D ARK SOLITON STEERING IN K ERR P T COUPLER WITH DISPERSIVE PERTURBATION
5.2 Theoretical Model
Loss Gain In 1
In 2
Out 1
Out 2 (b)
Gain Loss In 1
In 2
Out 1
Out 2 (a)
Figure 5.1:Schematic illustrations of two different configurations ofP T-symmetric coupler: (a) Type-1 and (b) Type-2P T-symmetric coupler.
each with different propagation constants. When optical power is launched into one of the two cores, both the modes insideNLDCare equally excited. Thus, different group delay between these two modes during propagation along the coupler length leads to a phenomenon called theIMD. In the coupled mode theory, this group delay between the modes is analogous to the coupling-coefficient dispersion. Previously it has been reported that both TODand IMDhave no effect on dark soliton [87,90,91] but they have considerable influence on bright soliton propagation inside a conventional coupler [92,93, 94, 95,96,97]. Recently a new kind of dark soliton has been introduced in a dual-core coupler [98] in the presence of such dispersive coupling. Also, it needs to be mentioned here that an optical system combined with IMDis known to be invariant underC P T symmetry, whereC symbolizes transverse spatial inversion related to the swapping of the two waveguides inside a coupler [99,100,101,102].
In this work, intrigued by the steady nature of the dark soliton, the steering dynamics of dark soliton inP T-symmetric couplers has been explored.
5.2 Theoretical Model
In general, soliton propagation inside anNLDCcan be represented by a pair of coupled NLSEs. Thus, the soliton dynamics inside aP T-symmetric coupler with balanced gain and loss can be expressed mathematically as [26]:
i∂Ψ1
∂ξ −1 2
∂2Ψ1
∂τ2 −iδ3∂3Ψ1
∂τ3 + |Ψ1|2Ψ1+κΨ2+iκ1∂Ψ2
∂τ =iΓΨ1 (5.1) i∂Ψ2
∂ξ −1 2
∂2Ψ2
∂τ2 −iδ3∂3Ψ2
∂τ3 + |Ψ2|2Ψ2+κΨ1+iκ1∂Ψ1
∂τ = −iΓΨ2 (5.2) where,ψ1andψ2denote the slowly varying complex-valued envelopes in the bar (core 1) and the cross channel (core 2) of theP T-symmetric coupler. The second and the third
Bar Channel Port 1
Port 2
Port 3
Port 4 ξ=0
Cross Channel ξ= Lc Loss
Gain Coupling length Lc
Figure 5.2:Schematic diagram portraying steering dynamics of dark solitons inside P T-symmetric couplers.
term of Eqs. (5.1) and (5.2) represent theGVDand theTOD, respectively. The fourth term is theSPMterm, where its coefficient is scaled to be one.κ,κ1,δ3are, respectively, the normalized linear coupling coefficient,IMDandTOD. Also,Γrepresents balanced gain (Γ>0) and loss (Γ<0) inside two channels making the system to beP T-symmetric.
It should be noted that the considered model is not applicable to coupler made of non- Kerr media such as organic, photorefractive and semiconductor materials. Again, the waveguides consisting of the coupler are assumed to be single-moded. If one has to consider coupler with multimoded waveguides, a different model needs to be considered.
In fact, this is a new and emerging field of research. However, in the current work we are primarily concerned with silica-based optical fiber or coupler made with materials exhibiting Kerr nonlinearity.
In the following, we illustrate the effect of inclusion of gain and loss by two different configurations. WhenΓ>0 in Eqs.(5.1) and (5.2), the system is termed as type-1 P T- symmetric coupler, having gain in the first waveguide and loss in the second Fig.5.1(a).
The other structure is termed as type-2P T-symmetric coupler withΓ<0 in the coupled equations, having loss in the first waveguide and gain in the second Fig.5.1(b). These two configurations bearing the same notion ofP T symmetry are introduced in order to highlight which model among these two (gain/loss and loss/gain) exhibits rich steering dynamics, as transmission property may differ since gain and loss have been swapped inside the waveguides [26]. It is worthwhile to stress again that the first observation of P T symmetry in optics was accomplished via a judicious design of a coupled waveguide which involved two-wave mixing process in order to provide gain and doping of transition metal ions to provide loss [6]. It could be possible to adopt the similar scheme to realize the present model for fiber coupler. Specifically, at 1550 nm, the gain can be provided by doping rare earth element like erbium in one channel and the loss by another dopant in
5.2 Theoretical Model
the second channel which shows absorption in the same wavelength regime. But as we are dealing with the dark soliton, a dispersion-shifted fiber or a dispersion-compensated fiber having normal dispersion at 1550 nmcan be optically pumped and utilized as the gain channel.
As theNLDCcan serve a potential candidate for all-optical switching, we mainly aim our attention to power-controlled steering dynamics of dark soliton insideP T-symmetric couplers following with the phase-controlled one. In general, anyP T-symmetric system exhibits two parametric regions, a region of unbrokenP T symmetry where all eigenval- ues are real and a region of broken symmetry where some of the eigenvalues are real and the rest are complex. Following the notion,P T-symmetric couplers are said to be operating in unbrokenP T-symmetric regime whenκ>Γand in brokenP T-symmetric regime whenκ<Γ[6]. If the value of coupling coefficient equals to the value of gain/loss parameter, i.e.,κ=Γ, the system is in singularity condition. In this paper, we limit our sys- tem to work in the unbrokenP T-symmetric regime (κ>Γ) and scale the linear coupling coefficientκand the gain/loss parameter,Γ, to be 1 and 0.5, respectively. As shown in Fig.5.2, we also assume that the second waveguide has been kept empty (Ψ2(ξ=0,τ)=0) and the input soliton pulse is always launched at port 1 of first waveguide, i.e.,
Ψ1(ξ=0,τ)=q tanh(qτ) (5.3)
where,q2=P0represents the input peak power, depending on which the dark soliton switches to any of the two output ports. To illustrate the effects of TOD andIMDon the dark soliton propagation inP T-symmetric couplers, Eqs.(5.1) and (5.2) are then numerically solved by the split-step Fourier method [90]. Here we calculate the fractional output power in the j-th core (j=1, 2) by the transmission coefficient which is represented as,
Tj=
R∞
∞ |Ψj(Lc,τ)|2dτ R∞
−∞(|Ψ1(Lc,τ)|2+ |Ψ2(Lc,τ)|2)dτ= Pj
P1+P2 (5.4)
where,P1=R∞
−∞|Ψ1(Lc,τ)|2dτand P2=R∞
−∞|Ψ2(Lc,τ)|2dτare the output powers of the transmitted pulse in the ports 3 and 4 of the coupler. The parameter Lc refers to the total coupling length of the system.