Theory of pulse propagation and four-wave mixing in a quantum dot semiconductor optical amplifier

Accepted Manuscript Theory of pulse propagation and four-wave mixing in a quantum dot semiconductor optical amplifier Ahmed H. Flayyih , Amin H. Al-Khursan PII:

S1567-1739(14)00131-X

DOI:

10.1016/j.cap.2014.04.014

Reference:

CAP 3627

To appear in:

Current Applied Physics

Received Date: 17 September 2013 Revised Date:

14 April 2014

Accepted Date: 29 April 2014

Please cite this article as: A.H. Flayyih, A.H. Al-Khursan, Theory of pulse propagation and four-wave mixing in a quantum dot semiconductor optical amplifier, Current Applied Physics (2014), doi: 10.1016/ j.cap.2014.04.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Theory of pulse propagation and four-wave mixing in a

Ahmed H. Flayyih, and Amin H. Al-Khursan*

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quantum dot semiconductor optical amplifier

Nassiriya Nanotechnology Research Laboratory (NNRL), Physics Department, Science College, Thi-Qar University, Nassiriya, Iraq.

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* Corresponding Author. E-mail: [email protected].

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Abstract

A theory, combining the relations of pulse traveling into quantum dot (QD) semiconductor optical amplifier (SOA) with the four-wave mixing (FWM) theory in these SOAs, is developed. Carrier density pulsation (CDP),

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carrier heating (CH), and spectral hole burning (SHB) contributions on FWM efficiency are discussed. Effect of QD ground state and wetting layer

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are included. An additional parameter appears in the gain integral relation of QD SOAs. An equation formulating pulses in the QD SOAs is introduced.

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We have found that FWM in QD SOAs is detuning and is pulse width dependent. For short pulses, CH is dominant at high detunings (10-100GHz) while at higher detunings (>100GHz) the SHB is the dominant one. Undesired paunch behavior is shown in QD SOAs then, CDP must be reduced.

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Keywords Quantum dot SOA, Four-wave mixing, Spectral hole burning.

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1. Introduction With the increasing of Internet traffic in networks, ultrafast optical networks becomes more demanding [1], [2]. This requires all-optical

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processing techniques to avoid electro-optical conversion that may create data-flow bottleneck. One of these techniques is four-wave mixing (FWM).

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It is a promising technique that can replace multi-wave converters by a single one [3]. It is typically realized in semiconductor optical amplifiers (SOAs) [4]. Low-dimensional SOA structures such as quantum dot (QD) in its active region gets considerable attention due to the possibilities offered

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by QDs. This includes: excellent controllability of intraband transitions which have been essential in optical devices, ultrafast response unlimited by

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carrier recombination lifetime [5]. In addition, the promising properties such as low threshold current, temperature insensitivity, high bandwidth, and

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low chirp [6]. All of these characteristics make QD SOAs a promising candidates for devices used in fast and all-optical manipulations [7]. FWM results from nonlinear interaction between two waves which are

different in frequency and intensity inside a semiconductor. The beating of two waves results in new waves from modulation of both gain and refractive

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index and a generation of diffraction grating [1]. The mechanisms that lead to FWM in semiconductors include carrier density pulsation (CDP), carrier

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heating (CH) and spectral hole burning (SHB) [8]-[10]. Since QD SOA active region has a totally quantized QDs grown on a two-dimensional wetting layer (WL). There are differences appearing in FWM processes in

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QD SOAs from that of bulk SOAs [11]. The SHB is governed by the carriercarrier scattering rate where the optical field digs a hole in the intraband

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carrier distribution due to stimulated emission [8]. Here in QD SOAs, to return to quasi-equilibrium in QDs, intersubband and interdot relaxations must occur. The relaxation from WL to QD is slow as a result of transition from two-dimensional WL to completely quantized QD states [12]. It is on

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the order of picoseconds [12], [13]. This is the well-known phonon bottleneck effect [14]-[16]. CDP is governed by the radiative recombination

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time which is on the order of nanosecond. It results due to beating between the pump and signal, which deplete carriers near the signal wavelength.

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Thus, reducing the overall gain spectrum [17]. CH is governed by carriercarrier and carrier-phonon scattering times [12], [17]. Since QDs show a reduced carrier density due to discrete energy subbands, thus, it is demonstrated experimentally that QDs have a reduced CH compared with bulk and quantum-wells [18].

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Several theoretical models and experimental works deal with FWM in QDs. Gain saturation mechanisms including SHB and CDP are modeled

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using the density matrix theory by shifting of the global quasi-Fermi level [17]. FWM in QDs is shown to be efficient on high-speed signals greater than 160 Gb/s [12], [18]. QDs show a negligible patterning effect at this rate

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[16]. It is demonstrated experimentally that wavelength conversion can be enhanced by cavity resonance [4]. The effect of the pulse on the FWM

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efficiency in QD SOAs does not take an enough length in researches. Thus, a detailed study combines the theory of pulse impinging on the QD SOA and its effects on the FWM in SOAs is required. This work combining FWM model presented in [12] with the pulse propagation effect in QD SOAs

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resulting in a new model formulating pulse effect on FWM in QD SOAs. It is organized as follows: in section 2, theory of pulse propagation in SOAs is

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discussed. Section 3 describes FWM relations in these SOAs. Relation of FWM efficiency in QD SOAs is states in section 4 while the QD SOA

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structure used in this study is described in section 5. The results are stated and discussed in section 6, while the conclusions drawn from this work are presented finally in section 7.

2. Theory of pulse propagation into QD SOAs

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The propagation of an electromagnetic field inside SOA is governed by the wave equation ε ∂2E c 2 ∂t 2

(1)

=0

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∇2E +

E (x, y, z, t) is the electric field vector of the wave, c is the light velocity and ε is the dielectric constant of the amplifier medium. The dielectric constant is

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given by [6]

( 2)

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ε = nb2 + χ ( N )

where the background refractive index nb is generally a function of the transverse coordinates x and y to account for the dielectric wave guiding in SOAs. The susceptibility represents the contribution of the charge carriers

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inside the active region of the amplifier and it is a function of the carrier density N. The optical gain (g) approximately varies as g ( N ) = Γa ( N − N 0 )

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where a is the differential gain constant, N is the injected carrier density and

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N 0 is the carrier density needed for transparency. Assuming that the input

light is linearly polarized during propagation, the electric field inside the amplifier can be written as [19]

1 E (x , y , z , t ) = εˆ  F (x , y ) A (z , t ) e i ( k 0 z −ω0t ) + c .c   2 

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(3)

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where εˆ is the polarization unit vector, F ( x, y ) is the waveguide-mode distribution, k 0 =

ω0 n c

, ω0 is the frequency of the emitted photon, n is the

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refractive index and A(z,t) is the slowly varying amplitude of the propagating wave. Using Eqs. (1) and (2), neglecting the second derivatives of A(z,t) with respect to t and z, and integrating over the transverse

dA 1 dA iωo Γ 1 + = χ A − α in A dz υ g dt 2nc 2

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2 d 2F d 2F 2 2 ωo   n n + + − b  c2 F = 0 dx 2 dy 2 

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dimensions x and y, one obtains [19]

(4)

(5)

Γ is the optical confinement factor, α int is the loss coefficient, υ g is the

given by [19]

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 ∂n  ng = n + ω0    ∂ω 

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group velocity (υ g = c / n g ) , while ng is the group refractive index and is

(6)

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The rate equations system for the two-level QD system, see Fig. 1, is

given by [12]

dN W L f N (1 − f ) N W L I = D − WL − + dt τe τc τs eV

(7)

df f 1 NWL (1 − f ) f =− + − − a (2 f − 1) S dt τe D τc τs

(8)

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f is the occupation probability of electrons in the ground state (GS) of the

dot, N W L is the WL carrier density. The electron emission and capture times

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from GS to WL are τ e , τ c , the spontaneous radiative lifetime is τ s , e is the electric charge, V is the active region of the volume, I is the injected current, D is total number of states in the dots which is determined from the areal

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density per dot layer, a is the differential gain and S is the photon density.

Combining Eqs. (7) and (8) with the definition of the optical gain,

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neglecting the spontaneous radiative term, one obtains dg g o − g = + κQD f τc dt

κQD =

D Γa

τD

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With

(9)

(10)

Where the SHB time constant for QDs [12] is −1

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1 N  τ D =  + WL   τe D τc 

The 2nd term in Eq. (9) represents the difference from that of the bulk

SOA. It is an entity of QD SOAs and as a rate results from the contribution of both WL and GS in the QDs. It recognizes the system equations of QD SOA from the bulk SOA. Eq. (9) shows the controllability of τ c on the gain

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change in QD SOAs. Here, the occupation probability of GS can be

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calculated from [20]

 1 g f =  + 1 2  g max 

(11)

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While [6]

(12 )

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I  g o = ΓaN o  − 1  Io 

I 0 is the current required for transparency and g 0 is the small-signal gain.

Eqs. (5) and (9) governing the pulse propagation in SOAs. They can be further simplified by making the following transformation z

υg

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τ =t −

(13)

the reduced time ( τ ) is measured in a reference frame moving with the

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using [19]

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pulse. It is also useful to separate the amplitude and phase of the pulse by

(14 )

A = Pe i ϕt

Where P(z,τ) and φ(z, τ) are the instantaneous power and the phase of

the propagating pulse, respectively. From the susceptibility and Eqs. (5) and (9), one obtains [19]

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(15)

d φ −1 = αg dz 2

(16 )

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dP 1 = ( g − α int )P dz 2

The evolution of the pulse inside the amplifier generally requires a numerical solution of Eqs.(15)-(16). However, if α int << g , it is possible to

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solve these equations in a closed form. This condition is often satisfied in

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practice. In the following, we set α int =0. Eqs. (15) and (16) are then readily integrated over the amplifier length to provide [19] Pout (τ ) = Pin (τ ) e h (τ ) 1 2

φout (τ ) = φin (τ ) − α h (τ )

(17 ) (18)

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where Pin (τ ) and φin (τ ) are the power and phase of the input pulse. The function h (τ ) is defined as [19] L

0

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h (τ ) = ∫ g (z ,τ ) dz

(19 )

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Physically, it represents the integrated gain at each point of the pulse profile. Integrating it over the amplifier length yields

 d h (τ ) g o L − h (τ ) κQD  h (τ ) = + +L  dτ 2  g max τc 

(20)

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This equation plays a milestone in formulating pulses inside QD SOA. It is the rate of the integral gain in QD SOAs.

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3. FWM pulses

If we have two injected pulses (assuming transform-limited Gaussian

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pulses) represented by [21]

 1  τ + φ 2  E 1 (τ ) = A1 exp  −     2  τ 1  

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 1  τ 2  E 0 (τ ) = A 0 exp  −  2    2  τ 0 

( 21)

( 22 )

where E0 is the pump signal at ω0 , E1 is the probe signal at ω1 , and φ is the

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delay time between the two pulses. The FWHM of pump and probe pulses are τ 0 and τ 1 , respectively. The optical field at the input facet (z = 0) is [21]

( 23)

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E ( z = 0,t ) = E 0 + E 1e i δ t

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Note that δ is the phase between E 0 and E 1 . Integrating Eq. (23) yields [21] E out (L ,τ ) = E in

  h   2 1 − i α ( )  (0,τ ) e 

( 24 )

The small signal analysis can be applied to Eqs. (23) and (24), this

treatment leads to [21] E0 ( L) = e

h(τ ) 2(1−iα )

( 25)

1 + F− (δ ) E1 2  E0  

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E2 ( L) = e

h(τ ) 2(1−iα )

1 + F+ (δ ) E0 2  E1  

( 26 )

F− (δ ) E02 E1*

( 27 )      

( 28)

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  h τ (1 − i α x ) κ x e ( ) −1  1− i α F± (δ ) = −C + ∑ 2 2  1 ± i δτ x E 1 + 0 ± i δτ x =SHB ,CH 2  E sat

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E1 ( L) = e

h(τ ) 2(1−iα )

2

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In the expressions above, the field intensity, E0 , is normalized to the 2 , C is the phenomenological parameter to amplifier saturation intensity Esat

compensate for the nonplanar nature of the waveguide [22]. Eq.(27) describes the wave mixing product, E2, whose frequency is ω2 = ω0 + δ , and

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τ is the gain recovery time. Other products are also created but all are much

smaller than E2 and therefore, they are subsequently neglected. The set of

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Eqs. (25)-(27) represents the relations of the three output fields to the two input fields. The nature of these relations is determined by the function

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F± (δ ) , Eq. (28), which contains all the physical details of the various

nonlinearities. In Eq. (28), the first term in the square brackets describes the wave mixing due to carrier density pulsations (CDP). The second term was added to account for the summation of three intraband processes. They are carrier heating of electrons (holes) in the conduction (valence) band, and spectral hole burning (SHB). Each process has a characteristic time constant, 11

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τ x , a corresponding nonlinear gain coefficient κ x , and a unique linewidth

enhancement factor, α x , associated with it. The formula that describes

( 29 )

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    hτ 1 − iα x ) κ x  ( e ( ) − 1  1 − iα N QD F± (δ ) = −C +∑ 2 2  1 ± iδτ x  E0 x ζ +  2   Esat

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travailing the electric field in QD SOA can be written as

where −1

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 D N  1− f 1  1 N   D N  1 1 − f 1 ζ = Dτ s  + w    + − iδ  + w + − iδ  −  + w  τs τe τc   τc  τ e Dτ c τ s   τ e τ c  D τ c

   

δ is the pump-probe detuning. The nonlinear gain coefficient for SHB is µk

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 2τ d  N w − 1  χ k (ω0 ) − χ k* (ω )  h  Dτ c  2   µ  2τ   ∑ k2  d N w − 1 ( χ k (ω ) − χ k* (ω ) )    h  Dτ c   

∑

χ k (ω ) 

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    2i ω0 τ d  κ SHB (1 − i α SHB ) =  dg      (c n )ε 0τ s dN    

(30)

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k

3

while that of CH is given by 3τ CH N ∆E h 2l

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κ CH =

( 31)

π τ 1r ( kβ T ) m* hc 3

∆E is the energy difference between the chemical potential, the energy

needed to add one electron to the continuum, and the energy of an electron in a QD GS. τ CH is the time at which the electron gas cools back to the

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lattice temperature, m ∗ is the effective mass for the electrons or holes and l is the effective height of the quantum-dot layer. hc is the heat capacity of

hc =

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the free electrons assuming a two-dimensional (2D) electron-gas model [12], π kβ2 T m*

( 32 )

3 h2l

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Inclusion of excited state (ES) is important in the study of QD phenomena. A lot of works deals with FWM in QDs including ES, see [18]

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for example. Note that, the interdot processes between the dot the WL reservoir are much slower than the intradot processes between GS and ES which are in the order of 100-250 fs. The slower processes results in deep

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spectral hole burning and then, efficient FWM in QDs [12]. Taking this fact into account, one can neglect ES effect at this stage. This is also helps as to make our results simpler. Including ES in the examination of pulse effect on

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near future.

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FWM in QDs is one of the main works must be done by our group in the

4. Wavelength conversion The conversion efficiency of SOA is the ratio between the power of the

converted signal at the device-output and the probe-power at the input [23]. In QD SOA, FWM efficiency is given by

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η =e

h (τ )

QD −

F

 E 02  (δ )  2   E sat 

( 33)

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5. The quantum dot structure used in this study The QD structure used in this theoretical study is a tenfold InAs QD layer with 5 × 1010 cm −2 dot density per layer grown by molecular beam

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epitaxy at NanoSemiconductor GmbH in Germany [17, 24]. Each QD array

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is covered with InGaAs WL and 33nm thick GaAs barrier layer. Each QD active layer is sandwiched between 1.5µ m thick AlGaAs cladding layer. Note that, the input peak power of all input pulses used is (1mW ) .

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6. Results and discussion

To check that the nonlinear behavior in QD SOAs is pulse width

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dependent, three pulse widths (1, 2, and 3ps) are studied and plotted in Fig. 2. With increasing pulse width, the shape becomes asymmetric due to the

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variation in the population of the electronic level. This is due to sharpening of the leading edge compared with the trailing edge [25]. Fig. 2 shows also an increasing output peak power when the input pulse width is reduced. This is also demonstrated earlier for bulk SOAs for pulses with short durations in the range of carrier lifetime [22]. Fig. 3 shows the reduction of gain integral with increasing pulse width. Gain curves are coincides till~2ps for all pulses 14

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used. GS occupation probability is shown in Fig. 4. An absorption appears during the pulse excitation, then it is recovered fast. Time recovery is very

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fast but it becomes longer with the increase of FWHM. This fast recovery can be attributed to the reduction in the WL carrier density, N WL , which is shown in Fig. 5. This behavior in QD SOAs is also seen earlier [26]. Here,

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N WL curves are reduced with increasing FWHM pulse. After a time of ~

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(1ps) the reduction becomes discriminated. For each curve, N WL is reduced, then it is clamped at some value. The reduction shown in Figs. 4 and 5 is related to passing the input pulse inside QD SOA, see Fig. 2. The parameter κQD was plotted in Fig. 6. Here, κQD is reduced with increasing pulse width.

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κQD changes few for wider pulses as can be seen for (6 and 9ps) pulses. Note

that the pulse widths studied are (3, 6, and 9 ps) to see the behavior of this parameter obviously.

parameter is specified at (τ = 8 ps ) for all pulse widths studied here.

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κQD

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FWM efficiency (η ) is calculated for different pulses using Eq. (33).

Then, the parameters required for calculation (h (τ ), f GS , N W L ) are specified from Figs. 3, 4, and 5, respectively. Other parameters like κ SHB and κCH are calculated from Eqs. (30) and (31). Then, κQD is calculated using these

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parameters. The specified parameters are listed in Table 1. Other parameters, used in the calculations, are set in Table 2.

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calculations FWM efficiency is plotted in Fig. 7 and compared with experimental measurements from [27] which is done for InAs/In0.17Ga0.83As QD SOA. A good agreement is obtained with our theoretical calculations.

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Fig. 8 (a) and (b) show FWM efficiency with its components (CDP, CH and SHB) for negative and positive detunings, respectively for 1ps pulse. CDP

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gives a higher contribution at low detuning (< 10GHz), then it falls, while both SHB and CH contributions become the dominant at higher detunings. This is with the evidenced result that the detuning of the pump and probe wavelengths relative to the amplifier gain peak plays a key role in setting the

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relative strength of the CH and SHB processes [22]. In Fig. 8 (b), a plateau behavior appears in the positive detuning (between ~10-100GHz) for total

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FWM curve which refers to the asymmetry of FWM branches. This is related to the inclusion of linewidth enhancement factor in the theory. This is

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a result of inhomogeneity which originates in QDs from their manufacture imperfections [7]. The effect of pulse width on FWM asymmetry is examined in Fig. 9, where the positive branch is shown lower than the negative one. This is with the recently observed experimental results for QD SOA [12] and also with theoretical one for bulk SOA [28]. When the input

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pulse becomes shorter (1ps), the difference between the two branches increases with a paunch that appears in the positive branch at high detuning

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between (10-100 GHz), then at higher detuning (>100GHz) it ascends above the negative branch. Fig. 10 shows a FWM efficiency when a SHB is neglected from FWM curve. For wider pulses, a paunch in the positive

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branch becomes big and shifts to a higher detuning, see vertical lines in Fig. 10. The paunch behavior is not seen in the behavior of bulk and QW SOAs.

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Fig. 11 shows FWM efficiency when CH is neglected. At high detuning, here a plateau behavior is seen but there is no paunch, while at higher detunings (>100GHz) the branches coincide. Fig. 12 shows FWM efficiency (CH and SHB components).

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From Figs. 10, 11, and 12 one can concludes that SHB contribution is reduced for shorter pulses at high detuning (10-100GHz) while CH

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dominates. After 100GHz, the SHB contribution is increased with reducing the pulse width, causing the ascending behavior of positive branch for

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shorter pulses. No paunch is seen in Fig. 12 (CH and SHB) while a small (and approximately constant) paunch is seen in Fig. 11 (CDP and SHB). Thus, the paunch is appear due to CDP while SHB reduces it. Since paunch behavior is not required (asymmetric FWM), thus it is preferable to reduce

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CDP but not neglect it. High FWM efficiency is obtained with CDP, compare Figs. 10, 11 with Fig. 12.

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7. Conclusions

This work discusses FWM in QD SOAs including its structure specialty

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where WL, GS, and ES are included in the formulation. This formulation is combined with the effect of pulse traveling in the QD SOA which results in

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a new theory in a pulse effect on FWM in QD SOAs.

It is found that FWM in QD SOAs is detuning and pulse width dependent. For short pulses, it is shown that carrier heating is dominant at high detuning (10-100GHz), and a paunch is result, while at higher detuning

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(>100GHz) the spectral hole burning is dominant. The paunch behavior is not seen in the behavior of bulk and QW SOAs. To reduce the undesired

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paunch behavior CDP must be reduced, but not neglected.

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[16] M. Sugawara , T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe,"Quantum-dot semiconductor optical amplifiers", Proceedings of SPIE

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4905, 259-275, 2002.

[17] J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, "Static

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IEEE J. Quantum Electron., 44, 658-666, 2008.

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gain saturation model of quantum-dot semiconductor optical amplifiers",

[18] O. Qasaimeh, "Theory of four-wave mixing wavelength conversion in quantum dot semiconductor optical amplifiers", IEEE Phot. Tech. Lett.

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16, 993-995, 2004.

[19] G. P. Agrawal and N. A. Olsson, “Self phase modulation and spectral

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broadening of optical pulses in semiconductor laser amplifiers” , IEEE J.

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Quantum Electron., 25, 2297–2306, 1989.

[20] O. Qasaimeh, “Characteristics of cross-gain (XG) wavelength conversion in quantum dot semiconductor optical amplifiers”, IEEE Photon.Technol. Lett., 16, 542–544, 2004.

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[21] N. Dutta & Q. Wang , “Semiconductor Optical Amplifiers”, University

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of Connecticut, Ch.6 , pp. 129-135, 2006.

[22] M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier” , Appl.

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Phys. Lett. 66 , 1458-1460, 1995.

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[23] S. Scotti, L. Graziani, A. D’Ottavi, F. Martelli, A. Mecozzi, P. Spano, R. Dall’Ara, F. Girardin, and G. Guekos, “Effects of Ultrafast Processes on Frequency Converters Based on Four-Wave Mixing in Semiconductor Optical Amplifiers“, IEEE J. selected topics Quantum Electronics 3, 1156-

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[24] S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V.

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Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperatureinsensitive 1.3

µ m InAs/InGaAs/GaAs quantum dot lasers”, Semicond.

Sci. Technol., 20, 340–342, 2005.

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[25] R. Maram, H. Baghban, H. Rasooli, R. Ghorbani, A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers”:

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dynamic behaviour and saturation properties, J. Opt . A Pure Appl. Opt. 11, 105205(1-8), 2009.

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[26] T. W. Berg, Svend Bischoff, Ingibjorg Magnusdottir, and Jesper Mørk, "Ultrafast gain recovery and modulation limitations in self-assembled

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quantum-dot devices", IEEE Phot. Technol. Lett., 13, 541-543, 2001.

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Technol. Lett. 14, 2002, 1139.

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[28] A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due

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Quantum Electron. 30, 1769-1781, 1994.

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Tables

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Table 1. Four wave mixing parameters.

1

2

NWL (×1023 m −3 )

0.8584

0.6282

f Gs

0.9931

h

2.959

κ SHB κ CH

0.4176

0.9904

0.9671

2.638

2.416

0.3915

0.5098

0.7219

0.07

0.05196

0.03433

2.334

1.712

1.143

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κ 0 (×1013 m−1. s −1 )

3

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τ 0 (ps)

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FWM Parameters

Pulse width at half maximum τ 0

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Table 2. Life Time Constants.

Symbol

Value

Capture time from WL

τc

1 ps

Spontaneous radiative lifetime

τs

200 ps

CH time constant

τ CH

the characteristic time of SHB

τ SHB

0.2 ps

Linewidth enhancement factor due CDP

αN

0.1

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Parameter

2.5 ps

α SHB

0.013

α CH

3

Thickness of dot layer

Lw

0.1 nm

Amplifier width

W

2µ m

L

2 mm

Linewidth enhancement factor due SHB

Amplifier length

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Linewidth enhancement factor due CH

−1

g max

1400 m

surface QD density

NQ

1015 m −2

No. of layers

l

10

pump wavelength

λ0

optical Confinement factor

Γ

0.027

Differential gain

a

10−19 m 2

Energy separation between WL and GS

∆E

0.7 eV

Temperature

T

300 k

thickness of the dot layer

Lw

10nm

nonlinear gain coefficient for SHB

κ SHB

0.11

nonlinear gain coefficient for CH

κCH

0.08

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Maximum gain value

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1.28

µm

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τc

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τe

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Figures

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τs

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Figure 1: Quantum-dot band-structure and carrier relaxation processes.

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20 τ =1 ps 0 ps ττ =1 00 = 2 ps ττ0 =1 = 3ps ps

18

0

16

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Output Power (mW)

14 12 10 8

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6 4

0 -5

-4

-3

-2

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2

-1

0

τ / τ0

1

2

3

4

5

Figure 2: Output power for pulses with different widths.

τ0=1 ps

2

τ0=2 ps τ0=3 ps

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The Gain Integral

2.5

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3

1.5

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1

0.5

0 0

L=3 ps gmax=31 dB

2

4

τ (ps)

6

Figure 3: The gain integral for pulses with different widths.

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8

10

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1 0.98

0.94

τ0=1 ps

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Occupation probability of Gs

0.96

0.92 0.9 τ0=2 ps

0.88

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0.86 0.84 τ0=3 ps

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0.82

L=3 ps gmax=31 dB

0.8 0

2

4

τ (ps)

6

8

10

Figure 4: Time series of GS occupation probability for pulses with different widths.

0.9

L=3 ps gmax=31 dB

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τ0=1 ps

0.7

τ0=2 ps

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NWL x 10

-23

m

-3

0.8

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1

0.6

0.5

0.4 0

1

τ0=3 ps

2

3

4

5

τ (ps)

6

7

8

9

10

Figure 5: Time series of WL carrier density for pulses with different widths.

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13

3

x 10

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2.5

κQD (m.sec) -1

2

1.5 τ0=3 ps

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1

τ0=6 ps

τ0=9 ps

0 0

0.1

0.2

0.3

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0.5

0.4

0.5

τ (s)

0.6

0.7

0.8

0.9 x 10

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Figure 6: Variation of κQD with time for pulses with different widths.

30

1 -11

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5 Positive detuning: Experimental Theoretical

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-5

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-10

-15

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Conversion efficiency (dB)

0

-20

-25 1 10

2

10

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δ (GHz)

3

10

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[26].

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Figure 7: A comparison between our results with the experimental measurements from

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Negative Detuning -10 τ0=1 ps

-20

CDP SHB CH Total

(a)

η (d B )

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-30

-40

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-50

-70 0 10

1

10

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-60

2

10

δ (GHz)

3

10

Positive Detuning -10

τ0=1 ps

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-50

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η (d B )

-40

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-20

-30

CDP SHB CH Total

(b)

-60

-70 0 10

1

10

2

10

δ (GHz)

3

10

Figure 8: Conversion efficiency versus (a) negative and (b) positive detunings for the conjugate signal. Components of CDP, SHB, and CH are shown.

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-10 τ0=1 ps τ0=2 ps τ0=3 ps

-ve δ

+ve δ

-20

η (d B )

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-30

-40

-50

-70 0 10

1

2

δ (GHz)

10

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10

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-60

3

10

Figure 9: Conversion efficiency versus detuning for the conjugate signal for pulses with

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different widths.

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10 -ve δ

0

τ0=0.5 ps

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+ve δ

-10

-20 Max

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η (dB)

-30

-40

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-50

-60

τ0=2.5 ps

-70

-80

-90 0 10

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Max 1

10

2

10

3

10

δ (GHz)

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Figure 10: Conversion efficiency (CDP and CH components) vs. detuning of the conjugate signal for pulses with different widths. Both positive and negative detunings

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are shown.

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Figure 11: Conversion efficiency (CDP and SHB component) vs. detuning of the conjugate signal for pulses of different widths. Both positive and negative detunings are

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shown.

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Figure 12: Conversion efficiency (CH and SHB component) vs. detuning of the conjugate

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signal for pulses of different widths. Both positive and negative detunings are shown.

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1. A theory combines the pulse traveling into QD SOA with FWM is discussed. 2. FWM in QD SOAs is pulse width dependent.

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3. For short pulses, carrier heating is dominant at high detuning (10-100GHz)

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4. at higher detuning (>100GHz) the spectral hole burning is

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dominant.