3 B. Optical analogue of population trapping of the optical analogue of population trapping, let us con- sider the monochromatic case, with the only nonvanish- ing spectral component at Ω = 0, and two different input excitations, corresponding the former to single waveguide excitation and the latter to simultaneous excitation of the two channel waveguides. In the former case, assum- ing for instance c$_{1}$ (0 , Ω) = √$_{2}$$_{πδ}$$_{(Ω) and}$ c$_{2}$ (0 , Ω) = 0, one obtains If one of the two channel waveguides, say e.g. W$_{2}$, were removed, light initially injected into waveguide W$_{1}$ would decay into the slab waveguide, a phenomenon fully analogous to the quantum mechanical decay of a bound state coupled to a continuum. In the markovian approxi- mation, valid for weak coupling and for a nearly unstruc- tured continuum, the decay is well described by an ex- ponential law. The presence of waveguide W$_{2}$ generally modifies the decay behavior and, under certain condi- tions, the decay can be suppressed owing to a destructive Fano-like interference of different decay channels. Full or fractional suppression of the decay is related to the ap- pearance of a trapped (or dark) state in the continuum. To derive the decay laws of light waves in the channel waveguides W$_{1}$ and W$_{2}$, we follow a standard procedure [13], detailed for instance in Ref.[9], and eliminate the amplitudes c$_{k}$ of continuous modes by a formal integra- tion of Eq.(7) with the initial condition c$_{k}$ (0 , Ω) = 0. This yields a set of two coupled integro-differential equa- tions for the amplitudes c$_{1}$ and c$_{2}$ of discrete modes. In the weak coupling limit and assuming a nearly unstruc- tured continuum, the markovian approximation can be made and the following reduced equations for amplitudes c$_{1}$ and c$_{2}$ are derived ψ ( ρ, z ) = S$_{11}$ ( z ) u$_{1}$ ( ρ )+ S$_{12}$ ( z ) u$_{2}$ ( ρ )+ S$_{13}$ ( z ) θ ( ρ, z ) (15) where the last term on the right hand side of Eq.(15) accounts for the light field tunnelled into the slab waveg- uide and θ ( ρ, z ) ia a suitable superposition of continuous modes u$_{k}$ ( ρ ) normalized such that ∫ dρ | θ ( ρ, z ) | 2 = 1. For power conservation, the relation | S$_{11}$ | $^{2}$+ | S$_{12}$ | $^{2}$+ | S$_{13}$ | 2 = 1 then holds. Note that, after a propagation distance z a few times the decay length l$_{d}$ ≡ (1 /σ ), one has | S$_{11}$ | 2 = | S$_{12}$ | 2 = 1 / 4 and | S$_{13}$ | 2 ≃ 1 / 2, i.e. half of the injected light power has decayed into the slab, whereas the other half of light power is equally dis- tributed into the two channel waveguides. The fact that the decay is not complete (fractional decay) indicates that a bound state embedded in the continuum does ex- ist. When both waveguides W$_{1}$ and W$_{2}$ are initially ex- cited with coherent fields of equal amplitudes but op- posite sign, i.e. c$_{1}$ (0 , Ω) = − c$_{2}$ (0 , Ω) = √$_{2}$$_{πδ}$$_{(Ω), one}$ obtains ψ ( ρ, z ) = [ u$_{1}$ ( ρ ) − u$_{2}$ ( ρ )] exp( − i Δ β$_{0}$z ), i.e. the decay into the slab waveguide is fully suppressed. This is due to a destructive interference effect between dif- ferent decay channels when c$_{2}$ = − c$_{1}$ [see Eqs.(7) and (8)] and to the existence of a trapped state embedded in the continuum. Numerical examples of fractional light decay for single waveguide excitation, and of full decay suppression for simultaneous waveguide excitation in the trapped state, as obtained by a direct numerical analy- sis of Eq.(1) in the monochromatic regime, are shown in Fig.2. In the simulations, we assumed circular channel waveguides with a Gaussian index core profile of radius r$_{c}$ (at 1 /e ), and a step-index slab waveguide of thick- ness a . Equation (1) has been integrated by a standard split-step pseudospectral method with absorbing bound- ary conditions [9]. Generalization of light propagation in the non- monochromatic case simply follows from the superposi- tion principle. For instance, if waveguides W$_{1}$ and W$_{2}$ are excited at the input plane by two pulses with en- velopes r$_{1}$ ( t ) and r$_{2}$ ( t ), from Eqs.(4), (10) and (11) it follows that the field envelope ψ ( ρ, z, t ) can be cast in the form Note that, in the quasi-monochromatic approximation assumed in this work, the matrix coefficients S$_{n,l}$ are in- dependent of frequency Ω. To understand the appearance is the decay rate of the single channel waveguide into the continuum. The solution to Eqs.(8) reads explicitly ∂c$_{1}$ ∂z = − i Δ β$_{0}$c$_{1}$ − σ ( c$_{1}$ + c$_{2}$ ) , ∂c$_{2}$ ∂z = − i Δ β$_{0}$c$_{2}$ − σ ( c$_{1}$ + c$_{2}$ ) , (8) where where σ = ∫ ∞ 0 dτ ∫ dk | g ( k ) | $^{2}$exp {− i [Δ β ( k ) − Δ β$_{0}$ ] τ } (9) c$_{1}$ ( z, Ω) = S$_{11}$ ( z ) c$_{1}$ (0 , Ω) + S$_{12}$ ( z ) c$_{2}$ (0 , Ω) (10) c$_{2}$ ( z, Ω) = S$_{21}$ ( z ) c$_{1}$ (0 , Ω) + S$_{22}$ ( z ) c$_{2}$ (0 , Ω) (11) S$_{11}$ ( z ) = 1 2 exp( − i Δ β$_{0}$z ) [1 + exp( − 2 σz )] (12) S$_{12}$ ( z ) = − 1 2 exp( − i Δ β$_{0}$z ) [1 − exp( − 2 σz )] (13) S$_{22}$ ( z ) = S$_{11}$ ( z ) , S$_{21}$ ( z ) = S$_{12}$ ( z ) . (14)