A study of pulse shape in electromagnetically induced transparency based slow light

Physics Letters A 374 (2010) 2183–2187

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Physics Letters A www.elsevier.com/locate/pla

A study of pulse shape in electromagnetically induced transparency based slow light Wenzhuo Tang, Bin Luo, Yu Liu, Hong Guo ∗ CREAM Group, State Key Laboratory of Advanced Optical Communication Systems and Networks (Peking University) and Institute of Quantum Electronics, School of Electronics Engineering and Computer Science (EECS), Peking University, Beijing 100871, China

a r t i c l e

i n f o

Article history: Received 12 January 2010 Received in revised form 26 February 2010 Accepted 5 March 2010 Available online 11 March 2010 Communicated by R. Wu Keywords: Electromagnetically induced transparency Slow light Pulse shape Time delay Broadening

a b s t r a c t Slow light of four different pulses are demonstrated and analyzed in cesium vapor based on electromagnetically induced transparency. Each pulse, generated with an identical temporal width, experiences very different slow light effects due to its temporal and spectral distributions simultaneously. Aiming at the applications such as timing, instead of communication, we obtained two optimized ones among the four different shaped pulses. Firstly, the single-exponential pulse is more appropriate to maximize the time delay than other three pulses; secondly, the cosine pulse shows advantage to minimize the distortion (broadening) of the slowed pulse. Finally, using optical filter model, we present a convenient simulation method in theory; moreover, the theoretical results show excellent agreement with experiments. © 2010 Elsevier B.V. All rights reserved.

Slow light has drawn significant attention over the past few decades for its numerous potential applications in all-optical processing, such as optical buffer and data resynchronization [1]. Usually, the bandwidth–delay product (BDP) is considered as the key performance for slow light systems [2]. Therefore, various schemes and methods were proposed to improve the BDP of slow light systems [3–6]. Note that, in a real system, the BDP is often limited by the distortion (broadening) of the slowed optical pulse, especially when its spectral width is comparable to or larger than the system bandwidth [7]. Usually, a Gaussian shaped pulse was used to demonstrate the slow light. Nevertheless, it is worth noting that some non-Gaussian pulses (shaped by non-Gaussian functions) can show very different characteristics (especially the BDP) in slow light. For example, S. Chin and L. Thevenaz studied the slow light of a Gaussian and two non-Gaussian shaped pulses in Brillouin fiber system [8]. This work shows that the single-exponential pulse obtains the largest time delay and suffers smaller broadening than the Gaussian pulse in the slow light system based on stimulated Brillouin scattering. However, in their experiment the temporal duration of pulses was so short [with full width at half maximum (FWHM) ∼ 14 ns] that the sharply rising/falling edges of generated rectangular and single-exponential pulses experienced severe distortions, which is

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caused by the limited response bandwidth of optoelectronic devices. These distortions can be clearly found in Fig. 3 of [8], where the sharply rising/falling edges are significantly smoothed with duration of ∼ 10 ns. Obviously, it can minimize the influence of limited response bandwidth of optoelectronic devices by using longer pulses in experiment. Therefore, we demonstrated the slow light of four different pulses (all with the typical shapes and FWHMs of 2 μs) using an EIT window with FWHM of several hundred kilohertz. If the delay line is used for timing applications, the information would be extracted when the intensity of the arrived pulse reaches a preset level (typically, half of its peak value). So, we characterize the time delay and broadening (distortion) of pulse according to its peak and FWHM values, respectively. In this way, in a single-Lorentzian EIT system in cesium vapor, we obtained two optimized pulses both in experiment and in theory. In experiment, four commonly used functions (Gaussian, cosine, double-exponential and single-exponential) were selected to shape the amplitudes (thus the intensities) of the input probe pulses. Their temporal (spectral) amplitudes E (t ) [E (ω)] and time– bandwidth products K are shown in Table 1. In addition, the spectral intensity profiles of four pulses are shown in Fig. 1. Generally speaking, when the spectral bandwidth (∼ 1/τ ) of pulse is much smaller than the width of EIT window, it would experiences almost no broadening and meanwhile the time delay of pulse has nothing to do with its shape. By contrast, when its spectral bandwidth is comparable to or even larger than the width of EIT window, both the time delay and distortion of pulse will severely depend on its

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Table 1 Temporal amplitudes and (Fourier) spectral amplitudes of four pulses. Pulse shape

E in (t ) (normalized) t2

E in (ω) 2 2

Gaussian

exp(− τ 2 )

cosine

cos( πτt ), |t |  τ2

double-exponential

exp(− τ )

√τ exp(− ω τ 4 2 √ τ 2π cos(ωτ /2) 2 2 2 √π −ω τ τ 2/π 1+ω2 τ 2

single-exponential

u (t ) exp(− τt )

√

|t |

iτ 2π (i +ωτ )

)

ν (kHz)

t (μs)

K

220.5

2

0.441

297.5

2

0.595

71.0

2

0.142

55.0

2

0.110

u (t ) is the unit step function at the origin, and the time–bandwidth product K = ν · t of pulses are calculated in the transform-limited case. ν and t are defined as the FWHMs of the spectral and temporal intensity distributions of pulse, respectively.

Fig. 1. Normalized intensity spectrum of pulses. These four shaped pulses, possessing an identical intensity temporal FWHM width ∼ 2 μs, show different spectrums. υ is the FWHM of intensity spectral distribution.

shape. Moreover, with a certain constraint of broadening (distortion), the corresponding BDP will reach its maximum in the latter case. According to its time–bandwidth product (K ) of 0.441, a Gaussian pulse with 2 μs intensity FWHM possesses the spectral FWHM of 220 kHz [which is a typical EIT window bandwidth in Fig. 2(c)]. The similar setup and atomic scheme with [9] are used in experiment as shown in Fig. 2(a) and (b), and all the four pulses are shaped with the intensity FWHM of 2 μs. A frequency-stabilized 852 nm external cavity diode laser is tuned to the 133 Cs D2 transition: F = 3 → F  = 2, 4. The laser with beam diameter of ∼ 2.0 mm is divided into the coupling and probe by PBS1. Through AOM1 and AOM2 respectively, two lasers are red-shifted ∼ 175 MHz to be resonant with transition F = 3 → F  = 2 as shown in Fig. 2(b). A weak (< 10 μW) continuous-wave laser is shaped as the input probe pulse and its temporal amplitude E (t ) is precisely determined by the arbitrary waveform generator (AWG). After being combined at PBS2, the overlapped coupling and probe lasers co-propagate through the two serial cesium cells placed in the MS at room temperature. Therein, a certain portion of the input probe is split to APD1 as reference, and its intensity is set

to be equal with the output probe when the laser frequency is fardetuned from all the atomic transitions. Therefore, we can treat the reference as input signal in the following analysis to eliminate the background absorption caused by experimental apparatus. At last, the coupling laser is filtered out by PBS3 and then only the slowed probe pulse is detected by APD2. By decreasing the coupling power from 0.78 mW to 0.14 mW, we obtained the EIT window with decreasing bandwidth, as shown in Fig. 2(c). Then, the corresponding slow light of four pulses are obtained as shown in Fig. 3, when the coupling powers are tuned at 0.78 mW, 0.61 mW, 0.45 mW, 0.27 mW, and 0.14 mW, respectively. Here, all the experimental curves of reference and slowed pulses are recorded as an average of 16 measured values, so the noises are very small. To characterize the pulse broadening, we defined the broadening factor as the FWHM ratio of the output pulse to the input one. We compared both the time delay and broadening factor of the four pulses as shown in Fig. 4. For all pulses, both the larger time delay [Fig. 4(a)] and larger broadening factor [Fig. 4(b)] are obtained with the narrower EIT window using the smaller coupling power. However, a properly shaped pulse can show specific advantages, e.g., the large time delay or the small pulse broaden-

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Fig. 2. (a) Experiment setup. λ/2, half-wave plate; PBS, polarizing beam splitter; AOM, acousto-optic modulator; AWG, arbitrary waveform generator; MS, magnetic shield; APD, avalanche photodetector. (b) Schematic configuration of a single-Λ type 133 Cs atom interacting with the coupling and probe lasers. Ground states |x and | y  represent two magnetic sublevels. (c) Linear fitting of EIT bandwidth versus the coupling power. In the inset, the bandwidth (FWHM) of EIT window is obtained by a Lorentzian fitting.

Fig. 3. Normalized time traces (an average of 16 measured results) of reference and slow light of four pulses with an identical intensity temporal width (FWHM ∼ 2 μs), the coupling powers are varied at 0.78 mW, 0.61 mW, 0.45 mW, 0.27 mW, and 0.14 mW, respectively.

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Fig. 4. The experimental results of (a) time delay and (b) broadening factor of four different shaped pulses are plotted as the power of coupling laser (determines the EIT bandwidth) increases.

ing. Concretely speaking, the single-exponential pulse achieves the largest delay of 2.78 μs (corresponds to the BDP of 1.39); and at the same time, it experiences the largest broadening [much larger than Gaussian pulse, see Fig. 4(b)]. This result is very different from [8] where the single-exponential pulse experiences smaller broadening than the Gaussian one, even though the Brillouin gain spectrum is also considered as the Lorentzian profile. Obviously, this divergence is mainly due to that the duration of the sharply rising edge of single-exponential pulse in [8] is too long (∼ 10 ns for a ∼ 14 ns FWHM pulse). That is to say, the input pulse has been significantly smoothed when it is generated so that the broadening factor of output pulse is reduced compared to the ideal case. Moreover, in the following, it will be shown that our experimental results agree with the theoretical results well. Therefore, it is reasonable to say that our results are more accurate. This is mainly owing to the much longer pulses (with 2 μs FWHM) used in our experiment, where the input pulses can be generated with higher fidelity (less distortion). In fact, the rising edge of the 2 μs singleexponential pulse is only approximately several tens of nanoseconds in our experiment, as shown in Fig. 3. On the other hand, the cosine pulse experiences an extremely small broadening factor, which is typically 1.03 (while 1.00 means no broadening) and is still less than 1.09 even when its intensity spectral bandwidth (FWHM ∼ 297.5 kHz) is larger than the width of EIT window [with ∼ 220 kHz FWHM at the coupling power of 0.14 mW, see Fig. 2(c)]. Suppose that the EIT medium can be treated as a narrow bandpass optical filter, the optical pulse passing through it will experience different effects due to its temporal and spectral distributions simultaneously. Due to the single Lorentzian EIT spectrum [10], we used a unitary transfer function to describe this EIT-based slow light system as

 H (ω) =

a

2

ω + ia

,

where 2a represents the FWHM of EIT window and the amplitude (phase) of H (ω) corresponds to the absorption (dispersion) of this EIT system. Then, we analyze the time delay and broadening of a pulse after passing through this optical filter. For the singleexponential pulse, its input temporal and spectral amplitudes (see Table 1) are

E in (t ) = u (t ) · e −t /τ

(t > 0) and E in (ω) = √

iτ 2π (i + ωτ )

,

where the time parameter satisfies τ > 0. Then, we obtained the spectral and temporal amplitudes of the output pulse as

 E out (ω) = H (ω) E in (ω) =

a

ω + ia

2 √

iτ 2π (i + ωτ )

,

and

1

E out (t ) = √

 =

2π

∞ E out (ω)e i ωt dω −∞

a2 τ 2 [et /τ (at +1− τt )−eat ]

0,

(aτ −1)2

, t > 0; t < 0.

Since the temporal intensity of pulse is I in (t ) = | E in (t )|2 = u (t ) · e −(2t /τ ) and its FWHM equals 2 μs, so that the time parameter is chosen as τ = 5.77 μs. In our experiment, the bandwidth 2a of EIT window varies from 220 kHz to 482 kHz, which almost linearly corresponds to the coupling power varies from 0.14 mW to 0.78 mW [see Fig. 2(c)]. The simulation results for the single-exponential pulse are shown in Fig. 5, whose inset shows that the time delay and broadening factor decrease as the FWHM of EIT window increases (corresponds to the coupling power increases linearly). Compared with the experimental results in Figs. 3 and 4, the simulation results showed excellent agreement, except that both the time delay and broadening factor are a little bit larger than that of experimental results. This is due to that the single-exponential pulse used in experiment is generated with the non-ideal rising edge (with several tens of nanoseconds), which is caused by the limited response bandwidth (thus response time) of the optoelectronic devices. Obviously, the experimental results can be improved to agree with theory better by using devices with larger response bandwidth. In addition, a similar simulation was applied to the cosine pulse. The results showed that the cosine pulse experiences a moderate time delay with extremely small broadening, as shown in Fig. 6. Compared with the corresponding experimental results in Figs. 3 and 4, it showed that the simulation results indeed exhibit an almost perfect agreement with the experiment, which is much better than the case for single-exponential pulse. This is because that the input cosine pulse can be generated with much higher fidelity (less distortion) in experiment owing to its smoothly rising and falling edges. In this Letter, the isolated input pulses are used for the timing applications, while the interferences between adjacent pulses do not exist and the FWHM values of pulses are considered. So, it is distinct from the study on how to optimize the pulse shape of a continuous data stream for communication applications [11]. We studied the dependence of time delay and broadening of pulses on their own shapes in a single-Lorentzian EIT system. Consequently, with excellent agreement between experiment and theory, four typical pulses exhibit significantly different characteristics due to their temporal and spectral distributions simultaneously. In other words, the EIT medium acts as a narrow bandpass optical filter, so it changes the spectral distribution and especially smoothes

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Fig. 5. Left: slow light simulation of the single-exponential pulse. All the slowed pulses are normalized. Right: time delay (top) and broadening factor (bottom) versus the FWHM bandwidth of EIT (2a), respectively.

Fig. 6. Left: slow light simulation of the cosine pulse. All the slowed pulses are normalized. Right: time delay (top) and broadening factor (bottom) versus the FWHM bandwidth of EIT (2a), respectively.

the sharply rising/falling edges of the output (slowed) pulses. For the single-exponential pulse, its sharply rising edge is significantly smoothed and hence the pulse peak is delayed more. Therefore, it exhibits the largest broadening and the largest time delay. Further, we have confirmed that the single-Gaussian and single-cosine pulses (with sharply rising edge) also show similar characteristics in experiment. By comparison, with both smoothly rising and falling edges, the cosine/Gaussian/double-exponential pulses experience both moderate delay and broadening. As a special case, the double-exponential pulse shows a relatively small delay and large broadening owing to the discontinuity at peak. Among them, the cosine pulse shows the smallest broadening, and so it is feasible to improve the time delay through using multiple delay lines serially, e.g., using two serial cells in this Letter. To summarize, we presented a convenient method to study the pulse shape in slow light and obtained two optimized pulses for the large delay and small broadening, respectively. Further, the theoretical model of optical filter is extremely convenient to simulate and understand the slow light, and shows excellent agreement with the experiments. Note that, the analytical solutions of (inverse) Fourier transformation do not exist in some cases, and then the numerical solutions are needed. Finally, in principle, all meth-

ods and conclusions presented in this Letter can be extended to other slow light systems. Acknowledgement This work was supported by the Key Project of the National Natural Science Foundation of China (Grant No. 60837004). References [1] R.W. Boyd, D.J. Gauthier, ‘Slow’ and ‘Fast’ Light, in: E. Wolf (Ed.), Progress in Optics, vol. 43, Elsevier, Amsterdam, 2002, pp. 497–530, Chapter 6. [2] R.W. Boyd, D.J. Gauthier, A.L. Gaeta, A.E. Willner, Phys. Rev. A 71 (2005) 023801. [3] Z. Deng, D. Qing, P. Hemmer, C.H. Raymond Ooi, M.S. Zubairy, M.O. Scully, Phys. Rev. Lett. 96 (2006) 023602. [4] Z. Shi, R.W. Boyd, Phys. Rev. A 79 (2009) 013805. [5] M.D. Stenner, M.A. Neifeld, Z. Zhu, A.M. Dawes, D.J. Gauthier, Opt. Express 13 (2005) 9995. [6] R. Pant, M.D. Stenner, M.A. Neifeld, D.J. Gauthier, Opt. Express 16 (2008) 2764. [7] R.W. Boyd, P. Narum, J. Mod. Opt. 54 (2007) 2403. [8] S. Chin, L. Thevenaz, Opt. Lett. 34 (2009) 707. [9] W. Tang, B. Luo, Y. Liu, H. Guo, IEEE J. Quantum Electron. 46 (2010) 579. [10] R.M. Camacho, M.V. Pack, J.C. Howell, A. Schweinsberg, R.W. Boyd, Phys. Rev. Lett. 98 (2007) 153601. [11] M.D. Stenner, M.A. Neifeld, Opt. Express 16 (2008) 651.