## Abstract

We theoretically and experimentally evaluate the propagation, generation and amplification of signal, harmonic and intermodulation distortion terms inside a Semiconductor Optical Amplifier (SOA) under Coherent Population Oscillation (CPO) regime. For that purpose, we present a general optical field model, valid for any arbitrarily-spaced radiofrequency tones, which is necessary to correctly describe the operation of CPO based slow light Microwave Photonic phase shifters which comprise an electrooptic modulator and a SOA followed by an optical filter and supplements another recently published for true time delay operation based on the propagation of optical intensities. The phase shifter performance has been evaluated in terms of the nonlinear distortion up to 3* ^{rd}* order, for a modulating signal constituted of two tones, in function of the electrooptic modulator input RF power and the SOA input optical power, obtaining a very good agreement between theoretical and experimental results. A complete theoretical spectral analysis is also presented which shows that under small signal operation conditions, the 3

*order intermodulation products at 2Ω*

^{rd}_{1}+ Ω

_{2}and 2Ω

_{2}+ Ω

_{1}experience a power dip/phase transition characteristic of the fundamental tones phase shifting operation.

© 2010 OSA

## 1. Introduction

The last years have witnessed an increasing interest in the study and development of novel materials, devices and technologies, with the aim of controlling the speed of light [1–4]. This field of activity, now known within the photonics community as Slow and Fast Light (SFL), offers the potential of direct application to a wide variety of signal processing tasks which are currently needed both in digital and analog systems [1,2]. Several technologies have been reported in the literature to implement slow and fast light devices including those based on optical fibers [3], photonic crystals [4] and semiconductors [5]. This last approach is particularly interesting since semiconductor technology is very mature, offers the potential of velocity control by means of external electric signals and last, but not least, can be integrated.

In digital applications the key desired functionality is the implementation of a delay line [2] which can, in turn, be employed as a building block for optical buffers. Currently there is ongoing research to circumvent the bandwidth-delay tradeoff that SFL devices offer for this application outlined in [1]. Another field of potential interest is Microwave Photonics (MWP) [6–8]. Here two basic functionalities are desired [8]; the implementation of tunable true time delays and the implementation of broadband tunable microwave phase shifters. The most successful approach reported so far is based on the so-called Coherent Population Oscillations (CPO) in Semiconductor Optical Amplifiers (SOAs) [9–11]. It has been shown both theoretically and experimentally that CPO in SOAs can support both applications [12–19], offering different degrees of freedom (input optical pump, bias voltage or electrical injection current). In particular, time delay applications have been addressed in [12,13,18] showing the potential for operation at GHz frequencies [13]; while microwave phase shifting has been demonstrated in [14–17,19,20] including several applications such as microwave photonic filtering [13,14] and optoelectronic oscillators [20].

While time delay applications usually exploit CPO effects in standalone SOA configurations (SOA + detector) [12], it has been found that microwave phase shifting can be greatly improved by optically filtering the red-shifted radiofrequency sideband at the output of the SOA prior to photodetection (SOA + Optical filter + detector) [14,15]. Furthermore, by cascading several SOA + optical filtering stages a full tunable 360° phase shift can be achieved at very high frequencies [17].

As with any element in a microwave photonic link, the impact of the SOA-CPO device must be evaluated, especially in terms of nonlinear distortion leading to harmonic and intermodulation distortions, which can degrade the systems performance [21]. Recently, a complete model has been published in the literature that accounts of this analysis in the case of true time delay applications (SOA + detector configuration) [22]. This model is based on the propagation of the optical intensities within the SOA devices and has outlined the interplay that the contributions from direct amplification and the π-phase shifted component generated by CPO have on the evolution of third order nonlinear distortion terms. A detailed study of the Spurious Free Dynamic Range (SFDR) based on the model has also been reported which, as a main conclusion has outlined that the CPO in SOA device can cooperate to reduce the initial nonlinear distortion terms arising in the signal Mach-Zehnder modulator.

For Microwave Phase shifters (SOA + optical filter + detector configuration), a full model on nonlinear distortion is missing. In this context, the model developed in [22] is not applicable since the location of an optical filter after the SOA requires a description based on the propagation of optical fields and not of optical intensities. In [10] an optical field model was developed by Agrawal for multichannel amplification in SOAs, based on the propagation of optical field. Intermodulation distortion was considered, however for the particular case of *M* equispaced RF channels and distortion-free input signal to the amplifier. Other published contributions based on the optical field propagation have been particularly focused on harmonic distortion [23,24]. The model developed in this paper supplements and generalizes those of [10] and [22] since it accounts for arbitrary spaced RF channels, input distorted signal to the SOA, and takes into account all the beating terms (harmonic and intermodulation).

The paper is organized as follows. In section 2 we develop the optical field model that contains the equations accounting for any general order of both harmonic and intermodulation distortions when the modulating signal is composed of an arbitrary number *N* of RF tones. Section 3 involves the experimental and theoretical analysis of the nonlinear distortion introduced in a phase shifter composed of an electro-optic modulator (EOM), a SOA device under the CPO effect and an optical filtering stage. This is carried by evaluating its performance as a function of the SOA input optical power, the electrical input power (dynamic range) and the modulation frequency (spectral response). With that purpose we specialize our general model to 3* ^{rd}* order distortion when the modulating signal is composed of two radiofrequency (RF) tones. Our main conclusions and summary are provided in section 4.

## 2. Theoretical model

With the objective of analyzing the harmonic and intermodulation distortion up to order *M* produced in the SOA based microwave photonic phase shifter, we assume an electrical modulating signal composed of a general number *N* of tones of angular frequencies Ω_{1}, Ω_{2}, … Ω* _{N}*. The schematic of the phase shifter under analysis is shown in Fig. 1
, where we have stressed the consideration of the presence of harmonics and intermodulation products at the input of the SOA device

*E*|

^{in}*instead of assuming an ideal electrooptic modulator [10,23].*

_{SOA}Within the range of modulation frequencies and power level of interest, the saturation due to intraband transitions, arising as a consequence of the spectral hole-burning, can been neglected, as justified in [10,23]. This means that the carrier density is assumed to be linearly dependent on the optical intensity. Thus, we will consider that CPO is the main nonlinear mechanism that contributes to an oscillating optical gain in the SOA. Assuming the carrier density *N* to be spatially homogeneous [9], we can state that the dynamics of CPO is governed by the well known carrier rate equation

*I*the injection current,

*e*the unit electron charge,

*V*the volume of the active region,

*τ*the carrier lifetime, Г the confinement factor,

_{s}*a*the SOA differential gain and

*N*the transparency carrier density.

_{tr}The optical field inside the SOA cavity can be expressed as:

*E*

_{(}

_{k1}_{, ...}

_{kN}_{)}defines the field complex amplitude and

*β*the propagation constant, both related to the angular frequency${\omega}_{0}+{\displaystyle {\sum}_{i=1}^{N}{k}_{i}{\mathrm{\Omega}}_{i}}$, being

_{kiΩi}*ω*

_{0}the angular frequency of the optical carrier while Ω

*the angular frequency of the*

_{i}*i*-th modulating tone. This optical beam interacts with carriers inside the semiconductor, through stimulated emission, and modulates the carrier density, introducing carrier population oscillations of RF frequencies$\sum}_{i=1}^{N}{m}_{i}{\mathrm{\Omega}}_{i$. These oscillations generate, in turn, a temporal grating, causing dispersive gain and index modifications on the two sidebands of the travelling optical field while inducing wave mixing between them. We can therefore assume that the carrier density inside the SOA, and thus the gain coefficient, will oscillate following a similar variation.

From the slowly varying envelope approximation of the propagation equation through the SOA device [9], and following a similar procedure as that described in [10,24], we get the following set of coupled field differential equations

*γ*

_{int}represents the internal waveguide losses,

*α*the SOA linewidth enhancement factor or SOA chirp parameter, while the coefficients for the dynamic evolution of the gain can be identified from Eq. (1) and (2) as, (similarly to [24]):

*P*the saturation power [24]. The term

_{s}*S*

_{(}

_{m1}_{, ...}

_{mN1}_{)}represents the Fourier component of the optical intensity at frequency $\sum}_{i=1}^{N}{m}_{i}{\mathrm{\Omega}}_{i$, from Eq. (2):

*M*-1 terms on both sides of the boundary channels.

Secondly, our description relies also on the consideration of the distortion introduced by an EOM, instead of presuming, as customary in the literature, an ideal [10,23] or a zero-chirp [22] EOM. For the determination of the output optical field of a dual-drive EOM, that is, the initial conditions required by Eq. (3), we will consider a general case applicable for any value of chirp in the EOM when the modulating signal is composed of two RF tones of angular frequencies Ω_{1} and Ω_{2}. We assume that the voltage signals applied on both electrodes, *V _{a}*(

*t*) and

*V*(

_{b}*t*), are each one composed of a DC bias term, V

*and V*

_{DCa}*, and that the RF tones are characterised by the same amplitude*

_{DCb}*V*and different initial phases, ${\varphi}_{ia}$and ${\varphi}_{ib}$, for

_{RFi}*i*= 1,2:

*E*comes from the input intensity provided by the laser and

_{s}*V*is the quadrature voltage. Expanding the cosine and the exponential terms in Eq. (7) in terms of Bessel functions of the first kind, we finally get:

_{π}*V*+

_{DCa}*V*)

_{DCb}*π*/(2

*V*),

_{π}*φ*represents the normalized bias voltage

*φ*= (

*V*-

_{DCa}*V*)

_{DCb}*π*/(2

*V*) while the modulation indices are defined as

_{π}*m*=

_{i}*V*/

_{RFi}π*V*, for

_{π}*i*= 1,2.

## 3. Results and discussion

#### 3.1. Experimental Setup

In order to validate our model and subsequently study the impact of the nonlinear distortion in the microwave phase shifter we have assembled the experimental setup schematically depicted in Fig. 2
. A Distributed-Feedback (DFB) laser operating at 1547.6 nm was used as a light source. Its output was modulated after passing through an external dual-drive zero-chirp EOM with two RF synthesizers. Both tones have the same RF power while their spectral location has been selected in order to compare different optical filtering levels, that is, one case for *f*
_{1} = 10.5 GHz and *f*
_{2} = 11 GHz and another one for *f*
_{1} = 9 GHz and *f*
_{2} = 12 GHz. These frequency spacings Δ*f* = *f*
_{2} - *f*
_{1} of 0.5 and 3 GHz are representative of a broadband signal transmission scenario where different analogue or digital services can be located within the whole microwave or millimeter wave region, covering different MWP applications. Note that [22] deals with Δ*f* = 10 MHz, typical of antenna feed systems for radar applications where a jammer signal located at Ω_{2} close to the radar emission frequency Ω_{1}.The harmonics and intermodulation product terms at the input of the SOA device are illustrated in the left upper part of the figure (*E ^{out}|_{MZM}* ).

The microwave phase shifter is formed by a commercial SOA and a filtering stage which has been implemented by a Fiber Bragg Grating (FBG) operating in transmission. The SOA saturation power is 5 dBm at a bias of 350 mA and is characterized by a bandwidth over 10 GHz. The inset in Fig. 2 illustrates the FBG magnitude and phase responses. The magnitude notch presents a depth of −33 dB and a spectral −3dB width of 11 GHz, while the maximum phase excursion of the spectral response has a value around 300 degrees.

The MZM output signal is amplified by a 20dB-gain Erbium Doped Fiber Amplifier (EDFA) to assure the SOA operates under saturation conditions. It must be noted that the phase shift can be optically controlled [14,23], by means of the optical CW power at the SOA input, which requires the presence of an optical variable attenuator (VOA). The diagram of the frequency components at the phase shifter output are depicted on the top right corner of Fig. 2 as (*E ^{out}|_{SOA + FBG}* ), where we appreciate how the notch filter blocks not only the fundamentals at the red-shifted sideband (-Ω

_{1}and -Ω

_{2}) yielding an attenuation around 32 dB for

*f*

_{1}= 10.5 GHz and

*f*

_{2}= 11 GHz, while 28 dB for 9 and 12 GHz; but also the 3

*order IMD terms placed at Ω*

^{rd}_{1}-2Ω

_{2}and Ω

_{2}-2Ω

_{1}, providing an attenuation of around 32 dB and 5 dB respectively. Finally, the magnitude and the experienced phase shift of the fundamental, harmonic and intermodulation terms were measured using a Vector Network Analyzer (VNA). For better understanding, the measured magnitude and phase shift responses of the FBG are shown in Fig. 3 .

#### 3.1. Spurious Free Dynamic Range evaluation

To analyze the impact of the nonlinear distortion in the phase shifter performance we have first evaluated the RF photodetected power, defined as *P _{d}*

_{(}

_{m}_{1, ...}

_{mN}_{)}= |

*S*

_{(}

_{m}_{1, ...}

_{mN}_{) }

*ℜ*|

^{2}

*Z*, being

_{o}*S*

_{(}

_{m1}_{, ...}

_{mN}_{)}the Fourier component of the optical intensity specified in Eq. (5), while

*ℜ*and

*Z*are respectively the photodiode responsivity and impedance. This theoretical and experimental evaluation has been carried out for every fundamental tone, harmonic and intermodulation distortions up to 3

_{o}*order, named as HD*

^{rd}_{3}and IMD

_{3}, for

*f*

_{1}= 9 GHz and

*f*

_{2}= 12 GHz.

We have applied the general expression given by Eq. (8) to compute the optical field at the EOM output (initial conditions). In particular we have considered the case of a zero-chirp modulator, for which *V _{a}*(

*t*) = -

*V*(

_{b}*t*). Under this assumption and particularizing for initial RF phases${\varphi}_{ia}={\varphi}_{ib}=0$, the SOA initial condition for the complex field ${E}_{({k}_{1},{k}_{2})}$results in:

*τ*= 110 psec, length

_{s}*L*= 1 mm, unsaturable loss

*γ*

_{int}= 4/L (1/m), unsaturated modal gain $\mathrm{\Gamma}\overline{g}$ = 4.5/L (1/m), saturation power

*P*= −6 dBm and a linewidth enhancement factor

_{s}*α*= 6. The modeling of the optical filtering stage has been performed meticulously applying the pertinent measured values of attenuation and phase shift applied not only to the fundamental tones but also for all the DC, HD

_{2}, IMD

_{2}, HD

_{3}and IMD

_{3}terms. Note that in order to account for the effect of such filter we need the optical field (and not the optical power) signal as obtained by solving Eq. (3). For a properly evaluation of the phase shifter performance, we have computed the results for the photodetected RF power [by solving Eq. (4)] versus the RF input power to the EOM, which is related to the common electrical modulation index

*m*, defined as

*m*=

*m*

_{1}=

*m*

_{2}, by

*order intermodulation products of interest; that is the fundamental [*

^{rd}*P*

_{d}_{(1,0)}at frequency Ω

_{1}and

*P*

_{d}_{(0,1)}at frequency Ω

_{2}] and relevant IMD

_{3}terms [

*P*

_{d}_{(2,-1)}at 2Ω

_{1}-Ω

_{2}and

*P*

_{d}_{(−1,2)}at 2Ω

_{2}-Ω

_{1}]. The 2

*order harmonic and intermodulation products are shown in Fig. 5 , which comprise HD*

^{nd}_{2}[

*P*

_{d}_{(2,0)}at 2Ω

_{1}and

*P*

_{d}_{(0,2)}at 2Ω

_{2}] and IMD

_{2}[

*P*

_{d}_{(1,1)}at Ω

_{1}+ Ω

_{2}and

*P*

_{d}_{(−1,1)}at Ω

_{2}-Ω

_{1}]. In both figures we have also included (dashed curves) the equivalent detected RF power that would be collected if a photodetector were placed at the SOA input. The optical intensity provided by the CW laser has been set to obtain a high input optical CW power (0 dBm) to the SOA, in order to guarantee that the amplifier is working under saturation regime.

A very good level of agreement can be observed between the theoretical results and those rendered by the measurements. A first feature of interest in both figures is the slight degradation produced in the theoretical curves as the RF electrical input power is risen to levels above 6 dBm (which corresponds to modulation index of *m* = 0.56), consequence of the distortion already introduced in the EOM for the optical power level provided by the laser.

Another interesting feature that can be observed from Fig. 4 and Fig. 5 is related to the fact that the CPO effect produces mainly second order distortion as compared to third order contributions. This feature can be explained by analyzing Eq. (3), for the IMD_{2} and HD_{2} optical fields where the dominant wave mixing terms in the double summation are due to the product of the fundamental component of the SOA dynamic gain (first-order CPO) and the fundamental signal contribution. For example, in the case of the equation describing the evolution of *dE*
_{(1,1)}/*dz* the dominant terms are { *g*
_{(0,1)}
*E*
_{(1,0)} + *g*
_{(1,0)}
*E*
_{(0,1)} }, the magnitude of which is higher than those of the dominant wave mixing terms contained in the propagation equations for the IMD_{3} contributions, which are proportional to 2* ^{nd}* order gain coefficients.

For the evaluation of the SFL phase shifter performance we will resort to a common figure of merit, the Spurious Free Dynamic Range (SFDR) widely used to simultaneously characterize the linearity and noise characteristics of microwave devices (e.g. amplifiers, mixers, etc.), analog-to-digital converters and optical devices such as laser diodes and external modulators. Two-tone SFDR is defined as the carrier-to-noise ratio when the noise floor in the signal bandwidth equals to the power of a given order intermodulation product (usually the 3* ^{rd}* order IMD

_{3}). The noise floor level

*P*in (dBm/Hz) has been derived taking into account the different noise contributions present in the microwave phase shifter illustrated in Fig. 1,

_{noise}^{2}

*is the intensity noise of the laser,*

_{RIN}*σ*

^{2}

*the ASE noise generated in the SOA by the carrier-spontaneous beat-note, σ*

_{ASE}^{2}

*the thermal noise and σ*

_{Thermal}^{2}

*the shot noise contribution.*

_{Shot}In this context, the SFDR of an RF link limited by 2* ^{nd}* (

*SFDR*

_{2}) or 3

*(*

^{rd}*SFDR*

_{3}) intermodulation distortion can be computed from:

*IP*

_{2}and

*IP*

_{3}are the linearly extrapolated input powers in dBm at which respectively the fundamental and 2

*or 3*

^{nd}*intermodulation output powers would be equal and*

^{rd}*P*is the electrical noise power in dBm contained in 1-Hz bandwidth, calculated from Eq. (11). To obtain the values of the SFDR from Fig. 4 and Fig. 5, we will denote the intersection points of the power in dBm delivered to the load resistance by the signal and every harmonic distortion term at frequency (

_{noise}*m*

_{1}Ω

_{1}+

*m*

_{2}Ω

_{2}) with the noise floor level as

*P*

_{d}^{i}_{(}

_{m}_{1,}

_{m}_{2)}. Thus, the SFDR referred to the fundamental term at frequency Ω

_{1},

*P*

_{d}^{i}_{(1,0)}, can be calculated as:

_{2}, [

*P*

_{d}_{(1,1)}and

*P*

_{d}_{(−1,1)}], and IMD

_{3}, [

*P*

_{d}_{(2,-1)}and

*P*

_{d}_{(−1,2)}]. The computed values of SFDR from each pair of both 2

*and 3*

^{nd}*intermodulation products are 72 dB/Hz*

^{rd}^{1/2}and 101 dB/Hz

^{2/3}, respectively. It can be observed that a reasonably good margin of SFDR

_{3}is achieved, high enough for the majority of applications where the phase shifter can be integrated. Note that while the 2

*order intermodulation distortion can be significant, its effect can be neglected if the two subcarriers are placed within an octave in the RF spectrum.*

^{nd}The distortion evaluation performed in Fig. 4 and Fig. 5 provides the electrical linear operation regime that will assure a proper functionality of the SOA based MWP phase shifter. A conservative limit is imposed by the point where the IMD_{3} equals the noise level, given in the worst case by a level of *P _{RF}* |

*of around −31 dBm, equivalent to an electrical common modulation index*

_{in}*m*= 0.008, Eq. (10).

#### 3.2. Performance analysis of the MWP phase shifter versus SOA optical input power

The measured and computed results of both the photodetected RF power and experienced microwave phase shift, in function of the optical power at the SOA input, for the case where the spectral separation between the modulating tones is 0.5 GHz,(i.e. *f*
_{1} = 10.5 and *f*
_{2} = 11 GHz), are shown in Fig. 6
for the fundamental tones and the IMD_{3} terms placed at 2Ω_{1}-Ω_{2} and 2Ω_{2}-Ω_{1}, while the IMD_{3} contributions at 2Ω_{1} + Ω_{2} and 2Ω_{2} + Ω_{1} are plotted in Fig. 7
and the 2* ^{nd}* order distortion elements, that is HD

_{2}and IMD

_{2}, are plotted in Fig. 8 . The corresponding results for the case where

*f*

_{1}= 9 and

*f*

_{2}= 12 GHz are illustrated in Figs. 9 -11 .

A very good level of agreement between the theoretical results and those rendered by the measurements can be observed, which consequently confirms the validation of the model developed in section 2. Some comments are applicable independently of the spectral separation between the evaluated electrical signals. At a first glance, we observe the expected power dip/phase shift sharp transition of around 140°, 150° when filtering the red-shifted frequency sideband in the fundamental [8,14,15,17,23,24] components, (see Figs. 6 and 9). In the contrary, a smooth power/phase-shift behavior is obtained for all the 2* ^{nd}* order terms, (see Figs. 8 and 11), with a maximum phase-shift level of around 65°, typical of the situation where no optical filtering stage is introduced after SOA propagation. It gains peculiar interest the comportment of the IMD

_{3}contributions placed at 2Ω

_{1}+ Ω

_{2}and 2Ω

_{2}+ Ω

_{1}, (see Figs. 7 and 10 ), since their phase shift response tends to follow the sharp excursion characteristic of the fundamental signals, although no optical filtering has been implemented at their locations.

If we focus on the comparison between the scenarios for Δ*f* = 0.5 and 3 GHz, we clearly appreciate the effect of the FBG response on the IMD_{3} terms placed at 2Ω_{1}-Ω_{2} and 2Ω_{2}-Ω_{1}. For Δ*f* = 0.5 GHz, the high level of attenuation produced at these frequencies (around 33 dB) results in a power notch similar to that occurred for the modulating signals connected to an even sharper phase-shifter excursion. In the other hand, when Δ*f* = 3 GHz, the photodetected power for these IMD_{3} terms experiences a visible smoothed behavior correlated with a reduction, (although still considerable if comparing with HD_{2} and IMD_{2}), of the phase shift level from approximately 160° to 120°. Here, the attenuation suffered from these tones has been reduced to values of around 3 and 5 dB. For the above observations, we can state than the 3* ^{rd}* order intermodulation products are characterized for a phase shift excursion of at least 100° although no optical filtering has been applied above them.

It becomes also remarkable when comparing both cases, that more noisily phase shift data points have been measured for Δ*f* = 3 GHz. This is a consequence of some thermal drifts experienced in the FBG response which leads to an appreciable fluctuation in the level of attenuation/phase shift applied by the filter. These fluctuations are more perceptible for the major separation Δ*f*, as in this case the fundamentals and, above all, both products at 2Ω_{1}-Ω_{2} and 2Ω_{2}-Ω_{1}, are located in the slopes of the FBG notch response (see Fig. 3). Since at those locations the red shifted optical field components are only attenuated by 3-5 dB, the variations in the phase shift applied by the FBG become more sensible to possible optical wavelength drifts.

Finally it is worth mentioning that a similar behavior can be achieved by tuning the electrical injection current instead of varying the input optical power [16].

#### 3.3. Theoretical evaluation of the MWP phase shifter frequency response

With the purpose of evaluating the influence of the nonlinear distortion in the frequency response of every spectral contribution taken into account in the previous section, simulations for different values of the common electrical modulation index m have been performed, comparing both the small and large signal operation conditions. It should be clear however that the large signal regime although interesting for other applications is discarded for linear phase shifting. Same values chosen for the proper adjustment of the theoretical and experimental results carried out in the previous sections have been now applied for the modeling of the EOM as well as the phase shifter comprised by the SOA and the FBG. The frequency response of the photodetected RF power from every fundamental tone, HD_{2}, IMD_{2} and IMD_{3} are plotted in Fig. 12
while the corresponding microwave phase shifts have been plotted in Fig. 13
. For the correct interpretation of the results it must be taken into account that every photodetected power and microwave phase shift has been plotted in terms of a common frequency *f* which has been chosen such that *f*
_{1}
*= f*. This is a generalized practice in the evaluation of the amplitudes and electrical powers of nonlinear RF terms.

A different spectral behavior can be generally appreciated when comparing the fundamental photodetected signal with the rest of oscillating terms. When filtering the red-shifted frequency component, a maximum phase shift transition of around 150 degrees is obtained as expected for the tones at Ω_{1} and Ω_{2}. This 150° phase transition can be physically explained as follows. The CPO effect occurs mainly at frequencies below 1/*τ _{s}* while for higher frequencies the SOA pure amplification phenomenon dominates. The power dip occurs thus around 1/

*τ*where the signal is suffering a transition between these two regimes, being the one due to amplification in phase with the incident beat-note, while the CPO effect is in anti phase.

_{s}However, in the case of HD and IMD_{2} they mainly experience a low-pass frequency response (with the exception of the case of operation under very large signal input (*m* = 0.5)), with no sharp transition in their respective phase-shifts, in concordance with the results presented in function of the SOA input optical power. In this context, it is worth highlighting, as it was performed in the previous subsection, the sharp phase shift transition that characterizes the components at 2Ω_{1} + Ω_{2} and 2Ω_{2} + Ω_{1}, which reaches a remarkable maximum value of around 280° and 190° respectively. It must be noted that we have also simulated the same scenarios in the absence of selective optical filtering (operation as a true time delay), confirming as in [22] (and similar to the performance observed in the previous subsection), that the 3* ^{rd}* order IMD frequency response is characterized by the same power dip/phase transition observed for the fundamental tones.

The case of very high input signal (*m* = 0.5), although, in principle, of no practical interest shows, some distinctive features. As expected, the microwave phase shift performance is degraded since for very high *m* values there exists a nonegligible interaction between the fundamental and the different nonlinear contributions within the SOA, which consequently leads to important power and phase fluctuations. Particularly for Fig. 13 we can observe a remarkable change in the behaviour of the spectral of the 2* ^{nd}* order contributions, especially in ${\varphi}_{(0,2)}$, ${\varphi}_{(2,0)}$ and ${\varphi}_{(-1,1)}$, with the presence of a power dip/

*π*-phase transition (characteristic of the fundamental tones) which can be explained as follows. The photodected power at these frequencies results from the mixing between the different optical field contributions as described in the definition of the optical intensity

*S*

_{(}

_{m}_{1,}

_{m}_{2)}at Eq. (5). At low modulation indices, the main contribution in the double summation for

*S*

_{(}

_{m}_{1,}

_{m}_{2)}is due to the product between the DC and the respective

*E*

_{(}

_{m}_{1,}

_{m}_{2)}components; while for large signal condition, the energy transfer between the optical fields within the SOA leads to different dominant contributions in the wave mixing experienced in

*S*

_{(}

_{m}_{1,}

_{m}_{2)}. As a consequence, the beat-tones where the fundamental tone is present become more relevant and the response of the distortion terms in question tends to follow the response of the fundamental one.

Finally, we should stress again, despite the particular characteristic of the above results for large RF signal transmission, that if we are interested in the application of the SOA device as a MWP linear phase shifting element, the device must be operated under small signal regime. The minor phase shift degradation in this case can even, if required, be counteracted by properly filtering the high-order harmonics and intermodulation products either at the SOA input (band pass filter) or after SOA propagation modifying the selective notch filter that is required for assuring the MWP phase shifter functionality [24].

## 4. Conclusion

In summary, we have presented a theoretical and experimental evaluation of the propagation, generation and amplification of signal, harmonic and intermodulation distortion terms in a Slow Light Microwave Photonic phase shifter, comprised of an electrooptic modulator and a Semiconductor Optical Amplifier under Coherent Population Oscillation regime followed by an optical filter. With that objective, we have described a general optical field based model applicable to any number *N* of arbitrarily spaced modulating tones that accounts for nonlinear distortion up to order *M*. This model supplements another recently published for true time delay operation based on the propagation of optical intensities and widens the range of applicability of other optical field propagation methods previously available.

The performance of the microwave phase shifter has been evaluated for up to 3* ^{rd}* order distortion when the modulating signal is composed of two RF tones, distinguishing two different spectral separations between them, namely 0.5 and 3 GHz, with the purpose of covering different MWP applications. We have evaluated the influence of the magnitude and phase shift responses of the optical filter implemented by a Fiber Bragg Grating (FBG) operating in transmission on the overall performance of the phase shifter. The dynamic range analysis has shown the feasibility of obtaining SFDR values limited by third order nonlinearities in the range of 100 dB/Hz

^{2/3}. It has also confirmed that CPO based phase shifters are mainly limited by second order nonlinear distortion which can, nevertheless be overcome by pacing all the relevant input signals within an octave in the RF spectrum. The evaluation of the phase shifter operation as a function of the SOA input optical power has shown that the intermodulation terms placed at 2Ω

_{1}-Ω

_{2}and 2Ω

_{2}-Ω

_{1}experience a power dip/phase excursion similar to that experienced by fundamental signal that is reduced when the spectral separation is increased (as expected due to the implied filter attenuation reduction). On the contrary, the intermodulation terms at frequencies 2Ω

_{1}+ Ω

_{2}and 2Ω

_{2}+ Ω

_{1}, although not suffering any optical attenuation after SOA propagation, experience also the phase excursion characteristic of the microwave phase shifter operation, as it is corroborated by the performed spectral analysis.

## Acknowledgements

The authors wish to acknowledge the financial support of the European Commission Seventh Framework Programme (FP 7) project GOSPEL; the Generalitat Valenciana through the Microwave Photonics research Excellency award programme GVA PROMETEO 2008/092 and also the Plan Nacional I + D TEC2007-68065-C03-01.

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