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where A 1 and A 2 are two areas in the transverse ( x, y ) plane surrounding waveguides W$_{1}$ and S, respectively. The correlation function P ( t$_{1}$, t$_{2}$ ) is proportional to the joint probability of detecting one photon in waveguide W$_{1}$ at time t$_{1}$ , and one photon in waveguide S at time t$_{2}$ , after a propagation distance z from the input plane. In fact, 〈Q| ˆ ψ $^{†}$( ρ$_{2}$, t$_{2}$ ) ˆ ψ $^{†}$( ρ$_{1}$, t$_{1}$ ) ˆ ψ ( ρ$_{1}$, t$_{1}$ ) ˆ ψ ( ρ$_{2}$, t$_{2}$ ) |Q〉 is pro- portional to the joint probability of detecting one photon at point ρ$_{1}$ and time t$_{1}$ , and one photon at point ρ$_{2}$ and time t$_{2}$ at the same propagation distance z . The integral over the areas A 1 and A 2 thus gives the joint probabil- ity of detecting one photon in waveguide W$_{1}$ at time t$_{1}$ , and one photon in waveguide S at time t$_{2}$ . For photons generated by spontaneous parametric down conversion in type-I nonlinear crystal using a monochromatic pump beam at frequency ω$_{p}$ = 2 ω , the two-photon spectrum C (Ω$_{1}$ , Ω$_{2}$) of a pair of signal and idler photons can be expressed as [18, 34] 

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where G (Ω$_{1}$ , Ω$_{2}$) is the phase matching function. The last exponential term on the right hand side in Eq.(51) has been introduced to account for a possible time de- lay δ between signal and idler wave packets introduced by different optical paths from the crystal to waveguides W$_{1}$ and W$_{2}$. The phase matching function G (Ω$_{1}$ , Ω$_{2}$) is assumed to be a real-valued and symmetric function [i.e. G (Ω$_{1}$ , Ω$_{2}$) = G (Ω$_{2}$ , Ω$_{1}$)], peaked at around Ω$_{1}$ = Ω$_{2}$ = 0, with e.g. a Gaussian profile [18]. Taking into account that ˆ ψ ( ρ, t ) = (2 π ) − 1 / 2 ∫ d Ω ˆ φ ( ρ, Ω) exp( − i Ω t ), substitution of Eqs.(48) and (51) into Eq.(50), using the commutaion relations [ ˆ φ ( ρ, Ω) , ˆ a † 1 , $_{2}$(Ω $^{′}$)] = u$_{1}$$_{,}$$_{2}$ ( ρ ) δ (Ω − Ω $^{′}$), [ ˆ φ ( ρ, Ω) , ˆ a † $_{3}$(Ω $^{′}$)] = θ ( ρ, z ) δ (Ω − Ω $^{′}$) and the re- lations ∫ A 1 dρ | u$_{1}$ ( ρ ) | 2 ≃ 1, ∫ A 2 dρ | θ ( ρ, z ) | 2 ≃ 1, ∫ A 1$_{,}$2 dρ | u$_{2}$$_{,}$$_{1}$ ( ρ ) 2 = ∫ A 1 dρ | θ ( ρ, z ) | 2 = ∫ A 2 dρ u$_{2}$ ( ρ ) | 2 ≃ 0, after some lengthy but straightforward calculations one obtains 

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where τ = t$_{2}$ − t$_{1}$ and where we introduced the real-valued correlation function r ( τ ) defined by 

In practice, coincidence measurements correspond to an integration of P ( t$_{1}$, t$_{2}$ ; z ) with respect to the time dif- 

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ference τ = t$_{2}$ − t$_{1}$ over the resolving coincidence time, which is typically much longer than the correlation time τ$_{c}$ of g ( τ ). Integrating Eq.(52) with respect to τ from −∞ to ∞ and taking into account that S$_{21}$ = S$_{12}$ and | S$_{23}$ | 2 = | S$_{13}$ | 2 = 1 − | S$_{11}$ | $^{2}$− | S$_{12}$ | $^{2}$, the following expres- sion for the correlation function P versus time delay δ is finally obtained 

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where α = ∫ ∞ −∞ dτ r $^{2}$( τ ) and the expression of the co- efficients S$_{11}$ ( z ) and S$_{12}$ ( z ) are given by Eqs.(12) and (13). The behavior of P ( δ ; z ) versus δ shows a charac- 

teristic dip at δ = 0 of width ∼ τ$_{c}$ , which is analogous to the Hong-Ou-Mandel dip observed in two-photon inter- ference from a beam splitter [18]. Far from the dip, P ( δ ) 

