## Abstract

Dynamic characteristics of semiconductor optical amplifiers (SOAs) with certain facet reflection in different operation conditions are theoretically investigated with a detailed wideband model. Influences of different facets reflectivities are numerically simulated for different lengths of active regions. The results indicate that the gain recovery time can be reduced to 50% of the initial value while the other related characteristics are optimized for appropriate facets reflections. A half reflective semiconductor optical amplifier (HR-SOA) with a cleaved facet on rear facet and an antireflection coating on front facet can speed up the gain recovery with easy realization and low cost. The related characteristics of this structure are evaluated. It’s also indicated that the gain recovery has further potential to be reduced as low as twenties picoseconds for a long active region.

© 2007 Optical Society of America

## 1. Introduction

Semiconductor optical amplifiers (SOA) as important wavelength conversion and switching devices used in optical networks have been theoretically and experimentally studied in last decades [1–3]. In high-speed optical communication systems, the performance is restricted by the dynamic characteristics of the SOA related to carrier lifetime. To overcome this bottleneck, structural and operational designs are suggested. For structural improvement, quantum well SOAs (QW-SOAs) [4] have been already commercialized. Using p-type-doped multiple quantum wells is newly advised and fabricated [5]. Recently, a theoretical designed carrier reservoir SOA (CR-SOA) [6] may become a transitional design for quantum dot SOAs (QD-SOAs) [7] which still have some fabrication problems and won’t be commercialized in a short time. For further improvement, operational designs can be used to reduce effective carrier lifetime. It is suggested by using an external [8–10] (an assist light or a holding beam e.g.) or an internal [11, 12] (GC-SOA e.g.) light. A scheme of a detuned band-pass filter cascaded after an SOA can satisfy with the switching speed of 160Gbps optical communication system [13]. However, the trade-off between low patterning and high optical signal to noise ratio (OSNR) must be balanced. A simple principle suggested in this paper is that a properly designed facet reflection may increase the optical intensity as well as saturation in the amplifier while decrease the effective carrier lifetime without special schemes or additional devices simultaneously. An optimal design for facets reflections to accelerate gain recovery and improve cross-gain modulation (XGM) and cross-phase modulation (XPM) related dynamic characteristics are the main purpose of this paper. A half reflective semiconductor optical amplifier (HR-SOA) with a cleaved rear facet and an antireflection (AR) coating on the front facet of the SOA can satisfy with this demand. Comparing with the reflective SOA (RSOA) [14] in which input and output share one port, this proposed HR-SOA is different, as that its input and output ports are separated. A detailed model is necessary in order to precisely evaluate the impact of facets reflection on the dynamic characteristics of SOA. It is worth to be noticed that few of the models described in [15–18] account for facet reflection but amplified spontaneous emission (ASE) noise is considered to be wavelength independent. In this paper, the impacts of different reflectivities are analyzed based on numerical simulation with a wideband multisection model which is similar to but more detailed than Connelly described in literature [17] in treating ASE noise. In our model, two aspects different from usual description must be emphasized. The model is still efficient and not intensive in computation by using Newton iteration method and modified TMM [19] that the spectrum property of spontaneous emission was enhanced. The wavelength dependent spontaneous emission is calculated by a quasi-analytic method rather than an equivalently phenomenal treatment for accurate evaluation with considerable facet reflections.

The paragraphs will be presented as follows. The detailed multisection model containing the interaction of photons and electrons in direct bandgap bulk-material is described in section 2. The effect of residual facet reflectivity will be discussed in the following section 3. Further more, the pulse induced variation of phase and gain change, and ER of switched channel are numerically simulated and analyzed respectively. The conclusion is presented in section 4.

## 2. Wideband model

#### 2.1 Optical parameters

To investigate the effect of the residual facet reflectivity on the dynamic characteristics of the SOA quantitatively, a detailed wideband model should account for the wavelength dependent ASE that can be greatly stimulated at a considerable value of facet reflectivity. To describe the wavelength dependent spontaneous emission and gain preciously, the material gain coefficient (*m*
^{-1}) of the active region in a InGaAsP direct bandgap bulk-material is given by [17, 20]

$$\times \left({f}_{c}\left(\omega \right)-{f}_{v}\left(\omega \right)\right)\frac{{T}_{2}}{\pi \left[1\text{}+{\left(\omega -{\omega}_{0}\right)}^{2}{T}_{2}^{2}\right]}\mathrm{d\omega}$$

$$\phantom{\rule{3.5em}{0ex}}=\frac{{c}^{2}}{2{n}_{1}^{2}{\omega}_{0}^{2}\tau}{\left(\frac{2{m}_{e}{m}_{\mathrm{hh}}}{\u0127\left({m}_{e}+{m}_{\mathrm{hh}}\right)}\right)}^{\frac{3}{2}}$$

$$\times {\left({\omega}_{0}-\frac{{E}_{g}}{\u0127}\right)}^{\frac{1}{2}}\left({f}_{c}\left({\omega}_{0}\right)-{f}_{v}\left({\omega}_{0}\right)\right)$$

where

c
| velocity of propagation of light in vacuum; |

n
_{1}
| active region refractive index; |

ω
_{0}
| stimulated recombination central angular frequency; |

τ
| radiative carrier recombination lifetime; |

ħ
| normalized Planck’s constant; |

m
_{e} | effective mass of an electron in conduction band; |

m
_{hh} | effective mass of an heavy hold in valence band; |

T
_{2}
| mean lifetime for coherent interaction of electrons with a monochromatic field in semiconductors and is the order of 1 ps. |

*f _{c}*(

*ω*

_{0}) and

*f*(

_{v}*ω*

_{0}) are the Fermi-Dirac distributions which determine the occupation probabilities for the electrons in the conduction band and the valence band respectively.

*E*is the bandgap energy.

_{g}In high speed communication system, an ultrashort pulse with the duration of few picoseconds may induce a gain compression which can be simply described by a compression factor, presented as [21]

where *ε* is the nonlinear gain suppression parameter with contribution from spectral-hole burning (SHB) and carrier heating (CH). *S* is the total photon density. However, the nonlinear gain suppression parameter in [10] is presented as a wavelength dependent function; the Eq. (2) is still accurate for the incident lights located near the gain peak

In quite a similar manner, the spontaneous emission rate per unit volume per unit frequency centered in angular frequency *ω _{j}* is expressed in

*m*

^{-3}

*s*

^{-1}

*Hz*

^{-1}as

The internal waveguide loss as a linear function of carrier density is expressed as [17]

The net gain coefficient can be expressed as

#### 2.2 Propagating of optical field

The propagating of incident optical fields is described as the Connelly’s model [17]. To further consider the contribution of phase change from CH, additional terms of $\frac{\partial \varphi}{\partial z}=-\frac{1}{2}\Gamma {\displaystyle \sum _{\beta =c,v}}{\alpha}_{{T}_{\beta}}{\epsilon}_{{T}_{\beta}}{g}_{c}S$ [21] are taken into account. However, the propagating of spontaneous emission field must be further discussed. The propagating equation of spontaneous emission fields is presented as

where *S _{j}*

^{ASE±}(

*z*) accounts for the amplified spontaneously emitted photon density per unit frequency spacing centered in angular frequency

*ω*. Γ is the optical confinement factor.

_{j}*R*

_{j}^{ASE±}is the amplified spontaneously emitted noise coupled into

*S*

_{j}^{ASE±}.

Equation (6) is subject to boundary conditions [22]

where *β* is the spontaneous emission coupling factor. *v _{g}* is the group velocity.

Assuming that the carrier density is spatial homogeneity, the solution of (6) is

where

#### 2.3 Multisection model & numerical simulation

The carrier density at z is governed by the rate equation

where *I* is the SOA bias current. *e* is the elementary charge of the electron. *V* is the active region volume. The second term to fourth term on the right hand of the Eq. (14) account for the spontaneous recombination rate, stimulated emission recombination rate from pump (*i* = 1) and probe (*i* = 2) lights and ASE optical field, respectively, which are expressed as

$$\phantom{\rule{3.4em}{0ex}}=\int \Gamma {g}_{c}\left(\omega \right){v}_{g}\frac{{R}_{j}^{\mathrm{ASE}}}{g}\left({C}^{+}\mathrm{exp}\left(\mathrm{gz}\right)+{C}^{-}\mathrm{exp}\left(-\mathrm{gz}\right)-2\right)\mathrm{d\omega}$$

The parameters of the cubic polynomial on the right hand of the Eq. (15) are as shown in table I.

The numerical simulation is based on the multisection model as shown in Fig. 1. The carrier density in each section is supposed as spatial homogeneity; however the optical intensity is exponentially varied along the propagating direction. The pump-probe scheme is used to detect the gain recovery process.

The stimulated emission rate induced by incident optical fields and amplified spontaneous emission in section *m*(*m* = 1,2,…,*M*) can be expressed as

$$\phantom{\rule{3.8em}{0ex}}=\Gamma {g}_{c}\left(\omega \right){v}_{g}\Delta \omega \sum _{j}[\frac{\mathrm{exp}\left[g({N}_{m},{\omega}_{j})\Delta l\right]-1}{g({N}_{m},{\omega}_{j})\Delta l}\left({S}_{j,m}^{\mathrm{ASE}+}+{S}_{j,m+1}^{\mathrm{ASE}-}\right)$$

$$+\frac{2{R}_{j}^{\mathrm{ASE}+}\left({N}_{m}\right)}{g({N}_{m},{\omega}_{j})}]-\frac{2{R}_{j}^{\mathrm{ASE}+}\left({N}_{m}\right)}{g({N}_{m},{\omega}_{j})}$$

The numerical simulation is combined with the static and successively dynamic processes as shown in Fig. 2 where the two steps are referring to the dash and dot box respectively. Every element in carrier array is the value of carrier density in each section. Every column of optical field matrix consists of optical intensities (or electric intensities for incident lights) of different frequencies in the same section; while every row is composed by optical (electric) intensities of a monochromatic frequency light in different sections. The static simulation starts with an arbitrary initial value of one-dimensional carrier density array. The static simulation based on the Newton iteration method aims at the right hand of the Eq. (14) (ordering to f (*n*)) approaching to zero for each section. The iteration continues until the maximum percentage change of the carrier density array between successive iteration is less than the desired tolerance.

The results of static simulation, including the distribution of optical (electrical) fields and carrier density, are set as the initial value of dynamic simulation. For dynamic simulation, the update of carrier density and optical (electrical) fields in each section after every timeslice Δ*t* is governed by the coupling rate equations described in the preceding section. The change of effective refractive index in each section is proportional to carrier density, expressed as
$\Delta {n}_{\mathrm{eq}}=\frac{d{n}_{\mathrm{eq}}}{\mathrm{dN}}\Delta N$
. After the pump pulse is turned off, the incident electrical fields and ASE optical field output from the front and rear facet of the SOA and the average carrier density in the SOA for each timeslice are recorded. The simulation based on the parallel calculation, including two incident lights, 588 discrete frequencies of spontaneous emission fields and 1,000 timeslices of pump pulse sequence can finish within five minutes.

## 3. Characteristics evaluation of facet reflectivity

The design on facets reflections may include two coating schemes, reflection coating on two facets or AR coating on front facet and reflection coating on rear facet. The reflection coating on two facets makes the amplifier as a high-finesse resonator. As a result, the carrier density will be modulated by the resonated pulse which introduces significant patterning. To address this problem, we suggest antireflection coating on the front facet to minimize the resonance. However, the pulse reflected from the rear facet can still induce significant impact on the distribution of carrier density throughout the amplifier as the reflectivity is high enough. Thereby, the acceleration of gain recovery is limited by the two-pass traveling time of signal pulse, the duration of which is less than twenty picoseconds for long SOA.

#### 3.1 Dependence of fundamental XM characteristics on facet reflectivity

Our reflection design focus on the rear facet under the condition that the reflectivity of the front facet is as low as 10^{-6}. The reflection or reflectivity mentioned below is corresponding to rear facet. The gain recovery time is defined as the duration time needed for the gain compression recover to 1/*e* of the initial compression on the assumption that the gain recovers to the initial state exponentially. The phase change is always referring to the maximum phase change induced by pump pulse during the pulse duration. The parameters used in simulation are listed in table I. The injection density of SOA is set to 10*kA*/*cm*
^{2} for the following simulation. The wavelength of probe and pump lights are fixed at 1550nm and 1559nm respectively.

The spectral properties of spontaneously emitted noises output from front and rear facet of the SOA in absence of incident light are shown in Fig. 3 for two different resonant cavities, respectively. The length of SOA is set to 1mm here. Figure 3 shows an apparent red-shift of ASE peak dependent on the facet reflectivity by reason of red-shift of gain spectrum as well as the spontaneous emission source for low carrier density in the high reflective cavity. The ASE output from front facet is lager than rear facet in the two cavities under the condition *R*
_{1} > *R*
_{2}.

The chip gain versus facet reflectivity for different probe powers and lengths of SOA is shown in Fig. 4 (output from rear facet) and Fig. 5 (output from front facet). The chip gain in long SOA is larger than that in short one for low reflectivities and week incident optical fields. The reason can be explained as sufficient amplification in a long active waveguide without saturation. However, the derivation of chip gain between two lengths of SOAs can be hardly distinguished if the gain already saturates after the first half of amplification in the long SOA for high facet reflectivities.

The gain recovery times at front and rear facet versus reflectivity for two different lengths of SOA are shown in Figs. 6(a)-6(c). The peak power and full width at half maximum (FWHM) of pump pulse are set to 5mW and 8.3ps respectively. The power of probe light is set to -20dBm, -10dBm and 0dBm in Figs. 6(a)-6(c) respectively to evaluate the gain recovery of SOA for different facet reflectivities

The gain recovery time can be approximately reduced by 50% of average in Figs. 6(a)-6(c). The long SOA proceeds over the short one because of shorter effective carrier lifetime induced by higher optical intensity. However, the gain recovery time doesn’t change significantly as deep saturation in the SOA when the reflectivity is high enough. In other words, a reflectivity higher than a critical value takes little effect on reducing gain recovery time; however, the other important characteristics must be optimized simultaneously which will be discussed in following paragraphs.

The phase changes as a function of reflectivity for three powers of probe lights are shown in Figs. 7(a)-7(c), respectively. In XPM systems, a large phase variation is expected for being induced by a small energy of modulation pump pulse. The long SOA also reveals higher performance than short one in this aspect. The phenomenon can be attributed to the length of the active waveguide which is proportional to the phase change. The phase change of output from front facet almost duplicates the value output from rear facet because of two-pass amplification. As a result of deep saturation, the phase change keeps similarity for different reflections when a large probe power injects into the SOA [Fig. 7(c)]. Further more, sufficient facet reflection produces the reflected pulse with a considerable power concomitantly. As the competition of carrier depletion from probe and pump lights, the phase change decease to a bottommost level before arising, as shown in Figs. 7(a)-7(c).

Usually the quality of the switched signal of XGM is related to its extinction ratio (ER). The spontaneously emitted noise filtered by a Gauss band-pass filter is a small fraction of the power in the switched channel. The ER characteristics for different lengths of SOA and powers of probe light are shown in Fig. 8(a)-8(c). It can be shown that the ER in long SOA is much higher than that in short one. It can be simply explained as deeper modulation for XGM signals and suppression for ASE in long SOA. The increase of ER at front facet is influenced a lot by the ASE noise when the reflectivity of rear facet and incident probe power is low. The degradation of ER for a moderate reflectivity is the result of saturation induced by the probe light. The slight improvement of ER for high reflectivity is caused by the strong pump pulse reflected from the rear facet of the SOA. However, the ER maintains at a very low level when the SOA is deeply saturated by an intensive continuous wave, which can be obtained from Fig. 8(c).

#### 3.2 Dependence of fundamental XM characteristics on the wavelength of probe wave for HR-SOA

It’s noticed in the preceding comparison that the reflectivity beyond a critical value contributes less additional improvement to the dynamic characteristics of SOA. Therefore, it is supposed that the SOA with AR coating on front facet (*R*
_{1}= 10^{-6}) and cleaved rear facet (*R*
_{2} = 0.3) can attain great improvement of dynamic characteristics with low cost. In following simulations, the wavelength and FWHM of pump pulse are set to 1559nm and 8.3ps respectively. The power of probe light is set to -10dBm. The length of SOA is 1mm.

In terms of XPM, the phase change of *π* induced by a small pulse energy is preferable. The probe light at gain peak wavelength achieves low gain recovery time while introducing high saturation. The required peak power of pump pulse for XPM-induced phase change of *π* and gain recovery time at front and rear output facet versus the probe wavelength are as shown in Fig. 9. For XGM scheme, the required pump power for XGM-induced ER of 10dB and gain recovery time at front and rear output facet versus the probe wavelength are as shown in Fig. 10.

From Fig. 9 and Fig. 10, it’s demonstrated that the gain recovery time can be reduced to forty picoseconds while the probe light is operated near gain peak wavelength, and accordingly a large pump power is necessary. The results reflect the property of gain spectrum. Owning to the asymmetry of the curves, it is suggested that up-conversion for XPM and down-conversion for XGM are preferable. For the practical operation, a fast dynamic response of chip gain and moderate operation condition must be balanced.

## 4. Conclusion

The impact of facet reflection of SOA has been analyzed by a newly reformed TMM which is accurate in describing the spectral property of ASE noise and efficient in calculation for analyzing the fast dynamic characteristics of the device, including gain recovery time, XPM-induced phase change and XGM-induced ER for different facet reflectivities and operational schemes. A structure of half reflective SOA (HR-SOA) which can be operated as traveling mode or reflective mode is designed and the fundamental XPM and XGM related characteristics are evaluated and discussed. This structure will not only show high performance in high-speed optical communication systems but also can be easily realized in practical designs with low cost.

## 5. Acknowledgments

This work was partially supported by National Natural Science Foundation of China (Grant No. 60407001), National High Technology Developing Program of China (Grant No. 2006AA03Z0414), the Science Fund for Distinguished Young Scholars of Hubei Province (Grant No. 2006ABB017), the Program for New Century Excellent Talents in Ministry of Education of China (Grant No. NCET-04-0715).

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