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Slow Light of an Amplitude Modulated Gaussian Pulse in Cesium Vapor

Wenzhuo Tang, Bin Luo, Yu Liu, and 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

∗Corresponding author. Tel.: +86 10 62757035; fax: +86 10 62753208. E-mail address: hongguo@pku.edu.cn (H. Guo).

Abstract

Slow light of an amplitude modulated Gaussian (AMG) pulse incesium vapor is demonstrated and studied, as an appropriateamplitude modulation to a single pulse can expand its spectrum and thus increase the utilization efficiency of the bandwidth of aslow light system. In a single-Λ type electromagnetically induced transparency (EIT) system, the slowed AMG pulse experiencessevere distortion, mainly owing to the frequency dependenttransmission of medium. Additionally, due to its spectral distribution,the frequency dependent dispersion of the medium causes simultaneous slow and fast light of different spectral components and thusa certain dispersive distortion of the AMG pulse. Further, apost-processing method is proposed to recover the slowed (distorted)pulse, which indicates that by introducing a linear opticalsystem with a desired gain spectrum we can recover the pulse in an“all-optical” way. Finally, we discuss the limitations during this compensation procedure in detail. Although it is demonstrated inthe cesium vapor using EIT, this method should be applicableto a wide range of slow light systems.

Key words: Electromagnetically induced transparency (EIT), slow light, Gaussian pulse, distortion, cesium vapor, amplitudemodulation, delay line.PACS: 42.50.Gy, 42.50.Ct, 42.30.Lr

1. Introduction

Recently, slow light has attracted tremendous attention for itspotential applications in optical communication. Group veloc-ity of an optical pulse can be slowed down using various ef-fects, such as electromagnetically induced transparency (EIT)[1, 2], double absorption line [3, 4], coherent population oscil-lations [5, 6, 7], stimulated Brillouin scattering [8, 9] and stim-ulated Raman scattering [10], etc. Note that, most of the slowlight experiments are performed using a Gaussian pulse, whilea few non-Gaussian (such as rectangularly [11] and exponen-tially [12] shaped) pulses are studied. These studies are basedon the pulse shaping, while an additional amplitude modulationto the pulse has not been well investigated yet. Moreover, thestudy on the slow light of a pulse with more complex spectrumshould be very promising to the applications. So, we believethat an amplitude modulated pulse has a potential applicationin the optical communication. In this paper, we demonstratethe slow light of a cosine-type AMG pulse experimentally, andstudy both the absorptive and the dispersive properties of thispulse in depth.

2. Experiment

Our experimental setup is schematically illustrated in Fig.1(a). An 852 nm narrowband (∼300 kHz) diode laser (out-put power∼30 mW) is stabilized to theD2 transition of133Csatom: F = 3 → F′ = 2, 4 crossover. By the combinationof λ/2 (half-wave plate) and PBS1, the laser is divided into

two beams, the coupling and the probe, with perpendicular po-larizations and adjustable proportions. The frequency of cou-pling laser is redshifted∼175 MHz by AOM2, with the power∼2.35 mW and 1/e2 beam diameter∼2.0 mm. The probe pulseis generated by passing a weak (<10µW) continuous-wavelaser through AOM1, with the frequency redshifted∼175 MHzand 1/e2 beam diameter∼2.0 mm. The amplitude of probe fieldE(t) (nonnegative for convenience in data processing) is propor-tional to the amplitude of the driving RF (radio frequency) waveto AOM1, which is directly controlled by an arbitrary wave-form generator (AWG). After that, both the coupling (verticallypolarized) and the probe (horizontally polarized) lasers are res-onant with theD2 transitionF = 3 → F′ = 2 of 133Cs atom[see Fig. 1(b)], and combined at PBS2. Then, the two over-lapped lasers copropagate through the cell in order to reducethe total Doppler broadening [13]. The beam splitter (BS) infront of the cell splits an appropriate portion of the probe toAPD1 as reference, whose intensity is set to be equal to that ofthe output when the probe is detuned far off-resonant from allatomic transitions. Thus, the background absorption in experi-ment is eliminated and the reference can be treated as “input”in the following analysis. A 10 cm-long paraffin-coated cesium(133Cs) vapor cell at room temperature (∼25 C) is placed inthe magnetic shield (MS) to screen out the earth magnetic fieldin lab. The exiting beams are separated by PBS3 and only theslowed probe pulse beam reaches APD2 as the “output” in thefollowing analysis.

In experiment, both the optical intensitiesI(t) of the refer-ence (treated as “input”) and output pulses are recorded on a

Preprint submitted to Optics Communications December 17, 2018

R f

PD1

Probe

PD2

Diode

laserλ/2

PBS1

AOM1

PBS2

Reference MS

C li

PBS3

Probe

BS

AWG

C C ll

Mirror

AOM1

AOM2

CouplingCs Cell

Mirror

F = 2, 4

Crossover~175 MHz 50

60

70

(%)

F = 2

Coupling Probe

175 MHz

20

30

40

50

ran

sm

iss

ion

F= 3

Coupling Probe

| >y| >x-1000 -500 0 500 10000

10Tr

Detuning kHz( )

(a)

(b) (c)

Figure 1: (color online). (a) Experimental setup.λ/2, half-wave plate; PBS, po-larizing beam splitter; AOM, acousto-optic modulator; AWG, arbitrary wave-form generator; MS, magnetic shield; APD, avalanche photodetector. (b) Asimplified single-Λ type system of133Cs atom interacts with the coupling andthe probe lasers. Two ground states|x〉 and |y〉 represent appropriate superpo-sitions of magnetic sublevels. (c) Probe transmission spectrum (an average of16 measured results) versus detuning from the two-photon resonance, with theFWHM bandwidth of∼350 kHz and the maximum (background) transmissionof ∼61.5% (∼10%).

100 MHz digital oscilloscope triggered by the AWG. Thus, theamplitudes of both input and output (slowed) pulses can be ob-tained from this relation:E(t) =

√I(t), which is the reason of

requiring the amplitudes of probe pulses to be nonnegative inexperiment. The intensities of two probe (input) pulses in timedomain are chosen as

Gaussian :I1(t) = |E1(t)|2 = exp

− (ln 2)t2

T 20

,

AMG : I2(t) = |E2(t)|2 = |E1(t) [1 + A cos(2πδt)] |2

= exp

− (ln 2)t2

T 20

[1 + A cos(2πδt)]2 ,

whereI1(t) and I2(t) are the intensities of two pulses, respec-tively; δ (= 700 kHz) is the modulation frequency,A (= 1) isthe modulation depth, andT0 (≈ 6.5 µs) is the temporal dura-tion (FWHM) of the intensity of the Gaussian pulse.

Experimental results are shown in Fig. 2. The Gaussian andthe AMG pulses exhibit quite different slow light effects. Inthe Gaussian case, the pulse [Fig. 2(a)] is delayed by∼0.47µs,with relatively low loss and little pulse distortion. Hence, wecan compensate for the loss by directly amplifying the inten-sity of the pulse with a constant magnitude determined by thesystem. However, in the AMG case [Fig. 2(b)], the outputpulse undergoes relatively high loss and significant distortion[meanwhile the delay time (∼0.14µs) is obviously decreased].Thus, the direct amplification is not feasible any more [see the“rescaled” pulse in Fig. 2(b)].

3. Distortion due to absorption and dispersion

It is known that the pulse distortion is caused by both ab-sorption and dispersion [14]. In an ideal EIT medium, whenthe coupling field (Ω) keeps unchanged both temporally andspatially, the frequency response function of the slow light sys-tem can be expressed asH(∆) = exp[−∆Z/(∆ − iΓEIT )], whereΓEIT = |Ω|2/γ (proportional to the intensity of coupling laser)is the bandwidth of EIT spectrum,Z (proportional to the atomicnumber and cell length) is the normalized propagation length,and∆ is the detuning of probe laser. Suppose the EIT slow-lightsystem is linear, we have

Eout(∆) = H(∆)Ein(∆)

= exp

[

−∆Z∆ − iΓEIT

]

Ein(∆)

= exp

[

−∆2γ2Z

∆2γ2+ |Ω|4

]

exp

[

−i∆γZ|Ω|2

∆2γ2+ |Ω|4

]

Ein(∆)

≡ A(∆)e−iΦ(∆)Ein(∆),

where Ein(∆)[Eout(∆)] represents the amplitude spectrumof the input (output) pulse. Real functionsA(∆) ≡exp[

−∆2γ2Z/(∆2γ2+ Ω

4)]

[corresponding to the measuredtransmission spectrum in Fig. 1(c)] andΦ(∆) ≡∆γZ|Ω|2/(∆2γ2

+ |Ω|4) are related to the absorption and disper-sion, respectively.

3.1. Absorptive distortion

In this section, we analyze the “absorptive distortion” us-ing the transmission spectrum [see Fig. 1(c)]. The intensityspectrums of the input and output pulses are calculated out us-ing discrete Fourier transform (DFT) based on the experimen-tal results. It can be seen that the intensity spectrum of theGaussian pulse keeps unchanged with the FWHM bandwidthof ∼74.4 kHz [Fig. 3(a)]; by contrast, the intensity spectrum ofthe AMG pulse has not only a resonant Gaussian component,but also two non-resonant Gaussian sidebands with±700 kHzdetunings symmetrically [Fig. 3(b)]. Since the amplitudesofthe optical fieldE(t) and the spectrumE(∆) satisfy the Fouriertransform relation, this effect of generating sidebands can beanalyzed through the Fourier intensity spectrums of the twoin-put pulses:

Gaussian :I1(∆) = |E1(∆)|2 = exp

− (ln 2)∆2

Ω20

,

AMG : I2(∆) = |E2(∆)|2

= I1(∆) +A2

4[I1(∆ − δ) + I1(∆ + δ)] ,

whereI1(∆) andI2(∆) are the intensity spectrums of two pulses,respectively; (A2/4) is the relative intensity of the two sidebandcomponents;Ω0 (determined byT0) is the FWHM bandwidthof intensity spectrum of the Gaussian pulse. From the spectrum[I2(∆)] of the AMG pulse, we can see that the “oscillation” of

2

intensity in time domain [I2(t), in Fig. 2(b)] is due to the inter-ference of its three spectral components:I1(∆), I1(∆ − δ), andI1(∆ + δ).

Note that, we haveIout(∆) = |A(∆)|2 · Iin(∆), where the phaseinformation is eliminated, and|A(∆)|2 is just the transmissionspectrum of the system [see Fig. 1(c)]. Three spectral compo-nents experience different transmission rates according to thedetuning (∆). In the AMG case, two sideband componentsof the pulse undergo much more severe absorptions than theresonant one. Thus, a compensation method is proposed torecover the output slowed pulse. As shown in Fig. 3, weobtain the compensated spectrums by simply amplifying theoutput spectrums according to the transmission spectrum, i.e.,Icomp(∆) = Iout(∆)/|A(∆)|2measured. As the compensated and theinput spectrums are well overlapped in Fig. 3, we can see thatit is the frequency dependent absorption that mainly causesthedistortion of the AMG pulse owing to the different loss of threespectral components.

(b)

(a)

Figure 2: (color online). Input (normalized) and output pulses in time domain.Each pulse is an average of 16 measured results, and the “rescaled” is the nor-malization of the “output”. (a) Slow light of the Gaussian pulse with low lossand little distortion. Pulse delay time∼0.47µs is measured according to thepeak of the pulse. (b) Slow light of the cosine-type AMG pulsewith high lossand significant distortion. Pulse delay time∼0.14µs is measured according tothe highest peak of the pulse.

It should be mentioned that, the noise is not considered in thispaper for its limited influence. Further, for a good compensa-tion, in experiment we use a scanning probe laser with slightlyhigher intensity to obtain the transmission spectrum with ahighSNR (signal-noise-ratio), which also causes the∼10% “back-ground transmission” as shown in Fig. 1(c). Note that, on theother hand, the probe laser (<10µW) is weak enough comparedto the intensive coupling laser (∼2.35 mW) and its influence onthe transmission spectrum can be ignored. To further confirmthe compensation result, we transform the compensated spec-trum back to time domain using the inverse discrete Fouriertransform (IDFT), and plot the input and the recovered pulses in

Figure 3: (color online). The intensity spectrums of the input (black solid),output (red solid) and compensated (blue solid with bar) pulses, and the trans-mission spectrum. (a) Gaussian case. The compensated spectrum overlaps withthe input one. (b) AMG case. The compensated spectrum overlaps with theinput one only with minor deviations.

time domain together as shown in Fig. 4. The recovered pulsesof Gaussian and AMG, with different time delays, are both ingood agreement with their corresponding input pulses in shapeand intensity. Especially in the AMG case, even with the sig-nificant loss and distortion, both the intensity and the shape ofthe slowed pulse are well recovered. In addition, although thispresented method is a post-processing one, a linear opticalsys-tem with the corresponding gain spectrum is able to achieve thisrecovered result in an “all-optical” way.

3.2. Dispersive dispersion

The different spectral components, according toΦ(∆), alsoexperience different phase shift, which leads to different dis-persion and introduces a certain distortion that is not severein our case. By separately transforming each spectral compo-nent back to time domain using IDFT, we obtain three Gaussianpulses with absolutely different “time-delays”. For the reso-nant component, we obtain the “slow light” of a Gaussian pulse[∼0.47µs, see Fig. 5] with an identical time delay to the non-modulated Gaussian one [∼0.47µs, see Fig. 4(a)]. On the con-trary, for each sideband (left or right) component, we obtaina Gaussian pulse even with a little time advancing [∼0.18µsor ∼0.20µs, see Fig. 5], namely “fast light”. It should bepointed out that this phenomenon, called “simultaneous slowand fast light”, is of different spectral components of a singlepulse while that presented in [15] is of two fields with the op-posite polarizations in the cold Rb atoms. The intensities of theleft and right sideband spectral components (fast light) inFig. 5have a minor difference, since they experience different absorp-tions due to the asymmetry of energy level structure [see Fig.1(c)]. As a result, the time delay of the slowed AMG pulse is

3

(a)

(b)

Figure 4: (color online). Input and recovered (calculated from the compensatedspectrum in Fig. 3) pulses in time domain. (a) Gaussian case.Delay time∼0.46µs is measured according to the peak of pulse. (b) AMG case. Delaytime∼0.11µs is measured according to the highest peak of pulse.

decreased since it is a result of the interfering superposition ofthese three separate spectral components, which also tellswhythe Gaussian and AMG pulses experience different time-delays[see Fig. 2].

On the other hand, this “simultaneous slow and fast light”effect in a single pulse will inevitably cause an additional pulsedistortion (“dispersive distortion”), but here it is negligiblysmall for such a single-Λ type system [16, 17]. Actually, agood compensating result has been obtained without eliminat-ing the dispersive distortion here. However, if the dispersivedistortion is so severe that a single AMG pulse splits into twoor three pulses, the compensation method we proposed is nolonger valid. In addition, the modulation frequency (δ), deter-mining the detuning and thus the absorption and dispersion ofthe sideband components, have a significant impact on the com-pensation result. Further, there is always a residual dispersivedistortion unlessδ is small enough that all the spectral compo-nents are within the linear dispersion region of EIT spectrum.As another influencing factor, the spectral bandwidth [Ω0, in-versely proportional to the temporal FWHM (T0)] of three com-ponents has a relatively small impact to the compensation, aslong asΩ0 (T0) is not too large (small). Finally, by using thetotally opposite dispersive regions, a single EIT medium cansimultaneously generate slow and fast light for a single cosine-type AMG pulse with the suitable modulation frequency. Inthis way, this amplitude modulation increases the utilization ef-ficiency of the bandwidth of the EIT-based slow light system.

Further, the delay time of the Gaussian pulse [∼0.47µs,see Fig. 2(a)] remains approximately unchanged after com-pensation [∼0.46µs, see Fig. 4(a)], because its spec-trum has relatively intensive distribution within the trans-mission spectrum. By contrast, the delay time of the re-covered AMG pulse [∼0.11µs, see Fig. 4(b)] is decreased

Figure 5: (color online). Simultaneous slow and fast light of the separate spec-tral components of a single cosine-type AMG pulse. Using IDFT, the resonantand two sideband components (left and right) are obtained from the correspond-ing spectral components of the output AMG pulse [Fig. 3(b)],while a commonreference pulse is obtained from the resonant spectral component of the inputAMG pulse. As a result, the resonant spectral component propagates as slowlight, while both the left and right sidebands propagate as fast light. The de-layed (advanced) time∼0.47µs (∼0.18µs for left sideband,∼0.20µs for rightsideband) is measured according to the peak of pulse. In the inset, all the pulsesare normalized for comparison.

after the compensating process [compared with the delay∼0.14µs of the output pulse in Fig. 2(b)]. That is to say, to acertain extent, the compensation method weakens the slow lighteffect of the AMG pulse. This is because that, these sepa-rate spectral components propagating with definitely differentgroup velocities (in our case, slow and fast light), are amplifiedby different magnitudes during the compensating process. Thefast light effect of the two sideband components are enlargedmore than the slow light effect of the resonant one, so that thedelay time of the recovered AMG pulse, which is the interfer-ing superposition of the three spectral components, is undoubt-edly reduced. In addition, obviously, a larger modulation depth(A) will lead to the larger delay reduction and more significantresidual distortion (caused by frequency dependent dispersion)after compensation. Therefore, when applying this compen-sation, two aspects need to be considered. Firstly, we shouldevaluate the impact of three parameters of the amplitude mod-ulated pulse, i.e., modulation frequency (δ), temporal FWHM(T0) and modulation depth (A), and then determine whether awell-recovered (the residual distortion is small enough) pulsecan be obtained. Secondly, we need to make a trade-off be-tween the pulse shape (intensity) and time delay, i.e., to de-termine whether it is worth while to compensate the loss anddistortion with such a reduction of time delay.

4. Conclusion

To our knowledge, the slow light of a cosine-type AMGpulse, which can be used to increase the utilization efficiencyof the bandwidth of slow light system by using the different

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dispersion regions of the EIT spectrum, is demonstrated andstudied for the first time. Taking the cosine-type AMG pulse asan example, we focus on studying the significant pulse distor-tion (loss) in a single-Λ EIT system. Then we point out that thedistortion is mainly caused by the frequency dependent trans-mission. Accordingly, we present a compensation method torecover the slowed pulses in shape and intensity with high fi-delity, and the results also indicate that, by introducing alin-ear optical system with desired gain spectrum, the pulse canbe recovered in an “all-optical” way. On the other hand, dueto the spectral distribution of the cosine-type AMG pulse, thefrequency dependent dispersion causes an (minor) additionaldistortion and a reduction of time delay. Meanwhile, distin-guished from [15], we reveals the “simultaneous slow and fastlight” effect of different spectral components in a single ampli-tude modulated optical pulse. Further, according to the absorp-tive and dispersive properties of slow light medium, the groupdelay and shape distortion of a pulse can be manipulated respec-tively, while the absorptive distortion is well compensated usingthe method presented in this paper. In principle, this method isapplicable to any other amplitude modulated pulses, and is alsoextendable to many other slow light systems. It is reasonableto believe that the methods of an additional amplitude modula-tion to a single pulse and recovery of slowed (distorted) pulsein slow light would have potential applications in the opticalcommunication and the low-distortion optical delay lines.

Acknowledgement

The authors appreciate A. Dang, S. Yu, C. Zhou, S. Gao, andW. Wei for the helpful discussions. This work is supported bythe Key Project of the National Natural Science Foundation ofChina (Grant No. 60837004), and the Open Fund of Key Labo-ratory of Optical Communication and Lightwave Technologies(Beijing University of Posts and Telecommunications), Min-istry of Education, P. R. China.

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