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Abstract
In this paper, the fast and slow light effect of semiconductor optical amplifiers (SOAs) under assist light injection is theoretically analyzed. Coherent population oscillation (CPO) is used to induce fast and slow light effect in the theoretical model. The effects of parameters such as modulation frequency, modulation current and assist optical power on phase delay are investigated under the assist light injection. These effects are then compared with the case without assist light injection. The simulation results show that the injection of assist light can increase the phase delay of the output optical signal and enhance the slow light effect. Furthermore, when the assist light optical power is in the range of 0–15 dBm, the slow light effect is strong.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
All-optical networks are employed for information processing in optical communication networks [1]. In all-optical networks, semiconductor optical amplifiers (SOAs) are used to realize all-optical wavelength conversion, all-optical logic gates, and other applications. SOAs offer advantages in all-optical signal processing because of their small size, simple structure, and easy integration [2]. Therefore, SOAs are important nonlinear devices for all-optical signal processing [3]. In the optical domain, fast and slow light can realize fast and precise synchronization of high-speed signals. Therefore, fast and slow light technology is one of the key technologies for all-optical signal processing. Fast and slow light is also introduced in SOAs to control the group delay of optical signals by using a certain physical mechanism [4].
Numerous studies have been performed recently on controlling the group velocity of light. When the propagating pulse interacts with the medium, the speed of the pulse decreases, and then slow light is obtained [5]. Although SOAs offer many advantages, the nonlinear effect of SOAs in the transmission process is a little weak. Optical fibers, crystalline materials and semiconductors can be used as the medium to reduce the group velocity of the optical signal for fast and slow light technology, thereby improving the nonlinearity in materials. Therefore, fast and slow light is introduced in the SOA to enhance the nonlinear effect in this study.
There are many methods and techniques to achieve slow light and fast light, and it is clear that each of them has different advantages. These methods include Stimulated Brillouin Scattering (SBS), Coherent Population Oscillation (CPO), Electromagnetically Induced Transparency (EIT) and Stimulated Raman Scattering (SRS) [6–7]. Among them, CPO has been investigated by many researchers during recent years because this method has some benefits such as greater compatibility with optical circuits, a wide range of working temperatures and ability to adjust the buffer parameters.
In this study, the SOA is used as the medium to generate fast and slow light, and the phenomenon of fast and slow light is induced via the CPO effect [8]. The four-wave mixing(FWM) is one of the main nonlinear mechanisms for signal processing in SOAs, and therefore FWM effect is considered in the calculation [9–10]. Slow light in a medium can be utilized for optical signal processing such as optical delay and delay lines, optical buffers, and also data storage. SOA can be used as a phase shifter,which helps in the design of an integrated version of a microwave photonic filter, and also gives some flexibility to control the amplitude coefficients [11–12]. This paper is organized as follows. In Section 2, a theoretical model for the injection of assist light into the SOA is established. In Section 3, the effect of assist light on the fast and slow optical performance of the SOA is analyzed.
2. Theoretical model
A current-driven SOA is presented here. In this SOA, the injected optical signal is assumed to consist of a direct current optical signal ${E_0}$, two sideband optical signals ${E_{ {\pm} 1}}$, and an assist optical signal ${E_2}$. The spectral components of the current-modulated SOA and the injected light field are shown in Fig. 1.
Fig. 1. Current-modulated SOA and spectral components of the injected optical field. (The angular frequency of the sidebands shifts from ${\omega _0}$ by $\delta $:${\omega _{ {\pm} 1}} = {\omega _0} \pm \delta$, angular frequency of the assist light shifts from ${\omega _1}$ by ${\delta _1}$:${\omega _2} = {\omega _1} + {\delta _1}$)
When a strong control beam and a weak signal beam propagate through a SOA, beating between the two beams causes oscillations of the carrier density. These oscillations create dynamical gain and index gratings in the device. Interaction of the signal beam with the dynamical gratings results in the group index change experienced by the signal. The group index can be controlled either electrically (by changing the bias current of the SOA) or optically (by changing the pump power).
To simulate light field propagation in a semiconductor, the light beam $\varepsilon (t)$ incident on the medium must be considered. The optical field can be expressed as follows [13]:
where $\omega $ is the angular frequency of the optical field, $E(t)$ is the slowly varying amplitude, $c.c.$ is the complex conjugate, and t is time. This field is assumed to comprise a direct current control beam ${E_0}$, two sideband optical beams ${E_{ {\pm} 1}}$ and an assist optical beam ${E_2}$. $E(t)$ can be expressed as follows:When this modulated beam passes through the SOA, the four components of the electric field interact with the carriers in the semiconductor through stimulated radiation. Modulation of the carrier density can cause beat frequencies between light waves, which trigger an exchange of energy between different fields. The rate equation for the carrier density N is:
In Eq. (4), $g(N )$ is the model gain, and $g(N )$ can be expressed as:
where $\alpha $ is the gain coefficient and ${N_t}$ is the carrier density, at which the active region becomes transparent.So, $\frac{{dN}}{{dt}}$ can be expressed as:
The normalized currents ${R_{0, \pm 1}}$ and the saturation power ${P_{sat}}$ in Eq. (6) can be defined as follows:
When analyzing the CPO effect in the SOA, the frequency of the modulating current is the same as the frequency $\delta $ between the direct current beam ${E_0}$ and the sideband beams ${E_{ {\pm} 1}}$ in the optical field. $I(t)$ can be expressed as:
The slowly varying envelope approximation of the propagation can be expressed as follows [14]:
where $\beta $ is the linewidth enhancement factor. The differentiation of Eq. (2) can be expressed as:Upon combining Eqs. (10) and (11), the following equations are obtained:
The carrier density can be expressed as:
Then, upon combining Eqs. (6) and (14), the following equations are obtained:
Here, ${\omega _c} = 1 + {q_0}$, ${q_i}$, $i = 1,2,3,4,5$ can be expressed as:
For fast and slow light in semiconductors, the group refractive index is typically used as a measurement parameter:
where ${n_g}$ is the group refractive index, and ${v_g}$ is the group velocity in waveguides. From Eq. (17), it can be seen that the group velocity is inversely proportional to the group refractive index. The group refractive index consists of two parts: the intrinsic refractive index of the material ${n_{bg}}$, and the change in the refractive index caused due to effects such as coherent population oscillations $\varDelta {n_g}$. Here, $\varDelta {n_g}$ can be described by the phase delay [15]: where $\varDelta t$ is the length of the transmission time, L is the length of the waveguide, and $\varDelta \varphi$ is the phase delay of the light wave. Equation (19) shows that the change in the group refractive index is proportional to the change in the phase, which implies that fast and slow light effects can be obtained by detecting the phase of the modulated signal.All equations of the model are solved using a numerical simulation. The SOA is divided into several equal sections along the direction of light transmission, and the carrier density, photon density, and phase for each section are determined.
3. Simulation results
By using the aforementioned theoretical model, the effect of assist light injection into the SOA on fast and slow light is analyzed. In this theoretical model, the confinement factor, carrier lifetime, and length of the SOA are set as $\varGamma = 0.5$, $\tau = 300ps$, and $L = 0.3mm$, respectively.
The variation in the phase delay with the modulation frequency is shown in Fig. 2. The injection current ${I_{ + 1}} = {I_{ - 1}} = 10mA$, the relative phase shift $\psi = 0^\circ $, the gain coefficient $\alpha = 3 \times {10^{ - 20}}{m^2}$, the linewidth enhancement factor $\beta {\rm{ = 0}}$, the modulation frequency ${{{\delta _1}} \mathord{\left/ {\vphantom {{{\delta _1}} {2\pi = 1GHz}}} \right. } {2\pi = 1GHz}}$, assist optical power ${E_2} = 20dBm$. As the modulation frequency increases, the phase delay curve increases firstly and then decreases. The maximum phase delay occurs at a modulation frequency of approximately 1 GHz. The maximum phase delay exhibites a gradually decreasing trend as the direct current increases. The phase delay changes from a positive to a negative value when a larger direct current is injected. In addition, the injection of the assist light increases the maximum phase delay when direct current is small. However, the injection of the assist light decreases the maximum phase delay when the direct current is large . For example, when ${I_0} = 400mA$, the phase delay changes from positive to negative, that is, from slow light to fast light.
Fig. 2. For different direct currents, the phase delay variation with modulation frequency with and without assist light injection. (The line segment with the circular marker is the result of the injection of assist light.)
The variation in the phase delay with modulation frequency in the case of injection current ${I_0} = 100mA$ is shown in Fig. 3. As the modulation current increases, the phase delay increases. For a given modulation current value, when assist light with an optical power of 20 dBm is injected, the phase delay increases. Thus, the injection of assist light can enhance the slow light effect.
Fig. 3. For different modulation currents, the phase delay variation with modulation frequency with and without assist light injection. (The line segment with the circular marker is the result of the injection of assist light.)
The phase delay variation with relative phase shift in the case of injection currents ${I_0} = 100mA,{I_{ {\pm} 1}} = 10mA$ is shown in Fig. 4. As the modulation frequency increases, the phase delay curve first exhibites an increasing trend and then a decreasing trend. When $\psi = 0^\circ ,45^\circ ,90^\circ$, the larger the relative phase shift, the larger the maximum phase delay. Furthermore, when $\delta > 1GHz$, the injection of assist light enhances the phase delay, which means that the slow light effect is more pronounced.
Fig. 4. For different relative phase shifts, the phase delay variation with modulation frequency with and without assist light injection. (The line segment with the circular marker is the result of the injection of assist light.)
The variation in the phase delay with modulation frequency in the case of relative phase shift $\psi = 0^\circ $ is shown in Fig. 5. As the modulation frequency increases, the phase delay curve first exhibites an increasing trend and then a decreasing trend. Moreover, as the gain coefficient $\alpha$ changes from $1 \times {10^{ - 20}}{m^2}$ to $3 \times {10^{ - 20}}{m^2}$, the phase delay increases. When the gain coefficient $\alpha = 3 \times {10^{ - 20}}{m^2}$, the phase delay of the assist light injection is larger than that without the assist light.
Fig. 5. For different gain coefficients, the phase delay variation with modulation frequency with and without assist light injection (The line segment with the circular marker is the result of the injection of assist light.)
The variation in the phase delay with the linewidth enhancement factor in the case of gain coefficient $\alpha = 3 \times {10^{ - 20}}{m^2}$ is shown in Fig. 6. When the linewidth enhancement factor $\beta = 0$, the phase delay of the direct current optical signal is equal to 0 and doesn’t change with the modulation frequency. In addition, the two sideband optical signals exhibite a symmetrical phase change relationship. When the linewidth enhancement factor $\beta = 0.5$, curves of several light signals are totally shifted upwards, and the phase delay of the sideband light signal is no longer in a symmetrical relationship. The phase delay of the signal light ${E_{ - 1}}$ changes from a negative to a positive value, thus indicating that the sideband signal light changes from fast light to slow light.
Fig. 6. For different linewidth enhancement factors, the phase delay variation with modulation frequency with and without assist light injection. (The line segment with the circular marker is the result of the injection of assist light.)
Next, the assist light with optical power ${E_2} = 20{\rm{ }}dBm$ is injected. The absolute value of the phase delay increases when $\beta = 0$ and decreases when $\beta = 0.5$. Therefore, it can be concluded that the injection of assist light causes the change of the phase delay.
When modulation frequency $\delta = 1GHz$, the assist light signal ${E_2}$ which has a distance modulation frequency ${\delta _1}$ from the sideband optical signal ${E_1}$ is injected. The variation in the phase delay with ${\delta _1}$ is shown in Fig. 7. It can be seen that the image is symmetrical about approximately ${\delta _1} ={-} 1GHz$. Because when ${\delta _1} ={-} 1GHz$, ${\delta _1} ={-} \delta $, which means that ${E_2}$ coincides with ${E_0}$. In addition, the curve exhibites a decreasing trend and then an increasing trend. For example, when ${\delta _1} > - 1GHz$, the phase delay increases sharply and then increases slowly at approximately ${\delta _1} = 1GHz$. Furthermore, the change in the phase delay isn’t obvious at range ${\delta _1} > 1GHz$.
Fig. 7. For different assist optical power, variation of phase delay with modulation frequency ${\delta _1}$ between sideband light signal ${E_1}$ and assist light signal ${E_2}$
Therefore, the slow light effect is better for modulation frequencies at ${\delta _1} < - 3GHz$ (indicating that ${E_2}$ is to the left of ${E_{ - 1}}$) and ${\delta _1} > 1GHz$ (indicating that ${E_2}$ is to the right of ${E_1}$). Moreover, the phase delay can be increased by reducing the assist optical power, thereby enhancing the slow light effect.
The variation in the phase delay with assist optical power in the case of modulation frequency ${\delta _1} = 1GHz$ is shown in Fig. 8. As the assist light power increases, the phase delay gradually decreases and is smoother at range $0 - 15dBm$. Thus, the assist light power can be set in this range. In addition, by adjusting the modulation frequency $\delta $, it is observed that the smaller the modulation frequency $\delta $, the larger the phase delay. As a result, the phase delay can be increased by reducing the modulation frequency $\delta $.
4. Conclusion
A theoretical model of an assist light injection SOA is established in this paper. The fast and slow light effect is caused due to the nonlinear CPO effect. The simulation results have shown that when without assist light injection, the phase delay increases with increasing modulation current, relative phase shift, gain coefficient and linewidth enhancement factor. In addition, the phase delay decreases with increasing direct current. When the assist light is injected, the phase delay first decreases and then increases with increasing modulation frequency ${\delta _1}$. Moreover, with the increase of the assist optical power, the phase delay gradually decreases. The phase change is not obvious in the range of $0 - 15dBm$, thus, it is appropriate for the assist light to be injected into this range. Conclusively, the injection of assist light increases the phase delay, and enhancs the slow light effect.
Funding
Natural Science Foundation of Jiangsu Province (BK20191012).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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