c. For pulse generation, an acousto-optic modulator, AOM (Isomet 1205C-1), was used, which was controlled by a Tektronix AFG3102 pulse generator. The widths of the burn and probe pulse were set to 200 μs and 125 ns, respectively, and the probe pulse was delayed by 2 μs. A 200 μs burn pulse yielded deep spectral holes. The probe pulse was detected by a Thorlabs photodiode (PDA 36A-EC) connected to a digital oscilloscope (LeCroy wave surfer 422). The averaged waveform was subsequently read out by a personal computer (PC). The pulse sequence and the corresponding transmitted laser light are shown in the inset of Fig. 2. We applied a linear filter theory to describe how the transient spectral hole in the absorption spectra causes a delay for a light pulse passing through the optical medium [22]. In our THB-based slow light experiments, a burn pulse created a deep spectral hole and a subsequent probe pulse was used to observe the resulting delay. If the probe pulse amplitude is so low that it has little effect on the excited state population, then the propagation of the probe pulse can be described by: E out ω GωE p ω;
(3)
where E p ω is the amplitude of the incident pulse in the frequency domain (as obtained from a Fourier transformation) and Gω is the complex frequency domain amplitude response function. These parameters are given by 1 E p ω 2π Gω Fig. 1. Polarized absorption spectra of the R-lines in emerald at 2.2 K. The inset shows the energy level diagram for the R-lines, and the arrows indicate the π-polarized R1 3∕2 transitions used for the slow light experiments.
Z
E p t0 exp−iωt0 dt0 ;
p T ω expiΔΦω:
(4)
(5)
The transmission spectrum, Tω, is calculated by using experimental parameters for the inhomogeneous width (30 GHz), the absorbance, A (2.5 before spectral hole-burning), the hole width (as determined by scanning
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OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Fig. 3. Dependence of pulse delay on B‖c. (1: 0.62; 2: 0.73; 3: 1.83; 4: 5.49; 5: 9.15; 6: 12.81; and 7: 15.2 mT.) The reference pulse is also shown normalized (dashed line) to the delayed pulse in B‖c 15.2 mT.
Fig. 2. Slow light in the R1 3∕2 line at 2.2 K for (a) zero field and (b) B‖c 15.2 mT. The probe pulse in the absence of holeburning is denoted as Ref. This pulse is also shown normalized (green dashed–dotted line) to the delayed pulse. The (red) dashed lines were calculated by the linear filter theory (see text). The inset shows schematically the pulse sequence (dashed lines) and the corresponding transmitted laser light (solid lines) for the experiment with (upper traces) and without (lower traces) a burn pulse.
the laser frequency after a burn period), and the hole depth, ΔA. With these empirical parameters the phase can be calculated by using the Hilbert transform given in Eq. (6) [29]: 1 ΔΦω π
Z
p In Tω0 0 0 dω . −∞ ω − ω ∞
(6)
Calculations were conducted by a Matlab code that incorporates equations (3–6). The line shapes of the probe pulse (in the time domain) and the spectral hole were approximated by Gaussian line shapes as obtained from nonlinear least squares fits of experimental data. Experimental results are compared with calculations based on the linear filter theory in Fig. 2. A delay of 40 ns corresponding to a group velocity of 63 km∕s was obtained in the zero field [Fig. 2(a)], while a group velocity of 16 km∕s was observed for a magnetic field B‖c of 15.2 mT [Fig. 2(b)]. Because of depolarization, the probe pulse intensity in the absence of the burn pulse was dominated (ca. 80%) by σ-polarized light that “leaks” through the crystal. This σ component also became visible as a tail at shorter delays in the B‖c 15.2 mT data. Figure 3 shows the delayed pulse for a range of low magnetic fields B‖c. The intensity of the delayed pulse seems to decrease initially with increasing magnetic
field. This may be caused by the rapid intensity redistribution with the side holes in the excited state due to spinlattice relaxation and field dependent spectral diffusion. The variation in delay and the corresponding group velocity with respect to the magnetic field B‖c is shown in Fig. 4(a), which clearly shows that the optical pulse delay can be smoothly tuned by varying the strength of a small external magnetic field B‖c. Figure 4(b) shows the corresponding hole-widths that were measured as is discussed below. An increase in magnetic field strength reduced the electron spin-electron spin flip–flop rate, leading to reduced hole-width. In order to determine the hole-width, Γhole , independently, spectral hole-burning experiments were conducted. The burn-probe pulse was controlled by three pulse generators in these experiments. The main pulse generator created a probe pulse with a width of 120 μs that was delayed by 100 μs from the burn pulse with a width of 200 μs. A second waveform generator provided delay and triggered a function generator that modulated the laser by a triangular wave. The spectral hole could only be read out at a delay of 100 μs after the burn pulse, and by this time, the hole was subject to decay and spectral diffusion. To eliminate the effects of spectral diffusion, the spectral hole-widths were extrapolated to zero delay by taking readouts at different delays (ramp frequency of 2500, 5000, or 7500 Hz). The laser jitter of 1 MHz was also taken into account. Importantly, we note here that the slow light experiment can be used as a very accurate spectroscopic method. In our slow light experiment, the delayed pulse can be read out at times as short as 0.5 μs after the burn pulse and hence, using the linear filter theory, it is possible to accurately determine spectral hole-widths as a function of time [red solid triangles in Fig. 4(b)]. This in turn can be used to measure the rate of spectral diffusion and other time-dependent phenomena. We note here that it is possible to probe spectral holes with broadband pulses. However, if the pulses are not coherent, then such probing has no inherent advantage over a direct measurement by scanning a narrowband laser. The key advantage of our coherent probing technique is related to the fact that we use not only the
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
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delay by light (depth ΔA of spectral hole) and by low magnetic fields (Γhole ), which, in turn, can be employed as a facile and highly accurate way to determine the spectral hole width. We have previously reported THB-based slow light in α polarization in ruby (Al2 O3 ∶CrIII) [25]. The present experiment shows much larger delays due to the fact that magnetic fields B‖c yielded much narrower hole-widths than in the particular ruby used in the previous experiments.
Fig. 4. (a) Pulse delay (red circles) and corresponding group velocity (blue squares) as a function of B‖c. The (green) solid triangles were calculated using experimental values for A, ΔA, Γhole , and the width of the pulse. The large error is due to the uncertainty in Γhole as determined by hole-burning. (b) Dependence of Γhole [used for calculations in (a)], as a function of B‖c as measured by hole-burning experiments and corrected for spectral diffusion and laser jitter (green solid circles). The (red) solid triangles show the hole-width obtained from the slow light experiment. The error bars show 10× the standard deviation.
absorption, but also the associated Kramers–Kronig related dispersion of the hole, which greatly increases the speed, accuracy, and fidelity of the measurement. Other pulse shapes could be used as well, provided that these pulses are phase coherent. However, Gaussianshaped pulses seem to be the most straightforward to implement and to analyze. The slow light experiment allowed for the accurate determination of hole-widths as a function of time. For example, the pulse delay for 15.2 mT of 154 ns can easily be measured to an accuracy of 0.2 ns, which yielded an uncertainty 0.01 MHz for a hole of about 4.8 MHz width. This accuracy is very hard to achieve in conventional hole-burning experiments. For narrower holes, e.g., sub-MHz, the slow light experiment becomes even more advantageous. In summary, we have generated a group velocity of 16 km∕s in emerald in a low magnetic field B‖c of 15.2 mT. Moreover, we were able to control the pulse
References 1. R. W. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2002), pp. 497–530. 2. T. F. Krauss, Nat. Photonics 2, 448 (2008). 3. R. W. Boyd, J. Mod. Opt. 56, 1908 (2009). 4. A. M. Akulshin, S. Barreiro, and A. Lezama, Phys. Rev. Lett. 83, 4277 (1999). 5. B. Segard and B. Macke, Phys. Lett. A 109, 213 (1985). 6. K. J. Boller, A. Imamolu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991). 7. J. E. Field, K. H. Hahn, and S. E. Harris, Phys. Rev. Lett. 67, 3733 (1991). 8. A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 74, 2447 (1995). 9. 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, Phys. Rev. Lett. 82, 5229 (1999). 10. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999). 11. D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, Phys. Rev. Lett. 83, 1767 (1999). 12. S. E. Schwarz and T. Y. Tan, Appl. Phys. Lett. 10, 4 (1967). 13. L. W. Hillman, R. W. Boyd, J. Krasinski, and C. R. Stroud, Opt. Commun. 45, 416 (1983). 14. R. Kolesov, Phys. Rev. A 72, 051801 (2005). 15. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Phys. Rev. Lett. 90, 113903 (2003). 16. A. C. Selden, Opt. Spectrosc. 106, 881 (2009). 17. V. S. Zapasskii and G. G. Kozlov, Opt. Spectrosc. 109, 407 (2010). 18. A. C. Selden, Opt. Express 18, 13204 (2010). 19. P.-C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S.-W. Chang, and S.-L. Chuang, Opt. Lett. 29, 2291 (2004). 20. H. Riesen, Coord. Chem. Rev. 250, 1737 (2006). 21. R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, Phys. Rev. A 71, 053811 (2005). 22. A. Rebane, R. N. Shakhmuratov, P. Mégret, and J. Odeurs, Phys. Rev. A 127, 22 (2007). 23. J. Hahn and B. S. Ham, Opt. Express 16, 16723 (2008). 24. R. Lauro, T. Chanelière, and J. L. Le Gouët, Phys. Rev. A 79, 063844 (2009). 25. H. Riesen, A. K. Rebane, A. Szabo, and I. Carceller, Opt. Express 20, 19039 (2012). 26. N. E. Rigby, N. B. Manson, L. Dubicki, G. J. Troup, and D. R. Hutton, J. Opt. Soc. Am. B 9, 775 (1992). 27. H. Riesen, Chem. Phys. Lett. 382, 578 (2003). 28. B. F. Hayward and H. Riesen, Phys. Chem. Chem. Phys. 7, 2579 (2005). 29. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972).
OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Controlling pulse delay by light and low magnetic fields: slow light in emerald induced by transient spectral hole-burning Rajitha Papukutty Rajan,1 Hans Riesen,1,* and Aleksander Rebane2,3 1
School of Physical, Environmental, and Mathematical Sciences, The University of New South Wales, UNSW Canberra, Canberra, ACT 2600, Australia 2 Physics Department, Montana State University, Bozeman, Montana 59715, USA 3
National Institute of Chemical Physics and Biophysics, Tallinn, Estonia *Corresponding author: [email protected]
Received July 4, 2013; revised September 5, 2013; accepted October 2, 2013; posted October 3, 2013 (Doc. ID 193112); published November 5, 2013 Slow light based on transient spectral hole-burning is reported for emerald, Be3 Al2 Si6 O18 ∶Cr3 . Experiments were conducted in π polarization on the R1 3∕2 line (2 E←4 A2 ) at 2.2 K in zero field and low magnetic fields B‖c. The hole width was strongly dependent on B‖c, and this allowed us to smoothly tune the pulse delay from 40 to 154 ns between zero field and B‖c 15.2 mT. The latter corresponds to a group velocity of 16 km∕s. Slow light in conjunction with a linear filter theory can be used as a powerful and accurate technique in time-resolved spectroscopy, e.g., to determine spectral hole-widths as a function of time. © 2013 Optical Society of America OCIS codes: (300.0300) Spectroscopy; (300.6240) Spectroscopy, coherent transient; (300.6320) Spectroscopy, high-resolution. http://dx.doi.org/10.1364/OL.38.004546
In recent years, the ability to produce slow light has generated considerable interest because of its potential in optical data storage, signal and quantum-data processing, optical switching, and pulse regeneration [1–3]. The propagation of an optical pulse through a medium is determined by its group velocity, vg . The group velocity is related to the group refractive index, ng , as, vg
c ng
1
with ng given by ng n ω
dn : dω
(2)
The first term in Eq. (2) is the refractive index and the second term is determined by dispersion. Under normal conditions, the dispersion dn∕dω is very small and ng ≈ n. Rapid variation in dn∕dω can produce extremely large, small, and negative group velocities. An increase in ng by increasing dn∕dω gives slow (subluminal) light, while fast (superluminal) light [4] is characterized by a group velocity beyond the speed of light in vacuum, c, or if ng is negative, even a negative group velocity [5]. A number of techniques have been used over recent years to slow down light. First, electromagnetically induced transparency [6,7] has been used widely in slow light experiments. In 1995, a group velocity of c∕165 was observed in a 10 cm Pb vapor cell by Kasapi et al. [8]. Experiments were carried out more rigorously since 1999 when Kash et al. [9], Hau et al. [10], and Budker et al. [11] measured group velocities of 90, 17, and 8 m∕s, respectively, by applying different methodologies. Slow light based on coherent population oscillation (CPO) has also been reported. Schwarz and Tan [12] were the first to predict CPO-based spectral holes and 0146-9592/13/224546-04$15.00/0
in 1983, Hillman et al. [13] observed this phenomenon for the first time via employing ruby. We note here that related coherent population trapping was also reported for ruby at room temperature [14]. Bigelow et al. [15] used the CPO-based spectral hole in ruby to generate slow light and observed a group velocity of 57 m∕s. We point out that drawing a distinction between slow light and a distortion of the pulse shape due to scattering remains rather subtle [16–18] as the whole notion of welldefined group velocity does not account for the detection of photons, which of course remains an essentially quantum-mechanical process. Ku et al. [19] reported a group velocity of 9600 m∕s in semiconductor quantumwell structures induced by CPO. Slow light based on (conventional) spectral hole-burning [20] was first proposed by Shakhmuratov et al. [21]. They also proposed a waveguide geometry to slow down light in an optical dense medium and presented calculations based on a linear filter theory [22]. Applying persistent hole-burning, Hahn and Ham [23] observed a group velocity of 75 m∕s in a Pr3 -doped Y2 SiO5 crystal. Subsequently, Lauro et al. obtained a group velocity reduction of 105 in Tm:YAG [24]. More recently, slow light induced by transient spectral hole-burning (THB) in ruby was observed and a group velocity of c∕1400 was reported [25]. THB involves the selective excitation of a subset of optical centers from the ground state to an excited state, which results in the depletion of the ground state at the laser frequency within the inhomogeneously broadened transition. The population storage in the excited state leads to a narrow spectral hole in the absorption spectrum, which can be read out by rapidly scanning the laser frequency. The spectral hole leads to low absorption and high dispersion at the laser frequency in the sample. There are a vast number of rare earth or transition metal ion-doped systems that display THB; moreover, THB allows the shaping of spectral holes, and © 2013 Optical Society of America
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
hence, there are numerous possibilities and opportunities for slow and fast light applications based on THB. Therefore, it is important to explore this potential. The present Letter reports our study of slow light in a laboratory grown pale-green emerald (Be3 Al2 Si6 O18 ∶Cr3 , 0.04 wt. %) by THB and its dependence on a low external magnetic field. The emerald crystal was cut parallel to the crystal c axis and polished to a path length of 2.5 mm. The optical properties of emerald have been studied for many years [26,27 and references therein]. The green color of emerald is due to the substitution of a small percentage of Al3 by Cr3 ions and the concentration of Cr3 determines the strength of the green color. The spin forbidden electronic transitions (ΔS ≠ 0) of Cr3 from the 4 A2 ground state to the lowest excited state 2 E (4 A2 → 2 E) are the most interesting features of emerald. At low temperatures, the radiative decay 4 A2 ←2 E results in two (relatively) sharp zero phonon lines, called R-lines, at ∼679 nm and ∼682 nm. The 4 A2 → 2 E transitions in emerald are strongly temperature dependent, subject to inhomogeneous broadening, and, naturally, display the Zeeman effect [27,28]. The crystal was cooled to 2.2 K by mounting the sample on the cold finger of a Janis/Sumitomo SHI-4.5 closed cycle refrigerator (CCR) using crycon grease. For polarized transmission spectra, the light from a 50 W tungstenhalogen light bulb was passed through a Spex 1402 monochromator (1200 grooves∕mm grating), a colored glass filter (FGL435), and a chopper blade (Thorlabs MC1000) before being focused onto the sample. A polarizer was also placed in front of the sample. The transmitted light was detected by a photomultiplier (Hamamatsu R928) and the signal was processed by a pre-amplifier (Femto DLPCA-200) and a lock-in amplifier (Stanford Research Systems SR810). Figure 1 shows the polarized absorption spectra for the R-lines (2 E←4 A2 transitions) for π and σ polarization. The inset shows the corresponding energy level diagram for the 4 A2 ground and 2 E excited states and indicates the π-polarized 2 E←4 A2 transitions used in the slow light
4547
experiments. The diagram also illustrates the linear Zeeman splitting upon the application of an external magnetic field B‖c for the 2 E and 4 A2 levels. We note that the apparent value of absorbance of 1.67 for the R1 line in π polarization was affected to a lesser extent by luminescence and to a greater extent by depolarization. Depolarization was clearly seen in the σ-polarized spectra since the R1 3∕2 line should be fully π polarized as unequivocally follows from spectra measured in α polarization [27]. The same experiment was conducted in nominally chrome-free emerald (0.0017 wt. % Cr3 ) to obtain the ratio between the R-lines; using this ratio an absorbance of 2.5 was calculated for the emerald sample at 2.2 K. We note here that the inhomogeneously broadened linewidth of the R1 3∕2) line was approximately 30 GHz. Slow light experiments by THB were conducted in the R1 3∕2 line in a zero field and for various low magnetic fields B‖c [25]. The 682.30341 nm laser light from an external cavity diode laser, ECDL (Toptica DL 100), was focused onto the emerald crystal by a 150 mm lens in π polarization. To avoid back reflection, an isolator was placed just after the ECDL. A pair of 250 mm diameter Helmholtz coils generated the external magnetic field B‖c. For pulse generation, an acousto-optic modulator, AOM (Isomet 1205C-1), was used, which was controlled by a Tektronix AFG3102 pulse generator. The widths of the burn and probe pulse were set to 200 μs and 125 ns, respectively, and the probe pulse was delayed by 2 μs. A 200 μs burn pulse yielded deep spectral holes. The probe pulse was detected by a Thorlabs photodiode (PDA 36A-EC) connected to a digital oscilloscope (LeCroy wave surfer 422). The averaged waveform was subsequently read out by a personal computer (PC). The pulse sequence and the corresponding transmitted laser light are shown in the inset of Fig. 2. We applied a linear filter theory to describe how the transient spectral hole in the absorption spectra causes a delay for a light pulse passing through the optical medium [22]. In our THB-based slow light experiments, a burn pulse created a deep spectral hole and a subsequent probe pulse was used to observe the resulting delay. If the probe pulse amplitude is so low that it has little effect on the excited state population, then the propagation of the probe pulse can be described by: E out ω GωE p ω;
(3)
where E p ω is the amplitude of the incident pulse in the frequency domain (as obtained from a Fourier transformation) and Gω is the complex frequency domain amplitude response function. These parameters are given by 1 E p ω 2π Gω Fig. 1. Polarized absorption spectra of the R-lines in emerald at 2.2 K. The inset shows the energy level diagram for the R-lines, and the arrows indicate the π-polarized R1 3∕2 transitions used for the slow light experiments.
Z
E p t0 exp−iωt0 dt0 ;
p T ω expiΔΦω:
(4)
(5)
The transmission spectrum, Tω, is calculated by using experimental parameters for the inhomogeneous width (30 GHz), the absorbance, A (2.5 before spectral hole-burning), the hole width (as determined by scanning
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OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Fig. 3. Dependence of pulse delay on B‖c. (1: 0.62; 2: 0.73; 3: 1.83; 4: 5.49; 5: 9.15; 6: 12.81; and 7: 15.2 mT.) The reference pulse is also shown normalized (dashed line) to the delayed pulse in B‖c 15.2 mT.
Fig. 2. Slow light in the R1 3∕2 line at 2.2 K for (a) zero field and (b) B‖c 15.2 mT. The probe pulse in the absence of holeburning is denoted as Ref. This pulse is also shown normalized (green dashed–dotted line) to the delayed pulse. The (red) dashed lines were calculated by the linear filter theory (see text). The inset shows schematically the pulse sequence (dashed lines) and the corresponding transmitted laser light (solid lines) for the experiment with (upper traces) and without (lower traces) a burn pulse.
the laser frequency after a burn period), and the hole depth, ΔA. With these empirical parameters the phase can be calculated by using the Hilbert transform given in Eq. (6) [29]: 1 ΔΦω π
Z
p In Tω0 0 0 dω . −∞ ω − ω ∞
(6)
Calculations were conducted by a Matlab code that incorporates equations (3–6). The line shapes of the probe pulse (in the time domain) and the spectral hole were approximated by Gaussian line shapes as obtained from nonlinear least squares fits of experimental data. Experimental results are compared with calculations based on the linear filter theory in Fig. 2. A delay of 40 ns corresponding to a group velocity of 63 km∕s was obtained in the zero field [Fig. 2(a)], while a group velocity of 16 km∕s was observed for a magnetic field B‖c of 15.2 mT [Fig. 2(b)]. Because of depolarization, the probe pulse intensity in the absence of the burn pulse was dominated (ca. 80%) by σ-polarized light that “leaks” through the crystal. This σ component also became visible as a tail at shorter delays in the B‖c 15.2 mT data. Figure 3 shows the delayed pulse for a range of low magnetic fields B‖c. The intensity of the delayed pulse seems to decrease initially with increasing magnetic
field. This may be caused by the rapid intensity redistribution with the side holes in the excited state due to spinlattice relaxation and field dependent spectral diffusion. The variation in delay and the corresponding group velocity with respect to the magnetic field B‖c is shown in Fig. 4(a), which clearly shows that the optical pulse delay can be smoothly tuned by varying the strength of a small external magnetic field B‖c. Figure 4(b) shows the corresponding hole-widths that were measured as is discussed below. An increase in magnetic field strength reduced the electron spin-electron spin flip–flop rate, leading to reduced hole-width. In order to determine the hole-width, Γhole , independently, spectral hole-burning experiments were conducted. The burn-probe pulse was controlled by three pulse generators in these experiments. The main pulse generator created a probe pulse with a width of 120 μs that was delayed by 100 μs from the burn pulse with a width of 200 μs. A second waveform generator provided delay and triggered a function generator that modulated the laser by a triangular wave. The spectral hole could only be read out at a delay of 100 μs after the burn pulse, and by this time, the hole was subject to decay and spectral diffusion. To eliminate the effects of spectral diffusion, the spectral hole-widths were extrapolated to zero delay by taking readouts at different delays (ramp frequency of 2500, 5000, or 7500 Hz). The laser jitter of 1 MHz was also taken into account. Importantly, we note here that the slow light experiment can be used as a very accurate spectroscopic method. In our slow light experiment, the delayed pulse can be read out at times as short as 0.5 μs after the burn pulse and hence, using the linear filter theory, it is possible to accurately determine spectral hole-widths as a function of time [red solid triangles in Fig. 4(b)]. This in turn can be used to measure the rate of spectral diffusion and other time-dependent phenomena. We note here that it is possible to probe spectral holes with broadband pulses. However, if the pulses are not coherent, then such probing has no inherent advantage over a direct measurement by scanning a narrowband laser. The key advantage of our coherent probing technique is related to the fact that we use not only the
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delay by light (depth ΔA of spectral hole) and by low magnetic fields (Γhole ), which, in turn, can be employed as a facile and highly accurate way to determine the spectral hole width. We have previously reported THB-based slow light in α polarization in ruby (Al2 O3 ∶CrIII) [25]. The present experiment shows much larger delays due to the fact that magnetic fields B‖c yielded much narrower hole-widths than in the particular ruby used in the previous experiments.
Fig. 4. (a) Pulse delay (red circles) and corresponding group velocity (blue squares) as a function of B‖c. The (green) solid triangles were calculated using experimental values for A, ΔA, Γhole , and the width of the pulse. The large error is due to the uncertainty in Γhole as determined by hole-burning. (b) Dependence of Γhole [used for calculations in (a)], as a function of B‖c as measured by hole-burning experiments and corrected for spectral diffusion and laser jitter (green solid circles). The (red) solid triangles show the hole-width obtained from the slow light experiment. The error bars show 10× the standard deviation.
absorption, but also the associated Kramers–Kronig related dispersion of the hole, which greatly increases the speed, accuracy, and fidelity of the measurement. Other pulse shapes could be used as well, provided that these pulses are phase coherent. However, Gaussianshaped pulses seem to be the most straightforward to implement and to analyze. The slow light experiment allowed for the accurate determination of hole-widths as a function of time. For example, the pulse delay for 15.2 mT of 154 ns can easily be measured to an accuracy of 0.2 ns, which yielded an uncertainty 0.01 MHz for a hole of about 4.8 MHz width. This accuracy is very hard to achieve in conventional hole-burning experiments. For narrower holes, e.g., sub-MHz, the slow light experiment becomes even more advantageous. In summary, we have generated a group velocity of 16 km∕s in emerald in a low magnetic field B‖c of 15.2 mT. Moreover, we were able to control the pulse
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