Cross gain modulation in quantum dot semiconductor optical amplifiers under the influence of the Probe

—The Pulse effect on the cross-gain modulation (XGM) in the quantum dot (QD) semiconductor optical amplifiers (SOAs) is investigated using the combining of the SOA power with the QD rate equations system. The QDs structure includes three regions: ground state (GS), excited state (ES) and wetting layer (WL). Thus, a set of rate equations for pump and probe signals is introduced for both steady-state and small-signal power values, which are solved numerically. The pulse shape was included in the analysis, for pump and probe signals. The theoretical results showed a good agreement with the experimental results. It was found that decreasing pulse width of pump/probe ratio is efficient to increase XGM efficiency and bandwidth .


I. INTRODUCTION
Recently, w wavelength conversion techniques attract a great interest due to their all-optical potential applications [1].One of the most semiconductor optical amplifiers (SOAs) processes, which is considered as the key device for all-optical convertors, is the cross-gain modulation (XGM) [2][3][4].In XGM, the SOA gain of a weaker probe is modulated by a strong pump signal, and then transferring information with the pump signal to the probe.Thus, the data are converted from pump to probe wavelength [5] and the conversion takes place without modifying the data content of the signal [6][7][8].An important advantage in XGM is that the input power of the optical signal has a large dynamic range.It is relying on the modulation of the carrier density and is thus limited by the relatively slow carrier generation and recombination rates [9].For bulk and quantum well (QW) SOAs, when the bit period is close to gain recovery time of the SOA, a crosstalk penalty results between the multiplexed input signals change in their active regions [10,11].Another XGM discontinuity is the degradation of the extinction ratio of the upconverted signal [5].In bulk and QW SOAs, the dominant gain recovery mechanism is the total carrier density depletion which is slow.This limitation can be overcome by the use of quantum dot (QD) SOAs which are predicted to have an ultrafast gain recovery time due to the dominance of spectral hole burning (SHB) mechanism [12][13].
It is demonstrated that the use of QD SOAs improves XGM characteristics.The pattern-free effect is demonstrated by the high gain or low dot density [13].The gain recovery is enhanced by the carrier relaxation from excited state (ES) to the ground state (GS) in the QD [14].Additionally, QDs have both homogenous and inhomogeneous broadenings of gain so that broadband XGM is possible [15].XGM has been discussed extensively during the last decades.XGM in pdoped QD SOAs was reported [12].Error-free 320 Gb/s operation is presented [16].XGM in columnar QD SOAs was also reported experimentally [17].Different approaches and wide structures were discussed [4,12,18].
Studies on theory of XGM in QD SOAs follows the method reported in [19,20].In their works, the small-signal analysis was s done for QD SOA and a rate equation for small-signal power in both pump and probe were calculated.The steady-state part for power was defined by a rate equation of power ( Eq.4) , but this is not the adequate case since there is a correlation between small-signal and steadystate parts of power, as we can see in Eq. ( 11) below.So, an adequate definition for the steady-state power rate equation was undertaken here.Additionally, pulse shape was introduced through our analysis of power which is important to be covered in SOA processes analysis.These point represents the difference between our work and others.Thus, in our analysis, here, a set of four rate equations system for steady-state and small-signal powers for both pump and probe were obtained.Then, they were solved numerically to get the efficiency.When these equations were connected with the QD SOA rate equations, a form covers the most important factors into QD SOA work was introduced.The results from our model were compared with the experimental result in [19] and a good a agreement was shown.This work was organized as follow: in section 2, XGM analysis is presented.In section 3, the conversion efficiency is defined while the small-signal theory for XGM is states.The simulated QD structure in this study was described in section 5. Section 6 presents the results and the discussion.Finally, The conclusions from this work are drown in section 7.

II. Theoretical analysis of XGM in QD SOAs
Theory of XGM in QD SOAs based on QD rate equations has been developed in different works [12,13,18,21].The carrier dynamics were described by the rate equations for electrons in GS, ES, and WL which is serving as a carrier reservoir [14].This is because of the larger effective mass of holes and lower quantization energies of QD levels in the valence band, and that led to a faster relaxation of holes.Therefore, electrons" behavior limits dynamics of the carrier.It was assumed that carriers were injected directly from the contacts into WL and thus, the barrier dynamics are ignored [18].The rate equations in the QD SOA can be written as [6] 22 (1 ) (1) (1 ) (1 ) 2 1 (3)  is the material loss, z is the distance in the longitudinal direction where (z=0) is the input facet and the output facet is at (z=L), L is the length of the SOA, ( , )  g z t is the gain and is expressed as , where max g is the maximum gain.The solution of Eq. ( 4) is obtained by integration as follows [6],  (1 ) 11 (1 ) (14) ( 1)

we need to know the variation ratio between GS and ES occupation probability
is the variation between carrier density and ES occupation probability.It was obtained from the small-signal analysis of Eqs.(1).It is given by,  can be obtained from Eq. ( 13), 0 f can be taken from GS time series curve when it reaches steady-state.A similar procedure was also done for 0 In Ga As WL, while the substrate is GaAs.This structure has been used by many research to study the different properties of QD [22,23].It was assumed here that the disk has a radius of (14 nm) and a height of (2nm).The sub-bands energy of this QD structure are shown in the Fig. 1.The structure has two conduction sub-bands and four heavy hole (HH) valence sub-bands.The parameters were used in the calculations are in Table 1.

VI. CALCULATIONS, RESULTS AND DISCUSSION
Figure 2 shows that was calculated from Eq. ( 15) which is reduced at wider detuning.increased at wider detuning.Figure 4 shows XGM conversion efficiency from QD SOA for different pulse widths for pump and probe pulses.For a short pump and probe pulses with FWHM equals (1ps) a high XGM efficiency was obtained.When the pump/probe width ratio increased,XGM is decreased, however, it increased with the reduction of pump/probe ratio.It is well known that QD SOAs can be driven to saturation by a short pulse.This increases the XGM efficiency.When the pulse width ratio between pump/probe increases, the saturation gain, and then efficiency increases.Accordingly, one can use the lower pump/probe ratio to obtain low (-3dB) bandwidth, i.e. low cross-talk.One can obtain high bandwidth with high pump/probe pulse width ratio.Note that increasing probe width is more efficient to increase bandwidth, as shown here, where an enough discrimination between curves is shown.Fig. 5 shows XGM with probe power as a parameter.Increasing probe power reduces the efficiency.Each efficiency curve decline first, then, at ~10GHz it becomes flat.Additionally, XGM bandwidth decreases with increasing the probe power.It was observed experimentally that XGM in QD SOAs behavior attributes to the saturation features [18].At low frequency, the modulation was due to the carrier density depletion in the WL carrier reservoir, where the overall gain spectrum is reduced.It was characterized by a slow recovery time.At high frequency, SHB, which is characterized by fast recovery, was the dominant mechanism.Fig. 6 shows a comparison between calculated XGM efficiency from this model with experimental results from [12] where a good agreement was obtained.

VII. CONCLUSIONS
Theory of XGM in QD SOAs was discussed by combining the QD rate equation system with the pulse propagation into QD SOAs including the three region of QD structure GS, ES and the WL carrier reservoir.It was found that both WL reservoir and QD states contributedinto slow and fast recovery, respectively.It was found that the pump/probe pulse width ratio can be adjusted to control both efficiency and bandwidth.In Ga As / GaAs )QD.
Fig. 2 The variation of (WL carrier density/ES occupation) versus detuning.Open circles are for experimental data which is taken from [12].
CONFLICT OF INTEREST Authors declare that they have no conflict of interest.

at frequency 2 
) are the input intensity of a pump and a probe pulses which are injected from the same facet of SOA while () p PL and () s PL are their values at the distance L from the input (at the end facet of the SOA).Let the frequency detuning between the two fields is larger than the inverse of stimulated carrier lifetime.Neglecting the four-wave mixing products induced by nonlinear interaction between the two fields.p p and s p are the small-signal harmonic modulations of p P and s P .At the input of the SOA, only the pump (0) QUANTUM DOT STRUCTURE USED IN THIS STUDYThe QD structure chosen in this study is (InAs) grown on 0.53 0.47

Fig. 6 .
Fig. 6.Comparison between theoretical and experimental XGM efficiency.Open circles are for experimental data which is taken from[12].
is the carrier density in WL.F and h are occupation probabilities in GS and ES respectively, e is the electron