Arab J Sci Eng (2014) 39:557–561DOI 10.1007/s13369-013-0868-7

RESEARCH ARTICLE - PHYSICS

Analytical Model of Four-Wave Mixing in SemiconductorOptical Amplifier

Sadiq Jaafar Kadhim · Ali Hadi Hassan ·Hassan Abid Yasser

Received: 16 March 2012 / Accepted: 10 August 2012 / Published online: 12 November 2013© King Fahd University of Petroleum and Minerals 2013

Abstract The four-wave mixing process in pump-probeconfiguration requires the solutions of propagation equationsof the three pulses: pump, probe and conjugate simultane-ously. Since the pump power is much larger than the otherpulses, this makes the differential equation of pump not cou-pled with the other equations, hence it can be solved alone. Inthis paper, we consider that the differential equation of pumpis linear by assuming that the amplifier is divided into smallsections to be the power change through each section is toosmall. Solving the resulting pump equation requires calculat-ing the integrated gain, where the integrated gain is calculatedafter each section numerically by an iterative equation thatwas derived. Simultaneously, the solution of the probe andconjugate equations were also calculated after each section,where a simple recursive formula was found. Finally, theconversion efficiency will be calculated, where the presentanalysis agrees very well with the numerical solution.

Keywords SOA · FWM · Conversion efficiency

S. J. Kadhim · H. A. Yasser (B)Physics Department, Science College, Thi-Qar University,Nasiriyah, Iraqe-mail: hahmha@yahoo.com

S. J. Kadhime-mail: sadiqKadhim@yahoo.com

A. H. HassanPhysics Department, Education College, Al-Mustansyria University,Baghdad, Iraqe-mail: d.alihadi@yahoo.com

1 Introduction

Four-wave mixing (FWM) is a third-order nonlinear processin which a polarization is created in a medium that dependson the product of three electric fields. The induced polariza-tion leads to the creation of new frequency components of theelectric field. In a semiconductor, the nonlinear polarizationcan be mediated by resonant interactions of the electric fieldswith the carriers in the medium [1,2]. Since resonant inter-actions lead to attenuation of light, the nonlinear productsthat are created have low intensities. One way to circum-vent this problem is to use population inversion to create again in the medium [3–5]. A semiconductor optical ampli-fier (SOA) can provide high gain (typically ∼30 dB) whichthen amplifies the nonlinear mixing products that are created.FWM in SOAs has been studied extensively for wavelengthconversion [6–8] and it has the advantage of being transpar-ent to bit format. For detuning frequencies less than a fewgigahertz, the largest contribution to the FWM susceptibil-ity is due to carrier density modulation (CDM) which arisesfrom the beating of the pump and probe waves [9,10]. The

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intensity beating due to the pump and the probe lead to apulsation of the population inversion in the medium. Thispulsation of the total carriers leads to a gain modulation asseen by the traveling waves, which gives rise to the FWMsidebands. Due to the slow recovery of the carrier density,determined by the carrier lifetime, which is on the order ofseveral hundred picoseconds, the efficiency of FWM medi-ated by CDM drops off for frequency detuning much largerthan 10 GHz [11]. At large detuning frequencies, the gainand index gratings are formed by intraband processes, suchas carrier heating (CH) and spectral hole burning (SHB) [12].

In this paper, we present a theoretical analysis in which thecoupled-amplitude equations are solved. The output pumpsignal is calculated using the numerical solution of the inte-grated gain equation. Subsequently, an implicit recursiverelation is obtained for the probe and conjugate amplitude,where the derivation is carried on a single section and thenthe result will be extended to any number of concatenationsections.

2 Pump-Probe Configuration

When two optical signals, a pump signal and a probe sig-nal, with frequency wp and wq are injected into the SOA, anew frequency component, the FWM signal with frequencywc = 2wp − wq will be generated at the output of the SOAdue to the nonlinearity of the SOA. To simplify the analy-sis, we adopt a phenomenological model to account for thecontribution of various nonlinear mechanisms to the FWMprocess. A detailed quantum-mechanical analysis can be usedfor calculating the SHB and CH effects but requires complexformalism [13]. The quasi-steady-state evolution of pump,probe, and conjugate wave amplitudes is given by the cou-pled equations [14]

dAp

dz= (a + ib)Ap (1)

dAq

dz= (a+ib)Aq + (c1 + id1)|Ap|2 Aq + (c1 + id1)A2

p A∗c

(2)dAc

dz= (a + ib)Ac+(c2 + id2)|Ap|2 Ac + (c2 + id2)A2

p A∗q

(3)

where

c1 = − g

2

∑

m=cdp,shb,ch

εm(1 − �βmτm)

1 + �2τ 2m

c2 = − g

2

∑

m=cdp,shb,ch

εm(1 + �βmτm)

1 + �2τ 2m

d1 = g

2

∑

m=cdp,shb,ch

εm(�τm + βm)

1 + �2τ 2m

d2 = g

2

∑

m=cdp,shb,ch

εm(�τm − βm)

1 + �2τ 2m

g = go

1 + P/Psat

εcdp = 1

Psat + P

τcdp = τc

1 + P/Psat

ε = εshb + εch

P =∑

i=p,q,c

|Ai|2 ≈ ∣∣Ap∣∣2

a = g

2(1 − ε P)

b = − g

2(βcdp − βshbεshb P − βchεch P)

A j , j = p, q, c are the pulses amplitude, τc is the carrierlifetime, Psat is the saturation power, g is the saturation gain,go = �ao N0(I/I0−1) is the small signal gain, � is the modeconfinement factor, ao is the material parameter (referred toas gain cross-section), N0 is the transparency carrier density,Io = qV N0/τco is the current required for transparency, τc0

is the carrier lifetime at transparency, V is the active vol-ume, q is the electronic charge, and � = wq − wp is thefrequency detuning. The parameter βx , x = cdp, shb, ch rep-resents the linewidth enhancement factors of the processesCDP, SHB, CH. The parameters εx and τx , x = shb, ch, arethe gain compression factors and the characteristic times ofthe respective process. Note that, the following approxima-tion are carried through the derivation of the above equations.First, the phase-match condition, �k = 2kp − kq − kc ≈ 0,is satisfied due to short device length and low dispersion ofthe SOA. Second, higher-order pump power is much strongerthan the input signal power. Third, the attenuation factor, α,is very small and may be neglected without any importanteffects on the resulted pulses.

Remember that the parameters εshb and εch are a real char-acteristics of the SOA device, but εcdp is not, which has differ-ent values in different distances due to the variation of P withdistance. That is εcdp will decrease along SOA and the samebehavior is hold for the parameter τcdp. Equations (1)–(3)describe how complex amplitude of pump, probe and conju-gate evolves along the amplifier by including the effects ofboth interband and intraband effects, where the former con-tribution becomes negligible for �τcdp >> 1 since carriers

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cannot respond fast enough to changes in the amplitudes if� exceeds much more than τ−1

cdp.

3 Analytical Analysis

Here, we will adopt the idea of dividing the SOA toNconcatenation sections, the coupled-propagation equationsare solved on one section and then the result is generalizedof any number of sections. The power P ≈ |Ap|2 can bedifferentiated as

d P

dz= Ap

dA∗p

dz+ A∗

pdAp

dz(4)

Using Eq. (1) and its complex conjugate into Eq. (4), applyingthe Taylor expansion, 1/(1 + ε P) ≈ 1 − ε P for ε P << 1,and rearrange the result to deduce

d P

dz= g(1 − ε P)P (5)

This equation has the solution

P(z, τ ) = Pp(0,τ )eh(τ ) (6)

where Pp(0, τ ) = Poe−τ 2/τ 2o (τo is the initial pulse width)

represents the initial pump power for a Gaussian pulse andh(τ ) is the integrated gain of a SOA within L length, whichdefines as

h(τ ) =L∫

0

g

1 + ε Pdz (7)

This equation must be solved numerically using the equa-tion [15]

dh

dτ= 1

1 + ε Poeh

[ho

τc− h

τc− (eh − 1)λ

](8)

where λ = ε d Podτ

+ ε Poτc

+ Poτc Psat

For N -sections, Eqs. (6) and (8) may be rewritten as

dhm

dτ= 1

1 + ε Pm−1ehm

[ho

τc− hm

τc− (ehm − 1)λm

](9a)

Pm = P(m�z, τ ) = Pm−1ehm = P((m − 1)�z, τ )ehm

(9b)

λm = εd Pm−1

dτ+ ε Pm−1

τc+ Pm−1

τc Psat(9c)

where ho = goL/N and m = 1, 2, ...., N . Equation (9b)means that, the power and integrated gain are calculated in arecursive manner.

Now, the segment must be short enough so that variationsin the incident optical power are small over the length ofsegment. As a consequence, Eq. (1) may be solved as

Ap(z, τ ) = Ap(0, τ )e(a+ib)z (10)

where 0 ≤ z ≤ �z, for the first section. Using Eq. (10) into(2) and (3), making the transformations

A j = B j exp[(a + ib)z + (ck + idk)Poe2az/2a] (11)

where k = 1, 2 for j = q, c, respectively, and simplified theresult, yields

dBq

dz= (c1 + id1) exp[(−c − id)Poe2az/2a]e2az PoC∗

c

(12)

dBc

dz= (c2 + id2) exp[(c − id)Poe2az/2a]e2az PoC∗

q

(13)

where c = c1 −c2 and d = d1 +d2. Using the Taylor expan-sion e2az ≈ 1 + 2az into Eqs. (12) and (13), and simplifiedthe result, one may be obtained

dBq

dz= M1eR1z Po B∗

c (14)

dBc

dz= M2eR2z Po B∗

q (15)

where

M1 = (c1 + id1)e−(c+id)Po/2a

M2 = (c2 + id2)e(c−id)Po/2a

R1 = 2a − (c + id)Po

R2 = 2a + (c − id)Po

Differentiating Eq. (14) and the complex conjugate of (15),and rearrange the result, we easily deduced

d2 Bq

dz2 − R1dBq

dz− Me4az Bq = 0 (16)

d2 B∗c

dz2 − R∗2

dB∗c

dz− Me4az B∗

c = 0 (17)

where M = M1 M∗2 .

4 Recursive Formula

It is important to note that: Eqs. (16) and (17) are a nonlineardifferential equations that cannot be solved in a closed formbecause of the factor e4az . However, since our calculation isbased on dividing the SOA into many concatenation sectionswith very small length, such that e4az ≈ 1. Moreover, we

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560 Arab J Sci Eng (2014) 39:557–561

are assumed that the coefficients R1 andR2 are constantsdue to very small variations with each SOA segment. Theinitial conditions for solving this equation are B j (0, τ ) anddB j/dz|z=0 where j = q, c. The general solutions of Eqs.(16) and (17) may be deduced for any number of sections asfollows

Bq((m + 1)�z, τ ) = k1 Bq(m�z, τ ) + k2 B∗c (m�z, τ )

(18)

B∗c ((m + 1)�z, τ ) = k3 Bq(m�z, τ ) + k4 B∗

c (m�z, τ )

(19)

where

k1 = eR1�z/2[

cosh(g1�z) − R1

2g1sinh(g1�z)

]

k4 = eR2�z/2[

cosh(g2�z) − R∗2

2g2sinh(g2�z)

]

k2 = M1eR2�z/2 sinh(g1�z)

g1

k3 = M∗2 eR∗

2�z/2 sinh(g2�z)

g2

g1 =√

R21

4+ M

g2 =√

(R∗2)2

4+ M

and m = 0, 1, 2, ....., N − 1.Now, the original amplitudes, i.e., Aq and Ac, may be

restored by applying the inverse transformations

B j = exp

[−(a + ib)z − ck + idk

2aPoe2az

]A j

where k = 1, 2 for j = q, c, respectively. Accordingly, theamplitudes presented in Eqs. (18) and (19) will be replacedby following

Bq((m + 1)�z, τ ) = exp[−χ1 − η1e2a�z/2a]Aq((m + 1)�z, τ )

B∗c ((m + 1)�z, τ ) = exp[−χ∗

1 − η2e2a�z/2a]A∗

c((m + 1)�z, τ )

Bq(m�z, τ ) = exp[−χ2 − η1/2a]Aq(m�z, τ )

B∗c (m�z, τ ) = exp[−χ∗

2 − η2/2a]A∗c(m�z, τ )

where

η1 = (c1 + id1)Pme2am�z

η2 = (c2 − id2)Pme2am�z

χ1 = (a + ib)(m + 1)�z

χ2 = (a − ib)m�z

Using the last equations into Eqs. (18) and (19), we will getthe final recursive formula as[

Aq(L , τ )

A∗c(L , τ )

]= TN TN−1 . . . . . . . . . T3T2T1

[Aq(0, τ )

A∗c(0, τ )

]

(20)

where Tm represents the transfer matrix of the m segment,which is defined as

Tm =⎡

⎣k(m)

1 k(m)2

k(m)3 k(m)

4

⎤

⎦ (21)

and

k(m)1 = k1 exp[(a + ib)�z − η1(1 − e2a�z)/2a]

k(m)4 = k4 exp[(a − ib)�z − η2(1 − e2a�z)/2a]

k(m)2 = k2 exp[(a + ib + 2ibm)�z − (η2 − η1e2a�z)/2a]

k(m)3 = k3 exp[(a − ib − 2ibm)�z − (η1 − η2e2a�z)/2a]

Consequently, the initial conditions Aq(0, τ ) = √Pqe−τ 2/τ 2

o

and A∗c(0, τ ) = 0 are enough to construct the resulted signals

at the output of SOA. Equations (9) and (20) are our mainresults. They describe in recursive manner how the complexamplitudes of pump, probe and conjugate evolves along theamplifier by including the effects of both interband and intra-band effects.

5 Results and Discussion

The SOA will be divided into 100 sections, each of them withlength L/100. Using Eq. (9) will determine the output pumppower after each section, which can be used in turn to deter-mine the amplitudes Aq and A∗

c through the use of recursiveformula presented in Eq. (20). After that, the probe and conju-gate powers are calculated, where the device parameters usedin the simulation are listed in Table 1. The parameters: A,B,Cand No do not appear explicitly in the equations derived, butthey are required to find Io and go. Figure 1 illustrates theconversion efficiency as a function of frequency detuning.Where, the efficiency is only calculated for detuning signif-icantly larger than the pulse spectral width to prevent theinterference of pulses spectrum. Also, the above expressions

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Arab J Sci Eng (2014) 39:557–561 561

Table 1 Typical parameters value of I nGa As P around 1,550 nm [14]

Symbol Value Symbol Value

� 0.4 εshb 0.91 W−1

ao 10.5 × 10−20 m2 εch 1.62 W−1

I 40 mA τshb 0.036 ps

V 2.83 × 10−16 m3 τch 0.52 ps

Psat 10 mW βcdp 10

Pp 0.2 mW βshb 0.21

Pq 0.02 mW βch 2.81

L 0.5 mm A 3.9 × 109 s−1

No 1.6355 × 1023 m−3 B 8.5 × 10−16 m3/s

τc 250 ps C 3.3 × 10−40 m6/s

Fig. 1 Conversion efficiency as a function of frequency detuning

are valid when phase matching between the wave-vectors ofthe different frequencies is satisfied. For broadband wave-length conversion (detuning in the terahertz regime), phasematching is not necessarily satisfied and has to be includedin the expressions derived above. However, the relationshipbetween the conversion efficiency and frequency detuningand, therefore, the interpretation of asymmetric gain are wellknown in the scientific references [1,2,7,15]. Here, our goalhere is the comparison between the proposed solution and theactual solution (numerical), where Fig. 1 shows an excellentfit for all values of frequency detuning whereas the error ratedoes not exceed 1 %.

6 Conclusions

In conclusion, the coupled propagation equations in SOA wassolved. A simple recursive formula is presented to calculatethe output power of pump, probe and conjugate signals. Thenumerical results show that our formula agrees well with thenumerical solution.

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