## Theoretical Study of the Spurious-Free Dynamic Range of a Tunable Delay Line based on Slow Light in SOA

Optics Express, Vol. 17, Issue 22, pp. 20584-20597 (2009)

http://dx.doi.org/10.1364/OE.17.020584

Acrobat PDF (313 KB)

### Abstract

We developed a predictive model describing harmonic generation and intermodulation distortions in semiconductor optical amplifiers (SOAs). This model takes into account the variations of the saturation parameters along the propagation axis inside the SOA, and uses a rigorous expression of the gain oscillations harmonics. We derived the spurious-free dynamic range (SFDR) of a slow light delay line based on coherent population oscillation (CPO) effects, in a frequency range covering radar applications (from 40kHz up to 30GHz), and for a large range of injected currents. The influence of the high order distortions in the input microwave spectrum is discussed, and in particular, an interpretation of the SFDR improvement of a Mach-Zehnder modulator by CPOs effects in a SOA is given.

© 2009 Optical Society of America

## 1. Introduction

1. J. Yao, “Microwave Photonics,” J. Lightwave Technol. **27**, 314–335 (2009).
[CrossRef]

3. J. Capmany, B. Ortega, and D. Pastor, “A Tutorial on Microwave Photonic Filters,” J. Lightwave Technol. **24**, 201–229 (2006).
[CrossRef]

4. G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, “Fiber-Based Slow-Light Technologies,” J. Lightwave Technol. **26**, 3752–3762 (2008).
[CrossRef]

5. Y. Chen, W. Xue, F. Ohman, and J. Mørk, “Theory of Optical-Filtering Enhanced Slow and Fast Light Effects in Semiconductor Optical Waveguides,” J. Lightwave Technol. **26**, 3734–3743 (2008).
[CrossRef]

6. M. González Herráez, K. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express **14**, 1395–1400 (2006).
[CrossRef] [PubMed]

8. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B **5**, 147–159 (1988).
[CrossRef]

11. J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, “Slow light in a semiconductor waveguide at gigahertz frequencies,” Opt. Express **13**, 8136–8145 (2005).
[CrossRef] [PubMed]

^{2/3}(ground based antennas) to 115dB/Hz

^{2/3}(airborne antennas). Most of the previously reported work on nonlinear frequency mixing in SOAs has been carried out in the frame of Radio-over-Fiber applications, or SOA-based in-line photodetection [13

13. J. H. Seo, Y. K. Seo, and W. Y. Choi, “Spurious-Free Dynamic Range Characteristics of the Photonic Up-Converter Based on a Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. **15**, 1591–1593 (2003).
[CrossRef]

13. J. H. Seo, Y. K. Seo, and W. Y. Choi, “Spurious-Free Dynamic Range Characteristics of the Photonic Up-Converter Based on a Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. **15**, 1591–1593 (2003).
[CrossRef]

13. J. H. Seo, Y. K. Seo, and W. Y. Choi, “Spurious-Free Dynamic Range Characteristics of the Photonic Up-Converter Based on a Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. **15**, 1591–1593 (2003).
[CrossRef]

## 2. Principle of harmonics calculation

*E*(

*z*,

*t*) which propagates along a traveling wave semiconductor optical amplifier (SOA). The interaction of light with carriers in the SOA is governed by the well known carrier rate equation and field propagation equation [8

8. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B **5**, 147–159 (1988).
[CrossRef]

*N*(

*z*,

*t*) is the carrier density,

*I*the current injected in the SOA,

*τ*the carrier lifetime,

_{s}*q*the elementary electric charge,

*V*the SOA active volume,

*g*the optical gain, Γ the confinement factor, and

*γ*stands for the internal losses. In order to find the level of the generated harmonics, we first consider that the input optical field is modulated at the RF frequency Ω. |

_{i}*E*|

^{2},

*g*and

*N*are hence all time-periodic functions with a fundamental frequency of Ω. They can therefore be written into Fourier harmonic decompositions:

*N̄*(

*z*) and

*g*̄(

*z*) respectively denote the DC components of the carrier density and of the optical gain.

*a*(

*z*) is the SOA differential gain, defined as

*a*(

*z*)=

*∂g*̄/

*∂N*̄. Defining

*g*as the oscillating component of the gain at frequency

_{k}*k*Ω, and considering only a finite number

*K*of harmonics, the carrier rate equation [Eq. (1)] can be written in an equivalent matrix formulation:

*α*=

_{k}*I*(1+

_{s}*M*

_{0}/

*I*-

_{s}*ik*Ω

*τs*), and

*α*

_{0}=

*M*

_{0}is the DC optical intensity.

*I*denotes the saturation intensity and is defined as

_{s}*I*=

_{s}*h̄ω*/

*aτ*. It is worth mentioning that

_{s}*α*is obtained at the first order of Eq. (1), when mixing terms are not considered. One can also notice that

_{k}*g*

^{*}

*=*

_{k}*g*

_{-k}. For the sake of clarity, we did not write the dependence on the propagation coordinate

*z*in Eq. (6). It is however important to note that in the following,

*N*̄,

*g*̄,

*a*,

*τ*,

_{s}*I*, and consequently the

_{s}*α*’s are all actually functions of

_{k}*z*. Their variations along the propagation axis is then taken into account, unlike most of the reported models in which effective parameters are used [8

8. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B **5**, 147–159 (1988).
[CrossRef]

10. H. Su, P. Kondratko, and S. L. Chuang, “Variable optical delay using population oscillation and four-wave-mixing in semiconductor optical amplifiers,” Opt. Express **14**, 4801–4807 (2006).
[CrossRef]

11. J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, “Slow light in a semiconductor waveguide at gigahertz frequencies,” Opt. Express **13**, 8136–8145 (2005).
[CrossRef] [PubMed]

## 2.1. Small RF signal formulation

*g*’s, as functions of the harmonic components of the optical intensity

_{k}*M*’s, of

_{k}*g*̄, and of

*I*and

_{s}*τ*. Under small RF signal approximation, i.e. considering a small modulation index of the optical carrier, one can assume that |

_{s}*M*

_{k-1}|≫|

*M*| for

_{k}*k*>0 (and |

*M*

_{k-1}|≪|

*M*| for

_{k}*k*<0). It can also be noticed that the same relations hold for the

*N*’s since the carrier density oscillations are induced by the illumination oscillations through gain saturation. On the basis of this assumption, the commonly used expression for the coefficients gk is [16

_{k}16. J. Herrera, F. Ramos, and J. Marti, “Nonlinear distortion generated by semiconductor optical amplifier boosters in analog optical system,” Opt. Lett. **28**, 1102–1104 (2003).
[CrossRef] [PubMed]

18. S. Ó Dúill, R. F. ODowd, and G. Eisenstein, “On the Role of High-Order Coherent Population Oscillations in Slow and Fast Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. **15**, 578–584 (2009).
[CrossRef]

*M*|’s, but the terms |

_{k}*M*

_{k-p}×

*g*|. Consequently, the hypothesis|

_{p}*M*

_{k-1}|≫|

*M*| leads to neglect only the terms for which |

_{k}*p*|+|

*k*-

*p*|>|

*k*|, the others being of the same order. Under these conditions, Eq. (6) is reduced into:

*g*̃ and

*g*

_{±1}as in Eqs. (7) and (8), indicating no change on the fundamental component of the output modulation, compared to common models. However, for |

*k*|>1, additional terms appear, whose importance will be emphasized later on. The expressions of

*g*

_{2}and

*g*

_{3}are hence:

*z*, through their dependence on the static carrier density

*N*̄(

*z*). Doing so, the relations between

*I*,

_{s}*τ*and the physical device constants are preserved, which ensures the predicting capability of the model when the operating conditions (optical input power or bias current) are changed. As reportes in [19], a simple measurement of the unsaturated gain as a function of the SOA bias current, gives a first relation between

_{s}*ḡ*and

*N*̄/

*τ*:

_{s}*α*and

*β*are empiric coefficients determined experimentally by measuring the SOA’s small signal gain. For our simulations, we used

*α*=5.88·10

^{3}m

^{-1}and

*β*=-1.84·10

^{37}m

^{2}.s

^{-1}, corresponding to the COVEGA InGaAsP/InP quantum well SOA available in our laboratory [19]. Then, solving the system constituted of Eqs. (8) and (12), we obtain

*g*̄(

*z*) and

*N*̄/

*τ*(

_{s}*z*) as functions of

*α*,

*β*,

*I*and the local DC optical intensity

*M*

_{0}(

*z*). Finally, we model the carrier lifetime in our SOA using the well known expression:

*A*,

*B*and

*C*are the carrier recombination coefficients of the semiconductor structure, and correspond respectively to the non-radiative, spontaneous, and Auger recombination coefficients. Equation (13) and the expressions of

*g*̄(

*z*) and

*N*̄/

*τ*(

_{s}*z*) then enable to derive

*τ*and

_{s}*I*as functions of

_{s}*I*,

*M*

_{0}(

*z*) and

*A*,

*B*and

*C*.

*A*,

*B*and

*C*are the only adjustment parameters of the model: they are determined by adjusting the simulated and measured fundamental RF transfer functions of the SOA. In the case of the COVEGA SOA we consider in this paper, we obtained

*A*=2·10

^{9}s

^{-1},

*B*=1.2·10

^{-10}cm

^{3}.s

^{-1},

*C*=1.8·10

^{-31}cm

^{6}.s

^{-1}. It is worthwhile to mention that these values were found to be valid for the full range of injected currents or optical input powers, proving the predictive capability of this approach.

## 2.2. Large RF signal formulation

*M*

_{k-1}|≫|

*M*| do no longer apply. Equation (6) has to be rigorously solved. Moreover, Eq. (8) is no longer valid, and consequently,

_{k}*g*̄,

*I*and

_{s}*τ*cannot be obtained as in the small signal case. We thus use the following iterative procedure: in a first step, we substitute

_{s}*N*̄/

*τ*,

_{s}*I*and

_{s}*τ*in Eq. (6) by their small signal values

_{s}*N*̄/

*τ*

^{(0)}

*,*

_{s}*I*

^{(0)}

*and*

_{s}*τ*

^{(0)}

*as obtained in section 2.1. The central matrix of Eq. (6), referred as*

_{s}*D*, can then be inversed. The coefficients of

*D*

^{-1}are denoted (

*δ*

_{k,p}). Equations (7) and (8) then becomes:

*N*̄/

*τ*

^{(1)}

*,*

_{s}*I*

^{(1)}

*and*

_{s}*τ*

^{(1)}

*as functions of*

_{s}*I*,

*A*,

*B*,

*C*and

*M*(

_{k}*z*). This procedure is repeated until convergence of

*N*̄/

*τ*

^{(n)}

*,*

_{s}*I*

^{(n)}

*and*

_{s}*τ*

^{(n)}

*, which typically occurs after a few tens of iterations. The propagation equation [Eq. (2)] can now be expressed in a matrix formulation similarly to [17*

_{s}17. T. Mukai and T. Saitoh, “Detuning characteristics and conversion efficiency of nearly degenerate four-wave mixing in a 1.5-m traveling-wave semiconductor laser amplifier,” IEEE Quantum Electron. **26**, 865–875 (1990).
[CrossRef]

## 2.3. Comparison with reported models

*g*

_{2}and

*g*

_{3}in Eqs. (10) and (11), we calculated the third harmonic power

*H*

_{3}at the output of a 1.5mm-long SOA. The optical carrier is modulated using a Mach-Zehnder modulator. The initial conditions for the resolution of Eq. (16) are hence [20

20. Y. Shi, L. Yan, and A. E. Willner, “High-Speed Electrooptic Modulator Characterization Using Optical Spectrum Analysis,” J. Lightwave Technol. **21**, 2358–2367 (2003).
[CrossRef]

*m*is the modulation index, and

*J*denotes the

_{k}*k*order Bessel function of the first kind. For this example, we chose a modulation index of 0.01, which ensures the validity of the small RF signal hypothesis.

^{th}*Is*in the model, which is a local parameter, varying along the propagation axis. The optical input power is 10dBm, for which the optical gain is strongly saturated, and which leads to the maximum RF phase shift that we obtained with this device [19]. Equation (16) is solved using the expression of the

*g*’s given first by the common Eqs.(7) and (8), and then by Eqs. (10) and (11) containing the additional terms. Figure 1(a) shows both experimental (red circles) and theoretical (solid lines) third harmonic photodetected power, normalized to the third harmonic power at the input of the SOA. The theoretical curves are evaluated according to:

_{k}*η*are respectively the photodiode resistive load (50Ω) and efficiency (equal to 0.8).

_{ph}*S*denotes the SOA modal area.

*GHz*, the two models are equivalent, and the third harmonic power naturally tends to the value it would have if no CPO effects were present (i.e.

*M*

_{3,out}is equal to

*M*

_{3,in}multiplied by the optical gain

*G*). However, at low frequencies, the difference between the two models reaches 20dB. The experimental measurement shows a very good agreement with our model including the additional terms, which confirms the validity of our approach. The large discrepancy with common models can be qualitatively understood considering the asymptotic case where Ω tends to zero. Using Eqs. (10) and (11) in the propagation equation [Eq. (16)], one obtains for the third harmonic term, with Ω

_{opt}*τ*≪1:

_{s}*M*

_{3}. The third one, respectively denoted

*A*and

*B*for Eqs. (19) and (20) describe the energy transfer from

*M*

_{1}and

*M*

_{2}to

*M*

_{3}through CPO process. Both

*A*and

*B*have a negative sign, which is consistent with the CPO’s oscillations being in antiphase with the illumination oscillations. In Fig. 1(b) are plotted the evolution along the propagation axis inside the SOA of terms

*A*and

*B*, and of

*M*

_{3}, calculated according to Eq. (19) in red, and according to Eq. (20) in green. These results indicate that |

*B*| is much larger than |

*A*|, and hence that conventional models overestimates the third harmonic generation term due to CPO for the low frequency part of the spectrum. Consequently, CPO contribution largely dominates the amplification term, and leads to a relatively high output level of third harmonic. On the other hand, when the gain harmonics are rigorously derived, one ends up with the term

*A*, with a smaller magnitude which balances the amplification process. The resulting output

*H*

_{3}level is therefore significantly reduced, and can even be under the input level, if the optical gain is saturated enough such as the overall amplification/generation terms are below the linear losses attenuation.

*g*even in the small signal situation, and especially for the low frequency part of the microwave spectrum. When Ω increases, since the 2

_{k}*and 3*

^{nd}*terms of Eq. (11) respectively evolve as Ω*

^{rd}^{-1}and Ω

^{-2}, then Eq. (11) progressively tends to the common expression given by Eq. (7). This can be seen on Fig. 1(a), where the two curves finally coincide for Ω> 4

*GHz*. Consequently, the present analysis does not question the results obtained for instance in [18

18. S. Ó Dúill, R. F. ODowd, and G. Eisenstein, “On the Role of High-Order Coherent Population Oscillations in Slow and Fast Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. **15**, 578–584 (2009).
[CrossRef]

*GHz*. However, as confirmed by experiments, a rigorous calculation is mandatory when the modulation frequency lies in the spectral region 1–3GHz where most of ground radars operate.

## 3. Intermodulation distortion

_{1}, and facing a jammer emitting at Ω

_{2}, close to Ω

_{1}. Both Ω

_{1}and Ω

_{2}are collected by the antenna and transferred to the optical carrier through a single electro-optic modulator. The point is then to determine the nonlinear frequency mixing due to the coherent population oscillations (CPO) inside the SOA. In particular, the mixing products at frequencies Ω

_{2}-Ω

_{1}(or Ω

_{1}-Ω

_{2}) and 2Ω

_{2}-Ω

_{1}(or 2Ω

_{1}-Ω

_{2}) — respectively called second (IMD

_{2}) and third (IMD

_{3}) order intermodulation distortions — have to be evaluated at the output of the SOA. The main difference with harmonic calculation is that the optical intensity, and hence the SOA carrier density

*N*, and the SOA gain

*g*are no longer time-periodic functions of period Ω, but of period

*δ*Ω=Ω

_{2}-Ω

_{1}. Then, using a Fourier decomposition of |

*E*|

^{2},

*N*, and

*g*, one has:

_{1}of 10GHz, and a frequency spacing

*δ*Ω of 10MHz. Here, for intermodulation distortion calculation, we assume that only the spectral components at Ω

_{1,2}, 2Ω

_{1,2}, and all their first order mixing products significantly contribute to the generation of IMD

_{2}and IMD

_{3}, as illustrated in Fig. 2. The

*M*’s and the

_{k}*g*’s are then reduced in 19 elements vectors, and gathered into blocks, denoted

_{k}*M*

_{block, j}and

*g*

_{block, j}. The

*j*block contains the mixing products with frequencies close to

^{th}*j*×Ω

_{1}. According to the notation of Fig. 2, the carrier rate equation [Eq. (1)] and the propagation equation [Eq. (2)] can respectively be written as:

*D*

_{i, j}and

*H*

_{i, j}are sub-blocks of the complete matrices of Eqs. (6) and (16) respectively. They describe the mixing of the spectral components around the

*i*and the

^{th}*j*harmonics. From now on, the full procedure described in Section 2 can be applied in the same iterative way to determine the

^{th}*g*’s,

_{k}*I*and

_{s}*τ*, and to finally numerically solve the Eq. (25). Similarly to Eq. (18), the photodetected RF power at 2Ω

_{s}_{2}-Ω

_{1}is then calculated through:

## 4. Simulation results

20. Y. Shi, L. Yan, and A. E. Willner, “High-Speed Electrooptic Modulator Characterization Using Optical Spectrum Analysis,” J. Lightwave Technol. **21**, 2358–2367 (2003).
[CrossRef]

*m*is the modulation index, and

*ϕ*is the modulator phase bias. When developed into first kind Bessel functions, and according to the notation of Fig. 2, the corresponding input distribution of the

*M*’s, at quadrature bias (

_{k}*ϕ*=

*π*/2), is:

_{1,2}-Ω

_{2,1}, denoted IMD

^{in}

_{3}, and equal to

*J*

_{1}(

*m*)

*J*

_{2}(

*m*). In a general way, when considering the propagation equation in its matricial form (Eqs. (16) and (25)), the propagation of the Fourier compounds of an optically carried microwave signal into the SOA can be seen as resulting from an amplification process (the diagonal terms of the matrix) and a generation process by frequency mixing through coherent population oscillations (non-diagonal terms of the matrix). In order to better understand the combination of these two effects in the case of IMD3 propagation, we conducted our simulations considering two initial conditions, with and without IMD

^{in}

_{3}. The latest case corresponds to the use of a perfectly linear modulator, or as in [13

**15**, 1591–1593 (2003).
[CrossRef]

_{1}and Ω

_{2}are produced by two distinct modulators on two incoherent optical carriers. An alternative way to avoid IMD

^{in}

_{3}would be to use a Single-Side-Band modulator, as proposed in [21, 22

22. M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and P. Thony, “Offset phase locking of Er:Yb:Glass laser eigenstates for radio-frequency photonics applications”, IEEE Photon. Technol. Lett. **13**, 367 (2001).
[CrossRef]

_{3}, calculated according to Eq. (26), as a function of the modulation frequency Ω

_{1}, and for a set of SOA injected currents, ranging from 75mA, slightly above the transparency (50mA), up to the maximal current of 550mA. IMD

_{3}is known to scale with (

*P*)3, where

^{in}_{RF}*P*the RF power applied to the Mach-Zehnder modulator. Moreover,

^{in}_{RF}*P*is proportional to

^{in}_{RF}*m*

^{2}. Thus, in order to get rid of the dependence on the modulation index, we chose to normalize IMD

_{3}by

*m*

^{6}. In Fig. 3(a) the simulation results are plotted for

*m*=0.01 and

*m*=0.5. As expected, the curves are perfectly coincident.

_{2}-Ω

_{1}is injected into the SOA (dashed lines) [16

16. J. Herrera, F. Ramos, and J. Marti, “Nonlinear distortion generated by semiconductor optical amplifier boosters in analog optical system,” Opt. Lett. **28**, 1102–1104 (2003).
[CrossRef] [PubMed]

^{in}

_{3}multiplied by

*G*

^{2}

_{opt}, for the same reason as the low-pass behavior mentioned above, i.e. that CPO effects are roughly restricted to frequencies below 1/

*τ*. Conversely, for the low frequency part of the spectrum, the intermodulation products can be seen as the result of the combination of two effects, namely the amplification of the incident beat-note at 2Ω

_{s}_{2}-Ω

_{1}, and the generation of a 2Ω

_{2}-Ω

_{1}mixing term due to the gain modulation induced by the CPO effects inside the SOA. The major difference between these two effects is that the amplified beat-note is roughly in phase with the incident one, whereas for the CPO effects, due to the π-phase shift between the intensity and gain modulations, the phase of the CPO-induced beat-note at 2Ω

_{2}-Ω

_{1}is also

*π*-shifted with respect to the incident beat-note. To better understand the results of Fig. 3(a), and particularly the dip observed in the IMD3 curves, we also computed the evolution of the phase of the beat-note term

*M*

_{2Ω2-Ω1}versus modulation frequency (see Fig. 3(b)). At low frequencies below 1/

*τ*, CPO effects dominates, as indicated by the phase curves distribution around -

_{s}*π*, whereas at high frequencies, the pure amplification dominates with phases distributed around 0 and -2

*π*. The dip observed in the IMD

_{3}curves around 1/

*τ*therefore corresponds to a transition between these two regimes, where the two contributions tend to cancel each other as they are of opposite signs. This is also confirmed by the π-phase shift at the dip frequency that is seen on Fig. 3(b). It can also be noticed that when the SOA gain gets weaker, as for the solid blue curve (

_{s}*I*=75mA), then the amplification process always dominates over the CPO effect, even if they are probably quite balanced at low frequencies, explaining the low level of IMD

_{bias}_{3}in this frequency domain.

## 5. Spurious-Free Dynamic Range

*P*, and the third order intercept point IP

_{Noise}_{3}. This point corresponds to the extrapolated fundamental RF output power

*P*

_{1}such as

*P*

_{1}=IMD

_{3}. The noise floor is defined as the electrical noise power contained in a 1Hz electrical analysis bandwidth. The values of

*P*we used in this paper were both measured and theoretically calculated for our SOA as a function of the bias current and the RF frequency [23]. Finally, the SFDR is obtained according to [12]:

_{Noise}_{3}) equals the noise floor, as illustrated in 4.

_{3}in the modulation frequency range from 40kHz up to 30GHz, and for various injected currents. Once again, we compared the situation of a perfectly linear modulation, namely without intermodulation products at the input of a SOA, with the case of an actual Mach-Zehnder modulation, with intermodulation products given by the Bessel functions expansion. These two situations are represented in Figs. 5(a) and 5(b) respectively in dotted and solid lines.

^{2/3}. These values reasonably match radar system requirements. Moreover, for modulation frequencies below or in the range of 1/

*τ*, the presence of an initial IMD

_{s}_{3}at the input of the SOA can even improve the SFDR of the link. This phenomenon is widely known among the analog optical transmission community, where the nonlinear gain transfer function of a SOA is used to linearize a Mach-Zehnder modulator and hence reduce the nonlinear distortion effects [24

24. D.-H. Jeon, H.-D. Jung, and S.-K. Han, “Mitigation of Dispersion-Induced Effects Using SOA in Analog Optical Transmission,” IEEE Photon. Technol. Lett. **14**, 1166–1168 (2002).
[CrossRef]

_{3}being in antiphase with the CPO gain gratings (see Section 4), cast new light on this effect, and gives it different perspectives when slow light applications are considered. Moreover, it is also important to notice that the frequency domain where a dip is observed in the IMD

_{3}, and consequently a peak in the SFDR, roughly corresponds to the domain where slow-light effects are the most efficient in the SOA, i.e. where the CPO-induced phase shift is maximal. This last point could be of importance from the operational point of view.

## 6. Conclusion

## Acknowledgments

## References and links

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12. | H. Zmuda and E. N. Toughlian, Photonic Aspects of modern radar, Artech House, 1994. |

13. | J. H. Seo, Y. K. Seo, and W. Y. Choi, “Spurious-Free Dynamic Range Characteristics of the Photonic Up-Converter Based on a Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. |

14. | A. Sharaiha, “Harmonic and Intermodulation Distortion Analysis by Perturbation and Harmonic Balance Method for In-Line Photodetection in a Semiconductor Optical Amplifier,” IEEE Photon. Technol. Lett. |

15. | E. Udvary, T. Berceli, T. Marozsak, and A. Hilt, “Semiconductor Optical Amplifiers in Analog Optical Links,” in Proc. of IEEE Transparent Optical Network Conf., 2003, paper ThC3. |

16. | J. Herrera, F. Ramos, and J. Marti, “Nonlinear distortion generated by semiconductor optical amplifier boosters in analog optical system,” Opt. Lett. |

17. | T. Mukai and T. Saitoh, “Detuning characteristics and conversion efficiency of nearly degenerate four-wave mixing in a 1.5-m traveling-wave semiconductor laser amplifier,” IEEE Quantum Electron. |

18. | S. Ó Dúill, R. F. ODowd, and G. Eisenstein, “On the Role of High-Order Coherent Population Oscillations in Slow and Fast Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. |

19. | P. Berger, J. Bourderionnet, F. Bretenaker, D. Dolfi, and M. Alouini, “Dynamic saturation in semiconductor optical amplifiers: accurate model, role of carrier density, and slow light,” to be published |

20. | Y. Shi, L. Yan, and A. E. Willner, “High-Speed Electrooptic Modulator Characterization Using Optical Spectrum Analysis,” J. Lightwave Technol. |

21. | N. Breuil, M. Dispenza, L. Morvan, A.-M. Fiorello, S. Tonda, D. Dolfi, M. Varasi, and J. Chazelas, “New optical modulation schemes applied to local oscillator distribution in radar systems,” in Proc. of IEEE Microwave Photonics conf.119–122, (2004). |

22. | M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and P. Thony, “Offset phase locking of Er:Yb:Glass laser eigenstates for radio-frequency photonics applications”, IEEE Photon. Technol. Lett. |

23. | P. Berger, J. Bourderionnet, F. Bretenaker, D. Dolfi, and M. Alouini, “Influence of slow light effect in semiconductor amplifiers on the dynamic range of microwave-photonics links,” Slow and Fast Light 2009, OSA conference proceed., in press. |

24. | D.-H. Jeon, H.-D. Jung, and S.-K. Han, “Mitigation of Dispersion-Induced Effects Using SOA in Analog Optical Transmission,” IEEE Photon. Technol. Lett. |

25. | C. Zmudzinski, E. Twyford, L. Lembo, R. Johnson, F. Alvarez, D. Nichols, and J. Brock, “Microwave optical splitter/amplifier integrated chip (MOSAIC) using semiconductor optical amplifiers”, Photonics and Radio Frequency, Proc. SPIE |

**OCIS Codes**

(070.1170) Fourier optics and signal processing : Analog optical signal processing

(250.5980) Optoelectronics : Semiconductor optical amplifiers

(190.4223) Nonlinear optics : Nonlinear wave mixing

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: July 29, 2009

Revised Manuscript: October 14, 2009

Manuscript Accepted: October 17, 2009

Published: October 23, 2009

**Citation**

Perrine Berger, Jérôme Bourderionnet, Mehdi Alouini, Fabien Bretenaker, and Daniel Dolfi, "Theoretical Study of the Spurious-Free Dynamic Range of a Tunable Delay Line based on Slow Light in SOA," Opt. Express **17**, 20584-20597 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-20584

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### References

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- H. Zmuda, E. N. Toughlian, Photonic Aspects of modern radar, Artech House, 1994.
- J. H. Seo, Y. K. Seo, and W. Y. Choi, "Spurious-Free Dynamic Range Characteristics of the Photonic Up- Converter Based on a Semiconductor Optical Amplifier," IEEE Photon. Technol. Lett. 15, 1591-1593 (2003). [CrossRef]
- A. Sharaiha, "Harmonic and Intermodulation Distortion Analysis by Perturbation and Harmonic Balance Method for In-Line Photodetection in a Semiconductor Optical Amplifier," IEEE Photon. Technol. Lett. 10, 421-423 (1998). [CrossRef]
- E. Udvary, T. Berceli, T. Marozsak, A. Hilt, "Semiconductor Optical Amplifiers in Analog Optical Links," in Proc. of IEEE Transparent Optical Network Conf., 2003, paper ThC3.
- J. Herrera, F. Ramos, and J. Marti, "Nonlinear distortion generated by semiconductor optical amplifier boosters in analog optical system," Opt. Lett. 28, 1102-1104 (2003). [CrossRef]
- T. Mukai, and T. Saitoh, "Detuning characteristics and conversion efficiency of nearly degenerate four-wave mixing in a 1.5-m traveling-wave semiconductor laser amplifier," IEEE Quantum Electron. 26, 865-875 (1990). [CrossRef]
- S. O´ Ó Dúill, R. F. O Dowd, G. Eisenstein, "On the Role of High-Order Coherent Population Oscillations in Slow and Fast Light Propagation Using Semiconductor Optical Amplifiers," IEEE J. Sel. Top. Quantum Electron. 15, 578-584 (2009).
- P. Berger, J. Bourderionnet, F. Bretenaker, D. Dolfi, M. Alouini, "Dynamic saturation in semiconductor optical amplifiers: accurate model, role of carrier density, and slow light," to be published
- Y. Shi, L. Yan, A. E. Willner, "High-Speed Electrooptic Modulator Characterization Using Optical Spectrum Analysis," J. Lightwave Technol. 21, 2358-2367 (2003). [CrossRef]
- N. Breuil, M. Dispenza, L. Morvan, A.-M. Fiorello, S. Tonda, D. Dolfi, M. Varasi, J. Chazelas, "New optical modulation schemes applied to local oscillator distribution in radar systems," in Proc. of IEEE Microwave Photonics conf. 119-122, (2004).
- M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, et P. Thony, "Offset phase locking of Er:Yb:Glass laser eigenstates for radio-frequency photonics applications", IEEE Photon. Technol. Lett. 13, 367 (2001). [CrossRef]
- P. Berger, J. Bourderionnet, F. Bretenaker, D. Dolfi, M. Alouini, "Influence of slow light effect in semiconductor amplifiers on the dynamic range of microwave-photonics links," Slow and Fast Light 2009, OSA conference proceed., in press.
- D.-H. Jeon, H.-D. Jung, S.-K. Han, "Mitigation of Dispersion-Induced Effects Using SOA in Analog Optical Transmission," IEEE Photon. Technol. Lett. 14, 1166-1168 (2002). [CrossRef]
- C. Zmudzinski, E. Twyford, L. Lembo, R. Johnson, F. Alvarez, D. Nichols, J. Brock, "Microwave optical splitter/amplifier integrated chip (MOSAIC) using semiconductor optical amplifiers", Photonics and Radio Frequency, Proc. SPIE 2844, 163 (1996)

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