Frequency-dependent electric dc power consumption model including quantum-conversion efficiencies in ultrafast all-optical semiconductor gates around 160 Gb / s

Based on nine up-to-date types of semiconductor-opticalamplifier (SOA) samples, we devised a power-consumption model of SOAbased all-optical gates as a tool to develop faster and more efficient OTDM systems for bitrates from 10 to 160 Gb/s and those over 160 Gb/s. The conventional effect of a continuous wave (cw) holding beam was included in the model. Furthermore, in this work we defined three step-wise quantum conversion efficiencies η1, η2, and η3 from current-injected carriers through photons. The dependence of each of the three efficiencies on the SOAstructure was studied. The total efficiency ηT observed for the nine SOAs ranged widely from 0.07 to 0.46. The validity of the power-consumption model was verified by systematically measuring the effective carrier recovery rate. According to our model, the power consumption of the best existing SOA-based gate for 160-Gb/s signals is 750 mW, and this increases at a rate approximately proportional to (bitrate), and decreases proportionally to (1/ηT). ©2007 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.1155) All-optical networks; (120.5060) Phase modulation; (190.5970) Semiconductor nonlinear optics including MQW; (190.7110) Ultrafast nonlinear optics; (250.5980) Semiconductor optical amplifiers; (320.7080) Ultrafast devices; (320.7160) Ultrafast technology. References and links 1. K. E. Stubkjaer, “Semiconductor optical amplifier-based all-optical gates for high-speed optical processing,” IEEE J. Selected Topics in Quantum Electron. 6, 1428-1435 (2000). 2. Y. Ueno, S. Nakamura, and K. Tajima, “Nonlinear phase shifts induced by semiconductor optical amplifiers with control pulses at repetition frequencies in the 40–160-GHz range for use in ultrahigh-speed all-optical signal processing,” J .Opt. Soc. Am. B19, 2573-2589 (2002). 3. S. Nakamura, Y. Ueno, K. Tajima, “Error-free all-optical demultiplexing at 336 Gb/s with a hybridintegrated symmetric-Mach-Zehnder switch,” presented at Optical Fiber Communications Conference (2002), FD3. 4. Y. Ueno, S. Nakamura, and K. Tajima, “Penalty-free error-free all-optical data pulse regeneration at 84 Gb/s by using a symmetric-Mach-Zehnder-type semiconductor regenerator,” IEEE Photonics. Technol. Lett. 13, 469-471 (2001). 5. S. Nakamura, Y. Ueno, and K. Tajima, “168-Gb/s all-optical wavelength conversion with a symmetricMach-Zehnder-type switch,” IEEE Photonics. Technol. Lett. 13, 1091-1093 (2001). 6. Y. Liu, E. Tangdiongga, Z. Li, S. Zhang, H. de Waardt, G. D. Khoe, and H. J. S. Dorren, “Error-free alloptical wavelength conversion at 160 Gb/s using a semiconductor optical amplifier and an optical bandpass filter,” J. Lightwave. Technol. 24, 230-236 (2006). 7. Y. Liu, E. Tangdiongga, Z. Li, H. de Waardt, A.M.J. Koonen, G.D. Khoe, X. Shu, I. Bennion and H.J.S. Dorren, “Error-free 320-Gb/s all-optical wavelength conversion using a single semiconductor optical amplifier,” J. Lightwave. Technol. 25, 103-108 (2007). #87129 $15.00 USD Received 31 Aug 2007; revised 23 Oct 2007; accepted 24 Oct 2007; published 26 Oct 2007 (C) 2007 OSA 29 October 2007 / Vol. 15, No. 22 / OPTICS EXPRESS 14887 8. E. Tangdiongga, Y. Liu, H. de Waardt, G. D. Khoe, A. M. J. Koonen, H. J. S. Dorren, X. Shu and I. Bennion, “All-optical demultiplexing of 640 to 40 Gbits/s using filtered chirp of a semiconductor optical amplifier,” Opt. Lett. 32, 835-837 (2007). 9. C. Schubert, R. H. Derksen, M. Moller, R. Ludwig, C.-J. Weiske, J. Lutz, S. Ferber, A. Kirstadter, G. Lehmann and C. Schmidt-Langhorst, “Integrated 100-Gb/s ETDM receiver,” J. Lightwave Technol. 25, 122-130 (2007). 10. R. J. Manning and D. A. O. Davies, “Three-wavelength device for all-optical signal processing,” Opt. Lett. 19, 889-891 (1994). 11. J. L. Pleumeekers, M. Kauer, K. Dreyer, C. Burrus, A. G. Dentai, S. Shunk, J. Leuthold and C. H. Joyner, “Acceleration of gain recovery in semiconductor optical amplifiers by optical injection near transparency wavelength,” IEEE Photonics Technol. Lett. 14, 12-14 (2002). 12. G. Talli and M.J. Adams, “Amplified spontaneous emission in semiconductor optical amplifiers: modelling and experiments,” Opt. Commun. 218, 161-166 (2003). 13. G. Talli and M.J. Adams, “Gain recovery acceleration in semiconductor optical amplifiers employing a holding beam,” Opt. Commun. 245, 363-370 (2005). 14. A. E. Siegman, Lasers (Oxford Univ. Press, 1986), Chap. 7 and Chap. 10. 15. T. Saitoh and T. Mukai, “Gain saturation characteristics of traveling-wave semiconductor laser amplifiers in short optical pulse amplification,” IEEE J. Quantum. Electron. 26, 2086-2094 (1990). 16. Y. Ueno, M. Toyoda, R. Suzuki and Y. Nagasue, “Modeling of the polarization-discriminating symmetricMach-Zehnder-type optical-3R gate scheme and its available degree of random amplitude-noise suppression,” Optics Express, 14, 348-360 (2006). 17. J. Sakaguchi, M.L. Nielsen, T. Ohira, R. Suzuki and Y. Ueno, “Observation of small sub-pulses out of the delayed-interference signal-wavelength converter,” Jpn. J. Appl. Phys 44, L1358-1360 (2005). 18. M. J. Connelly, “Wideband semiconductor optical amplifier steady-state numerical model,” IEEE J. Quantum. Electron. 37, 439-447 (2001). 19. M. L. Nielsen, J. Mork, R. Suzuki, J. Sakaguchi and Y. Ueno, “Experimental and theoretical investigation of the impact of ultra-fast carrier dynamics on highspeed SOA-based all-optical switches,” Optics Express 14, 331-347 (2006).


Introduction
All-optical signal-processing gates based on the nonlinearities of semiconductor optical amplifiers (SOAs) are promising devices for future OTDM networks because they are capable of ultrafast signal processing and low power consumption [1,2].All-optical demultiplexing, 3R regeneration, and wavelength conversion based on cross-phase modulation (XPM) in SOAs have been achieved for operating frequencies exceeding 100 GHz [3][4][5], and 320-to 640-GHz operation of all-optical SOA gates using ultrafast chirp dynamics has recently been demonstrated [6][7][8].These devices typically consumed 0.4 to 1.0 W electrical power, while the latest electrical demultiplexer consumes 5.5 W in 100-GHz operation [9].As the capabilities of SOAs for ultrafast gating have become clear, the lower limit of electric power consumption and its origin have become important design issues.A practical engineering model that describes dc power consumption has, however, not yet been reported to the best of the authors' knowledge.Because such a model has not been available, it has been very difficult for system researchers to design the total power consumption of the various largescale 160-Gb/s OTDM systems now under research.
The electrical power consumption of an SOA in an all-optical gate is dominated by two factors: the amount of optical-pulse-induced modulation as the optical phase shift in the case of XPM, and the recovery time of the depleted carrier density after optical modulation.It has long been recognized that the so-called optical holding beam effectively accelerates the carrier's recovery time [10,11], while the holding beam itself consumes a significant part of the injected carriers via the stimulated recombination process and consequently reduces the amount of optical phase shift in the co-propagating optical signal component.It has also been qualitatively well known that to accelerate the recovery time while maintaining the amount of optical phase shift in the 0.3-to-1.0π range, the carrier injection rate (i.e., the injection current density) must be increased, and this increases in the dc power consumption.
To discuss the power consumption, a detailed understanding of the quantum conversion efficiency from the number of injected carriers to the number of optical-pulse-induced stimulated recombinations in the SOA is needed.It is reasonably expected that an SOA with low conversion efficiency requires a large current injection for the gating operation.Furthermore, an increase in the injection current density might slightly degrade the quantum conversion efficiency.This quantum efficiency was not taken into account in most previous studies.Some researchers discussed SOA gain within continuous wave (cw) regime and carrier recovery using one conversion efficiency parameter [12,13], but no one has studied how conversion efficiency is influenced by the SOA properties.Furthermore, no study of the conversion efficiency in ultrafast optical gating has been reported, even though this has turned out to be significantly lower than that for cw-light amplification.
In this paper, we explain our development of a practical engineering model that scientifically describes the dc power consumption of an SOA-based all-optical gate as a function of its working frequency up to 160 Gb/s.The validity of this model is verified with one series of measured power consumption values.A model of the holding beam's contribution, which is a part of our power-consumption model, is verified through a series of measured results.In these verification processes, measured parameter values are put into most of the independent model parameters, such as several types of quantum efficiencies.The observed dependence of the measured quantum efficiencies on the SOA structural parameters are also discussed.

Our model of carrier conversion efficiency
We define three different carrier conversion efficiencies, and use them to evaluate different carrier losses in the SOA. Figure 1 shows the loss model for injected carriers.

Definition of carrier conversion efficiency
We start from the total carrier number N and carrier density n in the active region.When n is below the population inversion threshold n 0 , the SOA has a negative gain and shows no bandfilling nonlinearity.Thus, the injection current I op should be larger than the transparent current I 0 (where n=n 0 ) and the number of carriers available for the gating should be reduced to the excess carrier number N ex , where N ex =N-N 0 .For this process, we define the first conversion efficiency η 1 as η 1 =N ex /N.Next, we introduce the carrier number N cw , which is the maximum number of stored carriers available for amplification of saturating cw light.This can also be understood as the number of carriers available for optical gating at the low frequency limit.N cw is supposed to be smaller than N ex defined above, because of the effects of ASE, Auger recombination, carrier overflow, intervalence band absorption, and free carrier absorption.The second conversion efficiency η 2 is defined as η 2 =N cw /N ex .In this paper, we will not discuss the individual loss contributions, because of the experimental difficulty of determining these.
Finally, we introduce the carrier number N pulse , which is the maximum number of stored carriers available for amplification of ultrafast optical pulses.N pulse is supposed to be smaller than N cw , and the third efficiency η 3 is defined as η 3 =N pulse /N cw .Its potential causes are spectral hole burning and carrier heating, or increases in ASE recombination caused by the higher average-carrier density compared with the cw-light amplification case, and so on.Note that this difference in SOA reactions to cw light and ultrafast pulses only appears well above the small signal regime, where practical gating operation is carried out.We consider N pulse to have the largest influence on ultrafast gating performance.Therefore, we define the total carrier-conversion efficiency η T for ultrafast gating as η T =N pulse /N=η 1 η 2 η 3 .If the dominant factors of each efficiency can be revealed, we will be able to maximize the total efficiency by improving each contribution separately.

Characterization of the conversion efficiency
We suppose that the carrier numbers and conversion efficiencies defined above can be characterized from fundamental SOA parameters as follows.When the carrier loss rate is 1/τ C and the total carrier number is N, the carrier loss N/τ C should balance the injection flow I op /q (q: elemental charge).Thus, the total carrier number N can be estimated as (2.1) η 1 and the excess carrier number are supposed to be given by the transparency current I 0 as Here, we should mention that N and N ex obtained from these relations contain certain underestimations when the dependence of the carrier loss rate on the injection current is large.We will discuss it in Section 3.
N cw is supposed to be obtained from the gain-saturation measurement of the SOA for cw light.As the intensity of the input light increases, the SOA chip gain G decreases from the small signal gain (SSG) G 0 .For an SOA with saturation power P sat cw , G is suppressed by 3 dB when the output power reaches P 3dB = P sat cw •ln 2. Taking the conventional discussion [14,15] regarding the gain saturation into account, we relate N cw to P sat cw , G 0 and the carrier lifetime Similarly, N pulse is supposed to be obtained from the gain-saturation measurement of the SOA for ultrafast pulses.If the injected pulse train has a much longer interval than τ C , the net gain of the pulse will be as follows: (2.4) The pulse saturation energy E sat pulse can be obtained from the 2.35-dB suppression point, and we relate it to N pulse as follows: Wavelength ( μm)   The intrinsic difference between N cw and N pulse seems to be overlooked in the conventional gain-saturation theory.In the new rate equation model (see Section 4), both cw and pulse saturation are modeled as independent characteristics.

Conversion efficiency measurement and results
To evaluate the carrier conversion efficiency of actual SOAs, we measured the fundamental parameters of several custom-designed SOA chips and commercial modules from different manufacturers (A to D). Attributes of the SOA samples and examples of ASE spectra are shown in Table 1 and Fig. 2, respectively.For series A and B, we prepared chips with different (effective) active-region lengths L eff .For each custom-designed chip, except for the shortest one (A#3), more than two wires per electrode were bonded to enable injection currents over 500 mA.
The typical setup for SOA-chip characterization is shown in Fig. 3.We coupled lensed fibers to the input and output facets of the chip using precision stages, and obtained 2-to 5-dB coupling losses for each facet.Temperature was stabilized at close to 25°C.The ASE intensities from both facets were continuously monitored by power meters, and through careful adjustment of the couplings fluctuations were kept below 0.2 dB.First, we measured the cw-gain by using a DFB-LD (λ 2 = 1548 nm) and an optical spectrum analyzer.Figure 4 plots the measured SSG of each sample against I OP .The transparent currents I 0 were obtained from these results.Figure 5(a) shows typical gain-saturation profiles.We obtained P sat cw as a function of I OP from the 3-dB suppression points.As can be seen in Fig. 6(a), P sat cw increased Fig. 4. Measured small-signal gains of SOA samples versus current.Sample details are given in Table 1.Gains were measured with cw light, λ 2 =1548 nm.Input cw power into each chip was kept under -30 dBm.almost linearly with I OP without convergence.The SOA gains for ultrafast pulses were measured for each sample.To obtain ultrafast optical pulses with lower frequency than 1/τ C , we used a mode-locked fiber laser (MLFL: Pritel Inc., UOC-3) with an external LiNbO 3 modulator.Pulses with a 2-ps width, λ 1 =1555 nm, and 10.5-GHz frequency from the MLFL were modulated down to a 0.65-GHz pulse train.We monitored the auto-correlation trace of the pulse train so that we could keep its extinction ratio above 20 dB.Typical gain-saturation profiles are shown in Fig. 5(b).We see that for each I OP , the SSGs for the cw input are almost the same as for ultrafast pulses.We fitted the measured profiles to Eq. (2.4), and they showed good agreement up to E out ~ 3000 fJ.The difference at E out > 3000 fJ was probably caused by the slight residual of the 10.5-GHz pulses.From the fit results, we obtained E sat pulse of each SOA as a function of I OP (Fig. 6(b)).In contrast to the case of P sat cw , they converged as I OP increased.The carrier recovery rates 1/τ C of each SOA at several injection currents were determined from XGM measurements using a cross-correlator (Femtochrome Research Inc., FR-103XR) and mode-locked pulses modulated down to 1.3 GHz.External variable delays and an elaborate calibration process were used to expand the scan range to 600 ps.The intensity of the probe cw light was typically set to -25 dBm to suppress the holding beam effect.Pulse energies into each SOA chip were adjusted within 10~500 fJ so that no gain modulation due to residual 10.5-GHz pulses appeared.Figure 5(c) shows examples of measured XGM waveforms.We can see two recovery processes after gain depletion.From the slower process, we obtained 1/τ C (Fig. 6(c)).In one exception, we assumed τ C ~100 ps for sample A#3 without fitting, since its small gain seriously degraded the S/N ratio of the output signal and prevented a precise determination of τ C .From Fig. 6(c), we see that the SOA samples can be categorized into two types: the recovery rates of B#1 and D#1 increased rapidly to 30 GHz with I OP , while those of the others increased slowly around 5 to10 GHz.
Using the measured fundamental parameters, we calculated the conversion efficiencies (Fig. 7).The efficiency values for each sample at I OP = 200 mA were η 1 = 0.68 to 0.87, η 2 = 0.17 to 1.30, η 3 = 0.29 to 0.53, and η T = 0.07 to 0.40.These values indicate that total efficiency η T strongly depends on the SOA structure.This dependence resulted mostly from η 2 .The B-series MQW samples had a much larger η 2 than the other bulk samples.Among the A-series, the longer chips had a larger η 2 , whereas the B-series showed the opposite η 2  dependence on L eff .The reason for this difference is unclear.η 1 depends on I 0 , and decreases with L eff since I 0 is proportional to L eff .η 3 was significantly smaller than unity.The results in Fig. 7(c) indicate that η 3 decreases as I OP goes above the optimum condition, and that longer SOAs tend to have larger η 3 values at large I OP .The total efficiency η T of longer SOAs also tends to be higher at large I OP .Sample B#1 will be the most economical device for I OP ~ 400 mA (η T ~0.34), while B#3 has the best efficiency for I OP ~ 150 mA (η T ~0.46).There appears to be a problem with the definition of N ex using Eqs.(2.1) and (2.2), since the calculated η 2 sometimes exceeds unity.This underestimation of N ex will be reduced if we use another definition as taking into acount that the carrier lifetime τ C , which is related to the differential carrier injection rate, decreases with the injection current.Determination of N ex using Eq.(3.1) is difficult, however, because it requires precise measurement of τ C at small I OP .Then we think using Eqs.(2.1) and (2.2) is not a concern and appropriate since the idea is to get a rough understanding of the SOA-characteristics.Thus, our interpretation of the dependence of the efficiencies on the SOA structure can be summarized as follows.First, the MQW structure is advantageous for increasing η 2 .A longer active region decreases η 1 , increases or decreases η 2 , and increases both η 3 and total efficiency at large I OP .We could not obtain conclusive results for width, thickness or confinement factor, though some of these factors seemed to significantly affect efficiencies.Although these results are useful, we will need to perform more systematic characterizations before we can completely understand the SOA-structure effects.

Estimation of SOA electrical power consumption of all-optical gating
Here we propose a model for determining the electrical power requirements of SOAs used in ultrafast all-optical gates.First, we discuss the carrier dynamics of SOAs.Then, we estimate the power consumption from the calculated carrier dynamics.Finally, we compare the calculated power consumption with measured results.

Rate equation model including conversion efficiencies
In our preceding analyses, we used a single rate equation to calculate the carrier-density modulation in one SOA module [2,16].This is a simple but powerful tool to explain the gating characteristics in the time domain [17].To account for the effect of the three different conversion efficiencies, we expanded the rate equation to the following form: (4.1) (n pulse =N pulse /V: carrier density available for ultrafast gating, P pulse and P CW : input light intensities, L eff and V=L eff w A d A : active region effective length and volume, Γ: confinement factor, dg/dn: differential gain).Note that η 3 is multiplied by the cw-light term, in view of the difference between n pulse and n cw (the carrier density available for cw light).Both cw gain and pulse gain were supposed to be determined by n pulse , since we did not observe much difference between them unless gain saturation occurred (Fig. 5).We also supposed there are reserved carriers (n reserve =n cw -n pulse ) which do not explicitly affect gain, and (1-η 3 ) of the photons are converted from reserved carriers through intraband interactions in the case of cw-light amplification.This rate-equation model can explain the experimental results regarding the gain saturation for ultrafast pulses quite well, and those regarding the cw-gain saturation up to 3-dB suppression points (Fig. 8(a)).For stronger cw light, other theories may provide better predictions of the gain suppression [18].
The rate equation can be used to calculate the optical carrier modulation and nonlinear phase shift.Injection of P pulse causes carrier recombination Δn pulse , and the nonlinear phase shift is given by ( (k 0 : wave number in vacuum, n r : refractive index).
To verify the model, we simulated the SOA characteristics under holding-beam injection.
We used the measured values of η and τ C at each I OP , and extracted dg cw /dn and dg pulse /dn from the corresponding SSG values.Figure 8(b) shows measured and calculated results for the effective carrier recovery rate of sample B#3 accelerated by the holding-beam.Figure 8(c) shows measured and calculated results for pulse-gain saturation with and without a holding beam.We see that both results approximately agree with each other.(*3) Assumed in view of the values in Table 1.

1.4
Refractive index change, dn r /dn pulse (10 -20   We also compared measured and calculated nonlinear phase shifts for samples A#1 and B#1 to #3.These can be experimentally determined from XPM spectra out of the SOA [2,19].The chosen pulse frequency was either 2.6 GHz or 10.5 GHz, depending on the carrier recovery rate of the SOA.dn r /dn was estimated for each sample from the measured phase shifts at I OP = 200 mA (Table 2).The difference between samples seemed to be negatively correlated to the typical carrier densities.The SOA simulation could approximately reproduce the measured nonlinear phase shifts at various I OP , P CW and P pulse values (Fig. 8(d)), and our model seems feasible for power-consumption simulations.

Electrical power consumption
We calculated the correspondence between the electrical power consumption P OP and the operating frequency B for an SOA used in an XPM-based all-optical gate as follows.First, we fixed the control pulse energy at a realistic value (340 fJ).Then, for each I OP condition we calculated the induced ΔΦ as a function of the holding beam intensity.The required amount of ΔΦ depends on the function of the gate: for 3R regeneration or wavelength conversion it is ~0.3π [16].Therefore, for each I OP , the equation ΔΦ = 0.3π provides the maximum available holding-beam intensity P CW .The carrier recovery rate 1/τ eff for this condition can be interpreted as the approximate frequency limit B of the gate.P OP was calculated from I OP using empirical V-I characteristics.
Before discussing the power consumption of actual SOAs, we will look at calculated values using non-empirical parameter sets to find their dominant factors.Figure 9 shows our results.Here, the dependences of dg/dn, η and τ C on I OP were ignored, in contrast to the cases shown in Figs. 8 or 10.Calculations were done down to nearly the low power limits, where ΔΦ = 0.3π can be satisfied only when P CW = 0. We found that the maximum frequency is almost proportional to the total conversion efficiency η T and dg/dn, but not sensitive to the intrinsic carrier recovery rate 1/τ C .As for the conversion efficiencies of each process, η 3 affects the required holding-beam intensity as well as B. As long as we keep the SOA parameters constant, I OP increases almost linearly with B, so P OP increases with B 2 .
Table 3. Parameters assumed for Fig. 9 Fig. 9. Calculated power consumption of SOAs versus their maximum operating frequencies when SOA parameters are independent of I OP (Table 3) Fig. 10.SOA electrical-power consumption versus carrier recovery rate under holding-beam acceleration.Calculated results using the measured parameters for each sample and measured results for sample B#3 are shown.A control pulse with a 2-ps width and 340-fJ energy was used.
Figure 10 shows the calculated power consumption for the SOA samples.We used the SOA parameters from Section 4.1.The B-series shows better performance than the A-series.When P OP is small, B#3 has the best performance around 20 GHz.As the target frequency increases to over 100 GHz, B#1 becomes preferable.The power consumption of B#1 at 160 GHz is about 750 mW.This is comparable to the power consumption, about 400 mW, of the latest XPM-based 160-Gb/s wavelength conversion [5] or wavelength conversion using ultrafast chirp dynamics [6].For the A-series, the longer sample had a higher frequency, and the shortest sample A#3 could not achieve ΔΦ = 0.3π under the injection-current limit.These characteristics result from those of η T as discussed in Section 3. Except for the difference in Electrical Demux, ref [9] ( ) scale, most of the P OP -B profiles in Fig. 10 resemble the corresponding P OP -1/τ C profiles.This is due to the complicated correlations of SOA parameter changes.Consequently, P OP of A#1 and B#1 increase with B 2 within the range of about 300 to 1000 mW, and the corresponding values of the others likely increase faster than that.Further discussion requires measurement of the efficiencies at larger P OP .
To prove the validity of our power-consumption model, we measured P OP and 1/τ eff at ΔΦ = 0.3π for sample B#3 (Fig. 10).We obtained good agreement between the measured and calculated results.Hence, we believe that our model produces satisfactory results as an initial power-consumption model, although it can be improved.

High-frequency limit
To discuss the future of photonic networks, it is useful to express the B-P OP relation with a simple formula.In this section, we therefore derive an analytical expression for the highfrequency limit.
First, we assume that both the injection current I OP and holding-beam power P CW are large.P CW is chosen according to I OP so that ΔΦ stays at a fixed value, as in Section 4.2.Without control pulses the carrier density reaches an equilibrium at n pulse = n eq .The small difference in dg cw /dn and dg pulse /dn is ignored for simplicity, and the saturated SOA gain is expressed as ( 4 .3 ) Note that we can approximately suppose that G eq does not change with I OP , because of the constraint on ΔΦ.The reason is as follows: The number of carrier-to-photon conversions when a control pulse is injected into the SOA in the steady state is given from the net gain of the pulse G pe and E pulse : ( 4 .4 ) as long as the pulse width is short enough compared to the carrier relaxation time.Then, G pe can be roughly estimated from G eq .Since ΔΦ is given from Δn pulse by Eq. (4.2), we can transform Eq. (4.4) into ( 4 .5 ) Both ΔΦ and E pulse are regarded as constant when we obtain the B-P op relation.We also assume that dn r /dn does not depend on I OP .Therefore, we can approximately regard G eq as constant.For example, ΔΦ=0.3π and E pulse =340 fJ result in G pe = 2.5 for sample B#1.
By assuming a steady state and taking only terms including I OP or P CW in Eq. (4.1), we obtain the following relation from Eqs. (4.1) and (4.3): ( 4 .6 ) The operating frequency limit can then be obtained as follows: After first-order expansion of Eq. (4.1) in terms of n pulse around n eq , the relaxation of the carrier density deviation Δn(t)=n pulse (t)-n eq after the control-pulse injection can be expressed as When 1/τ C converges to a finite value as I OP increases, the second term will dominate.The applied voltage V op approaches RI OP (R: resistance).Then, when we assume both dg/dn and η T are independent of I OP , P OP is expressed as (4.9)Therefore, the B-P op relation will be quadratic under the above assumption.This explains the Fig. 9 results.The total conversion efficiency has an inverse-square contribution to P OP for the actual SOA, so it should be elaborately optimized to reduce the dc power consumption.

Conclusion
We developed a new model of the electrical power consumption of SOAs used in ultrafast alloptical gates.The dominant factors of this model are three different carrier-conversion efficiencies and the differential gain of the SOA.
To predict power consumption we measured the conversion efficiencies of current SOAs with our original method.They were η T = 0.07 to 0.46 in total.A systematic study of the efficiencies of SOAs with different structures revealed that a longer SOA with an MQW structure will have high conversion efficiency at a high injection current.Further study is needed with this method to understand the dependences on the SOA width, thickness, and confinement factor, and through such studies we hope to design a more efficient SOA in the near future.We calculated the power consumption P OP as a function of operating frequency limit B, based on our power consumption model and measured SOA parameters.We obtained good agreement between the calculated and measured results.A longer SOA with an MQW structure should consume less electric power when the operation frequency is about 100 GHz.The calculated power consumption at 160 Gb/s for our best SOA (B#1) was 750 mW, and it is comparable to experimental values in the literature.At high bit-rate, P OP will increase as B 2 or even faster, depending on the decrease in the conversion efficiencies and differential gain.We believe that the dc power-consumption model that we have developed in this work can be extrapolated to higher frequencies, for example to the 400 to 640 Gb/s range.Such design extrapolation will help material, device, and system researchers in these fields to develop their visions regarding ultrafast, all-optical signal-processing systems in the near future at and beyond the 160-Gb/s level.

Fig. 5 .
Fig. 5. Typical SOA chip characteristics (a) Gain-saturation profiles for cw light.(b) Gainsaturation profiles for ultrafast pulses (2-ps width, 0.65-GHz repetition).The dashed lines in (b) show the theoretical fit using Eq.(2.4).(c) Typical XGM profiles measured with a cross correlator.The cw-probe intensity and pulse energy into the chip were set to -25 dBm and 10 fJ, respectively.

Fig. 6 .
Fig. 6.Measured SOA parameters used for evaluation of the conversion efficiencies (a) Saturation power of SOA chips versus current for cw light.(b) Saturation energy for ultrafast pulses (2-ps width, 0.65 GHz).(c) Carrier-recovery rate 1/τ C .

Table 1 .
List of SOA samples and their structure (*1) Effective length, considering 50 % contribution from taper regions (*2) Sum of well thickness

Table 2 .
Estimated refractive index change of each sample, and related parameters.