## Abstract

We investigate variable optical delay of a microwave modulated optical beam in semiconductor optical amplifier/absorber waveguides with population oscillation (PO) and nearly degenerate four-wave-mixing (NDFWM) effects. An optical delay variable between 0 and 160 ps with a 1.0 GHz bandwidth is achieved in an InGaAsP/InP semiconductor optical amplifier (SOA) and shown to be electrically and optically controllable. An analytical model of optical delay is developed and found to agree well with the experimental data. Based on this model, we obtain design criteria to optimize the delay-bandwidth product of the optical delay in semiconductor optical amplifiers and absorbers.

©2006 Optical Society of America

## 1. Introduction

Variable optical delay lines are essential for optical signal processing and communication systems such as optical networks and lidar phase arrays. To control optical delay, two methods are generally exploited: 1) modify the group index of optical medium; 2) use dispersive devices [1–7]. The first approach makes use of nonlinear optics such as electromagnetically introduced transparency, coherent population oscillation, Raman and Brillouin amplification. The second one utilizes Fabry-Perot resonators, high Q cavities, and photonic crystals. Atomic vapors, solid state crystals, fibers, and semiconductors are considered for variable optical delay. Among these media, semiconductors are attractive due to their compactness, direct current control, large bandwidth and easy integration with electrical and photonic circuits. We recently studied slow light and fast light using population oscillation (PO) in quantum dot (QD) semiconductor optical amplifiers (SOA). We showed a room temperature delay of about 8 ps, or a slow-down factor of 6 with respect to the group velocity in vacuum, with a 2 GHz bandwidth (slow light) in a 0.7 mm QD SOA under zero bias [8]. When the QD SOA is under a large forward bias into the gain regime, we achieved a group index reduction of 10% with a 13 GHz bandwidth (fast light) in a 2.0 mm device [9]. In those experiments, we isolated four-wave-mixing effects by means of a pump-probe counter-propagating scheme. In this paper, we investigate slow light and fast light in a high-gain quantum well (QW) SOA with PO and NDFWM with a single microwave-modulated beam, similar to the slow light effect reported in solid-state crystals [3]. PO and NDFWM in semiconductor optical amplifiers have been intensely studied [10,11]. A mean-field approximation is required to avoid numerical calculations by ignoring the spatial dependence of the optical fields. This approximation is applicable for short or thin active media. For practical applications, however, long devices are necessary to obtain a large delay. Lack of analytical solutions generally hinders the understanding and implementation of devices with large optical delay. We present an analytical model for the gain and phase delay of an optical microwave signal, even though the optical fields themselves can only be solved numerically. The model is verified by experimental results of a semiconductor waveguide. Based on the model, a design criterion for maximum delay-bandwidth using PO and NDFWM in SOAs is proposed. Delay and delay-bandwidth product in semiconductor optical absorbers are then discussed.

## 2. Theory and Experiment

The experimental setup and physical picture of variable delay of a microwave modulated beam in SOAs are given in Fig. 1. A DFB laser is used as the pump beam going through the Network Analyzer with a continuously variable microwave modulation, which generates two side-bands in addition to the dc component on the output spectrum. The three components induce population oscillation and nearly degenerate FWM in the QW SOA. The total optical intensity is defined as:

where *E _{0}* is the DC component,

*E*and

_{-1}*E*are the amplitudes of the two side bands, and Ω is the beat frequency. For simplicity, the optical intensities are normalized to the saturation intensity of the SOA, which is defined as

_{1}*P*=

_{sat}*hν/(g’τ)*, where

*hν*is the photon energy,

*g*’ is the differential gain, and

*τ*is the carrier lifetime. Therefore, Eq. (1) becomes,

This optical beam will interact with carriers in semiconductors through stimulated emission and modulate the carrier density, introducing a carrier population oscillation with a beat frequency determined by Ω.

where *N* is the carrier density, *I* is the injection current, *q* is the unit electron charge, *V* is the volume of the active region, *hω _{0}* the photon energy,

*Γ*is the modal gain, N

_{g}_{0}is the DC carrier density, N

_{tr}is the transparency carrier density. Conversely, the population oscillation, functioning as a temporal grating, causes dispersive gain and index modifications on the two sidebands and effectively induces wave-mixings between the three optical fields as described by the following equations:

where α is the linewidth enhancement factor and *a* is the unsaturable loss of the semiconductor waveguide. Detailed derivation of Eq. (7–9) can be found in Ref. 10. As shown in Eq. (8) and Eq. (9), the effective gain and refractive index of the two sidebands are modified by the DC optical pump *p*
_{0}(*χ*
^{(1)} effect), and the sidebands are coupled to each other through the FWM term which is proportional to the square of *E*
_{0}(*χ*
^{(3)} effect) [10, 11]. Due to the large gain in SOAs, the three E-fields are strongly coupled and spatially dependent, leading to the nonexistence of analytical solutions. Fortunately, for variable optical delay, what we measure is the phase delay of the modulated signal (*p*
_{1} or its conjugate), instead of the E-fields. From Eq. (8) and (9), we derive the following equations describing the propagation of DC (*p*
_{0}) and AC (*p*
_{1}) optical signals along the waveguide:

Note that there is no dependence on the linewidth enhancement factor in Eq. (10)–(11). Although E-fields depend on the linewidth enhancement factor as shown in Eq. (7)–(9), this dependence is cancelled in the case of *p*
_{1} and *p*
_{0}. This is not surprising since slow light was realized in solid state crystals where atomic energy systems of active ions provide a zero linewidth enhancement factor. Therefore, we would like to point out that the small linewidth enhancement factor of quantum dots is not a disadvantage in term of the variable optical delay discussed in this paper.

To obtain the phase delay and gain of *p*
_{1}, however, it is unfeasible to integrate Eq. (11) directly over the spatial variation. Fortunately, if we divide Eq. (11) by Eq. (10) and integrate over the variable of *p*
_{0}, we obtain the expressions for the gain and optical delay of the optical signal *p*
_{1}:

$$\times \left\{\mathrm{ln}\left[\frac{{\left(1+{p}_{\mathit{out}}\right)}^{2}+{\left(\Omega \tau \right)}^{2}}{{\left(1+{p}_{\mathit{in}}\right)}^{2}+{\left(\Omega \tau \right)}^{2}}\right]-2\mathrm{ln}\left[\frac{1-\frac{a}{\Gamma g}\left(1+{p}_{\mathit{out}}\right)}{1-\frac{a}{\Gamma g}\left(1+{p}_{\mathit{in}}\right)}\right]-2\Omega \tau \frac{a}{\Gamma g}{\mathrm{tan}}^{-1}\left[\frac{\left({p}_{\mathit{out}}-{p}_{\mathit{in}}\right)\Omega \tau}{\left(1+{p}_{\mathit{out}}\right)\left(1+{p}_{\mathit{in}}\right)+{\left(\Omega \tau \right)}^{2}}\right]\right\}$$

$$\times \left\{\frac{a}{\Gamma g}\mathrm{ln}\left[\frac{{\left(1+{p}_{\mathit{out}}\right)}^{2}+{\left(\Omega \tau \right)}^{2}}{{\left(1+{p}_{\mathit{in}}\right)}^{2}+{\left(\Omega \tau \right)}^{2}}\right]-2\left(\frac{a}{\Gamma g}\right)\mathrm{ln}\left[\frac{1-\frac{a}{\Gamma g}\left(1+{p}_{\mathit{out}}\right)}{1-\frac{a}{\Gamma g}\left(1+{p}_{\mathit{in}}\right)}\right]+\frac{a}{\Omega \tau}{\mathrm{tan}}^{-1}\left[\frac{\left({p}_{\mathit{out}}-{p}_{\mathit{in}}\right)\Omega \tau}{\left(1+{p}_{\mathit{out}}\right)\left(1+{p}_{\mathit{in}}\right)+{\left(\Omega \tau \right)}^{2}}\right]\right\}$$

where the *p _{out}* and

*p*are the output and input power of

_{in}*p*

_{0}, respectively.

## 3. Experimental Results and Comparison with Theory

In Fig. 2, we present experimental data (symbols) of the gain and delay of the optical signal and theoretical models (lines) based on Eq. (12) and (13). The theory agrees well with the experimental data. The gain of the microwave-modulated optical signal in dB scale is proportional to the gain given by Eq. (12). In the case of optical delay, we use the curve of 100 mA current injection as a reference. We observe that the delay decreases as the current injection increases, leading to an optical advance (or negative delay) compared to the 100 mA case. However, it can still be considered as a delay if we choose the case of 400 mA as the reference. In real applications, only the relative delay is important. A maximum variable delay of 160 ps with about 1 GHz bandwidth is achieved, as shown in Fig. 2. From the measured bandwidths, we find that the carrier lifetime in the device decreases from 380 *p _{s}* to 240

*p*, which is reasonable for typical SOAs [11], as the current increases from 150 mA to 400 mA. At 400 mA, the loss-gain ratio,

_{s}*a/Γg*, is found to 0.1, and the normalized output power,

*p*, about 2.0. In the case of low current injection, these two parameters become more intervened with each other and, thus, have large uncertainties. However, the loss-gain ratio and output power are still within a reasonable range.

_{out}The dependence of the variable optical delay on optical pump power is given in Fig. 3. The optical delay is presented in phase delay, instead of time delay, for simplicity. Since the phase is proportional to the modulation frequency, it reduces toward zero as the frequency approaching DC as shown in Fig. 3. The optical attenuation is set to be 0, 3, 5, 7, 9 and 11 dB. The device is biased at 450 mA. By varying the optical pump power, a variable phase change of 25 degree at 1 GHz is measured by a network analyzer. We confirm these results with a direct time-domain measurement using a clock recovery module and a digital communication analyzer (Agilent DCA 86001 A) as an oscilloscope. The time domain result is shown in Fig. 4. Approximately, a 4-dB loss of the optical power is introduced in order to trigger the clock recovery module. The time domain result is consistent with the frequency domain measurements by the network analyzer as seen in Fig. 3. We also verify that the measured delay is not due to the saturation behavior of the detector built inside the network analyzer. By varying the mechanical variable optical attenuator (VOA) in front of the detector, we find a phase change less than 1.0 degree at 1.0 GHz when the optical power is varied over a range larger than 30 dB.

## 4. Generalization of the theory and discussion

For a waveguide with low loss, the optical delay is dominated by the last term in the parenthesis on the right side of Eq. (13). The arctangent function decreases more rapidly than the Lorentzian function given by the population oscillation model with mean-field approximation [10]. Note that, when the input optical power is small, the bandwidth increases as a function of (1+*p _{out}*)

^{0.5}as described by Eq. (13), while the mean-field theory gives a dependence of (1+

*p*) [10], where

_{ave}*p*is the normalized average optical power of the DC component.

_{ave}The second term in the parenthesis on the right side of Eq. (13) comes from the gain/loss saturation effects [12]. The input (*p _{in}*) and output (

*p*) of the DC optical power are related by [12]:

_{out}The right side of Eq. (14) is basically the difference between the saturated gain and unsaturated gain. For a long waveguide, *p _{out}* will saturate and the only term scaled with the cavity length is

*(Γg-a)L*. Therefore, from Eq. (13), the optical delay of a long device can be simplified into:

It is shown that variable delay in a long device actually depends on the unsaturable loss of the waveguide. Typically, the unsaturable loss is predetermined by the quality of the material and waveguide, while the gain is adjustable by current injection. To take advantage of the large gain (*Γg _{max}*~50-100 cm

^{-1}) of QW semiconductors, we need to increase the loss (typically 10–20 cm

^{-1}) either by ion implantation in the materials or by roughening the sidewalls of the waveguide. As predicted by Eq. (15), a maximum delay can be achieved when the unsaturable loss equals to the value of 2

*Γg*/3. In this case, the maximum delay, with a bandwidth of 3/(2

_{max}*τ*), is given as:

Furthermore, from Eq. (15), the delay-bandwidth product is given as:

To achieve the maximum delay bandwidth product of -*ΓgL*/4, the loss of the waveguide needs to be designed to be half of the maximum gain. For a length of 1 cm and a typical modal gain of 80 cm^{-1}, the delay-bandwidth product of 20 can be achievable with the unsaturable loss designed to be 40 cm^{-1}. To the best of our knowledge, it is for the first time that these criteria are established to optimize the variable delay in SOAs. Physically, for a long device, the optical power will be saturated at the value of *Γg/a*-1 as predicted by Eq. (10). Thus, under this situation, the solution of Eq. (11) under mean-field approximation is exactly the same as Eq. (15).

Slow light is expected when a semiconductor medium is biased as an absorber. In this case, *p _{out}* can be ignored under the circumstance of either strong absorption or long devices.

*p*is also a small value in real communication systems considering the few mW output from distributed feedback lasers (as light source) and the loss (typically 6–10 dB) in the coupling between fibers and SOAs. Under these approximations, the delay described by Eq.(13) can be simplified into:

_{in}For zero current injection, the absorption coefficient of the waveguide is typically much higher than the unsaturable loss.

Therefore, the delay-bandwidth product is roughly *p _{in}*, a linear dependence on the pump power and independent of the device length. To achieve maximum delay-bandwidth product in a semiconductor optical absorber, the loss of the waveguide needs to be minimized.

## 5. Summary

In summary, we demonstrate a variable delay of 160 ps in a semiconductor waveguide with 1.0 GHz bandwidth using population oscillation and nearly degenerate FWM. An analytical model including the spatial dependence of the optical fields is derived and found consistent with the experimental data. We also propose the design criteria for the maximum delay and maximum delay-bandwidth product given the gain of semiconductor waveguides. Finally, the delay and bandwidth of an absorptive waveguide are estimated based on the model.

## Acknowledgments

The authors would like to thank Prof. Connie Chang-Hasnain at UC Berkeley and Prof. Hailin Wang at University of Oregon for the technical discussions. We also acknowledge the support from the DARPA under grant No. AF SA3631-22549 and grant No. DARPA UCB SA4472-32446.

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