- Email: [email protected]
Efficiency and noise performance analysis of four-wave mixing between short optical pulses in semiconductor optical amplifiers
15 June 1999
Optics Communications 164 Ž1999. 211–217 www.elsevier.comrlocateroptcom
Efficiency and noise performance analysis of four-wave mixing between short optical pulses in semiconductor optical amplifiers Chongjin Xie ) , Peida Ye Optical Communications Center, Beijing UniÕersity of Posts and Telecommunications, P.O. Box 57, Beijing 100876, China Received 10 February 1999; received in revised form 7 April 1999; accepted 14 April 1999
Abstract Efficiency and noise performance of four-wave mixing ŽFWM. between short optical pulses in semiconductor optical amplifiers ŽSOA. are analyzed, taking into account the effects of fast gain saturation. A simple theoretical model is presented to evaluate the efficiency and noise properties of the FWM between short optical pulses. Optimum conditions for maximum conversion efficiency and signal-to-noise ratio ŽSNR. of output FWM signal are also investigated. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Four-wave mixing; Semiconductor optical amplifiers; Nonlinearities
1. Introduction Four-wave mixing ŽFWM. in semiconductor optical amplifiers ŽSOA. has wide application in all-optical signal processing and all-optical networks, and it has been studied extensively over the past several years w1–9,11–15x. Generally, there are two types of FWM in SOA, that is, continuous-wave ŽCW. FWM or quasi-CW FWM, where the pump signal is a CW, and pulsed FWM, where both pump and probe signals are short optical pulses. CW FWM can be used for wavelength conversion and mid-span spectral inversion ŽMSSI. w1,2x, and pulsed FWM can be
) Corresponding author. Tel. q86-10-622-82-205; Fax: q8610-622-83-728; E-mail: [email protected]
used for optical time division demultiplexing and optical sampling w3,4x. Compared with the CW FWM, there are relatively few investigations about the pulsed FWM in SOA. The pulsed FWM was first investigated by Shtaif and Eisenstein w5x and Shtaif et al. w6x, and an analytical solution was given by them, then Eiselt studied the optimum parameters for demultiplexer application of FWM in SOA w7x. However, because they neglected the fast gain saturation due to the intraband processes of carrier heating ŽCH. and spectral-hole burning ŽSHB., their results were limited to pulse widths on the order of a few tens of picoseconds. Recently, Mecozzi and Mørk w8x presented a rigorous approach to describe the pulsed FWM in SOA taking into account the effects of fast gain saturation, which can be applied to pulses shorter than a few picoseconds. In this paper, we will study
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 8 6 - 8
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
212
both the conversion efficiency and the noise properties of the FWM between short optical pulses in SOA with the consideration of the fast gain saturation. A simple model to evaluate the conversion efficiency will be presented, and an approximate expression of the output noise power density of the SOA under the injection of a sequence of pulses will be derived taking into account the spatial dependence of both the gain coefficient and the inversion parameter. The optimum conditions for pulsed FWM in the SOA will also be investigated. In our model, the effect of gain dispersion is neglected.
2. FWM conversion efficiency 2.1. The new model of FWM between short optical pulses in SOA When two optical signals, a pump signal and a probe signal, with frequency v p and vs are injected into the SOA, a new frequency component, the FWM signal with frequency 2 v p y vs , will be generated at the output of the SOA due to the nonlinearity of SOA. We assume that the field of optical pulses at the input of the SOA is E Ž 0,t . s Ep Ž t . q Es Ž t . exp Ž i V t . ,
V s v p y vs Ž 1.
where Ep Ž t . and EsŽ t . are the slowly varying amplitude of the pump and probe pulse field, and V is the pump-probe detuning. For FWM between short optical pulses, the pump-probe detuning must be large enough to avoid the overlap of the bandwidth of the pump and probe pulses, and < V
Ep2 Ž t . Psat
exp Ž h . y 1
FŽ V . s
2
=
1yia 1 q exp Ž h . < Ep Ž t . < 2rPsat y i Vts
qÝ x
dh s
h0 y h
Ž 3.
1 y i Vt x
y exp Ž h . y 1
ts
dt
Ž 1 y i a x . ´ x Psat < E Ž 0,t . < 2 psatts
Ž 4.
L
where h s
H
0
g Ž z,t .d z Ž g Ž z,t . is the gain coeffi-
cient and g 0 is its small signal value, L is the length of the SOA, G s expŽ h. is the power gain., h 0 s g 0 L, Psat is SOA saturation power, a is the line width enhancement factor, x represent the intraband processes, which are carrier heating of electrons in the conduction band ŽCH c . of holes in the valence band ŽCH v ., and spectral hole burning of electrons ŽSHB c . and holes ŽSHB v ., b x , t x and ´ x are the line width enhancement factors, characteristic time constants and nonlinear gain coefficients associated with them, C is the phenomenological parameter to compensate for the non-planewave nature of the waveguide. The above model does not include the effect of fast gain saturation due to intraband processes, so it can only be applied to pulses about tens of picoseconds. In the following, we will develop a new model to describe the pulsed FWM in SOA, which can be extended to pulses shorter than a few picoseconds. Since the influence of the intraband processes on FWM has already been included in Eq. Ž3. Žthe second term in the square bracket., so we only need to consider the effect of intraband processes in the calculation of h. For pulse widths larger than the time constants of intraband processes, the propagation of pulses in the SOA can be described by the following equations with the consideration of fast gain saturation w10x E g Ž z ,t . g 0 y g Ž z ,t . g Ž z ,t . P Ž z ,t . s y , ts 1 q ´ P Ž z ,t . Psatts Et
´ s Ý ´x
Ž 5.
x
EsU Ž t .
Ž 2.
EP Ž z ,t . Ez
s
g Ž z ,t . 1 q ´ P Ž z ,t .
P Ž z ,t .
Ž 6.
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
213
where P Ž z,t . s < EŽ z,t .< 2 . Let h s H0L g Ž z,t .rw1 q ´ P Ž z,t .xd z, and after some algebra derivation, we get dh
1 s
dt
1 q ´ exp Ž h . P Ž 0,t .
y´ exp Ž h . y 1
½
h0 y h
ts
d P Ž 0,t . dt
y exp Ž h . y 1 P Ž 0,t .
ž
´ ts
1 q
Psatts
/5
Ž 7.
Eqs. Ž2., Ž3. and Ž7. are our model of pulsed FWM in SOA. It is simpler than the rigorous approach, and it can be easily applied to analyze FWM in SOA between a sequence of optical pulses, which is the case for optical time demultiplexing application. In this case, the pump pulses are injected into the SOA with a repetition period T. We only need to solve Eq. Ž7. in the time period 0–T with the boundary condition hŽ0. s hŽT .. 2.2. Numerical results In the following, we present some numerical results to prove our new model. The same parameters of SOA in Ref. w8x are used: differential gain a s 3.0 = 10y2 0 m2 , cross-sectional area A s 0.4 mm2 , confinement factor G s 0.3, SOA length L s 200
Fig. 1. Comparison of the FWM efficiency dependence on pump energy calculated with different models. Solid lines are calculated with our model, dash–dotted lines with the rigorous model, and dotted lines with Shtaif and Eisenstein model. Tp is the pulse width of the pump pulses.
Fig. 2. The dynamic gain response of the SOA to different width pump pulses calculated with different method. Solid lines are calculated with our method, dash–dotted lines with the rigorous method, and dotted lines with Shtaif and Eisenstein method. The input pump energy is 250 fJ, Tp is the pulse width of the pump pulses. Ža. Pulse width is 1 ps, Žb. pulse width is 20 ps.
mm, carrier density at transparency N0 s 1.0 = 10 24 my3 , carrier lifetime ts s 200 ps, saturation power Psat s 28.4 mW, ´ SHB c s 0.35 Wy1 , ´ CH c s 0.70 Wy1 , t SHB c s 70 fs, t CH c s 700 fs, the line width enhancement factors are a s 4.0, a CH c s 3.0, the unsaturated gain G 0 s 25 dB. The wavelength of the pump signal is 1550 nm and the beating frequency V s 2.5 THz. The effects of SHB and CH for the holes are neglected, since the heavier hole mass typically leads to much shorter relaxation times than found for the electrons. In the calculation, the pump and probe powers are in the ratio of 10:1, and the pulses are all Gaussian pulses. Fig. 1 shows the results of the FWM efficiency dependence on pump energy calculated for different pulse widths. For comparison, the results of three
214
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
models are given, which are our model, the rigorous model described by Mecozzi and Mørk, and the model presented by Shtaif and Eisenstein. In this figure, the pulse widths of the pump and probe pulses are the same, and there is no time delay between the pump and probe pulses. It shows that the results of our model agree well with that of rigorous model, and only when the SOA is deeply saturated, there exist a little disagreement. It also shows that the model of Shtaif and Eisenstein is only valid for the pulse width on the order of tens of picoseconds. Unlike the one showed by Shataif and Eisenstein, when the effect of fast gain saturation is considered, the optimum pump energy is dependent on the pulse width. The shorter the pulse width, the lower the optimum pump energy. The difference of the three models can be explained by Fig. 2, which shows the effects of intraband processes on the gain dynamics of SOA for different pump pulse widths with the same energy. We see that the intraband processes have a significant effect on the SOA gain dynamics for narrow pump pulses, but for longer pulses, the effects can be neglected. In Fig. 2a, the pulse width is 1 ps. In this case, there is rapid gain depletion near the pulse center, and the gain dynamics with and without the consideration of fast gain depletion is significantly different. However, when the pulse width is 20 ps, the results of the three models are almost the same,
which is shown in Fig. 2b. The figure also shows that the difference of the gain dynamics between our method and the rigorous method is very small. Fig. 3 shows the FWM between non-overlapping pulses. Optimum pump pulse energy is used in the figure. The energy is 63 fJ for 1 ps pulse width, and 250 pJ for 20 ps pulse width. The figure shows that the FWM efficiency peaks when the probe pulse proceeds the pump, as pointed out by Shtaif and Eisenstein, and that the optimum time delay becomes smaller for narrower pulses.
Fig. 3. The FWM efficiency dependence on pump-probe delay time. Solid line is for 1 ps pulse width, and dotted line is for 20 ps pulse width. Optimum pulse energies are used in the two cases. Td is the time delay between the pump and probe pulses, Tp is the pulse width of the pump pulses.
EEASE Ž z ,t .
3. Noise properties Another important issue of the FWM in SOA is the noise. It is well known that the successful application of FWM in SOA relies on both the FWM conversion efficiency and the noise properties of the output FWM signal. For practical applications, the noise properties of the output FWM signal are even of higher significance than the FWM conversion efficiency. Recently, some investigations of the noise properties of FWM in SOA were reported w11–15x, for example, Obermann et al. w11x analyzed the noise characteristics of the CW FWM in SOA in detail, Shtaif and Eisenstein w12,13x reported an investigation of the noise performances of pulsed FWM in SOA. Here, we will discuss the noise properties of the FWM between the short optical pulses in SOA, taking into account the effects of fast gain saturation. It was shown by Shtaif and Eisenstein w12x that under nonlinear operating conditions, the SOA noise contains a narrow-band noise and a broadband spontaneous noise. The narrow-band noise comes from the nonlinear coupling of the noise and the gain and it can be neglected in the cases of short pulse amplification and nondegenerate FWM. So for FWM between short pulses in SOA, only the broadband amplifier spontaneous emission ŽASE. noise need to be considered. The dynamics of the ASE noise in the SOA can be described by the following equation
Ez
1 s
1yia
2 1 q ´ P Ž z ,t . q n Ž z ,t .
g Ž z ,t . EASE Ž z ,t .
Ž 8.
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
where EASE Ž z,t . is the ASE noise field, nŽ z,t . is a spatially uncorrelated, white noise term of the zero average and the correlation functions ² n Ž z ,t . n Ž zX ,tX . : s 0 Ž 9. ² n Ž z ,t . nU Ž zX ,tX . : s"v
g Ž z ,t . 1 q ´ P Ž z ,t .
n sp Ž z ,t . d Ž z y zX . d Ž t y tX .
Ž 10 . where d are Dirac delta functions, n sp Ž z,t . is the inversion factor which equals w g Ž z,t . q G aN0 xr g Ž z,t ., with G , a, and N0 being the mode confinement factor, the differential gain and the carrier density at transparency. The solution of Eq. Ž8. gives the noise field at the output of SOA
215
where T is the input optical pulse period. It is not possible to get an analytical solution of Eq. Ž14., and the accurate noise power spectral density can only be obtained by solving Eqs. Ž5., Ž6. and Ž14. numerically. However, with some assumptions, an approximate analytical expression of noise power spectral density can be obtained. In calculation of the second term in Eq. Ž14., we assume that the gain coefficient is a constant over the length of SOA, that is hŽ z,t . s ln GŽ t .rLz. After some algebra derivation, we obtain WASE s " v
1
T
H T 0
½
GŽ t . y 1 q
½
= ´ P Ž 0,t . G Ž t . ln
G aN0 L ln G Ž t .
1 q ´ P Ž 0,t . G Ž t . 1 q ´ P Ž 0,t .
EASE Ž L,t . s
L
H0 exp Hz
L
1yia
1
yln G Ž t . q G Ž t . y 1
g Ž zX ,t . d zX
X
2 1 q ´ P Ž z ,t .
=n Ž z ,t . d z
Ž 11 .
Its autocorrelation function is U ² EASE Ž L,t . EASE Ž L,t q t . : g Ž z ,t .
L
s"v
H0
1 q ´ P Ž z ,t .
=exp
Hz
n sp Ž z ,t .
g Ž zX ,t .
L
d zX d z d Ž t .
1 q ´ P Ž zX ,t .
SNR s
½
s " v G Ž t . y 1 q G aN0 G Ž t . =
exp yh Ž z ,t .
L
H0
1 q ´ P Ž z ,t .
5
d z d Žt .
Ž 13 .
L
where hŽ z,t . s
H
0
g Ž zX ,t .rw1 q ´ P Ž zX ,t .xd zX . Under
the injection of a sequence of periodic pulses, the output ASE noise power spectral density is WASE s " v =
1
T
H T 0 L
H0
½
G Ž t . y 1 q G aN0 G Ž t .
exp yh Ž z ,t . 1 q ´ P Ž z ,t .
5
d z dt
Ž 14 .
dt
Ž 15 .
Eq. Ž15. is our analytical expression of the output ASE noise power spectral density of the SOA under the injection of a sequence of optical pulses. The noise power spectral density itself is not an appropriate measure in order to quantify the noise properties of devices based on FWM in SOA. Generally a more system related quantity signal-to-noise ratio ŽSNR. is used, which is defined as
Ž 12 .
Substitution of n sp Ž z,t . into Eq. Ž12. yields U ² EASE Ž L,t . EASE Ž L,t q t . :
55
Pc Ž L . PASE
s
Pc Ž L . WASE B
Ž 16 .
where Pc Ž L. is the average output FWM signal power and B is the bandwidth of the optical bandpass filter. Here, we present some calculation results. To evaluate the noise performance of the FWM in SOA between a sequence of pulses, it is convenient to use the average optical power instead of pulse energy. Fig. 4 shows the output noise power vs. the average input optical power. For comparison, the results of the CW input and the pulse input without the consideration of the fast gain saturation are also given. The input pulses are 3 ps Gaussian pulses with 100 ps period, and the bandwidth of the optical bandpass filter is 3 nm. The solid and the dashed lines are the results of the analytical expression Ž15., and the solid and open circles are the numerical results of
216
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
Fig. 4. The output ASE noise power vs. the average input optical power. Solid and dashed lines are results of Eq. Ž15. for pulse input with and without the effects of fast gain saturation, the solid and open circles are their numerical results, dotted line is for the CW input. The input pulses are 3 ps Gaussian pulses with 100 ps period.
Eq. Ž14.. It is shown that our analytical results agree very well with the numerical results. From Fig. 4, we see that the output noise power under the pulse injection is larger than that under the CW injection, and that when the fast gain saturation is included, the noise power increases. This is because that for equal average input optical powers, the average gain of the SOA under the pulse injection is larger than that under the CW injection, and the average gain increase for pulse injection when the fast gain satura-
Fig. 5. The output SNR vs. the average input pump power for different pulse widths. The pump and signal pulses are Gaussian pulses with 100 ps period. The input signal power is y10 dBm. Solid line is for 3 ps pulse width, dotted line is for 10 ps pulse width.
tion is considered. In addition, it is shown in Fig. 4 that when the input power is very large, the output noise power tend to approach a constant value, as indicated by Obermann et al. w11x. Figs. 5 and 6 are the SNR of the output FWM signal vs. the average input pump and signal powers for pulsed FWM with different pulse widths. Both the pump and signal pulses are Gaussian pulses with 100 ps period and there is no time delay between them. The solid line is for 3 ps pulse width, and dotted line is for 10 ps pulse width. In Fig. 5, we see that for FWM between short optical pulses, the SNR of the output FWM signal does not increase monotonically with the pump power, and there exists an optimum value for maximum SNR, unlike that indicated by D’Ottavi et al. w14x for CW FWM. In addition, Fig. 5 shows that the optimum pump power for maximum SNR depends on pulse width. Fig. 6 shows that the optimum value of the input signal power is larger than the pump power when only SNR is concerned. However, large signal power may introduce intersymbol interference due to the finite gain recovery time of the SOA. In practice, we generally let signal power 10 dB lower than pump power w15x. Fig. 7 gives the dependence of the output SNR on the unsaturated gain. The results of three period pulses with the same powers and pulse widths are
Fig. 6. The output SNR vs. the average input signal power for different pulse widths. The pump and signal pulses are Gaussian pulses with 100 ps period. The input pump power is 10 dBm. Solid line is for 3 ps pulse width, dotted line is for 10 ps pulse width.
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
217
taken into account. The optimum conditions for maximum conversion efficiency and the SNR of the output FWM signal are also investigated. The results show that when the effects of fast gain saturation are considered, both the optimum pump pulse energy and the optimum time delay between pulses are dependent on the pulse width for maximum conversion efficiency. In addition, we show that in order to get the maximum SNR of the output FWM signal, we should choose pump power, signal power and the gain of SOA properly. Fig. 7. The output SNR vs. the unsaturated gain. The pump and signal pulses are 3 ps Gaussian pulses. The average powers of input pump and signal are 10 dBm and 0 dBm. Solid line is for 100 ps period, dashed line is for 200 ps period, and dotted line is for 400 ps period.
given. It is shown that there exists an optimum unsaturated gain for maximum SNR, and that the optimum unsaturated gain decreases for large pulse period. This is because the FWM signal power depends mainly on the gain in the saturated duration, while the ASE noise power depends on the average gain over the pulse period, and that the difference of the saturated gain and the average gain increases with the pulse period. In Figs. 5–7, the bandwidth of the optical bandpass filter is 3 nm. 4. Conclusion In conclusion, we have analyzed the FWM conversion efficiency and noise performances of the FWM between short optical pulses in SOA. A simple model is presented to evaluate the FWM conversion efficiency for pulsed FWM in SOA with the consideration of fast gain saturation. The numerical results show that our simple model agrees well with the rigorous model. An approximate analytical expression of the ASE noise power spectral density of the SOA under the injection of a sequence of pulses is derived, in which both the fast gain saturation and the spatial dependence of the inversion parameter are
Acknowledgements This work was supported by 863 High Technology Research and Development Project under contract 863-317-9602-03-2. References w1x M.C. Tatham, G. Sherlock, L.D. Westbrook, IEEE Photon. Technol. Lett. 5 Ž1993. 1303. w2x A. Ellis, M. Tatham, D. Davies, D. Nesset, D. Moodie, G. Sherlock, Electron. Lett. 31 Ž1995. 299. w3x R. Ludwig, G. Raybon, Tech. Dig. ECOC’93 Ž1993. 57. w4x S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J.M. Jacob, M. Saruwatari, Electron. Lett. 30 Ž1994. 981. w5x M. Shtaif, G. Eisentein, Appl. Phys. Lett. 66 Ž1995. 1458. w6x M. Shtaif, R. Nagar, G. Eisenstein, IEEE Photon. Technol. Lett. 7 Ž1995. 1001. w7x M. Eiselt, IEEE Photon. Technol. Lett. 7 Ž1995. 1312. w8x A. Mecozzi, J. Mørk, IEEE J. Sel. Top. Quantum Electron. 3 Ž1997. 1190. w9x G.P. Agrawal, J. Opt. Soc. Am. B 5 Ž1988. 147. w10x A. Mecozzi, J. Mørk, J. Opt. Soc. Am. B 14 Ž1997. 761. w11x K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, H.G. Weber, IEEE J. Quantum Electron. 33 Ž1997. 81. w12x M. Shtaif, G. Eisenstein, IEEE J. Quantum Electron. 32 Ž1996. 1801. w13x M. Shtaif, G. Eisenstein, J. Lightwave Technol. 14 Ž1996. 2069. w14x A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, J. Eckner, G. Guekos, IEEE Photon. Technol. Lett. 7 Ž1995. 357. w15x M.A. Summerfield, R.S. Tucker, IEEE Photon. Technol. Lett. 8 Ž1996. 1316.
Optics Communications 164 Ž1999. 211–217 www.elsevier.comrlocateroptcom
Efficiency and noise performance analysis of four-wave mixing between short optical pulses in semiconductor optical amplifiers Chongjin Xie ) , Peida Ye Optical Communications Center, Beijing UniÕersity of Posts and Telecommunications, P.O. Box 57, Beijing 100876, China Received 10 February 1999; received in revised form 7 April 1999; accepted 14 April 1999
Abstract Efficiency and noise performance of four-wave mixing ŽFWM. between short optical pulses in semiconductor optical amplifiers ŽSOA. are analyzed, taking into account the effects of fast gain saturation. A simple theoretical model is presented to evaluate the efficiency and noise properties of the FWM between short optical pulses. Optimum conditions for maximum conversion efficiency and signal-to-noise ratio ŽSNR. of output FWM signal are also investigated. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Four-wave mixing; Semiconductor optical amplifiers; Nonlinearities
1. Introduction Four-wave mixing ŽFWM. in semiconductor optical amplifiers ŽSOA. has wide application in all-optical signal processing and all-optical networks, and it has been studied extensively over the past several years w1–9,11–15x. Generally, there are two types of FWM in SOA, that is, continuous-wave ŽCW. FWM or quasi-CW FWM, where the pump signal is a CW, and pulsed FWM, where both pump and probe signals are short optical pulses. CW FWM can be used for wavelength conversion and mid-span spectral inversion ŽMSSI. w1,2x, and pulsed FWM can be
) Corresponding author. Tel. q86-10-622-82-205; Fax: q8610-622-83-728; E-mail: [email protected]
used for optical time division demultiplexing and optical sampling w3,4x. Compared with the CW FWM, there are relatively few investigations about the pulsed FWM in SOA. The pulsed FWM was first investigated by Shtaif and Eisenstein w5x and Shtaif et al. w6x, and an analytical solution was given by them, then Eiselt studied the optimum parameters for demultiplexer application of FWM in SOA w7x. However, because they neglected the fast gain saturation due to the intraband processes of carrier heating ŽCH. and spectral-hole burning ŽSHB., their results were limited to pulse widths on the order of a few tens of picoseconds. Recently, Mecozzi and Mørk w8x presented a rigorous approach to describe the pulsed FWM in SOA taking into account the effects of fast gain saturation, which can be applied to pulses shorter than a few picoseconds. In this paper, we will study
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 8 6 - 8
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
212
both the conversion efficiency and the noise properties of the FWM between short optical pulses in SOA with the consideration of the fast gain saturation. A simple model to evaluate the conversion efficiency will be presented, and an approximate expression of the output noise power density of the SOA under the injection of a sequence of pulses will be derived taking into account the spatial dependence of both the gain coefficient and the inversion parameter. The optimum conditions for pulsed FWM in the SOA will also be investigated. In our model, the effect of gain dispersion is neglected.
2. FWM conversion efficiency 2.1. The new model of FWM between short optical pulses in SOA When two optical signals, a pump signal and a probe signal, with frequency v p and vs are injected into the SOA, a new frequency component, the FWM signal with frequency 2 v p y vs , will be generated at the output of the SOA due to the nonlinearity of SOA. We assume that the field of optical pulses at the input of the SOA is E Ž 0,t . s Ep Ž t . q Es Ž t . exp Ž i V t . ,
V s v p y vs Ž 1.
where Ep Ž t . and EsŽ t . are the slowly varying amplitude of the pump and probe pulse field, and V is the pump-probe detuning. For FWM between short optical pulses, the pump-probe detuning must be large enough to avoid the overlap of the bandwidth of the pump and probe pulses, and < V
Ep2 Ž t . Psat
exp Ž h . y 1
FŽ V . s
2
=
1yia 1 q exp Ž h . < Ep Ž t . < 2rPsat y i Vts
qÝ x
dh s
h0 y h
Ž 3.
1 y i Vt x
y exp Ž h . y 1
ts
dt
Ž 1 y i a x . ´ x Psat < E Ž 0,t . < 2 psatts
Ž 4.
L
where h s
H
0
g Ž z,t .d z Ž g Ž z,t . is the gain coeffi-
cient and g 0 is its small signal value, L is the length of the SOA, G s expŽ h. is the power gain., h 0 s g 0 L, Psat is SOA saturation power, a is the line width enhancement factor, x represent the intraband processes, which are carrier heating of electrons in the conduction band ŽCH c . of holes in the valence band ŽCH v ., and spectral hole burning of electrons ŽSHB c . and holes ŽSHB v ., b x , t x and ´ x are the line width enhancement factors, characteristic time constants and nonlinear gain coefficients associated with them, C is the phenomenological parameter to compensate for the non-planewave nature of the waveguide. The above model does not include the effect of fast gain saturation due to intraband processes, so it can only be applied to pulses about tens of picoseconds. In the following, we will develop a new model to describe the pulsed FWM in SOA, which can be extended to pulses shorter than a few picoseconds. Since the influence of the intraband processes on FWM has already been included in Eq. Ž3. Žthe second term in the square bracket., so we only need to consider the effect of intraband processes in the calculation of h. For pulse widths larger than the time constants of intraband processes, the propagation of pulses in the SOA can be described by the following equations with the consideration of fast gain saturation w10x E g Ž z ,t . g 0 y g Ž z ,t . g Ž z ,t . P Ž z ,t . s y , ts 1 q ´ P Ž z ,t . Psatts Et
´ s Ý ´x
Ž 5.
x
EsU Ž t .
Ž 2.
EP Ž z ,t . Ez
s
g Ž z ,t . 1 q ´ P Ž z ,t .
P Ž z ,t .
Ž 6.
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
213
where P Ž z,t . s < EŽ z,t .< 2 . Let h s H0L g Ž z,t .rw1 q ´ P Ž z,t .xd z, and after some algebra derivation, we get dh
1 s
dt
1 q ´ exp Ž h . P Ž 0,t .
y´ exp Ž h . y 1
½
h0 y h
ts
d P Ž 0,t . dt
y exp Ž h . y 1 P Ž 0,t .
ž
´ ts
1 q
Psatts
/5
Ž 7.
Eqs. Ž2., Ž3. and Ž7. are our model of pulsed FWM in SOA. It is simpler than the rigorous approach, and it can be easily applied to analyze FWM in SOA between a sequence of optical pulses, which is the case for optical time demultiplexing application. In this case, the pump pulses are injected into the SOA with a repetition period T. We only need to solve Eq. Ž7. in the time period 0–T with the boundary condition hŽ0. s hŽT .. 2.2. Numerical results In the following, we present some numerical results to prove our new model. The same parameters of SOA in Ref. w8x are used: differential gain a s 3.0 = 10y2 0 m2 , cross-sectional area A s 0.4 mm2 , confinement factor G s 0.3, SOA length L s 200
Fig. 1. Comparison of the FWM efficiency dependence on pump energy calculated with different models. Solid lines are calculated with our model, dash–dotted lines with the rigorous model, and dotted lines with Shtaif and Eisenstein model. Tp is the pulse width of the pump pulses.
Fig. 2. The dynamic gain response of the SOA to different width pump pulses calculated with different method. Solid lines are calculated with our method, dash–dotted lines with the rigorous method, and dotted lines with Shtaif and Eisenstein method. The input pump energy is 250 fJ, Tp is the pulse width of the pump pulses. Ža. Pulse width is 1 ps, Žb. pulse width is 20 ps.
mm, carrier density at transparency N0 s 1.0 = 10 24 my3 , carrier lifetime ts s 200 ps, saturation power Psat s 28.4 mW, ´ SHB c s 0.35 Wy1 , ´ CH c s 0.70 Wy1 , t SHB c s 70 fs, t CH c s 700 fs, the line width enhancement factors are a s 4.0, a CH c s 3.0, the unsaturated gain G 0 s 25 dB. The wavelength of the pump signal is 1550 nm and the beating frequency V s 2.5 THz. The effects of SHB and CH for the holes are neglected, since the heavier hole mass typically leads to much shorter relaxation times than found for the electrons. In the calculation, the pump and probe powers are in the ratio of 10:1, and the pulses are all Gaussian pulses. Fig. 1 shows the results of the FWM efficiency dependence on pump energy calculated for different pulse widths. For comparison, the results of three
214
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
models are given, which are our model, the rigorous model described by Mecozzi and Mørk, and the model presented by Shtaif and Eisenstein. In this figure, the pulse widths of the pump and probe pulses are the same, and there is no time delay between the pump and probe pulses. It shows that the results of our model agree well with that of rigorous model, and only when the SOA is deeply saturated, there exist a little disagreement. It also shows that the model of Shtaif and Eisenstein is only valid for the pulse width on the order of tens of picoseconds. Unlike the one showed by Shataif and Eisenstein, when the effect of fast gain saturation is considered, the optimum pump energy is dependent on the pulse width. The shorter the pulse width, the lower the optimum pump energy. The difference of the three models can be explained by Fig. 2, which shows the effects of intraband processes on the gain dynamics of SOA for different pump pulse widths with the same energy. We see that the intraband processes have a significant effect on the SOA gain dynamics for narrow pump pulses, but for longer pulses, the effects can be neglected. In Fig. 2a, the pulse width is 1 ps. In this case, there is rapid gain depletion near the pulse center, and the gain dynamics with and without the consideration of fast gain depletion is significantly different. However, when the pulse width is 20 ps, the results of the three models are almost the same,
which is shown in Fig. 2b. The figure also shows that the difference of the gain dynamics between our method and the rigorous method is very small. Fig. 3 shows the FWM between non-overlapping pulses. Optimum pump pulse energy is used in the figure. The energy is 63 fJ for 1 ps pulse width, and 250 pJ for 20 ps pulse width. The figure shows that the FWM efficiency peaks when the probe pulse proceeds the pump, as pointed out by Shtaif and Eisenstein, and that the optimum time delay becomes smaller for narrower pulses.
Fig. 3. The FWM efficiency dependence on pump-probe delay time. Solid line is for 1 ps pulse width, and dotted line is for 20 ps pulse width. Optimum pulse energies are used in the two cases. Td is the time delay between the pump and probe pulses, Tp is the pulse width of the pump pulses.
EEASE Ž z ,t .
3. Noise properties Another important issue of the FWM in SOA is the noise. It is well known that the successful application of FWM in SOA relies on both the FWM conversion efficiency and the noise properties of the output FWM signal. For practical applications, the noise properties of the output FWM signal are even of higher significance than the FWM conversion efficiency. Recently, some investigations of the noise properties of FWM in SOA were reported w11–15x, for example, Obermann et al. w11x analyzed the noise characteristics of the CW FWM in SOA in detail, Shtaif and Eisenstein w12,13x reported an investigation of the noise performances of pulsed FWM in SOA. Here, we will discuss the noise properties of the FWM between the short optical pulses in SOA, taking into account the effects of fast gain saturation. It was shown by Shtaif and Eisenstein w12x that under nonlinear operating conditions, the SOA noise contains a narrow-band noise and a broadband spontaneous noise. The narrow-band noise comes from the nonlinear coupling of the noise and the gain and it can be neglected in the cases of short pulse amplification and nondegenerate FWM. So for FWM between short pulses in SOA, only the broadband amplifier spontaneous emission ŽASE. noise need to be considered. The dynamics of the ASE noise in the SOA can be described by the following equation
Ez
1 s
1yia
2 1 q ´ P Ž z ,t . q n Ž z ,t .
g Ž z ,t . EASE Ž z ,t .
Ž 8.
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
where EASE Ž z,t . is the ASE noise field, nŽ z,t . is a spatially uncorrelated, white noise term of the zero average and the correlation functions ² n Ž z ,t . n Ž zX ,tX . : s 0 Ž 9. ² n Ž z ,t . nU Ž zX ,tX . : s"v
g Ž z ,t . 1 q ´ P Ž z ,t .
n sp Ž z ,t . d Ž z y zX . d Ž t y tX .
Ž 10 . where d are Dirac delta functions, n sp Ž z,t . is the inversion factor which equals w g Ž z,t . q G aN0 xr g Ž z,t ., with G , a, and N0 being the mode confinement factor, the differential gain and the carrier density at transparency. The solution of Eq. Ž8. gives the noise field at the output of SOA
215
where T is the input optical pulse period. It is not possible to get an analytical solution of Eq. Ž14., and the accurate noise power spectral density can only be obtained by solving Eqs. Ž5., Ž6. and Ž14. numerically. However, with some assumptions, an approximate analytical expression of noise power spectral density can be obtained. In calculation of the second term in Eq. Ž14., we assume that the gain coefficient is a constant over the length of SOA, that is hŽ z,t . s ln GŽ t .rLz. After some algebra derivation, we obtain WASE s " v
1
T
H T 0
½
GŽ t . y 1 q
½
= ´ P Ž 0,t . G Ž t . ln
G aN0 L ln G Ž t .
1 q ´ P Ž 0,t . G Ž t . 1 q ´ P Ž 0,t .
EASE Ž L,t . s
L
H0 exp Hz
L
1yia
1
yln G Ž t . q G Ž t . y 1
g Ž zX ,t . d zX
X
2 1 q ´ P Ž z ,t .
=n Ž z ,t . d z
Ž 11 .
Its autocorrelation function is U ² EASE Ž L,t . EASE Ž L,t q t . : g Ž z ,t .
L
s"v
H0
1 q ´ P Ž z ,t .
=exp
Hz
n sp Ž z ,t .
g Ž zX ,t .
L
d zX d z d Ž t .
1 q ´ P Ž zX ,t .
SNR s
½
s " v G Ž t . y 1 q G aN0 G Ž t . =
exp yh Ž z ,t .
L
H0
1 q ´ P Ž z ,t .
5
d z d Žt .
Ž 13 .
L
where hŽ z,t . s
H
0
g Ž zX ,t .rw1 q ´ P Ž zX ,t .xd zX . Under
the injection of a sequence of periodic pulses, the output ASE noise power spectral density is WASE s " v =
1
T
H T 0 L
H0
½
G Ž t . y 1 q G aN0 G Ž t .
exp yh Ž z ,t . 1 q ´ P Ž z ,t .
5
d z dt
Ž 14 .
dt
Ž 15 .
Eq. Ž15. is our analytical expression of the output ASE noise power spectral density of the SOA under the injection of a sequence of optical pulses. The noise power spectral density itself is not an appropriate measure in order to quantify the noise properties of devices based on FWM in SOA. Generally a more system related quantity signal-to-noise ratio ŽSNR. is used, which is defined as
Ž 12 .
Substitution of n sp Ž z,t . into Eq. Ž12. yields U ² EASE Ž L,t . EASE Ž L,t q t . :
55
Pc Ž L . PASE
s
Pc Ž L . WASE B
Ž 16 .
where Pc Ž L. is the average output FWM signal power and B is the bandwidth of the optical bandpass filter. Here, we present some calculation results. To evaluate the noise performance of the FWM in SOA between a sequence of pulses, it is convenient to use the average optical power instead of pulse energy. Fig. 4 shows the output noise power vs. the average input optical power. For comparison, the results of the CW input and the pulse input without the consideration of the fast gain saturation are also given. The input pulses are 3 ps Gaussian pulses with 100 ps period, and the bandwidth of the optical bandpass filter is 3 nm. The solid and the dashed lines are the results of the analytical expression Ž15., and the solid and open circles are the numerical results of
216
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
Fig. 4. The output ASE noise power vs. the average input optical power. Solid and dashed lines are results of Eq. Ž15. for pulse input with and without the effects of fast gain saturation, the solid and open circles are their numerical results, dotted line is for the CW input. The input pulses are 3 ps Gaussian pulses with 100 ps period.
Eq. Ž14.. It is shown that our analytical results agree very well with the numerical results. From Fig. 4, we see that the output noise power under the pulse injection is larger than that under the CW injection, and that when the fast gain saturation is included, the noise power increases. This is because that for equal average input optical powers, the average gain of the SOA under the pulse injection is larger than that under the CW injection, and the average gain increase for pulse injection when the fast gain satura-
Fig. 5. The output SNR vs. the average input pump power for different pulse widths. The pump and signal pulses are Gaussian pulses with 100 ps period. The input signal power is y10 dBm. Solid line is for 3 ps pulse width, dotted line is for 10 ps pulse width.
tion is considered. In addition, it is shown in Fig. 4 that when the input power is very large, the output noise power tend to approach a constant value, as indicated by Obermann et al. w11x. Figs. 5 and 6 are the SNR of the output FWM signal vs. the average input pump and signal powers for pulsed FWM with different pulse widths. Both the pump and signal pulses are Gaussian pulses with 100 ps period and there is no time delay between them. The solid line is for 3 ps pulse width, and dotted line is for 10 ps pulse width. In Fig. 5, we see that for FWM between short optical pulses, the SNR of the output FWM signal does not increase monotonically with the pump power, and there exists an optimum value for maximum SNR, unlike that indicated by D’Ottavi et al. w14x for CW FWM. In addition, Fig. 5 shows that the optimum pump power for maximum SNR depends on pulse width. Fig. 6 shows that the optimum value of the input signal power is larger than the pump power when only SNR is concerned. However, large signal power may introduce intersymbol interference due to the finite gain recovery time of the SOA. In practice, we generally let signal power 10 dB lower than pump power w15x. Fig. 7 gives the dependence of the output SNR on the unsaturated gain. The results of three period pulses with the same powers and pulse widths are
Fig. 6. The output SNR vs. the average input signal power for different pulse widths. The pump and signal pulses are Gaussian pulses with 100 ps period. The input pump power is 10 dBm. Solid line is for 3 ps pulse width, dotted line is for 10 ps pulse width.
C. Xie, P. Ye r Optics Communications 164 (1999) 211–217
217
taken into account. The optimum conditions for maximum conversion efficiency and the SNR of the output FWM signal are also investigated. The results show that when the effects of fast gain saturation are considered, both the optimum pump pulse energy and the optimum time delay between pulses are dependent on the pulse width for maximum conversion efficiency. In addition, we show that in order to get the maximum SNR of the output FWM signal, we should choose pump power, signal power and the gain of SOA properly. Fig. 7. The output SNR vs. the unsaturated gain. The pump and signal pulses are 3 ps Gaussian pulses. The average powers of input pump and signal are 10 dBm and 0 dBm. Solid line is for 100 ps period, dashed line is for 200 ps period, and dotted line is for 400 ps period.
given. It is shown that there exists an optimum unsaturated gain for maximum SNR, and that the optimum unsaturated gain decreases for large pulse period. This is because the FWM signal power depends mainly on the gain in the saturated duration, while the ASE noise power depends on the average gain over the pulse period, and that the difference of the saturated gain and the average gain increases with the pulse period. In Figs. 5–7, the bandwidth of the optical bandpass filter is 3 nm. 4. Conclusion In conclusion, we have analyzed the FWM conversion efficiency and noise performances of the FWM between short optical pulses in SOA. A simple model is presented to evaluate the FWM conversion efficiency for pulsed FWM in SOA with the consideration of fast gain saturation. The numerical results show that our simple model agrees well with the rigorous model. An approximate analytical expression of the ASE noise power spectral density of the SOA under the injection of a sequence of pulses is derived, in which both the fast gain saturation and the spatial dependence of the inversion parameter are
Acknowledgements This work was supported by 863 High Technology Research and Development Project under contract 863-317-9602-03-2. References w1x M.C. Tatham, G. Sherlock, L.D. Westbrook, IEEE Photon. Technol. Lett. 5 Ž1993. 1303. w2x A. Ellis, M. Tatham, D. Davies, D. Nesset, D. Moodie, G. Sherlock, Electron. Lett. 31 Ž1995. 299. w3x R. Ludwig, G. Raybon, Tech. Dig. ECOC’93 Ž1993. 57. w4x S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J.M. Jacob, M. Saruwatari, Electron. Lett. 30 Ž1994. 981. w5x M. Shtaif, G. Eisentein, Appl. Phys. Lett. 66 Ž1995. 1458. w6x M. Shtaif, R. Nagar, G. Eisenstein, IEEE Photon. Technol. Lett. 7 Ž1995. 1001. w7x M. Eiselt, IEEE Photon. Technol. Lett. 7 Ž1995. 1312. w8x A. Mecozzi, J. Mørk, IEEE J. Sel. Top. Quantum Electron. 3 Ž1997. 1190. w9x G.P. Agrawal, J. Opt. Soc. Am. B 5 Ž1988. 147. w10x A. Mecozzi, J. Mørk, J. Opt. Soc. Am. B 14 Ž1997. 761. w11x K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, H.G. Weber, IEEE J. Quantum Electron. 33 Ž1997. 81. w12x M. Shtaif, G. Eisenstein, IEEE J. Quantum Electron. 32 Ž1996. 1801. w13x M. Shtaif, G. Eisenstein, J. Lightwave Technol. 14 Ž1996. 2069. w14x A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, J. Eckner, G. Guekos, IEEE Photon. Technol. Lett. 7 Ž1995. 357. w15x M.A. Summerfield, R.S. Tucker, IEEE Photon. Technol. Lett. 8 Ž1996. 1316.