1. Introduction

The demonstration of slow light propagation in ultra-cold atomic gasses [1] has spurred significant interest in this field, due both to the fundamental aspects as well as the possibility to implement all-optical buffers [2]. Recently, slow light propagation was also demonstrated in a ruby crystal at room temperature [3] as well as in a quantum well at low temperature [4]. These latter demonstrations used the effect of coherent population oscillations (CPO), which may be viewed as a wave-mixing interaction where the effective group velocity of a probe field is affected by a strong pump field. It is an important feature of CPO that the dispersion profile of the spectral hole seen by the probe field is determined by the population lifetime rather than the deplaning time [3], which governs experiments based on electromagnetically induced transparency (EIT) [1]. This leads to significantly relaxed conditions under which light slow-down may be observed.

For practical applications, semiconductor waveguides are attractive and we present, for the first time we believe, the experimental demonstration of slow-down in a short (100 μm) waveguide at room temperature. Furthermore, the degree of slow-down can be controlled optically as well as electronically. The degree of slow-down observed is modest, on the order of a factor of three, but is carried out at a record-high frequency of more than 15 GHz. In contrast, experimental observations of ultra-slow light in Refs. [1, 3, 4] were carried out at frequencies in the Hz to a few MHz range.

We also derive a theoretical model for the modification of the effective group velocity due to coherent population oscillations. The model includes the effect of the spatial dependence of the light intensity along the waveguide, but can yet be solved to arrive at a semi-analytical results. The model well accounts for the experimental results and is further used to analyze the limitations in the use of coherent population oscillations to control the velocity of light. Furthermore, it is shown that for the case of intensity modulated signals, as considered here, a comprehensive wave mixing model, including the effect of a dynamical index grating as well as the gain grating, leads to the same result as a rate equation model accounting only for absorption (gain) dynamics. This leads to additional insight into the physical mechanisms at play and provides a link to earlier results investigating the properties of pulses propagating in saturable media.

2. Experimental set-up and results

We use the experimental technique of Ref. [3] to measure the changes in propagation time through a reverse biased semiconductor waveguide, see Fig. 1.

 

Fig. 1. Experimental set-up for measuring the propagation speed of a modulated optical signal in a semiconductor waveguide device (electro-absorption modulator – EAM).

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In short, a weak intensity modulated signal is generated by modulating a CW laser beam. The signal is transmitted through the device and the phase delay is measured relative to a reference beam which has traversed an optical path length corresponding to the background group refractive index of the device. In the present case we use a network analyzer to generate the intensity modulated signal as well as measuring the phase delay relative to the internal drive signal. The system is carefully calibrated by measuring the phase response when the device is operated in the linear regime (e.g. for a very low optical input power, or in a situation where the device is excited close to the transparency point). The experimental controls are the frequency of the intensity modulated signal, the absorption of the device, which is readily controlled though the applied external bias voltage, and the input power to the device. The set-up incorporates an optical fiber amplifier (EDFA) in order to explore a larger range of input power levels, and the power is controlled by an optical attenuator in front of the device and after the fiber amplifier. In this way any influence of intensity induced changes of the propagation speed in the amplifier, due to dynamics in that medium, may be calibrated out.

The device investigated is a commercial InGaAsP bulk p-i-n electro-absorption modulator, the absorption of which is controlled by a reverse voltage. The device operates in the wavelength region of 1550 nm and has a waveguide length of approximately 100 μm. Polarization controllers in the set-up ensure that the device is excited in the TE mode.

Figure 2 presents the experimental results in a contour plot showing the measured group index versus input optical power and reverse bias for a fixed frequency of f₀ = 15 GHz. The different contour lines are labeled by the induced change the of group index, which were calculated from the measured temporal delay Δt, which in turn is obtained from the measured phase delay Δφ (in radians):

where Ω = 2πf₀ is the angular modulation frequency. The absolute group index change of Δn_g = 7.2 thus corresponds to a measured change in time delay of Δt = 2.4 ps. The background group refractive index of the device is n_gb ≅ 3.4, corresponding to a transit time of 1.1 ps, so the minimum group velocity observed is a factor of (3.4+7.2)/3.4 = 3.1 lower than for an unperturbed waveguide. Fig. 2 clearly illustrates that the group velocity can be changed all-optically, through the input intensity, or electrically, by the applied bias voltage. Previous measurements on a Ruby crystal [3] and semiconductor quantum wells [4] have already established the possibility to control the delay optically, whereas the electrical control has not, to our knowledge been demonstrated earlier. However, in many applications, e.g. as phase-shifters for phased array antennas, the electrical control may be the most convenient.

 

Fig. 2. Contour plot of measured absolute change in group refractive index versus input intensity and reverse voltage for a 100 μm long semiconductor waveguide. The measurements were carried out at a modulation frequency f₀ = 15 GHz.

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Figure 3 presents further details of the measurements by plotting the measured average power transmission (Fig. 2(a)) and the measured phase shift (or time delay, right axis) versus reverse bias for different input intensities. The curves have been normalized to 1 (0 dB) at an applied reverse bias of 0V; the insertion loss at this voltage (increased by the built-in field across the pn-junction) is 9 dB. For fixed input power, the transmission decreases with reverse bias due to the Franz-Keldysh effect, as is well-known, while the absorption saturates and the transmission increases when the input power is increased. The absorption saturation originates from bandfilling and carrier screening by the excited carriers. We notice that the saturation effect is quite modest compared to many other EAM devices, since the device has been designed to work as a linear electro-absorption modulator. The reversal of the intensity dependence of the transmission observed beyond 2.5 V is due to temperature effects induced by the photo-current [5] and plays no importance at the high modulation frequencies considered here. The phase shift, Fig. 3(b), is seen to display a maximum for a reverse voltage around 2 V, but increases monotonically with the input power in the power range that could be experimentally investigated in the set-up.

 

Fig. 3. Measured transmission (a) and phase shift (time delay) (b) versus voltage for different input power levels. Modulation frequency (detuning): 16.7 GHz.

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In the following sections we will derive a theoretical model that describes the effect of coherent population oscillations in extended semiconductor waveguides and compare with the measurement results above. In particular we will show that the obtainable time delay is limited by the carrier lifetime of the medium. In our case the delay is estimated to be as short as 10 ps at 2V reverse bias, and the experimentally observed delay relative to the lifetime, proportional to the product of time delay and bandwidth, is thus of the order of 20 %, which is comparable to other measurements [3]. This will be discussed in more detail later.

3. Theory

We theoretically analyze the effect of light slow-down in a semiconductor waveguide due to coherent population oscillations by using a four-wave mixing description. The intensity modulated signal may thus be considered, in the frequency domain, as composed of a central carrier component at the optical frequency ω₀ and two weak sidebands at frequencies ω₁ = ω₀+Ω and ω₂ = ω₀-Ω. In the analysis we shall consider the signal at ω₀ as the pump beam and calculate the change of the susceptibility seen by the probe beam at frequency ω₁. Physically, the pump and probe fields beat, leading to modulation of the carrier density. The pump beam subsequently scatters on the corresponding temporal gain and index grating and the component scattered to the probe frequency modifies the susceptibility seen by the probe. We notice that the same effect of wave mixing leads to the generation of a field at the mirror frequency ω₂, usually referred to as the conjugate four-wave contribution. In the analysis of experiments using double-sideband modulation, corresponding to the use of an intensity modulated signal as in Ref. [3] and here, one also needs to take into account the initial signal at the mirror frequency ω₂, which, by the mechanism just discussed, also leads to contributions at both of the sidebands. In order to make as simple and transparent an analysis as possible we shall first analyze the system for a pump-probe (single-sideband) configuration and subsequently make the (simple) modifications necessary for the actual experimental configuration.

We use the results of Ref. [6] to calculate the susceptibility including four-wave mixing effects. That model was primarily derived for an active semiconductor waveguide operated under forward-bias conditions leading to gain. However, the model equally applies for a reverse-biased waveguide operated in the absorption regime if one assumes that the dominant effect leading to optically induced change of the absorption is bandfilling, or that any field-screening can effectively be described in this way, and that the corresponding recovery of the carrier density change can be described by an effective carrier lifetime. A similar approach has been effective in analyzing all-optical signal processing using saturable absorber waveguides [7].

Wave mixing in semiconductor waveguides has contributions from carrier density modulation, carrier heating, spectral holeburning as well as Kerr and two-photon absorption effects [8]. At detuning frequencies ranging up to several hundred GHz, however, the dominant mechanism mediating wave interactions is pulsations of the carrier density. From Ref. [6] we have the following expression for the (resonant) complex third-order susceptibility at the sideband ω₁ = ω₀+Ω

where P is the power, P_sat is the saturation power, τ _s is the carrier lifetime, α is the Henry alpha-factor (linewidth enhancement factor), g_sat , is the modal saturated gain (negative for a device operating in absorption), n_b is the background refractive index, ω₀ is the optical carrier frequency and c is the velocity of light in vacuum. The saturated gain, g_sat , can be calculated from the mean power exciting the material; in steady-state one has

where g₀ is the small-signal (unperturbed) material gain and Γ is the confinement factor. The evolution of the optical intensity along the waveguide is governed by the propagation equation

where α _int is the internal waveguide loss, which can often be neglected compared to the contribution from the resonant transition.

Separating the susceptibility of the medium into a background contribution, χ _b , and the above resonant contribution, χ _r , we have for the refractive index (with χ′ ≡ Re{χ})

and the group index becomes

In this expression we have introduced the background index, $n_b^2$ =1 + χ′_b , and the corresponding group index, n_gb , and have neglected the contribution from the resonant susceptibility to the phase refractive index itself, which is usually a good approximation. An analytical expression for n_g can easily be written down using the expression for χ _r in Eq. (2).

Using parameters typical for InP or GaAs based optoelectronics devices, Fig. 4 shows examples of the real and imaginary parts of the resonant susceptibility due to wave mixing as well as the resulting group index. Notice that the group index is calculated at the frequency of the probe and thus governs the propagation speed of a probe pulse at that detuning with respect to the pump signal. The figure shows that the alpha-parameter strongly influences the susceptibilities as well as the group index. The physical reason is that for α≠0 the carrier density modulation also induces an index grating which leads to scattering of the pump. At zero detuning, however, the group index is independent of alpha. This is confirmed by the following simple expression for the group index at zero detuning, which is easily derived from Eqs. (6) and (2):

 

Fig. 4. Imaginary (a) and real (b) parts of the third-order probe susceptibility due to coherent population oscillations and corresponding group refractive index (c) calculated for different values of α. Other parameters are: P = 10 mW, P_sat , = 10 mW, τ _s = 200 ps, and Γg₀ = -5∙10⁴ m^-1.

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As pointed out in Ref. [3], a generalized form of Eq. (6) is needed when performing experiments using a modulation technique that leads to a discrete frequency spectrum with a carrier and two sidebands. We thus introduce the modulation refractive index [3]

and using Eq. (2) we find

We have here included a factor of two due to the excitation of the system by a double sideband signal, since it can be shown from Ref. [6] that the four-wave mixing contribution of the “mirror” sideband adds an equal contribution. This expression is identical to that of Ref. [3] with the identification of τ_s being equal to the T₁ lifetime, except for the additional factor of two. We notice that the modulation refractive index is independent of alpha for all modulation frequencies, which can be attributed to the double sideband excitation of the system and the fact that the single sideband contribution from the index grating is antisymmetric with respect to detuning frequency.

When comparing to experiment one needs to take into account the intensity variation along the waveguide, which leads to a spatially varying index. The integrated delay and the corresponding average modulation refractive index, n̄_mod, are given by

In the experiments we measure the phase delay, Δφ _mod, of the intensity modulated pattern relative to a reference signal, which is unaffected by the nonlinear interactions in the waveguide. The phase delay is given by

If the internal waveguide loss is neglected, α_int=0, it turns out that the integral in Eq. (10), with n _mod given by Eq. (8) can be carried out analytically. We get

where T_sat is the saturated value of the transmission

From the propagation equation we get the following implicit relation for the saturated value of the transmission [9]

which can easily be solved numerically for T_sat , given the input power and small-signal material gain (absorption).

In the case of an angular modulation frequency which is small compared to the inverse of the carrier lifetime, Ωτ_s <<1, we get the approximate expression

This expression clearly shows that the induced change of the modulation index scales with the absolute value of the carrier lifetime, and that there is a trade-off between index change and bandwidth [10,11]. An approximate expression for the bandwidth, defined as the frequency at which the index change has been reduced by a factor of two from the zero-detuning limit, can easily be obtained from Eq. (15), but we notice that for frequencies that are large relative to the inverse of the carrier lifetime, the full expression (12) should rather be used.

The expression (12) for the average index describing the propagation of the sinusoidal envelope modulation, corresponding to a phase delay as expressed by Eq. (11), has been derived using a four-wave mixing approach. However, we should like to point out that an identical expression for the phase change can be obtained if one uses the results of Ref. [12], where cross-gain modulation in semiconductor optical amplifiers was analyzed on the basis of standard rate equations including propagation effects. The expression obtained in [12] for the small-signal modulation component at angular frequency Ω at the output of the waveguide relative to the input modulation is (neglecting internal loss as well as the presence of a second CW signal considered in [12])

It is easily seen that the induced phase change of the modulation component, obtained as the phase of the RHS of Eq. (16), is identical to the phase change implied by Eqs. (11) and (12). From Eq. (16) one can additionally calculate the amplitude change of the modulation component as the absolute value of the RHS of Eq. (16). The fact that identical expressions for the phase delay can be obtained in two different ways is of course satisfying, but also suggests that one may interpret the induced phase change as a simple consequence of dynamical absorption saturation. In the context of an optical pulse propagating through a medium with saturable absorption, the delay can thus be explained as a consequence of the temporal variation of the absorption across the pulse: The front part of the pulse experiences larger absorption than the rear, resulting in a net delay of the pulse. We have recently measured the induced delay of 150 fs optical pulses in a semiconductor quantum dot optical amplifier, and shown that the experimental observations are qualitatively explained by a simple model accounting only for this temporal absorption change [13].

While the expression for the modulation refractive index is relevant for the comparison of our model with the experimental results, the group refractive index is the relevant quantity when considering input signals with continuous spectra, e.g. pulses or data-modulated signals. By integrating Eq. (7) over the waveguide length we get the following simple expression

It can easily be shown that n̄_m - n_bg → 2(n̄_g - n_bg ) for Ω → 0, with the factor of two arising from the double sideband excitation corresponding to the intensity modulated signal. From Eq. (17) we see that the controllable part of the refractive index, i.e. the part that can be influenced by changing the excitation of the waveguide or by controls that influence the transmission of the waveguide, leads to a temporal (group) delay change given by

Before discussing the general implications of these theoretical results for the temporal shifts achievable in semiconductor waveguides, we first make a specific comparison of calculations based on the derived model with the experimental results.

4. Modeling results

Figure 5 depicts calculated results for the transmission [Eq. (13)] and the effective modulation index [Eq. (12)] versus voltage. The dependence on reverse bias arises through the values of absorption, saturation power and carrier lifetime. Physically, a larger reverse bias results in a red-shift of the absorption edge, which increases the absorption, as well as a faster rate of sweep-out of excited carriers and consequent reduction of carrier lifetime. We use a phenomenological model to relate the gain and carrier lifetime to the applied voltage:

where a_V , τ₀, V_on and V_ref are constants. Since the saturation power is inversely proportional to carrier lifetime, we further assume P_sat = P _sat,0 exp(V/V_ref ), with P_sat being a constant. The results of Fig. 5 were calculated using the following parameter values: λ=1.55 μm, n_b ≅ n_gb =3.4, Γ=0.3, a_V = 6.7∙10⁵ m^-1V^-2, V_on = 0 V, τ_s0 = 30 ps, P _sat,0 = 6 mW, V_ref = 2V. At 2V reverse bias the carrier lifetime has a value of 10 ps and the saturation power is 16 mW.

 

Fig. 5. Calculated transmission (a) and phase / time delay (b) versus reverse bias voltage for different input power levels. Frequency detuning: 16.7 GHz.

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The modeling results are seen to agree well with the measurements. The appearance of a local maximum for the time delay at a specific voltage can be traced back to the voltage dependence of the lifetime: As the voltage is increased the absorption increases and this initially leads to a larger local change of the modulation refractive index, cf. Eq. (9). However, at the same time the larger field across the active region leads to a shorter carrier lifetime due to carrier sweep-out and the saturation power increases. This in turn decreases the relative degree of saturation of the medium, which eventually dominates the voltage dependence.

5. Discussion

Experimentally, we were limited in the range of absorption and input power levels that could be investigated. Fig. 6 shows modeling results for a wider range of values; the saturated transmission (power out relative to power in) and time delay are plotted versus input power normalized by the saturation power, P/P_sat . The different curves correspond to different levels of absorption, indicated by the low-power transmission, T₀ . The time delay is here plotted relative to the carrier lifetime, i.e., ξ= Δt_mod /τ_s . The calculations were carried out in the limit of zero detuning frequency, where the absolute time delay is largest [3].

 

Fig. 6. Calculated (a) transmission and (b) relative time delay ξ = Δt_mod /τ (at zero detuning) versus input power for different levels of small-signal (unsaturated) transmission T₀ .

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First, we notice that the theory predicts a local maximum for the delay in dependence on the power level, which occurs since saturation of the absorption and power broadening of the induced dispersion dominates for high power levels. We also see that ξ < 1, i.e., the time delay remains smaller than the carrier lifetime. It is easy to see from Eq. (15) that this is a general property of the considered mechanism of CPO.

Figure 7 further investigates the dependence on device parameters by plotting the relative delay, Fig. 7(a), and length averaged group index, Fig. 7(b), versus waveguide length for different values of input power and two different carrier lifetimes. For fixed waveguide length there exists an optimum input intensity that results in the largest relative delay, while for fixed input intensity the delay saturates with increasing waveguide length due to absorption of the signal. Figure 7(b), in contrast, shows that there is an optimum length at which the average modulation index is maximized. However, for the applications it is the total induced delay that matters.

 

Fig. 7. Calculated variation of (a) delay relative to carrier lifetime and (b) length-averaged modulation refractive index versus waveguide length. The absorption is Γg₀ =-5∙10⁴ m^-1.

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As pointed out earlier, and also discussed in Refs. [10,11] there is a trade-off between the bandwidth and the magnitude of the induced delay, cf. Eq. (15). A very simple argument can be given to elucidate the buffering capacity of a slow-light device based upon CPO. Since the achievement of light slow-down requires a relatively strong saturation of the medium, the bit-period T=1/B, where B is the bitrate of an optical signal, needs to be comparable to or longer than the lifetime, T >τ_s in order to avoid memory or patterning effects. Since we have found that the time delay is limited by the carrier lifetime, Δt < τ_s , we find the condition Δt B < 1, and unless special precautions are taken to limit patterning effects, the storage capacity is therefore limited to one bit. This is a severe limitation for the application of CPO to buffering or storage. However, we want to emphasize that there are interesting narrow bandwidth applications, e.g. for optically fed phased-array antennas, where the realization of a compact device allowing optical or electrical control of the phase delay is highly interesting.

6. Conclusions

We have experimentally demonstrated slow-down of light in a semiconductor waveguide at room temperature and at frequencies beyond 15 GHz using the effect of coherent population pulsations. The maximum time delay observed reflects an approximately three-fold increase of the group refractive index, corresponding to a time delay of approximately 20 % of the carrier (population) lifetime. A semi-analytical model taking into account propagation effects has been developed and has been shown to well account for the experimental observations. The model can be used to optimize the time delay, but also shows that the obtainable time delay is limited by the carrier lifetime. This means that a straightforward application of the effect of coherent population oscillations is limited to the buffering of just one bit, and points rather to narrow-band applications where a controllable phase delay is needed.