## Abstract

In this paper we propose and evaluate the optical mixing of RF signals by means of exploiting the nonlinearity of a SLA modulator. The results show the potential for devices with low conversion losses (and even gain) and polarization insensitivity and reduced insertion losses.

© 2002 Optical Society of America

## 1. Introduction

The semiconductor laser amplifier (SLA) is a versatile device that has found multiple applications in the fields of telecommunications and optical signal processing [1]. Especially interesting are those pertaining to the later class, where the fast dynamics and highly nonlinear properties of the devices are exploited to implement cross gain, cross phase and four wave mixing based wavelength converters.

A novel application which has not been considered in the literature to the best of our knowledge is related to its application as a microwave signal mixer. Several authors have proposed and demonstrated the implementation of this function by means of electrooptic [2–4] and electroabsorption [5] modulators. In both cases flexible broadband mixers can be obtained however with high conversion losses, ranging between -60 and -20 dB due to the passive nature of the mixing process, and in the case of electrooptic mixers, their sensitivity to signal polarization. To overcome these limitations in the above configurations high RF input powers may be required [6]. An alternative that is currently under investigation is use of active semiconductor optoelectronic devices such as phototransistors [7] and as proposed here, the use of highly nonlinear SLAs modulators.

The principle of operation and is illustrated in figure 1. In this case, the SLA is employed in a non linear modulator configuration [8]. In this paper we propose and evaluate the mixing of RF signals by means of exploiting the non linearity of an SLA modulator. The results show that potential high levels of conversion efficiency are attainable with modest modulation indexes.

## 2. SLA Modulator propagation and rate equations

For the sake of consistency we follow the model for the SLA modulator provided by
Mork and co-workers [8] . The equations for the propagation of the photon density
*S*=*S*(*z*,*t*) and
the rate equation for the carrier density
*N*=*N*(*t*,*z*) are
given by:

$$\frac{\partial N}{\partial t}=\frac{I\left(t,z\right)}{eV}-\frac{N}{{\tau}_{s}}-{v}_{g}a\left(N-{N}_{o}\right)S$$

As usual, *z* represents the spatial coordinate along the SLA,
*t* is the local time measured in a coordinate system moving with
the group velocity *v _{g}*,

*I*(

*t*,

*z*) the modulation current,

*N*the carrier density at transparency,

_{o}*V*the active region volume,

*τ*

_{s}the carrier lifetime, Γ the confinement factor, α

_{L}the internal loss in the SLA and

*a*the differential gain.

We aim to investigate the SLA RF mixing response to an input current modulation given by:

$$+\Delta I\left(z,{\Omega}_{S}\right){e}^{-j{\Omega}_{S}t}+\Delta {I}^{*}\left({\Omega}_{S}\right){e}^{j{\Omega}_{S}t}\phantom{\rule{6.em}{0ex}}$$

Where *I*̅ represents the constant DC current ,
∆*I*(Ω_{i}) i=LO,S stand for the modulation current amplitudes at Rf frequencies Ω_{i} , i=LO,s corresponding to the Local oscillator and the signal respectively
and ^{*} denotes the complex conjugate. If a small signal modulation
approach is considered taking into account up to the second order terms in
*S* and *N* then the following approximation can be
taken :

$$+\Delta Y\left(z,{\Omega}_{s}\right){e}^{-j{\Omega}_{s}t}+\Delta {Y}^{*}\left({\Omega}_{s}\right){e}^{j{\Omega}_{s}t}+\phantom{\rule{8.2em}{0ex}}$$

$$+\Delta Y\left(z,{2\Omega}_{\mathit{LO}}\right){e}^{-j{2\Omega}_{\mathit{LO}}t}+\Delta {Y}^{*}\left(z,2{\Omega}_{\mathit{LO}}\right){e}^{j{2\Omega}_{\mathit{LO}}t}+\phantom{\rule{3.7em}{0ex}}$$

$$\Delta Y\left(z,2{\Omega}_{s}\right){e}^{-j{2\Omega}_{s}t}+\Delta {Y}^{*}\left(2{\Omega}_{s}\right){e}^{j2{\Omega}_{s}t}+\phantom{\rule{7.3em}{0ex}}$$

$$\phantom{\rule{2.em}{0ex}}+\Delta Y\left(z,{\Omega}_{\mathit{LO}}-{\Omega}_{S}\right){e}^{-j\left({\Omega}_{\mathit{LO}}-{\Omega}_{s}\right)t}+\Delta {Y}^{*}\left(z,{\Omega}_{\mathit{LO}}-{\Omega}_{s}\right){e}^{j\left({\Omega}_{\mathit{LO}}-{\Omega}_{s}\right)t}+$$

$$\Delta Y\left(z,{\Omega}_{\mathit{LO}}+{\Omega}_{s}\right){e}^{-j\left({\Omega}_{\mathit{LO}}+{\Omega}_{s}\right)t}+\Delta {Y}^{*}\left(z,{\Omega}_{\mathit{LO}}+{\Omega}_{s}\right){e}^{j\left({\Omega}_{\mathit{LO}}+{\Omega}_{s}\right)t}$$

Where *Y*(*z*, *t*) represents either
*S*(*t*,*z*) or
*N*(*t*,*z*) as required.
Substituting (3) and (2) into (1) and retaining up to second order terms yields the
following set of differential equations:

$$\phantom{\rule{6.2em}{0ex}}+\left(\frac{{g}_{\mathit{sat}}-\beta \left(2{\Omega}_{i}\right)}{{g}_{\mathit{sat}}}\right)\left\{\begin{array}{c}\Gamma a{\tau}_{S}\Delta S\left({\Omega}_{i}\right)Q\left({\Omega}_{i}\right)x\phantom{\rule{3.5em}{0ex}}\\ x\left[\frac{\Delta I\left({\Omega}_{i}\right)}{eV}-\frac{{g}_{\mathit{sat}}{v}_{g}}{\Gamma}\Delta S\left({\Omega}_{i}\right)\right]\end{array}\right\}$$

$$\Gamma a{\tau}_{S}\left(\frac{{g}_{\mathit{sat}}-\beta \left({\Omega}_{\mathit{LO}}\pm {\Omega}_{S}\right)}{{g}_{\mathit{sat}}}\right)\phantom{\rule{12.em}{0ex}}$$

$$\phantom{\rule{3.em}{0ex}}\left\{\begin{array}{c}\Delta S\left({\Omega}_{S}\right)Q\left({\Omega}_{\mathit{LO}}\right)\left(\frac{\Delta I\left({\Omega}_{\mathit{LO}}\right)}{eV}-\frac{{g}_{\mathit{sat}}{v}_{g}\Delta S\left({\Omega}_{\mathit{LO}}\right)}{\Gamma}\right)\\ +\Delta S\left({\Omega}_{\mathit{LO}}\right){Q}^{(*)}\left({\Omega}_{S}\right)\left(\frac{\Delta I\left({\Omega}_{S}\right)}{eV}-\frac{{g}_{\mathit{sat}}{v}_{g}\Delta {S}^{(*)}\left({\Omega}_{S}\right)}{\Gamma}\right)\end{array}\right\}$$

In the above i=LO, s, *g _{sat}* =
Γ

*g*/(1 +

_{o}*S*̅/

*S*) is the saturated gain,

_{sat}*S*=1 /(

_{sat}*av*) = Γ

_{g}τ_{s}*P*/(

_{sat}*ħwAv*) the saturation photon density with P

_{g}_{sat}being the SLA output saturation power and

*A*the active zone cross area,

*g*=

_{o}*a*(

*I*̅

*τ*/(

_{s}*eV*) -

*N*) the small signal material gain,

_{o}*β*(Ω) =

*g*(Ω)(

_{sat}Q*S*̅/

*S*) and finally,

_{sat}*Q*(Ω) = [-

*j*Ω

*τ*+ 1 + (

_{s}*S*̅/

*S*)]

_{sat}^{-1}. Equations (5–7) with the initial conditions

*∆S*(

*0,U*)=

*∆N*(

*0,U*)=0, for all

*U*values, being

*U*a frequency variable and

*S*(0,

*t*) =

*S*̅(0) =

*S*describe the process of second order signal mixing in a SLA modulator and have been tested by comparing with the direct numerical solution of (1)–(2). In all cases the agreement has been excellent. For example, figure 2 shows the results obtained for a L=5 mm SLA modulator with similar characteristics as those of [8]. The parameters employed in the calculations are: P

_{in}_{in}=1 mW, P

_{sat}=28.5 mW,

*τ*

_{s}=100 psec,

*a*=3.10

^{-20}m

^{2}, N

_{o}=1.1.10

^{24}m

^{-3}, Γ=0.3, A=0.2.10

_{-12}m

^{2}. Solid lines correspond to the results obtained by solving (4–7), whereas (

^{*}) curves represent the numerical solution of (1)–(2).The modulating RF carriers are separated by 100 MHz. The linear characteristic displays a resonant characteristic, the origin of which has been thoroughly explained [8].

Figure 2 has been plotted for a 100 MHz frequency difference between the two RF signals. Similar behavior and efficiency of the mixing process have been detected for differences ranging from 10 MHz to 10 GHz (the SOA modulator bandwidth). Note that, the frequency response of the nonlinear process depends on the modulation bandwidth of the SOA. Notice that the resonant behaviour is enhanced in second order nonlinear terms that correspond to the sum of frequencies while it is mitigated in the nonlinear term corresponding to the frequency difference. Furthermore, in this last case a notch characteristic is displayed. All second order terms are low pass with a roll-off characteristic showing a double slope than that corresponding to the linear term. We have also obtained results for shorter amplifiers L=1 mm, where the resonant behaviour is not present even in the linear term. In this cases all the second order show analogous low-pass characteristics. In any case it is worth pointing out that second order non linearities can show high values which may indicate that SLA modulators may find an interesting application in the field of microwave mixing.

## 3. Mixer performance

The mixing efficiency can be obtained as usual [2–5] by comparing the RF
powers of the input signal P_{RFin} and the output powers of the
up-converted P_{RF-up} and downconverted signal P_{RF-down}, taking
as a parameter the value of the DC RF power due to the LO.

These values are related to those obtained from (4–7) by the following expressions:

$${P}_{\mathit{RF}-\mathit{up}}=\frac{{\mid R\left(\mathit{\u0127}w{v}_{g}A\right)\Delta S\left({{\Omega}_{\mathit{LO}}+\Omega}_{s}\right)\mid}^{2}{Z}_{r}}{2}$$

$${P}_{\mathit{RF}-\mathit{down}}=\frac{{\mid R\left(\mathit{\u0127}w{v}_{g}A\right)\Delta S\left({{\Omega}_{\mathit{LO}}-\Omega}_{s}\right)\mid}^{2}{Z}_{r}}{2}$$

Where R is the detector responsivity and Z_{r}, Z_{m} represent the
impedances of the receiver and the SOA modulator respectively, which we will assume
equal (Z_{r}=Z_{m}=50 Ω).

To gain insight on the mixing potential of SLA modulators we have evaluated the
conversion efficiency for the upconversion term (P_{RF-up} /
P_{RFin}) for a device with P_{in}=0.5 mW, P_{sat}=28.5 mW,
*τ*
_{s}=100 psec,
*a*=3.10^{-20} m^{2},
N_{o}=1.1.10^{24} m^{-3}, L=1mm, Γ=0.45,
A=0.2.10^{-12} m^{2}. Other relevant parameters are the
unsaturated gain Go=23.45dB; the saturated gain G(Pin=0.5mW)=18.33 dB,
*I*̅ = 450*mA* ,. The frequencies of the RF
signal and the LO are 3.5 and 3.49 GHz, resulting in an up-converted frequency of
6.99 GHz. Similar results have been obtained for RF and LO signal frequencies near
15 and 20 GHZ.

Figure 3 shows the results for P_{RF-up} vs the value
of P_{RFin} both expressed in dBm,. Taking the LO Rf power as a parameter.
For moderate LO powers (12 dBm) the obtained conversion loss is 6 dB which is far
smaller than the values reported for electroabsorption modulators^{5} (19.8
dB for P_{LO}=10 dBm), single LiNbO_{3} MZ modulators ^{9}
(15.8dB for P_{LO}=30 dBm), and standard microwave technology mixers (21.95
dB. For P_{LO}= 16 dBm). Furthermore, for LO powers comparable to the two
later devices, the SOA based mixer produces almost no conversion loss
(P_{LO}=16 dBm) and even conversion gain (6 dB for P_{LO}=24 dBm).
Thus the proposed device can be advantageous in terms of this parameter in
comparison with other technology options.

An issue of importance when dealing with SOA based devices attains to their noise
performance. In our case, the limiting noise source has been found to be that caused
by the beating between the signal and spontaneous noise and its value has also been
plotted in figure 3. Compared to other options, the noise floor as
expected is higher, however since the conversion losses are lower, there is enough
margin to yet attain a considerable SNR value for the converted signal. The reader
must note that total noise is almost constant with ∆P_{RFin}
since the signal spontaneous beating noise is mainly due the average optical power
and not to the modulation RF sidebands.

The results shown are only preliminary, but outline the potential for devices with low conversion losses (and even gain) and polarization insensitivity and reduced insertion losses. It should be pointed out that this analysis has only considered SOAs with standard parameters. Enhanced performance can be expected with special designs incorporating longer active lengths and higher gain and also, with tandem SOA configurations. This is currently under investigation. To the knowledge of the authors, this is the first time this device has been proposed for this purpose and it can find applications in microwave and millimeter wave fiber-radio and fiber wireless systems, where flexible broadband RF mixing with low loss or even gain is required.

## 4. Summary and Conclusions

In summary, we have proposed for the first time to our knowledge and evaluated the use of a SLA modulator for the mixing of RF signals directly in the optical domain by means of exploiting the nonlinearity of the modulator. The results show that broadband devices with potential low conversion losses (and even gain), polarization insensitivity and low insertion losses can be expected with moderate LO Rf powers.

## Acknowledgements

The authors wish to acknowledge the financial support of the Spanish CICYT via projects TEL 99–0437 and TIC 98–0346 and the European Union COST 267 Action. Useful comments from Dr. José Miguel Fuster are also acknowledged.

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