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Four-wave mixing in quantum dot SOAs: Theory of carrier heating
Accepted Manuscript Four-Wave Mixing in Quantum Dot SOAs: Theory of Carrier Heating Ahmed H. Flayyih, Ali Gehad Al-Shatravi, Amin H. Al-Khursan PII: DOI: Reference:
S2211-3797(17)30330-3 http://dx.doi.org/10.1016/j.rinp.2017.03.036 RINP 643
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
27 February 2017 25 March 2017 29 March 2017
Please cite this article as: Flayyih, A.H., Al-Shatravi, A.G., Al-Khursan, A.H., Four-Wave Mixing in Quantum Dot SOAs: Theory of Carrier Heating, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.03.036
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Four-Wave Mixing in Quantum Dot SOAs: Theory of Carrier Heating
Ahmed H. Flayyih(1), Ali Gehad Al-Shatravi(2), and Amin H. Al-Khursan(1),(*) (1) Nassiriya Nanotechnology Research Laboratory (NNRL), Physics Department, Science College, Thi-Qar University, Nassiriya, Iraq. (2) Physics Department, Science College, Thi-Qar University, Nassiriya, Iraq. * Corresponding Author. E-mail: [email protected].
Abstract Carrier heating (CH) theory in a four-wave mixing quantum dot structure has been investigated. The impact of wetting layer (WL) carrier density, CH time constant, effective intraband relaxation time have been examined. The derived heat capacity for QD structure have (T
−1
) dependence. It is shown here that both WL
carrier density and QD excited state (ES) occupation controls the overall nonlinear contributions. Then inclusion of WL and ES in the CH induces a new equilibrium reached at a faster recovery time. The proposed model yields results in a line with experiments at high carrier density reflecting the efficiency of our model.
Keywords: Quantum dot; Carrier heating; Four-wave mixing; excited state.
1
1. Introduction
A large number of recent research has been devoted to improving the response efficiency of optical devices. Many of these studies have focused on the employment of semiconductor optical amplifier (SOA) nonlinearities [1]. In fourwave mixing (FWM), the main effective nonlinear processes are spectral-hole burning (SHB), carrier density pulsation (CDP) and carrier heating (CH) [2,3]. The theoretical basis of these well-known processes is now largely understood [4-7]. Due to their excellent properties, devices based on quantum dot (QD) nanostructures have received a great deal of attention [8] and the theory of FWM in QD structures is comprehensively detailed in several studies [3, 9-11]. QD SOAs may offer a solution to the major two issues facing FWM in bulk SOAs: low conversion at long wavelengths, which is increased with QDs; and the asymmetric conversion, which is overcome with detuning independence using QDs [12, 13]. The QD structure is composed of a wetting layer (WL) and QDs, and there are a number of differences between these two layers. Due to its large number of states, the WL is considered as a continuum state while QD layer is completely quantized. Thus, the dynamics inside and between layers are varied. Due to the large energy separation between the layers, the transition from WL directly to the ground state (GS) is neglected [6]. Additionally, due to the difficulty of energy conservation rules between the WL and QD layers, the excited state (ES) phonon2
bottleneck effect occurs and the transition between them takes a long time [14]. On the other hand the transitions inside QDs (between GS and ES) are very fast, around 100-250ps [9]. The major sources of heating effects in SOAs are carrier injection, stimulated emission, Auger recombination and free carrier absorption (FCA). The injected carriers must release their access energy before reaching the lowest energy subbands. This energy will contribute to increase carrier temperature [15]. In stimulated emission the "cold carriers" which are close to the band edge are removed [4]. The FCA mechanism includes photon absorption by the interaction of free carriers within the same band [16]. The carriers are transition into higher energy states due to this process and, consequently, the temperature and energy of the carriers will increase. As the temperature of the carriers is higher than that of the lattice, thermalization will occur where the carriers transfer their excess energies to the crystal lattice through the interaction with phonons [17]. In QD structures, both discrete energy states and conduction band offset can reduces CH [9]. Furthermore, there is an experimental evidences on the significance of CH in QDs. Carrier heating in QDs has been studied experimentally, but the impact of CH on FWM in QDs has not been properly investigated. Borri et. al. has been investigated CH, experimentally, in QDs [18]. Additionally, Zilkie et. al. [19] 3
studied CH dynamics in quantum well, quantum dash and QD SOAs. CH in lightemitting QD structures has also been studied [20], where the quantum well (QW) WL is modeled as an infinite QW while QD energy subbands are calculated using a simple harmonic oscillator model. This model is significant in describing the essential physics of these structures in a simple manner, but they are far from practical structures due to infinite potential of QW and the equidistant energy of the harmonic oscillator [21]. Uskov et. al. [22] presented a model for CH induced by Auger capture between WL and GS. However, the effect of CH on FWM in QD structures has not been modeled, yet. This requires the modeling of the contribution of CH to the nonlinear susceptibility of QDs. Then, it should be easy to model optical properties, such as gain, including CH. This is of great importance for the study of QD devices. Since much of the foregoing work on FWM in QDs deals with SHB and CDP [9], it is important to deal with the effect of CH on FWM in QD nanostructures. Accordingly, this study introduces a model for CH theory on FWM in QD SOAs for three-level rate equations system (WL, ES and GS) based on density matrix theory. It is found that the nonlinear gain coefficient for CH in QD structures has a (K βT )−1 dependence since the heat capacity, in the contrary to the result in [9], which gives (K βT )−3 dependence. Additionally, a new equilibrium state was 4
predicted due to the inclusion of both WL and QD ES. This paper is organized as follows: Section 2 explains the theory of CH, Section 3 discusses the nonlinear gain coefficient and Section 4 states the relations of FWM conversion efficiency. Section 5 then describes and discusses the calculations and results. Section 6 presenting the conclusions of this study.
2. Theory
The population of GS and ES levels in QD is described by their carrier densities. Dots are interconnected by the WL. Using density matrix theory, and based on previous studies [4, 23, 24], the equation for the motion of a GS population, ρGS , can be written as
c d ρGS ,i
dt
c c c c c c c ( ρ ES ( ρGS ( ρGS , i )[1 − ( ρGS , i − f GS , i )] , i − f GS , i )[1 − ρ ES , i ] ,i ) = − −
τ 21
τ 12
c ( ρGS ) i − c , i − µcv , i ( ρcv , i − ρvc , i ) E ( z ,t ) τ CH
τ 1r
(1)
c where ρ ES , i is the occupation probability of carriers in ES in the state i in the dot
ensemble. The superscript (c) denotes that the state is in the conduction band. τ 1r is the spontaneous radiative lifetime in QDs, τ 21 (τ 12 ) is the carrier relaxation time from the ES (GS) to GS (ES). f GSc , i is the Fermi distribution function, where the
5
c carriers are relaxed back to the equilibrium distribution. τ CH is the CH time
constant, which is the time needed to cool the carrier temperature toward the lattice temperature. Since carrier-carrier and carrier-phonon scattering are intraband processes which do not affect the total carrier density, then Eq. (1) is adequate in this form to describe CH [4]. Note that another equation can be written for ES which is similar to Eq. (1) [3]. The coherence term of the density matrix ρ cv , i is given by [4]
d ρcv ,i dt
where
= −(i ωi +
µ cv , i
1
i c v ) ρcv ,i − µ cv ,i ( ρGS , i − ρGS , i − 1) E ( z , t ) τ cv
(2)
is the dipole moment and τ cv is the decoherence time. The superscript
(v) signifies that the state is in the valence band. E(z,t) is the electric field of the interacting light. The carrier density relates to the occupation probability according to the relationship [4] N jx (t ) =
1 V
∑ρ
x j ,i
(t )
(3)
i
where the subscript (j) refers to GS or ES while the superscript (x) refers to the (c) or (v) bands. The intraband processes does not affect the total carrier density, so
6
c c the steady-state ( ρ ES , i ) and small-signal ( ρ ES , i ) occupation probabilities of ES are
derived from the rate equation of ES, which are similar to Eq. (2), as we refer above, it is given by [3]
1
c ρ ES , i = τw 2
τwr
c ρ ES ,i =
+
1
−
1
τw 2 τ 2w
−
e Lw N w J
( 4)
J τw 2 Nw e Lw N w2
(5)
where N w is the steady-state carrier density in the WL, e is the electron charge, τ w2 is the carrier relaxation time from WL to ES, τ 2w is the carrier escape time from the ES to the WL, τ wr is the spontaneous radiative lifetime in WL, J is the current density and Lw is the effective thickness of the active layer. Note that the WL carrier density N w is assumed to be the same in both the conduction and valence bands. N w is the small-signal value of the carrier density in WL. Taking into
account the modulation between the pump and probe, the occupations can be expanded as follows
ρ xj , i = ρ jx, i + ρ xj , i e −i δ t + ρ xj ,∗i e i δt
(6)
7
f jx, i (t ) = f j x, i +
∂f jx, i ∂N w
(
∂f x N w e − i δt + c .c + j x, i Tjx, i e − i δ t + c .c ∂T j , i
)
(
)
(7)
Writing Eq.(1) for conduction and valence bands, and then substituting Eqs. (7) in it, the following equation is then derived from the small-signal analysis of occupation probability
(1 − i δτ ) (ρ SHB
∂f c + GSc ,i ∂TGS ,i
c GS ,i
v GS ,i
+ ρ
c v ∂ρGS ∂ρGS ,i ,i ) = + ∂N w ∂ N w
c ∂f GSv ,i T + GS ,i v ∂TGS ,i
v TGS ,i
∂f GSc ,i ∂f GSv ,i + + ∂N w ∂N w
µcv ,i − {2i τ SHB 2
(
N w
2
(ρ
× χ i (ω1 ) − χ i* (ω0 ) E 1E 0* + χ i (ωo ) − χ i* (ω2 ) E 0 E 2*
c GS ,i
v + ρGS ,i − 1) }
)
(8)
Note that E 0 , E 1 , E 2 are the field components for the pump, probe and FWM signal, respectively. τ SHB is the spectral hole burning time constant, it is derived at steadyc v state value of the relation of ( ρGS ,i + ρGS ,i − 1) , and is given by [3]
τ SHB
1 1 = + + ρ ES τ τ 12 wr
1 1 − τ 21 τ 12
−1
(9)
The nonlinear part of the second-order polarization is obtained from the definition of the polarization which is related to the dipole terms ( ρcv , i − ρvc , i ) and is given by [3, 4, 9]
8
1 P2 = V
µcv ,i
∑
2 c v χ i (ω1 ) ( ρGS ,i + ρGS ,i ) E 0
i
(10)
Substituting Eq. (8) into Eq. (10) results in, 1 P1 = V
∑
µcv ,i
i
∂f GSc ,i + ∂TGSc ,i − 2i τ SHB
2
c ∂ρGS ∂ρ v ,i + GS ,i ∂N w ∂N w (1 − i δτ SHB )
χi (ω1 )
c ∂f GSv ,i TGS ,i + v ∂TGS ,i
µcv ,i
2
2
(ρ
c GS ,i
1
v TGS ,i
∂f GSc ,i ∂f GSv ,i + + ∂ N ∂N w w
N w
+ ρGv S ,i − 1) × ( χ i (ωo ) − χ i* (ω2 ) )( E 1E 0* + E 0 E 2* )
(
)
(11)
where χ i is the Lorentzian lineshape that is responsible for homogenous broadening. The first term in Eq. (11) is the carrier density pulsation (CDP) term, which results from the beating between the pump and probe powers. The second term is the CH contribution, while the last term represents the SHB contribution. The effect of CH in QD SOA can be derived from the concept of carrier energy density, which is given by [4]: Ux =
1 V
∑E
x GS ,i
x ρGS ,i
(12)
i
x where E GS ,i is the subband ground state energy of the QD in the x band. The rate
equation of energy density is estimated from Eq. (1), and the result is,
9
dU x ρ h = dt −
i 1 V
∑E
(E
− (U x − U fx )
x GS ,i
τ 21 x GS ,i
) − (U
x
v − U fx )(1 − ρGS ) − U x − (U x ) + K x E (z ,t ) 2 x
τ 12
τ 1r
τ CH
µcv ,i ( ρcv , i − ρvc , i ) E ( z ,t )
(13)
i
Here the term ( K x E (z ,t )
2
) is added phenomenologically to represent the
contribution of CH induced by the free carrier absorption (FCA), and K x is a coefficient that can be expressed by cross-section σ x for FCA in the conduction and valence bands, which is given by [4],
K x = ε 0 n n g υg σ x N w
(14)
where ε 0 , n , n g ,υ g are respectively the permittivity of free space, the refractive index of the QD material, the group refractive index of the active material, and group velocity. The energy density of carriers is expressed as [4], U x = U x + h x (T x e −i δ t + c .c )
(15)
∂U x where h = is the heat capacity of a free electron. The small-signal analysis ∂T x x
of Eq. (13) gives,
10
x x (1 − ρ ES ) ρ ES 1 1 x hT + + x − − i δ = 2K x ( E 0*E 1 + E 0 E 2* ) τ 12 τwr τ CH τ 21 x
i − V
∑
2
µcv ,i
x E GS ,i
2
i
{( ρ
c GS ,i
(
v * * * * + ρGS ,i − 1) χ i (ω1 ) − χ i (ωo ) E 1E 0 + χ i (ωo ) − χ i (ω2 ) E 0 E 2
)}
(16) Then, the small-signal value of the carrier temperature is,
T = x
2 K x τ inx ( E 0*E 1 + E 0 E 2* ) h
{( ρ
c GS ,i
τ inx
i − x x h (1 − i δτ in ) V
(1 − i δτ ) +ρ − 1) ( χ (ω ) − χ (ω ) E E x
x in
v GS ,i
i
1
* i
1
o
c * v + ( ρGS ,i + ρGS ,i ) χ i (ω1 ) − χ i (ω2 ) E 0
2
∑ i * 0
µcv ,i
2
2 x E GS ,i
+ χ i (ωo ) − χ i* (ω2 ) E 0 E 2*
)
}
(17 )
where: 1
τ inx
=
1
τ SHB
+
1 x τ CH
−
x 2 ρ ES
(18)
τ 21
τ inx can be considered as the effective intraband relaxation time. It has no
counterpart in the CH relation for bulk SOAs. Note that E jx,i is considered (in Eq. (17)) as E GSx ,i . Since χ i (ω0 ) ≈ χi (ω1 ), χi (ω2 ) , where the main contribution comes from the resonant terms with ωi ≅ ω0 , Eq. (17) is then reduced to,
11
Tx =
τ inx ( E 0*E 1 + E 0 E 2* )
µ 2K − i cv 2,i V
h x (1 − i δτ inx )
x
2
x c v * E ρ ρ χ ω χ ω + − 1 ( ) − ( ) ∑i GS ,i ( GS ,i GS ,i )( i 0 i o ) (19 )
According to density matrix theory, the gain is defined as [9], iω 1 g (ω ) = 2c n ε 0 V
µcv ,i
∑
2
(ρ
c GS ,i
i
v * + ρGS ,i − 1)( χ i (ω ) − χ i (ω ) )
( 20)
The substitution of Eqs. (14) and (20) into Eq. (19) gives,
T x =
τ inx ( E 0*E 1 + E 0 E 2* ) 2ε 0 n c h (1 − i δτ x
x in
)
x ( σ x N w ω − g ( ω ) E GS ,i ) ω
( 21)
The susceptibility of CH contribution is simply determined from Eq. (11), giving,
X
x CH
(σ
x
=
1
∑
V ε0
i
µcv ,i
2
2
τ inx E 0 ∂f x 2ε 0 n c χ i (ω1 ) (1 − i δτ SHB ) hx (1 − i δτ inx ) ∂T x ω
x N w ω − g (ω ) E GS ,i )
( 22 )
Depending on the definition of the susceptibility [4], X (ω ) = −
nc
ω
(α + i ) g (ω ) ,
the following equality [4] can be obtained, 1 V ε0
∑ i
µcv ,i
2
χ i (ω )
∂f x cn ∂g (ω ) = − (αT x (ω ) + i ) ω ∂T x ∂T x
which is then used in Eq. (22), to give,
12
( 23)
X
x CH
=
τ inx
1
(1 − i δτ ) h (1 − i δτ ) x
SHB
x in
cn ∂g (ω ) α ( ω ) + i ( x ) − x T ω ∂T
2ε 0 n c E 0 2 x x × ( σ N ω − g (ω ) E GS ,i ) ω αT
x
( 24 )
is the linewidth enhancement factor results from CH. The above relationship
represents the CH contribution to QD nonlinear susceptibility. It differs from the bulk value, mainly, by the time τ inx , which covers the time constants of both SHB and CH. The inhomogenously broadened gain, g ( ω ) , is another aspect of Eq. (24).
3. Nonlinear gain coefficients
The nonlinear gain coefficient depends on the analytical solution of pulse propagation inside QD SOA [3]. For the FWM contribution, we have CDP, SHB and CH. In general, the nonlinear gain coefficient due to CDP is assumed to be equal to unity [9]. Other nonlinear gain coefficients are deduced from the normalized nonlinear susceptibility. Then, the nonlinear gain coefficient for CH is given by,
13
x κCH (αT (ω ) + i ) = − x
X X
CH
CDP Normalized
τ inx
1
(1 − i δτ ) h (1 − i δτ ) =− x
x in
SHB
cn ∂g ( ω ) 2ε 0 n c E 0 α ω ( ) + i x − x T ω ω ∂T 2 (c n )2 τwr ε 0 dg (ω ) 2 E 0 g (ω ) ω0ωq dN w δ τ − i + 1 ( nm SHB )
(
)
2
x ( σ x N w ω − g ( ω ) E GS ,i )
or,
x κCH
σ x N w ω ∂g ( ω ) x x τ E 1 − ( )( ) in GS x ∂T x g (ω ) E GS =− x x dg (ω ) h τwr (1 − i δτ in ) dN w
The ratio (
∂g (ω ) ∂T
x
)/(
dg ( ω ) dN w
( 25)
) is given by [9]:
∂g ( ω ) x x ∂T = − N w E GS K β (T x )2 dg ( ω ) dN w
( 26 )
where K β is the Boltzmann constant. Substituting Eq. (26) into Eq. (25) gives, x N w (E GS )2 τ inx x 2 x x K β (T ) τwr h (1 − i δτ in )
x κCH =
σ x N ω 1 − x g (ω ) E GS
( 27 )
which is the relation of the nonlinear gain coefficient for CH in QD structures. It has a (K βT )−1 dependence since the heat capacity, (h x ) , here is (1 / T ) -dependent
14
x as in Eq. (21). In [9], κCH have (K βT )−3 dependence so, it decrease extensively
with temperature compared with (T
−1
) -dependence, here. Note also that the heat
capacity, here, is a variable dependent gain, QD GS, pump, probe, and conjugate fields, Eq. (21). In our preceding work [24], pulse effect on FWM is detailed. This work deals with derivation and study the main features of CH. The examination of these parameters, including pulse effect on CH and phase-dependent FWM can be taken as separate work dealt with in the near future. This is also justified since it required the study of the overall FWM components which is out of the aim of this work.
4. FWM conversion efficiency
The conversion efficiency of SOA is defined as the ratio between the power of the signal converted at the device-output and the probe-power at the input. In the complete theory of FWM in QD SOAs, as stated elsewhere [24], the specialty of QD structure is taken into account, including ES and inhomogeneity, but an empirical value is used for CH as formulated by Nielsen and Chuang [9]. Now CH in QD is formulated to cover QD specialty as in Eq. (27) above. FWM efficiency is given by [24],
15
ηeff = e
FQD
GQD (τ )
E2 FQD ( −δ , L ) 20 E sat
( 28)
G τ 1 − i α j κ xj e QD ( ) − 1 1 − i α + ∑ ( −δ ) = −C 2 2 1 − i δτ j E0 j = SHB , CH 2ξ + 2 E sat
(
)
(29)
Note that ξ is given by [24],
ξ = −i δ +
NW L D
1 τw 2
2 − D
+
1
τ 2w
2J τw 2τ 1r 2 e Lw N W L
1 1 − ρ ES τw 2 τw 2
+
1
τ 2w
τwr + ( i δτwr
+1
)
( 30 )
where ρ ES is the steady-state value of ES. E sat is the saturated field for the threelevel QD system which can be defined as [24], E sat =
ω
( 31)
dg c n ε 0τwr dN WL
L
The gain integral GQD (τ ) in QD SOAs [GQD (τ ) = ∫ g (z ,τ ) dz ] is then defined as 0
follows [24],
16
dGQD (τ ) dτ
=−
GQD (τ )
τ C ,QD
+Γ
dg dN W L
J N ∫0 e Lw − τC ,wQD0 L
dz
(32)
where N w 0 is the WL carrier density at transparency, Lw is the effective thickness of the active layer, and τ C ,QD represents the effective capture time from WL to the QD ES, τ C ,QD
1 1 = + − ρ ES τw 2 τwr
1 1 + τ 2w τw 2
−1
Note that, the contribution of SHB to FWM in QDs is discussed in detail according to our model elsewhere [24].
5. Calculations, Results and Discussion
The QD structure used in this study is a tenfold InAs QD layer with 5 × 1010 cm −2 per layer covered with InGaAs WL and grown by molecular beam epitaxy at NanoSemiconductor GmbH in Germany. The structural description and parameters are detailed elsewhere [10, 25, 26]. The parameters used in the calculations are given in Table.1 unless otherwise stated. Since the cross-section of the valence x band is zero [4], thus κCH is calculated for the conduction band only and then, κCH is
c refer to κCH and also τ in refers to τ inc , for simplification. Throughout this work the
input pulse used have a width of (1ps ) . 17
The contribution of the CH time constant τ CH was checked through the κCH − τ CH − δ plot in Fig. 1. It is shown that κCH depends on both detuning and τ CH .
Longer τ CH , increases κCH at low detuning (≤ 500 GHz ) , while it decreases at higher detuning. Fig. 2 shows κCH − N W relationship at three τ CH values, where longer τ CH gives higher κCH . At low carrier density, κCH is the same for the three τ CH times. At higher N W , curves of κCH are easily discriminated for the three τ CH times. Fig. 3 shows κCH − τ in relation at the same three τ CH values. For each τ CH value, the curves go to a higher value of κCH . The comparison shows that the curve arrangement in Figs. 2 are contradicted the curve arrangement in Fig. 3. Note that N W is not constant in Fig. 3. To find the reason for this arrangement we return to Eq. (18), where there are three terms which contribute to τ in . Two of them (τ SHB , ρES ) depend on N W . Thus, N W controls the overall nonlinear contributions of τ in . This can also be seen in Fig. 4, where κCH − τ in is plotted with ρ ES as a parameter. The curves show a high dependence on ρ ES values. To study the effect of CH on the occupation, the time series of GS occupation probability ( ρGS ) is shown in Fig. 5, where the rate equations (1) and (2) are solved numerically. Blue curve shows GS occupation dynamics when CH is neglected. The absorption appears in the GS occupation, in Fig. 5, is due to the excitation by an input pulse with a width of (1ps ) . A fast recovery is shown especially for the 18
curve with CH. This may be caused by the additional time added due to CH which increases the recovery. CH is shown to reduce the GS probability as expected [22], but here there are some differences from the previous results in [22], where the overall GS probability is reduced, here. Note that in the earlier study [22], the CH curve returns to the same value, it reaches without CH, after reaching the steadystate. The behaviour found here means that there is another steady-state reached, or another equilibrium is attained, after CH takes place. This is an expected result as we discuss below.
Stability under CH effect
In [22], the CH curve was reached the steady state of that had been reaching without CH although they assumed that the thermal distribution, between the QD and the reservoir, is broken and carriers are removed from QD. Thus, the expected result is that the occupation probability stays at its "new" steady state since carriers are removed from QD. Carrier relaxation from QDs helps in this new steady state but not makes QD return to the steady state which is predicted by the curve without CH. Although QD models neglects ES but, it is evidenced that ES couples WL and GS [27]. It is experimentally reported that intradot relaxation (between GS and ES) is the most relevant process for linear gain and gain compression [28]. Thus, the 19
inclusion of ES in our model helps to see this result. This new steady state is not a bistable regime. It is the "real" steady state reached by the QD system since the earlier steady state (without CH) is higher than this one. There is also another difference from the previous work in [22], where it is shown here that the CH curve recovers more quickly. This is in coincidence with the "first" conclusion that CH makes a new equilibrium reached at a faster time. This also can be reasoned to the inclusion of ES, as referred above. In the previous QW model [15], CH is taken as a constant parameter. The method of modeling CH in QWs is employed empirically for QD structures [9]. This is, of course, not the correct way for formulate CH, but it is adequate for cases of low CH. Fig. 6 shows a comparison between our CH model for FWM in QDs and the previous QW model which is employed into QDs [9]. It is shown that at low WL carrier density the two models coincide, where the earlier QW model is adequate for cases of low CH. This is shown for WL carrier density N W = 1022 m −3 where both curves are coincides. When N W = 3.16 ×1023 m −3 the previous QW model applied to QD (dashed line) is higher than our model (solid line), but they coincides at high detuning (> 100GHz ) . The reason for this coincidence is the Lorentzian approximation, which gives approximate results (at higher detuning) in both models. Increasing carrier density makes our model works well, as shown for the curve when N W = 1025 m −3 , it gives efficiency in the range of (−15dB ) while the 20
calculated efficiency using previous QW model is (> 30dB ) which is overestimates our model. Note that the (30dB ) efficiency is so high, and is not obtained at any QD experiment. The value obtained by our current model is in the range of (−15dB ) . Compared with experimentally obtained results of QDs [29], the results
of our model, as shown in Fig. 6, is in agreement with experiment. Accordingly, our model works within the range of experimental results obtained for QDs. This result can be discussed depending on the conclusion from Fig. 5, i.e. the new "lower" steady state. One can also refer to Eq. (27) where (K βT )−1 dependence reduces κCH in comparison with the dependence in [9] as we refer in section III above. Accordingly FWM efficiency obtained by our model is reduced.
6. Conclusions
The theory of carrier heating in the FWM in QD structures is discussed for the first time. The most influential parameters are examined. Both detuning and carrier heating time constant control CH, which shows more sensitivity to WL carrier density. The derived heat capacity for QD structure have (T
−1
) dependence. CH
has been shown to lead to a new equilibrium distribution, reducing the occupation probability and speeding up recovery in QD structures. Our model works well at high carrier density providing a prediction to experimental work.
21
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[2] C .Y. Tsai, F. P. Shih, T. L. Sung and C.H. Chen, "Effects of carrier heating, nonequilibrium LO phonons and lattice heating on small-signal modulation response of semiconductor lasers", IEEE processing 145, 1998, 13-20.
[3] A. H. Flayyih and Amin H. Al-Khursan, "Four-wave mixing in quantum dot semiconductor optical amplifiers", Applied Optics 52, 2013, 3156-3165.
[4] A. Uskov, J. Mork, and J. Mark, "Wave Mixing in Semiconductor Laser Amplifiers Due to Carrier Heating and Spectral-Hole Burning", IEEE J. Quantum Electronics 30, 1994, 1769-1781.
[5] J. Ming L. and T. B. Simpson, "Four-Wave Mixing and Optical Modulation in a Semiconductor Laser", IEEE J. Quantum Electronics 30, 1994, 951-956.
22
[6] A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, "Four-Wave Mixing in Traveling-Wave Semiconductor Amplifiers", IEEE J. Quantum Electronics 31, 1995, 689-699.
[7] T. Mukai and T. Saitoh, "Detuning Characteristics and Conversion Efficiency of Nearly Degenerate Four-Wave Mixing in a 1.5µ m
Traveling-Wave
Semiconductor Laser Amplifier", IEEE J. Quantum Electronics , IEEE J. Quantum Electronics 26, 1990, 865-875.
[8] Ahmed H. Flayyih, and Amin H. Al-Khursan, "Integral gain in quantum dot semiconductor optical amplifiers", Superlattices and Microstructures 62, 2013, 8187.
[9] D. Nielsen and S. L. Chuang, "Four-wave mixing and wavelength conversion in quantum dot", Physical Review B 81, 2010, 035305.
[10] T. W. Berg and J. Mørk, “Saturation and noise properties of quantum-dot optical amplifiers” IEEE J. Quantum Electron. 40, 2004, 152–159.
23
[11] M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers” , Phys. Rev. B 69, 2004, 235332.
[12] A. Bilenca, R. Alizon, V. Mikhelashhvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, "Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs–InP quantum-dash semiconductor optical amplifiers operating at 1550 nm", IEEE Photonics Technology Letters 15, 2003, 563-565.
[13] M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, "Quantum-dot semiconductor optical amplifiers", Proceedings of SPIE 4905, 2002, 259-275.
[14] Amin H. Al-Khursan, “Intensity Noise Characteristics in Quantum-Dot Lasers: Four-Level Rate Equations Analysis”, J. Luminescence 113, 2005, 129136.
[15] L. F. Lester and B. K. Ridleya, "Hot carriers and the frequency response of quantum well lasers", Journal of applied physics 72, 1992, 2579-25887. 24
[16] D. A. Clugston and P. A. Basore, "Modelling Free-carrier Absorption in Solar Cells", Progress in Photovoltaic 5, 1997, 229-236.
[17] J. Siegret, "Carrier dynamics in semiconductor quantum dots", PHD Thesis, Sweden, 2006.
[18] P. Borri, W. Langbein, J. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, "Spectral hole-burning and carrier-heating dynamics in InGaAs quantum-dot amplifiers", IEEE J. Sel. Top. Quantum Electron. 6, 544-551, 2000.
[19] A. J. Zilkie, J. Meier, M. Mojahedi, P. J. Poole, P. Barrios, D. Poitras, T. J. Rotter, C. Yang, A. Stintz, K. J. Malloy, P. W. E. Smith, and J. S. Aitchison, "Carrier Dynamics of Quantum-Dot, Quantum-Dash, and Quantum-Well Semiconductor Optical Amplifiers Operating at 1.55 µ m ", IEEE J. Quantum Electronics, 43, 982- 991, 2007.
[20] J. Wolters, M. Dachner, E. Malic, U. Woggon, and A. Knorr, "Carrier heating in light-emitting quantum-dot heterostructures at low injection currents", Phys. Rev. B 80, 2009, 245401. 25
[21] Amin H. Al-Khursan, "Gain of excited states in the quantum-dots", Physica E 35, 2006, 6–8.
[22] A. V. Uskov, C. Meuer, H. Schmeckebier1, and D. Bimberg, "Auger Capture Induced Carrier Heating in Quantum Dot Lasers and Amplifiers", Applied Physics Express 4, 2011, 022202.
[23] N. Ogasawara and R. Ito, “Longitudinal mode competition and asymmetric gain saturation in semiconductor injection lasers: Theory” Japan. J. Appl. Phys., 27, 1988, 615-626.
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[25] J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, "Static gain saturation model of quantum-dot semiconductor optical amplifiers", IEEE J. Quantum Electronics, 44, 2008, 658-666.
26
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[27] Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA”, IEEE J. Quantum Electronics 41, 2005, 1268.
[28] A. Fiore and A. Markus, "Differential gain and gain compression in quantum dot lasers", IEEE J. Quantum Electronics 43, 2007, 287.
[29] Z.G. Lu, J.R. Liu, S. Raymond, P. J. Poole1, P. J. Barrios, G. Pakulski, D. Poitras, F. G. Sun, S. Taebi, and T. Hal, "Ultra-broadband quantum-dot semiconductor optical amplifier and its applications", OSA, Paper JThA33, 2007.
27
Tables
Symbol
Value
Unit
0.4
ns
Capture time from WL
τ 1r τw 2
3
ps
Escape time from ES to WL
τ 2w
1
ns
Capture time from ES to GS
τ 21
0.16
ps
Escape time from GS to ES
τ 12
1.2
ps
Decoherence time
τ cv
100
fs
Amplifier length
L
3
mm
Amplifier width
w
2
µm
Thickness of dot layer
Lw
10
nm
Confinement factor
Γ
0.027
Wavelength of pump
λ0
1.3
µm
Injected Current
I
50
mA
Surface density
NQ
1015
m-2
T
300
K
σc
3.5 . 10-22
m-2
Linewidth enhancement factor due CDP
αCDP
0.1
Linewidth enhancement factor due SHB
α SHB
0.1
Linewidth enhancement factor due CH
α CH
1
Parameter
Recombination time
Temperature Conduction band Cross section
28
Figures
0.8
κCH
0.6
0.4
0.2
0 2.1 10
2 x 10
8
-13
6
1.9 τ CH (sec)
4 1.8
2 0
x 10
11
δ (Hz)
Figure 1: A 3-D plot between nonlinear carrier heating parameter κCH and CH time τ in versus detuning δ in a QD-SOA.
29
0.7
0.6
0.5
κ
CH
0.4
0.3
0.2 τ =180 fs CH τ =190 fs CH τ =200 fs
0.1
CH
0 22 10
10
23
10
24
NWL (m-3)
Figure 2: The nonlinear carrier heating parameter κCH versus WL carrier density N W L at three values of CH time τ CH .
30
10
κ
CH
10
10
10
0
-1
-2
-3
τ CH= 180 fs τ CH= 190 fs τ CH= 200 fs
10
-4
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
(sec) in
τ
x 10
2.4 -12
Figure 3: The nonlinear carrier heating parameter κCH versus the effective intraband relaxation time τ in at three values of CH time τ CH .
31
0.07
0.06
0.05
CH
0.04 κ
ρ =0.9984 h ρ =0.9844 h
0.03
0.02
0.01
0 0.9
1
1.1
1.2
1.3
(sec) in
τ
1.4
1.5
1.6 x 10
-12
Figure 4: The nonlinear carrier heating parameter κCH versus the effective intraband relaxation time τ in at two values of ES occupation probability ρ ES .
32
Figure 5: The time series of GS occupation probability ( ρGS ) is shown. GS occupation dynamics neglecting CH is also shown in the blue curve, for comparison. CH is shown to reduce the overall GS probability. The behavior here means that there is another equilibrium attained after CH is taking place- which is the expected result. CH curve has recovered in a faster time.
33
Figure 6: A relation between FWM efficiency ηFW M versus detuning δ at different WL carrier density N W L in a QD-SOA. Dotted curves are for QW model, for comparison.
34
S2211-3797(17)30330-3 http://dx.doi.org/10.1016/j.rinp.2017.03.036 RINP 643
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
27 February 2017 25 March 2017 29 March 2017
Please cite this article as: Flayyih, A.H., Al-Shatravi, A.G., Al-Khursan, A.H., Four-Wave Mixing in Quantum Dot SOAs: Theory of Carrier Heating, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.03.036
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.
Four-Wave Mixing in Quantum Dot SOAs: Theory of Carrier Heating
Ahmed H. Flayyih(1), Ali Gehad Al-Shatravi(2), and Amin H. Al-Khursan(1),(*) (1) Nassiriya Nanotechnology Research Laboratory (NNRL), Physics Department, Science College, Thi-Qar University, Nassiriya, Iraq. (2) Physics Department, Science College, Thi-Qar University, Nassiriya, Iraq. * Corresponding Author. E-mail: [email protected].
Abstract Carrier heating (CH) theory in a four-wave mixing quantum dot structure has been investigated. The impact of wetting layer (WL) carrier density, CH time constant, effective intraband relaxation time have been examined. The derived heat capacity for QD structure have (T
−1
) dependence. It is shown here that both WL
carrier density and QD excited state (ES) occupation controls the overall nonlinear contributions. Then inclusion of WL and ES in the CH induces a new equilibrium reached at a faster recovery time. The proposed model yields results in a line with experiments at high carrier density reflecting the efficiency of our model.
Keywords: Quantum dot; Carrier heating; Four-wave mixing; excited state.
1
1. Introduction
A large number of recent research has been devoted to improving the response efficiency of optical devices. Many of these studies have focused on the employment of semiconductor optical amplifier (SOA) nonlinearities [1]. In fourwave mixing (FWM), the main effective nonlinear processes are spectral-hole burning (SHB), carrier density pulsation (CDP) and carrier heating (CH) [2,3]. The theoretical basis of these well-known processes is now largely understood [4-7]. Due to their excellent properties, devices based on quantum dot (QD) nanostructures have received a great deal of attention [8] and the theory of FWM in QD structures is comprehensively detailed in several studies [3, 9-11]. QD SOAs may offer a solution to the major two issues facing FWM in bulk SOAs: low conversion at long wavelengths, which is increased with QDs; and the asymmetric conversion, which is overcome with detuning independence using QDs [12, 13]. The QD structure is composed of a wetting layer (WL) and QDs, and there are a number of differences between these two layers. Due to its large number of states, the WL is considered as a continuum state while QD layer is completely quantized. Thus, the dynamics inside and between layers are varied. Due to the large energy separation between the layers, the transition from WL directly to the ground state (GS) is neglected [6]. Additionally, due to the difficulty of energy conservation rules between the WL and QD layers, the excited state (ES) phonon2
bottleneck effect occurs and the transition between them takes a long time [14]. On the other hand the transitions inside QDs (between GS and ES) are very fast, around 100-250ps [9]. The major sources of heating effects in SOAs are carrier injection, stimulated emission, Auger recombination and free carrier absorption (FCA). The injected carriers must release their access energy before reaching the lowest energy subbands. This energy will contribute to increase carrier temperature [15]. In stimulated emission the "cold carriers" which are close to the band edge are removed [4]. The FCA mechanism includes photon absorption by the interaction of free carriers within the same band [16]. The carriers are transition into higher energy states due to this process and, consequently, the temperature and energy of the carriers will increase. As the temperature of the carriers is higher than that of the lattice, thermalization will occur where the carriers transfer their excess energies to the crystal lattice through the interaction with phonons [17]. In QD structures, both discrete energy states and conduction band offset can reduces CH [9]. Furthermore, there is an experimental evidences on the significance of CH in QDs. Carrier heating in QDs has been studied experimentally, but the impact of CH on FWM in QDs has not been properly investigated. Borri et. al. has been investigated CH, experimentally, in QDs [18]. Additionally, Zilkie et. al. [19] 3
studied CH dynamics in quantum well, quantum dash and QD SOAs. CH in lightemitting QD structures has also been studied [20], where the quantum well (QW) WL is modeled as an infinite QW while QD energy subbands are calculated using a simple harmonic oscillator model. This model is significant in describing the essential physics of these structures in a simple manner, but they are far from practical structures due to infinite potential of QW and the equidistant energy of the harmonic oscillator [21]. Uskov et. al. [22] presented a model for CH induced by Auger capture between WL and GS. However, the effect of CH on FWM in QD structures has not been modeled, yet. This requires the modeling of the contribution of CH to the nonlinear susceptibility of QDs. Then, it should be easy to model optical properties, such as gain, including CH. This is of great importance for the study of QD devices. Since much of the foregoing work on FWM in QDs deals with SHB and CDP [9], it is important to deal with the effect of CH on FWM in QD nanostructures. Accordingly, this study introduces a model for CH theory on FWM in QD SOAs for three-level rate equations system (WL, ES and GS) based on density matrix theory. It is found that the nonlinear gain coefficient for CH in QD structures has a (K βT )−1 dependence since the heat capacity, in the contrary to the result in [9], which gives (K βT )−3 dependence. Additionally, a new equilibrium state was 4
predicted due to the inclusion of both WL and QD ES. This paper is organized as follows: Section 2 explains the theory of CH, Section 3 discusses the nonlinear gain coefficient and Section 4 states the relations of FWM conversion efficiency. Section 5 then describes and discusses the calculations and results. Section 6 presenting the conclusions of this study.
2. Theory
The population of GS and ES levels in QD is described by their carrier densities. Dots are interconnected by the WL. Using density matrix theory, and based on previous studies [4, 23, 24], the equation for the motion of a GS population, ρGS , can be written as
c d ρGS ,i
dt
c c c c c c c ( ρ ES ( ρGS ( ρGS , i )[1 − ( ρGS , i − f GS , i )] , i − f GS , i )[1 − ρ ES , i ] ,i ) = − −
τ 21
τ 12
c ( ρGS ) i − c , i − µcv , i ( ρcv , i − ρvc , i ) E ( z ,t ) τ CH
τ 1r
(1)
c where ρ ES , i is the occupation probability of carriers in ES in the state i in the dot
ensemble. The superscript (c) denotes that the state is in the conduction band. τ 1r is the spontaneous radiative lifetime in QDs, τ 21 (τ 12 ) is the carrier relaxation time from the ES (GS) to GS (ES). f GSc , i is the Fermi distribution function, where the
5
c carriers are relaxed back to the equilibrium distribution. τ CH is the CH time
constant, which is the time needed to cool the carrier temperature toward the lattice temperature. Since carrier-carrier and carrier-phonon scattering are intraband processes which do not affect the total carrier density, then Eq. (1) is adequate in this form to describe CH [4]. Note that another equation can be written for ES which is similar to Eq. (1) [3]. The coherence term of the density matrix ρ cv , i is given by [4]
d ρcv ,i dt
where
= −(i ωi +
µ cv , i
1
i c v ) ρcv ,i − µ cv ,i ( ρGS , i − ρGS , i − 1) E ( z , t ) τ cv
(2)
is the dipole moment and τ cv is the decoherence time. The superscript
(v) signifies that the state is in the valence band. E(z,t) is the electric field of the interacting light. The carrier density relates to the occupation probability according to the relationship [4] N jx (t ) =
1 V
∑ρ
x j ,i
(t )
(3)
i
where the subscript (j) refers to GS or ES while the superscript (x) refers to the (c) or (v) bands. The intraband processes does not affect the total carrier density, so
6
c c the steady-state ( ρ ES , i ) and small-signal ( ρ ES , i ) occupation probabilities of ES are
derived from the rate equation of ES, which are similar to Eq. (2), as we refer above, it is given by [3]
1
c ρ ES , i = τw 2
τwr
c ρ ES ,i =
+
1
−
1
τw 2 τ 2w
−
e Lw N w J
( 4)
J τw 2 Nw e Lw N w2
(5)
where N w is the steady-state carrier density in the WL, e is the electron charge, τ w2 is the carrier relaxation time from WL to ES, τ 2w is the carrier escape time from the ES to the WL, τ wr is the spontaneous radiative lifetime in WL, J is the current density and Lw is the effective thickness of the active layer. Note that the WL carrier density N w is assumed to be the same in both the conduction and valence bands. N w is the small-signal value of the carrier density in WL. Taking into
account the modulation between the pump and probe, the occupations can be expanded as follows
ρ xj , i = ρ jx, i + ρ xj , i e −i δ t + ρ xj ,∗i e i δt
(6)
7
f jx, i (t ) = f j x, i +
∂f jx, i ∂N w
(
∂f x N w e − i δt + c .c + j x, i Tjx, i e − i δ t + c .c ∂T j , i
)
(
)
(7)
Writing Eq.(1) for conduction and valence bands, and then substituting Eqs. (7) in it, the following equation is then derived from the small-signal analysis of occupation probability
(1 − i δτ ) (ρ SHB
∂f c + GSc ,i ∂TGS ,i
c GS ,i
v GS ,i
+ ρ
c v ∂ρGS ∂ρGS ,i ,i ) = + ∂N w ∂ N w
c ∂f GSv ,i T + GS ,i v ∂TGS ,i
v TGS ,i
∂f GSc ,i ∂f GSv ,i + + ∂N w ∂N w
µcv ,i − {2i τ SHB 2
(
N w
2
(ρ
× χ i (ω1 ) − χ i* (ω0 ) E 1E 0* + χ i (ωo ) − χ i* (ω2 ) E 0 E 2*
c GS ,i
v + ρGS ,i − 1) }
)
(8)
Note that E 0 , E 1 , E 2 are the field components for the pump, probe and FWM signal, respectively. τ SHB is the spectral hole burning time constant, it is derived at steadyc v state value of the relation of ( ρGS ,i + ρGS ,i − 1) , and is given by [3]
τ SHB
1 1 = + + ρ ES τ τ 12 wr
1 1 − τ 21 τ 12
−1
(9)
The nonlinear part of the second-order polarization is obtained from the definition of the polarization which is related to the dipole terms ( ρcv , i − ρvc , i ) and is given by [3, 4, 9]
8
1 P2 = V
µcv ,i
∑
2 c v χ i (ω1 ) ( ρGS ,i + ρGS ,i ) E 0
i
(10)
Substituting Eq. (8) into Eq. (10) results in, 1 P1 = V
∑
µcv ,i
i
∂f GSc ,i + ∂TGSc ,i − 2i τ SHB
2
c ∂ρGS ∂ρ v ,i + GS ,i ∂N w ∂N w (1 − i δτ SHB )
χi (ω1 )
c ∂f GSv ,i TGS ,i + v ∂TGS ,i
µcv ,i
2
2
(ρ
c GS ,i
1
v TGS ,i
∂f GSc ,i ∂f GSv ,i + + ∂ N ∂N w w
N w
+ ρGv S ,i − 1) × ( χ i (ωo ) − χ i* (ω2 ) )( E 1E 0* + E 0 E 2* )
(
)
(11)
where χ i is the Lorentzian lineshape that is responsible for homogenous broadening. The first term in Eq. (11) is the carrier density pulsation (CDP) term, which results from the beating between the pump and probe powers. The second term is the CH contribution, while the last term represents the SHB contribution. The effect of CH in QD SOA can be derived from the concept of carrier energy density, which is given by [4]: Ux =
1 V
∑E
x GS ,i
x ρGS ,i
(12)
i
x where E GS ,i is the subband ground state energy of the QD in the x band. The rate
equation of energy density is estimated from Eq. (1), and the result is,
9
dU x ρ h = dt −
i 1 V
∑E
(E
− (U x − U fx )
x GS ,i
τ 21 x GS ,i
) − (U
x
v − U fx )(1 − ρGS ) − U x − (U x ) + K x E (z ,t ) 2 x
τ 12
τ 1r
τ CH
µcv ,i ( ρcv , i − ρvc , i ) E ( z ,t )
(13)
i
Here the term ( K x E (z ,t )
2
) is added phenomenologically to represent the
contribution of CH induced by the free carrier absorption (FCA), and K x is a coefficient that can be expressed by cross-section σ x for FCA in the conduction and valence bands, which is given by [4],
K x = ε 0 n n g υg σ x N w
(14)
where ε 0 , n , n g ,υ g are respectively the permittivity of free space, the refractive index of the QD material, the group refractive index of the active material, and group velocity. The energy density of carriers is expressed as [4], U x = U x + h x (T x e −i δ t + c .c )
(15)
∂U x where h = is the heat capacity of a free electron. The small-signal analysis ∂T x x
of Eq. (13) gives,
10
x x (1 − ρ ES ) ρ ES 1 1 x hT + + x − − i δ = 2K x ( E 0*E 1 + E 0 E 2* ) τ 12 τwr τ CH τ 21 x
i − V
∑
2
µcv ,i
x E GS ,i
2
i
{( ρ
c GS ,i
(
v * * * * + ρGS ,i − 1) χ i (ω1 ) − χ i (ωo ) E 1E 0 + χ i (ωo ) − χ i (ω2 ) E 0 E 2
)}
(16) Then, the small-signal value of the carrier temperature is,
T = x
2 K x τ inx ( E 0*E 1 + E 0 E 2* ) h
{( ρ
c GS ,i
τ inx
i − x x h (1 − i δτ in ) V
(1 − i δτ ) +ρ − 1) ( χ (ω ) − χ (ω ) E E x
x in
v GS ,i
i
1
* i
1
o
c * v + ( ρGS ,i + ρGS ,i ) χ i (ω1 ) − χ i (ω2 ) E 0
2
∑ i * 0
µcv ,i
2
2 x E GS ,i
+ χ i (ωo ) − χ i* (ω2 ) E 0 E 2*
)
}
(17 )
where: 1
τ inx
=
1
τ SHB
+
1 x τ CH
−
x 2 ρ ES
(18)
τ 21
τ inx can be considered as the effective intraband relaxation time. It has no
counterpart in the CH relation for bulk SOAs. Note that E jx,i is considered (in Eq. (17)) as E GSx ,i . Since χ i (ω0 ) ≈ χi (ω1 ), χi (ω2 ) , where the main contribution comes from the resonant terms with ωi ≅ ω0 , Eq. (17) is then reduced to,
11
Tx =
τ inx ( E 0*E 1 + E 0 E 2* )
µ 2K − i cv 2,i V
h x (1 − i δτ inx )
x
2
x c v * E ρ ρ χ ω χ ω + − 1 ( ) − ( ) ∑i GS ,i ( GS ,i GS ,i )( i 0 i o ) (19 )
According to density matrix theory, the gain is defined as [9], iω 1 g (ω ) = 2c n ε 0 V
µcv ,i
∑
2
(ρ
c GS ,i
i
v * + ρGS ,i − 1)( χ i (ω ) − χ i (ω ) )
( 20)
The substitution of Eqs. (14) and (20) into Eq. (19) gives,
T x =
τ inx ( E 0*E 1 + E 0 E 2* ) 2ε 0 n c h (1 − i δτ x
x in
)
x ( σ x N w ω − g ( ω ) E GS ,i ) ω
( 21)
The susceptibility of CH contribution is simply determined from Eq. (11), giving,
X
x CH
(σ
x
=
1
∑
V ε0
i
µcv ,i
2
2
τ inx E 0 ∂f x 2ε 0 n c χ i (ω1 ) (1 − i δτ SHB ) hx (1 − i δτ inx ) ∂T x ω
x N w ω − g (ω ) E GS ,i )
( 22 )
Depending on the definition of the susceptibility [4], X (ω ) = −
nc
ω
(α + i ) g (ω ) ,
the following equality [4] can be obtained, 1 V ε0
∑ i
µcv ,i
2
χ i (ω )
∂f x cn ∂g (ω ) = − (αT x (ω ) + i ) ω ∂T x ∂T x
which is then used in Eq. (22), to give,
12
( 23)
X
x CH
=
τ inx
1
(1 − i δτ ) h (1 − i δτ ) x
SHB
x in
cn ∂g (ω ) α ( ω ) + i ( x ) − x T ω ∂T
2ε 0 n c E 0 2 x x × ( σ N ω − g (ω ) E GS ,i ) ω αT
x
( 24 )
is the linewidth enhancement factor results from CH. The above relationship
represents the CH contribution to QD nonlinear susceptibility. It differs from the bulk value, mainly, by the time τ inx , which covers the time constants of both SHB and CH. The inhomogenously broadened gain, g ( ω ) , is another aspect of Eq. (24).
3. Nonlinear gain coefficients
The nonlinear gain coefficient depends on the analytical solution of pulse propagation inside QD SOA [3]. For the FWM contribution, we have CDP, SHB and CH. In general, the nonlinear gain coefficient due to CDP is assumed to be equal to unity [9]. Other nonlinear gain coefficients are deduced from the normalized nonlinear susceptibility. Then, the nonlinear gain coefficient for CH is given by,
13
x κCH (αT (ω ) + i ) = − x
X X
CH
CDP Normalized
τ inx
1
(1 − i δτ ) h (1 − i δτ ) =− x
x in
SHB
cn ∂g ( ω ) 2ε 0 n c E 0 α ω ( ) + i x − x T ω ω ∂T 2 (c n )2 τwr ε 0 dg (ω ) 2 E 0 g (ω ) ω0ωq dN w δ τ − i + 1 ( nm SHB )
(
)
2
x ( σ x N w ω − g ( ω ) E GS ,i )
or,
x κCH
σ x N w ω ∂g ( ω ) x x τ E 1 − ( )( ) in GS x ∂T x g (ω ) E GS =− x x dg (ω ) h τwr (1 − i δτ in ) dN w
The ratio (
∂g (ω ) ∂T
x
)/(
dg ( ω ) dN w
( 25)
) is given by [9]:
∂g ( ω ) x x ∂T = − N w E GS K β (T x )2 dg ( ω ) dN w
( 26 )
where K β is the Boltzmann constant. Substituting Eq. (26) into Eq. (25) gives, x N w (E GS )2 τ inx x 2 x x K β (T ) τwr h (1 − i δτ in )
x κCH =
σ x N ω 1 − x g (ω ) E GS
( 27 )
which is the relation of the nonlinear gain coefficient for CH in QD structures. It has a (K βT )−1 dependence since the heat capacity, (h x ) , here is (1 / T ) -dependent
14
x as in Eq. (21). In [9], κCH have (K βT )−3 dependence so, it decrease extensively
with temperature compared with (T
−1
) -dependence, here. Note also that the heat
capacity, here, is a variable dependent gain, QD GS, pump, probe, and conjugate fields, Eq. (21). In our preceding work [24], pulse effect on FWM is detailed. This work deals with derivation and study the main features of CH. The examination of these parameters, including pulse effect on CH and phase-dependent FWM can be taken as separate work dealt with in the near future. This is also justified since it required the study of the overall FWM components which is out of the aim of this work.
4. FWM conversion efficiency
The conversion efficiency of SOA is defined as the ratio between the power of the signal converted at the device-output and the probe-power at the input. In the complete theory of FWM in QD SOAs, as stated elsewhere [24], the specialty of QD structure is taken into account, including ES and inhomogeneity, but an empirical value is used for CH as formulated by Nielsen and Chuang [9]. Now CH in QD is formulated to cover QD specialty as in Eq. (27) above. FWM efficiency is given by [24],
15
ηeff = e
FQD
GQD (τ )
E2 FQD ( −δ , L ) 20 E sat
( 28)
G τ 1 − i α j κ xj e QD ( ) − 1 1 − i α + ∑ ( −δ ) = −C 2 2 1 − i δτ j E0 j = SHB , CH 2ξ + 2 E sat
(
)
(29)
Note that ξ is given by [24],
ξ = −i δ +
NW L D
1 τw 2
2 − D
+
1
τ 2w
2J τw 2τ 1r 2 e Lw N W L
1 1 − ρ ES τw 2 τw 2
+
1
τ 2w
τwr + ( i δτwr
+1
)
( 30 )
where ρ ES is the steady-state value of ES. E sat is the saturated field for the threelevel QD system which can be defined as [24], E sat =
ω
( 31)
dg c n ε 0τwr dN WL
L
The gain integral GQD (τ ) in QD SOAs [GQD (τ ) = ∫ g (z ,τ ) dz ] is then defined as 0
follows [24],
16
dGQD (τ ) dτ
=−
GQD (τ )
τ C ,QD
+Γ
dg dN W L
J N ∫0 e Lw − τC ,wQD0 L
dz
(32)
where N w 0 is the WL carrier density at transparency, Lw is the effective thickness of the active layer, and τ C ,QD represents the effective capture time from WL to the QD ES, τ C ,QD
1 1 = + − ρ ES τw 2 τwr
1 1 + τ 2w τw 2
−1
Note that, the contribution of SHB to FWM in QDs is discussed in detail according to our model elsewhere [24].
5. Calculations, Results and Discussion
The QD structure used in this study is a tenfold InAs QD layer with 5 × 1010 cm −2 per layer covered with InGaAs WL and grown by molecular beam epitaxy at NanoSemiconductor GmbH in Germany. The structural description and parameters are detailed elsewhere [10, 25, 26]. The parameters used in the calculations are given in Table.1 unless otherwise stated. Since the cross-section of the valence x band is zero [4], thus κCH is calculated for the conduction band only and then, κCH is
c refer to κCH and also τ in refers to τ inc , for simplification. Throughout this work the
input pulse used have a width of (1ps ) . 17
The contribution of the CH time constant τ CH was checked through the κCH − τ CH − δ plot in Fig. 1. It is shown that κCH depends on both detuning and τ CH .
Longer τ CH , increases κCH at low detuning (≤ 500 GHz ) , while it decreases at higher detuning. Fig. 2 shows κCH − N W relationship at three τ CH values, where longer τ CH gives higher κCH . At low carrier density, κCH is the same for the three τ CH times. At higher N W , curves of κCH are easily discriminated for the three τ CH times. Fig. 3 shows κCH − τ in relation at the same three τ CH values. For each τ CH value, the curves go to a higher value of κCH . The comparison shows that the curve arrangement in Figs. 2 are contradicted the curve arrangement in Fig. 3. Note that N W is not constant in Fig. 3. To find the reason for this arrangement we return to Eq. (18), where there are three terms which contribute to τ in . Two of them (τ SHB , ρES ) depend on N W . Thus, N W controls the overall nonlinear contributions of τ in . This can also be seen in Fig. 4, where κCH − τ in is plotted with ρ ES as a parameter. The curves show a high dependence on ρ ES values. To study the effect of CH on the occupation, the time series of GS occupation probability ( ρGS ) is shown in Fig. 5, where the rate equations (1) and (2) are solved numerically. Blue curve shows GS occupation dynamics when CH is neglected. The absorption appears in the GS occupation, in Fig. 5, is due to the excitation by an input pulse with a width of (1ps ) . A fast recovery is shown especially for the 18
curve with CH. This may be caused by the additional time added due to CH which increases the recovery. CH is shown to reduce the GS probability as expected [22], but here there are some differences from the previous results in [22], where the overall GS probability is reduced, here. Note that in the earlier study [22], the CH curve returns to the same value, it reaches without CH, after reaching the steadystate. The behaviour found here means that there is another steady-state reached, or another equilibrium is attained, after CH takes place. This is an expected result as we discuss below.
Stability under CH effect
In [22], the CH curve was reached the steady state of that had been reaching without CH although they assumed that the thermal distribution, between the QD and the reservoir, is broken and carriers are removed from QD. Thus, the expected result is that the occupation probability stays at its "new" steady state since carriers are removed from QD. Carrier relaxation from QDs helps in this new steady state but not makes QD return to the steady state which is predicted by the curve without CH. Although QD models neglects ES but, it is evidenced that ES couples WL and GS [27]. It is experimentally reported that intradot relaxation (between GS and ES) is the most relevant process for linear gain and gain compression [28]. Thus, the 19
inclusion of ES in our model helps to see this result. This new steady state is not a bistable regime. It is the "real" steady state reached by the QD system since the earlier steady state (without CH) is higher than this one. There is also another difference from the previous work in [22], where it is shown here that the CH curve recovers more quickly. This is in coincidence with the "first" conclusion that CH makes a new equilibrium reached at a faster time. This also can be reasoned to the inclusion of ES, as referred above. In the previous QW model [15], CH is taken as a constant parameter. The method of modeling CH in QWs is employed empirically for QD structures [9]. This is, of course, not the correct way for formulate CH, but it is adequate for cases of low CH. Fig. 6 shows a comparison between our CH model for FWM in QDs and the previous QW model which is employed into QDs [9]. It is shown that at low WL carrier density the two models coincide, where the earlier QW model is adequate for cases of low CH. This is shown for WL carrier density N W = 1022 m −3 where both curves are coincides. When N W = 3.16 ×1023 m −3 the previous QW model applied to QD (dashed line) is higher than our model (solid line), but they coincides at high detuning (> 100GHz ) . The reason for this coincidence is the Lorentzian approximation, which gives approximate results (at higher detuning) in both models. Increasing carrier density makes our model works well, as shown for the curve when N W = 1025 m −3 , it gives efficiency in the range of (−15dB ) while the 20
calculated efficiency using previous QW model is (> 30dB ) which is overestimates our model. Note that the (30dB ) efficiency is so high, and is not obtained at any QD experiment. The value obtained by our current model is in the range of (−15dB ) . Compared with experimentally obtained results of QDs [29], the results
of our model, as shown in Fig. 6, is in agreement with experiment. Accordingly, our model works within the range of experimental results obtained for QDs. This result can be discussed depending on the conclusion from Fig. 5, i.e. the new "lower" steady state. One can also refer to Eq. (27) where (K βT )−1 dependence reduces κCH in comparison with the dependence in [9] as we refer in section III above. Accordingly FWM efficiency obtained by our model is reduced.
6. Conclusions
The theory of carrier heating in the FWM in QD structures is discussed for the first time. The most influential parameters are examined. Both detuning and carrier heating time constant control CH, which shows more sensitivity to WL carrier density. The derived heat capacity for QD structure have (T
−1
) dependence. CH
has been shown to lead to a new equilibrium distribution, reducing the occupation probability and speeding up recovery in QD structures. Our model works well at high carrier density providing a prediction to experimental work.
21
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27
Tables
Symbol
Value
Unit
0.4
ns
Capture time from WL
τ 1r τw 2
3
ps
Escape time from ES to WL
τ 2w
1
ns
Capture time from ES to GS
τ 21
0.16
ps
Escape time from GS to ES
τ 12
1.2
ps
Decoherence time
τ cv
100
fs
Amplifier length
L
3
mm
Amplifier width
w
2
µm
Thickness of dot layer
Lw
10
nm
Confinement factor
Γ
0.027
Wavelength of pump
λ0
1.3
µm
Injected Current
I
50
mA
Surface density
NQ
1015
m-2
T
300
K
σc
3.5 . 10-22
m-2
Linewidth enhancement factor due CDP
αCDP
0.1
Linewidth enhancement factor due SHB
α SHB
0.1
Linewidth enhancement factor due CH
α CH
1
Parameter
Recombination time
Temperature Conduction band Cross section
28
Figures
0.8
κCH
0.6
0.4
0.2
0 2.1 10
2 x 10
8
-13
6
1.9 τ CH (sec)
4 1.8
2 0
x 10
11
δ (Hz)
Figure 1: A 3-D plot between nonlinear carrier heating parameter κCH and CH time τ in versus detuning δ in a QD-SOA.
29
0.7
0.6
0.5
κ
CH
0.4
0.3
0.2 τ =180 fs CH τ =190 fs CH τ =200 fs
0.1
CH
0 22 10
10
23
10
24
NWL (m-3)
Figure 2: The nonlinear carrier heating parameter κCH versus WL carrier density N W L at three values of CH time τ CH .
30
10
κ
CH
10
10
10
0
-1
-2
-3
τ CH= 180 fs τ CH= 190 fs τ CH= 200 fs
10
-4
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
(sec) in
τ
x 10
2.4 -12
Figure 3: The nonlinear carrier heating parameter κCH versus the effective intraband relaxation time τ in at three values of CH time τ CH .
31
0.07
0.06
0.05
CH
0.04 κ
ρ =0.9984 h ρ =0.9844 h
0.03
0.02
0.01
0 0.9
1
1.1
1.2
1.3
(sec) in
τ
1.4
1.5
1.6 x 10
-12
Figure 4: The nonlinear carrier heating parameter κCH versus the effective intraband relaxation time τ in at two values of ES occupation probability ρ ES .
32
Figure 5: The time series of GS occupation probability ( ρGS ) is shown. GS occupation dynamics neglecting CH is also shown in the blue curve, for comparison. CH is shown to reduce the overall GS probability. The behavior here means that there is another equilibrium attained after CH is taking place- which is the expected result. CH curve has recovered in a faster time.
33
Figure 6: A relation between FWM efficiency ηFW M versus detuning δ at different WL carrier density N W L in a QD-SOA. Dotted curves are for QW model, for comparison.
34