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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 22 — Oct. 26, 2009
  • pp: 20584–20597
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Theoretical Study of the Spurious-Free Dynamic Range of a Tunable Delay Line based on Slow Light in SOA

Perrine Berger, Jérôme Bourderionnet, Mehdi Alouini, Fabien Bretenaker, and Daniel Dolfi  »View Author Affiliations


Optics Express, Vol. 17, Issue 22, pp. 20584-20597 (2009)
http://dx.doi.org/10.1364/OE.17.020584


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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

A continuously tunable optical delay line is a key element for a large variety of microwave photonics applications, including the control of optically fed phased array antennas, the filtering of microwave signals or the synchronization of optoelectronic oscillators [1

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

3

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

]. Slow light based tunable delay lines have been therefore intensively studied over the past few years. The main focus has been to understand and modelize the underlying phenomena [4

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]

], in order to increase the delays and/or the bandwidth of the studied effect [5

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

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]

]. Among the different slow and fast light architectures, coherent population oscillations (CPOs) in a Semiconductor Optical Amplifier (SOA) offer attractive operational advantages in terms of compactness, integrability and possible parallelism, as well as a continuous tunability of the delay, or phase shift, through the injected current. Moreover, a recent demonstration of 360° phase shift obtained at up to 19 GHz using this technology, makes it very promising towards integration in radar systems.

The principle of slow light in SOAs can be understood as follows. An optical carrier with a sinusoidal modulation envelope that propagates in a SOA induces a carrier population oscillation, via gain saturation in the semiconductor material. Since gain oscillations are in antiphase with the modulation envelope, and basically occur for modulation frequencies below the inverse of the carrier lifetime, they induce a dip in the RF gain spectrum of the SOA. Consequently, according to the Kramers-Kronig relations, the gain dip is associated with a large and positive refractive index dispersion, and hence a large group index, thus slowing down the sinusoidal modulation propagation velocity. This property, as well as corresponding phase and amplitude change of the modulation envelope, have been extensively studied, both experimentally or theoretically [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]

11

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]

]. In particular, this has provided the RF transfer function of the SOA-based delay line, similarly to any two-port electronic device.

In this paper, we first present a predictive model to determine the harmonic generation level, which takes into account the spatial evolution of the saturation parameters along the SOA length. We also discuss the expression of the gain modulation harmonics that is used in common models, and show that it can lead, under certain conditions, to a significant error on the harmonic’s estimate. Then, we generalize our model to end up with the third order intermodulation distortion (IMD3), and compute the IMD3 level and the SFDR over the frequency range of interest for radar applications (up to 20GHz), and for a large range of injected currents. Finally, we discuss the influence of the input optical spectrum on the dynamic range of the delay line by comparing the simulation results obtained for an ideal spectrum, i.e., a perfect sinusoidal modulation of the optical beam at the input of the SOA, and for a realistic spectrum, namely, when the RF signals to be delayed are transferred on the optical carrier using a standard Mach-Zehnder modulator.

2. Principle of harmonics calculation

We consider an optical field 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]

]:

dN(z,t)dt=IqVN(z,t)τs(z)g(z,t)E(z,t)2ω,
(1)
d(E(z,t)2)dz=(γi+Γg(z,t))E(z,t)2
(2)

where N(z, t) is the carrier density, I the current injected in the SOA, τs the carrier lifetime, q the elementary electric charge, V the SOA active volume, g the optical gain, Γ the confinement factor, and γi 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. |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:

E(z,t)2=Σk=+Mk(z)eikΩt,
(3)
N(z,t)=N̅(z)+Σk=+k0Nk(z)eikΩt,
(4)
g(z,t)=g̅(z)+a(z)Σk=+Nk(z)eikΩtk0
(5)

where (z) and ḡ(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)=∂ḡ/∂N̄. Defining gk as the oscillating component of the gain at frequency kΩ, and considering only a finite number K of harmonics, the carrier rate equation [Eq. (1)] can be written in an equivalent matrix formulation:

oe-17-22-20584-i001.jpg

where αk=Is(1+M 0/Is-ikΩτs), and α 0=M 0 is the DC optical intensity. Is denotes the saturation intensity and is defined as Is=h̄ω/s. It is worth mentioning that αk is obtained at the first order of Eq. (1), when mixing terms are not considered. One can also notice that 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, N̄, ḡ, a, τs, Is, and consequently the αk’s are all actually functions of 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

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

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

In order to solve the propagation equation [Eq. (2)], one first has to determine the expressions of the gain harmonic components gk’s, as functions of the harmonic components of the optical intensity Mk’s, of ḡ, and of Is and τs. Under small RF signal approximation, i.e. considering a small modulation index of the optical carrier, one can assume that |M k-1|≫|Mk| for k>0 (and |M k-1|≪|Mk| for k<0). It can also be noticed that the same relations hold for the Nk’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

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

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]

]:

gk=g̅MkIs1+M0Is+jkΩτs,
(7)

where :

g̅=h̅ω(IqVN̅τs)M0.
(8)

In other words, this is equivalent to consider as nonzero only the diagonal and center column of the matrix in Eq. (6). However, the quantities to be compared in the inversion of Eq. (6) are not the |Mk|’s, but the terms |M k-p×gp|. Consequently, the hypothesis|M k-1|≫|Mk| leads to neglect only the terms for which |p|+|k-p|>|k|, the others being of the same order. Under these conditions, Eq. (6) is reduced into:

oe-17-22-20584-i002.jpg

The resolution of this equation gives identical expressions for g̃ 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:

g2=g̅M2IS1+M0IS+2jΩτS+(M1IS)2(1+M0IS+jΩτS)(1+M0IS+2jΩτS),
(10)
g3=g̅(M3Is1+M0Is+3jΩτs+M1M2Is2(1+M0Is+jΩτs)(1+M0Is+3jΩτs)
+M1M2Is2(1+M0Is+2jΩτs)(1+M0Is+3jΩτs)
(M1Is)3(1+M0Is+jΩτs)(1+M0Is+2jΩτs)(1+M0Is+3jΩτs)).
(11)

Once again, Is and τs in Eqs. (7) to (11) are functions of the propagation coordinate z, through their dependence on the static carrier density N̄(z). Doing so, the relations between Is, τs 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

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

], a simple measurement of the unsaturated gain as a function of the SOA bias current, gives a first relation between and N̄/τs:

g̅=α+βτsN̅,
(12)

where α and β are empiric coefficients determined experimentally by measuring the SOA’s small signal gain. For our simulations, we used α=5.88·103m-1 and β=-1.84·1037m2.s-1, corresponding to the COVEGA InGaAsP/InP quantum well SOA available in our laboratory [19

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

]. Then, solving the system constituted of Eqs. (8) and (12), we obtain ḡ(z) and N̄/τs(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:

1τs=A+BN̅+CN̅2,
(13)

where 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 ḡ(z) and N̄/τs(z) then enable to derive τs and Is as functions of I, M 0(z) and A, B and C.

According to [19

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

], 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=2·109s-1, B=1.2·10-10cm3.s-1, C=1.8·10-31cm6.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

g̅=h̅ω(IqVN̅τs)δ0,0,
(14)
gk=g̅δk,0δ0,0.
(15)

Similarly to the small signal case, using Eqs. (12), (13) and (15), we obtain N̄/τ (1) s, I (1) s and τ (1) s as functions of I, A, B,C and Mk(z). This procedure is repeated until convergence of N̄/τ (n) s, I (n) s and τ (n) s, 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

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]

] and numerically solved.

oe-17-22-20584-i003.jpg

2.3. Comparison with reported models

To demonstrate the importance of the additional terms in the right hand side of the expressions of 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]

]:

{M0,in=PinM1,in=Pin×J1(m)M2,in=0M3,in=Pin×J3(m)Mk,in=Mk,in,
(17)

where m is the modulation index, and Jk denotes the kth 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.

Table 1. List of studied SOA parameters

table-icon
View This Table

On the other hand, we measured experimentally the third harmonic level at the input and at the output of a commercial SOA, whose main physical parameters are listed in Table 1. These parameters were also used in the simulations. It is worth mentioning that the saturation power given in the table is the measured output power for which the unsaturated optical gain is reduced by 3dB. Therefore it does not correspond to 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

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

]. Equation (16) is solved using the expression of the gk’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:

H3=2Rηph2M3,out×S2
(18)

As can be seen on Fig. 1(a), for a modulation frequency Ω above 4GHz, 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 Gopt). 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 Ωτs≪1:

dM3dz=γiM3+g̅IsM0+Is[M32M1M2M0+Is+M13(M0+Is)2A],
(19)

whereas, using the standard models based on Eq. (7), one has:

dM3dz=γiM3+g̅IsM0+Is[M32M1M2IsB].
(20)
Fig. 1. (a) Third harmonic generation as a function of the RF frequency. Green solid line: common expressions of gk (Eqs. (7) and (8)) are used. Red solid line: use of the rigorous expressions of gk (Eqs. (10) and (11)). The red circles represent experimental measurements. (b) Asymptotic case when Ω→0: Evolution of terms A and B along the SOA, and third harmonic level calculated according to Eq. (19) (red solid line) and Eq. (20) (green solid line).

Basically, the right-hand side of these two equations consists in three terms. The first two respectively describe the linear losses and amplification of 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.

This shows the importance of considering accurate expressions for gk even in the small signal situation, and especially for the low frequency part of the microwave spectrum. When Ω increases, since the 2nd and 3rd terms of Eq. (11) respectively evolve as Ω-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> 4GHz. 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]

], where harmonics calculations are carried out at a fixed frequency of 10GHz. 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

Intermodulation distortion (IMD) calculation is slightly different from what has been discussed in the above section. Indeed, the number of mixing terms that must be taken into account is significantly higher. For radar applications a typical situation where the IMD plays a crucial role is that of a radar emitting at a RF frequency Ω1, and facing a jammer emitting at Ω2, close to Ω1. Both Ω1 and Ω2 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 Ω21 (or Ω12) and 2Ω21 (or 2Ω12) — respectively called second (IMD2) and third (IMD3) 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 δΩ=Ω21. Then, using a Fourier decomposition of |E|2, N, and g, one has:

E(z,t)2=k=+Mk(z)eikδΩt,
(21)
N(z,t)=N̅(z)+k=+k0Nk(z)eikδΩt,
(22)
g(z,t)=g̅(z)+k=+k0gk(z)eikδΩt,
(23)

We consider a typical radar frequency Ω1 of 10GHz, and a frequency spacing δΩ of 10MHz. Here, for intermodulation distortion calculation, we assume that only the spectral components at Ω1,2, 2Ω1,2, and all their first order mixing products significantly contribute to the generation of IMD2 and IMD3, as illustrated in Fig. 2. The Mk’s and the gk’s are then reduced in 19 elements vectors, and gathered into blocks, denoted M block, j and g block, j. The jth block contains the mixing products with frequencies close to j×Ω1. According to the notation of Fig. 2, the carrier rate equation [Eq. (1)] and the propagation equation [Eq. (2)] can respectively be written as:

(00h̅ω(IqVN̅τs)00)=(D2,2D2,1D2,000D1,2D1,1D1,0D1,10D0,2D0,1D0,0D0,1D0,20D1,1D1,0D1,1D1,200D2,0D2,1D2,2)×(gblock,2gblock,1gblock,0gblock,1gblock,2),
(24)
ddz(Mblock,2Mblock,1Mblock,0Mblock,1Mblock,2)=(H2,2H2,1H2,000H1,2H1,1H1,0H1,10H0,2H0,1H0,0H0,1H0,20H1,1H1,0H1,1H1,200H2,0H2,1H2,2)×(Mblock,2Mblock,1Mblock,0Mblock,1Mblock,2),
(25)

where 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 ith and the jth harmonics. From now on, the full procedure described in Section 2 can be applied in the same iterative way to determine the gk’s, Is and τs, and to finally numerically solve the Eq. (25). Similarly to Eq. (18), the photodetected RF power at 2Ω21 is then calculated through:

IMD3=2Rηph2M2Ω2Ω1out×S2.
(26)
Fig. 2. Set of significant spectral components of |E|2, N and g, and associated index k in their Fourier decompositions. n is defined such as Ω1=Ω.

4. Simulation results

In the general situation depicted in Section 3, the optical intensity at the output of a chirp-free Mach-Zehnder modulator, that is, at the input of the SOA, will be of the following form [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]

]:

Iopt,in=I0[1+cos(m(cos(Ω1t)+cos(Ω2t))+ϕ)],
(27)

where 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 Mk’s, at quadrature bias (ϕ=π/2), is:

Mblock,0=00100;Mblock,1=J1(m)J2(m)J0(m)J1(m)J0(m)J1(m)J1(m)J2(m);Mblock,2=000;Mblock,j=Mblock,j.
(28)

In particular, one can notice that the optical intensity at the input of the SOA contains a term at 2Ω1,22,1, denoted IMDin 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 IMDin 3. The latest case corresponds to the use of a perfectly linear modulator, or as in [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]

] for Radio-over-Fiber applications, where modulations at Ω1 and Ω2 are produced by two distinct modulators on two incoherent optical carriers. An alternative way to avoid IMDin 3 would be to use a Single-Side-Band modulator, as proposed in [21

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

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]

]. In a tunable delay line based on slow-light effects in a SOA, the optical group delay can be typically tuned either by varying the optical input power, or the SOA bias current. Although both situations can be easily simulated, in this paper, we only present the latter one, using the bias current, which is the most suitable for an implementation in a real radar system. Figure 3(a) represents the simulated IMD3, calculated according to Eq. (26), as a function of the modulation frequency Ω1, and for a set of SOA injected currents, ranging from 75mA, slightly above the transparency (50mA), up to the maximal current of 550mA. IMD3 is known to scale with (PinRF)3, where PinRF the RF power applied to the Mach-Zehnder modulator. Moreover, PinRF is proportional to m 2. Thus, in order to get rid of the dependence on the modulation index, we chose to normalize IMD3 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.

Fig. 3. (a) Computed third order intermodulation power IMD3, normalized to m 6, as a function of the modulation frequency, for various injected currents. (b) Corresponding phase of the beat-note at 2Ω21. Dashed lines: case of a perfectly linear modulator (IMDin 3=0). Solid lines: actual Mach-Zehnder modulator (IMDin 3≠0).

The expected low-pass filter behavior of the SOA is observed on Fig. 3(a), when no beat-note at 2Ω21 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]

]. The situation is completely different when the nonlinearities of the modulator used to transfer the RF signal to the optical carrier are taken into account in the simulations (solid lines). First, as for the third harmonic generation (see Section 2.3), the asymptotic values for the high frequency part of the spectrum naturally corresponds to IMDin 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/τs. 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 2Ω21, and the generation of a 2Ω21 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 2Ω21 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/τs, CPO effects dominates, as indicated by the phase curves distribution around -π, whereas at high frequencies, the pure amplification dominates with phases distributed around 0 and -2π. The dip observed in the IMD3 curves around 1/τs 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 (Ibias=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 IMD3 in this frequency domain.

5. Spurious-Free Dynamic Range

SFDR=(IP3Pnoise)23,
(29)

and corresponds to the situation where the spurious (here the IMD3) equals the noise floor, as illustrated in 4.

Fig. 4. SFDR determination for Ibias=250mA, Pin=10mW, and IMDin 3=J 1(m)J 2(m). In blue: low modulation frequency (≪1/τs); in red: high modulation frequency (≫1/τs); in green: frequency around the IMD3 dip (≈1/τs).

We calculated both the SFDR and the third order intercept point IP3 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.

Fig. 5. (SFDR (a) and 3rd order intercept point IP3 (b), as a function of the modulation frequency, for various injected currents, and in the two configurations of a perfectly linear modulation and of a realistic Mach-Zehnder modulator (respectively in dotted and solid lines).

6. Conclusion

Acknowledgments

The authors acknowledge the partial support from the GOSPEL European project and from the French “Délégation Générale pour l’Armement”. The authors would also like to acknowledge Reynald Boula-Picard for early discussions.

References and links

1.

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

2.

D. Dolfi, P. Joffre, J. Antoine, J-P. Huignard, D. Philippet, and P. Granger, “Experimental demonstration of a phased-array antenna optically controlled with phase and time delays,” Appl. Opt. 35, 5293–5300 (1996). [CrossRef] [PubMed]

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]

7.

W. Sales, S. Xue, J. Capmany, and J. Mørk, “Experimental Demonstration of 360° Tunable RF Phase Shift Using Slow and Fast Light Effects”, Slow and Fast Light 2009, OSA conference proceed., paper SMB6.

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]

9.

C. J. Chang-Hasnain and S. L. Chuang, “Slow and Fast Light in Semiconductor Quantum-Well and Quantum-Dot Devices,” J. Lightwave Technol. 24, 4642–4654 (2006). [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]

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. 15, 1591–1593 (2003). [CrossRef]

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. 10, 421–423 (1998). [CrossRef]

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. 28, 1102–1104 (2003). [CrossRef] [PubMed]

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]

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]

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, 2358–2367 (2003). [CrossRef]

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. 13, 367 (2001). [CrossRef]

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. 14, 1166–1168 (2002). [CrossRef]

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 2844, 163 (1996)

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

  1. J. Yao, "Microwave Photonics," J. Lightwave Technol. 27,314-335 (2009). [CrossRef]
  2. D. Dolfi, P. Joffre, J. Antoine, J-P. Huignard, D. Philippet, and P. Granger, "Experimental demonstration of a phased-array antenna optically controlled with phase and time delays," Appl. Opt. 35,5293-5300 (1996). [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, A. E. Willner, "Fiber-Based Slow-Light Technologies," J. Lightwave Technol. 26, 3752-3762 (2008). [CrossRef]
  5. Y. Chen, W. Xue, F. Ohman, 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]
  7. W. Xue, S. Sales, J. Capmany, J. Mørk, "Experimental Demonstration of 360? Tunable RF Phase Shift Using Slow and Fast Light Effects", Slow and Fast Light 2009, OSA conference proceed., paper SMB6.
  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]
  9. C. J. Chang-Hasnain and S. L. Chuang, "Slow and Fast Light in Semiconductor Quantum-Well and Quantum-Dot Devices," J. Lightwave Technol. 24, 4642-4654 (2006). [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).
  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]
  12. H. Zmuda, 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. 15, 1591-1593 (2003). [CrossRef]
  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. 10, 421-423 (1998). [CrossRef]
  15. 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.
  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]
  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]
  18. 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).
  19. 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
  20. Y. Shi, L. Yan, A. E. Willner, "High-Speed Electrooptic Modulator Characterization Using Optical Spectrum Analysis," J. Lightwave Technol. 21, 2358-2367 (2003). [CrossRef]
  21. 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).
  22. 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]
  23. 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.
  24. 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]
  25. 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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