1. Introduction

Understanding the time evolution of an optical pulse, particularly in the presence of the coherent media, is very important in many applications ranging from all-optical switching, optical delay lines, to quantum-storage devices and to communications [1–3]. The studying of optical pulse propagation inside nonlinear dispersive media such as silica glass, for example, leads to the development of modern fiber optical telecommunication systems, which are crucial for the Internet. As the ultrafast lasers developed, by tailoring the transient properties of a short pulse, the energy loss can be reduced and the penetration depth in water can be increased [4], which could be applied in biomedical imaging or underwater communication. The propagation of the pulse in the medium is governed by the dispersive properties, which have led to both slow (subluminal) and fast (superluminal) light [5, 6]. Typically, the descriptions of transient pulse propagation have followed two different paths: optical precursors and coherent interactions [7]. The precursors are first studied by Sommerfeld and Brillouin in 1914 [8] aimed to investigate the causality. Recently, the optical precursors have been experimentally observed in the anomalously dispersive and normal dispersive atomic medium [9, 10]. The most previous studies on the short pulse propagation dynamics are carried out in the coherent atomic medium in order to manipulate the linear and nonlinear dispersion. In this paper, we investigate propagation dynamics of a wavepacket through a high finesse optical cavity. The optical cavity has a large linear dispersion which is similar as a two-level atom in the weak excitation. We demonstrate that the wavepacket propagation greatly depends on the spectral width of the incident pulse. The evolution dynamics shows distinctive behavior for narrow-band and quasi-percussional excitation of a cavity field. For narrow-band excitation where the pulse width in time of incident wavepacket is larger than the cavity decay time, except the slight increasing of the pulse width, the transmitted wavepacket preserves its initial shape and shows the same time evolution as the incoming pulse. For quasi-percussional excitation where the pulse duration in time is much shorter than the cavity decay time, the wavepacket increases over the time interval on the order of time duration of the incident pulse at the rising edge and exponentially decreases at the falling edge with a characteristic time of the cavity decay. We also measured the time delay of the pulse peak and found that, in contrast to the constant delay predicted [11, 12], it is greatly related to the pulse width of the incident wavepacket. Finally we investigate the cavity excitation of a step-modulation pulse and observe the optical precursor by the reflected field of the cavity. The measured results show a good agreement with a simple theoretical calculation. We believe that the observations could be important in the future optical information storage and all-optical transistor with the cavity.

2. Experimental results

To understand experimental results, the dynamics evolution of the wavepacket through a cavity need to be determined. The linear response theory or the cavity transfer function [13–15] is used to derive the theory of optical pulse propagation in the cavity. Here we consider a typical Fabry-Perot (FP) cavity formed by two mirrors with the identical intensity reflectivity R and transmittivity T, separated by a distance L. The transmitted electric field can be written as

The reflected electric field is given by

Near the cavity resonances, Eq. (1) can be approximately written as

If the initial incident field E_in is a one-photon wavepacket with the amplitude of the form ∫ dkg(k − k₀) (k₀ is the average wave vector and the central frequency is resonant with the cavity resonant frequency, i.e., k₀ = ω₀/c), the transmitted electric field E_T (t) becomes

For a Gaussian incoming wavepacket $f(t^{\prime })=1/\left(\sqrt{2\pi }\sigma \right)e^{-{(t^{\prime }-t_0)}^2/{\sigma }^2}$, the transmitted electric field in the time domain is analytically obtained

The schematic diagram of the experimental excitation of an optical cavity by a wavepacket is shown in Fig. 1. The two high reflectance cavity mirrors with the identical intensity reflectivity 99.96% (ATFilms) are mounted on an invar spacer of length 7.5 cm which is placed inside an ultrahigh vacuum chamber (3 × 10⁻¹¹ Torr). The measured finesse and a free spectral range are about 3770 and 2 GHz, respectively. A diode pumped solid state laser (Mephisto MOPA) at 1064 nm is served as a locking laser. The standard Pound-Drever-Hall (PDH) feedback method [17] is used to keep the cavity locked to one of its resonant modes. The phase modulation is produced by a resonant electro-optic modulator (EOM) driven at 20 MHz. The power of locking beam is kept as small as possible to minimize the frequency shift caused by light heating and optical feedback. The cavity linewidth and a fractional frequency drift after locking are about 530 kHz and 2 × 10⁻¹⁰ at 1 s, respectively. The probe light (excitation light) is from a commercial extended cavity diode laser (ECDL) at 671 nm (Toptica-TA Pro). We chop this light into pulses with a Gaussian or square temporal shape using an acousto-optic modulator (AOM) and a fast arbitrary waveform generator. The peak intensity of the pulse is operated in the weak excitation limit. The cavity transmission and reflectance of the pulses are detected by the fast avalanche photodiodes (APDs). Figure 1(b) and (c) are the illustration of transient pulse propagations in the temporal domain for the cavity excitation of Gaussian and a step-modulated pulse, respectively.

 

Fig. 1 (a) The experimental setup. PBS, polarized beam splitter; ISO, optical isolator; M1-M3, mirrors; L, lenes; D, dumper; BM1-BM2, Dichromatic mirrors; BS, beam splitter; AOM, acousto-optic modulator; EOM, a resonant electro-optic modulator; APDs, fast avalanche photodiodes; GP, Glan-Laser Polarizer; HW, half waveplate; QW, quarter wave-plate. (b) Transient pulse propagations in the temporal domain for narrow-band and quasi-precursor excitation. (c) The pulse dynamics in the temporal domain at the excitation of a step-modulation pulse.

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Figure 2 shows the temporal responses of the cavity transmission obtained on the APD1 for different temporal widths of the Gaussian excitation wavepackets at the cavity resonance. Each temporal response represents an average over 4 individual wavepackets having experienced the same sequence described. The red curves are fitted with a Gaussian function to extract the temporal pulse widths and the arrival time of the incoming wavepackets. The green curves are the fits based on Eq. (6). Figure 2(a) is the propagation of the Gaussian wavepacket with σ = 200 ns. For the case, σ < Γ⁻¹ and then the frequency width of the excitation wavepacket Δω > Γ. The transmission of the wave packet shows an obvious asymmetry in this broadband resonant excitation. The rising edge has a width on the order of σ and the falling edge decreases with a time constant determined by the cavity decay. Figure 2(b) is the propagation of the wavepacket with σ = 1500 ns. For such a narrow-band excitation, the frequency width of the incoming wavepacket is much smaller than Γ and the different waves making up the wavepacket will be transmitted in the same way, therefore the propagation of the wavepacket almost preserves its shape and the cavity transmission shows a standard Gaussian.

 

Fig. 2 The propagation of resonantly Gaussian wavepackets with broadband resonant excitation (a) σ = 200 ns and the narrow-band resonant excitation (b) σ = 1500 ns. The black dots are the measured data for the incoming pulses and blue dots are the data of the cavity transmission. The red solid curves are the Gaussian fits to get the temporal width of the incoming Gaussian wavepackets and the green solid curves are the fits from Eq. (6). τ_c represents the peak delay between the incoming pulses and the cavity transmitted pules.

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Figure 3 shows the widths (e⁻¹ of the transmitted intensity) for the rising and falling edge as a function of the widths of the incoming Gaussian wavepackets. The red and blue dots are the experimental measured data. The solid curves come from the calculations of Eq. (6) with the experimental parameters. The dashed line represents the decay time of the cavity. They clearly show that the propagation of the Gaussian wavepackets by the cavity is greatly different for the short and long pulses. For the broad pulses excitation, the wavepack is long in time σ ≫ Γ⁻¹, and as a result in frequency domain Δω ≪ Γ, the different frequency components will be scattered by the cavity in the same form. E_T (t) in Eq. (6) then clearly resembles f (t) and the cavity has little influence on the incoming pulses. The transmission intensity varies as the intensity of |f (t)|² of the incident wavepacket. However, for the short pulses excitation where the wavepack is short in time σ ≪ Γ⁻¹ and long in frequency Δω ≫ Γ, the only frequency components close to the resonant frequency ω_ba will be effectively scattered by the cavity. Correspondingly, the rising edge of the transmitted pulse basically follows the incoming Gaussian pulse. In contrast to the rising edge, the falling edge loses the information of the incoming excitations and just experiences an exponential decay related to the intracavity photon lifetime, as shown for the data dots in Fig. 3. The extrapolating red data to σ = 0 in Fig. 3 gives the characteristic time (295 ns) of falling edge, which is very close to the cavity decay time Γ⁻¹ = 300ns. The measured results are agreement with the simple theory.

 

Fig. 3 The widths (e⁻¹ of the transmitted intensity) for the rising (blue dots) and falling edge (red dots) as a function of the widths of the incoming wavepackets. The dots are the measured data. The solid curves are the calculations based on Eq. (6). Dashed line represents the cavity decay time. Error bars denote one standard deviation of the statistic. The central frequency of the pulse is resonant with the cavity resonant frequency.

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From Fig. 2, except for the change of the pulse shape, there obviously exists the peak time delay for the transmitted light pulses compared with the incident pulses. A remarkable feature is that the observed peak time delay greatly depends on the pulse width of the incident pulse in contrast to the prediction from the cavity resonant dispersion [11, 12]. Figure 4 summarizes the dependence of the peak time delay τ on the width of the incident pulse, where we have taken into account of the delay from all electronics and optical cables. The time delay of the pulse propagation through the cavity is defined being referred to the propagation time in free space τ₀ to be τ = τ_c − τ₀, where τ_c is the time delay between incoming pulse and transmitted pulse and τ₀ = L/c (L is the cavity length and c is the light speed). The dots are the measured data. It is interesting to see that, for the very narrow incoming pulses, the pulses can propagate through the cavity with very small delay. The peak delay time increases with increasing input pulse duration. For the large input pulse duration, the pulse spends longer time inside the cavity and the delay time becomes large correspondingly and eventually saturates to the maximum value τ = 590ns which just related to mirror reflectivity and optical length of the cavity. The observed delay times are in good agreement with the theory outlined above, as shown for the solid blue curve in Fig. 4, which notably contains no free parameter.

 

Fig. 4 Time delay of the transmitted light pulse at the resonant excitation versus the width of the incident pulse. The dots are the measured data. Arrow denotes the time delay prediction from the cavity resonant dispersion theory [11, 12]. Solid line: the solution of Eq. (6) with the experimental parameters. Error bars denote one standard deviation of the statistic.

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Now we turn our attention to the cavity excitation of a step-modulated optical pulse. The edge of rising (falling) time (10%–90%) of the square probe pulse is about 20 ns. Figure 5 shows the temporal evolution of the weak resonant square probe light reflected by and transmitted the cavity. When the pulse is turned on the reflected intensity immediately raises to the full intensity of the incident intensity and then decays to the steady state intensity, as shown in Fig. 5(a). Correspondingly, the cavity transmission is almost zero and gradually increases to its steady state value, as shown in Fig. 5(b). The initial cavity high-reflected transient spike is so-called optical precursors and very similar with the experimental observations in the atomic systems [9, 10]. The time scale of the spike, defined as the time from the initial turn-on to the e⁻¹ decay of the precursor, is about 400ns, which is about 1.3 times the cavity decay. The time scale of optical precursor is depended on the cavity mirrors reflectivity and the internal losses. Without the internal losses, the time scale is equal to the decay time of the internal cavity field. When the intracavity losses become large, it becomes larger than the cavity decay time and asymptotically approaches two times the decay time based on the theoretical calculations. When the pulse is turned off, the precursor transient spike is also appeared for the reflected field by the cavity at the falling edge, as shown in the inset of Fig. 5(a). The field quickly drops to almost zero and then increases to a value determined by the cavity parameters, following a long tail of the decay which is determined by the lifetime of the photons in the cavity. This feature is very like the case of two-level atoms at the lower optical depth [10]. Compared to the precursor transient spike at the rising edge the amplitude of the spike at the falling edge is small as expected because of the existence of the constructively interference of the optical precursor and the reflected field at the rising edge.

 

Fig. 5 Cavity reflection (a) and transmission (b) at the excitation of a step-modulated optical pulse. The blue dots are the measured data. The central frequency of the pulse is resonant with the cavity resonant frequency. Inset shows the pulse dynamics at the falling edge. Dashed lines represent the incoming pulse. Solid red lines are the numerical calculations of Eqs. (1) and (2) using the experimental parameters.

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3. Conclusion

We have experimentally studied propagation dynamics of the wavepackets through the optical cavity at the broadband and quasi-percussional excitation. The observation shows the profound dynamics even in such a simple optical system. The pulse evolution greatly depends on the spectral width of the incident wavepacket. The long pulse follows the same time evolution with the input wavepacket through the cavity and experiences the considerable peak time delay which is determined by the cavity parameters. The short pulse shows the distinctive behaviors that the pulse propagates with small peak time delay (even without delay) and the rising edge and falling edge are determined by the incoming pulse and the cavity decay time, respectively. We also investigate the cavity excitation of a step-modulation pulse and observe the optical precursor by the reflected field of the cavity. The observations can be generalized to an impulse excitation with ultrafast lasers for which the input pulse is short compared with the light round trip time inside the cavity. In this case, the destructive interference phenomena among the different time components of the injected pulses should be negligible. The transmission can show the cavity ring-down-like spectroscopy [14, 15]. Moreover, the controllable delay of the resonant optical pulses in the cavity is important in the storage of the optical information and can be utilized in realization of optical delay devices. The cavity, for example, has been utilized as a delay line arm in the detection of gravitational waves [18]. In addition, it is very interesting to study the propagation dynamics with the coherent multi-level atoms placed inside the cavity. The manipulation of the linear and nonlinear dispersion in such the system has been used to control the subluminal and superluminal pulse propagation in the cavity [19]. Many interesting spectral features in such systems with/without the coherent medium in an optical cavity need to be further investigated. Understanding and controlling propagation dynamics of wavepacket could find applications in studying cavity-QED effects and in designing novel optical devices.