Abstract

We investigate parametric down-conversion in a hexagonally poled nonlinear photonic crystal, pumped by a dual pump with a transverse modulation that matches the periodicity of the χ(2) nonlinear grating. A peculiar feature of this resonant configuration is that the two pumps simultaneously generate photon pairs over an entire branch of modes, via quasi-phase matching with both fundamental vectors of the reciprocal lattice of the nonlinearity. The parametric gain of these modes depends thus coherently on the sum of the two pump amplitudes and can be controlled by varying their relative intensities and phases. We find that a significant enhancement of the source conversion efficiency, comparable to that of one-dimensionally poled crystals, can be achieved by a dual symmetric pump. We also show how the four-mode coupling arising among shared modes at resonance can be tailored by changing the dual pump parameters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Since their proposal by Berger [1] and their first realization [2], χ(2) nonlinear photonic crystals (NPC) with a two-dimensional poling pattern have attracted great interest due to their potential applications in nonlinear optics [3–9], as well as in quantum optics for the generation and engineering of entangled states [10–15]. Considering parametric down-conversion (PDC) from an intense pump beam into twin photons or twin beams of lower frequencies, the vectors of the 2D reciprocal lattice associated to the nonlinear grating indeed offer multiple quasi-phasematching (QPM) possibilities not encountered in more conventional one-dimensional structures. Specifically, the source emission spectrum is characterized by special points, the so called “shared modes” lying at the interception of different QPM branches associated to non collinear lattice vectors [15–19]. In the stimulated regime of PDC, these shared modes are cross-seeded by two coupled modes simultaneously, with a significant enhancement of the parametric gain [15, 16]. These three-mode interaction processes involving the shared modes appear therefore in the source far field as localized bright spots against a more diffused background coming from standard two-mode PDC.

A recent analysis for a hexagonally poled crystal [14, 15] demonstrated a peculiar spatial resonance, reached by tilting the pump angle till its phase modulation matches the periodicity of the poling pattern: this enforces a transition from three- to four-mode entanglement among shared modes, the latter being dominated by the Golden Ratio of the segment ϕ=(1+5)/2. In the stimulated PDC regime, a sudden boost of the intensity of shared modes takes place as the pump incidence angle is tuned to resonance [15]. The quantum aspects of the underlying entangled state have been investigated in [14].

In this work we shall explore the possibility of modulating the intensity of the pump beam in the transverse plane, rather than only its phase, through the use of a dual pump. Coherent coupling in χ(2) materials via dual pumping has been investigated both in a bulk crystal [20], and in a one-dimensionally poled QPM structure [21], with the realization of self-diffraction in the first case, and resonant cascaded four-wave mixing in the second case.

In the 2D nonlinear photonics crystal we consider here, the addition of a second pump wave increases the complexity of the source spectrum, due to the increased QPM opportunities involving the two pump modes and the two vectors of the hexagonal reciprocal lattice. We shall focus on the condition of spatial resonance between the pump modulation and the nonlinear pattern, and demonstrate some unique features not encountered in the conventional single pump configuration.

A first noticeable feature is the existence of an entire branch of modes (a 3D surface in the Fourier domain) in which photon pairs are down-converted from both pumps simultaneously, quasi-phase matched by two fundamental vectors of the reciprocal lattice. We shall show that the parametric gain of this QPM branch depends on the coherent sum of the two pump amplitudes |α1+α2| and can be controlled by varying their relative intensities and phases. In particular, the use of a dual symmetrical pump (α1=α2) leads to a substantial increase of the gain compared to a single pump beam of the same energy. As a result, the dual pump configuration may in principle bring the source conversion efficiency close to the level of 1D poled crystals of the same material, compensating at least partially for the lower effective nonlinear coefficients associated to two-dimensional poling patterns [22].

We also show that a dual pumping allows a control over the four-mode processes characterizing the spatial resonance. Maximum enhancement of the shared mode intensity is obtained with two symmetric pumps, while the four-mode coupling degenerates into two independent two-mode processes of reduced gain for anti-symmetric pumps (α2=α1). A dual pump with controlled relative phases and intensities can therefore be used to tailor the multimode coupling among shared modes and, in the quantum domain, to engineer interesting quantum states, which will be the subject of a related investigation [23].

2. Model

Our description is based on a model similar to that in [14]. Even if our analysis may apply to various configurations, we focus here on degenerate down-conversion around λs=1064nm from a pump at λp=532nm, taking place in a hexagonally poled Lithium Tantalate (LiTaO3) slab similar to the one in [10].

Figure 1 shows the arrangement: the χ(2) crystal is hexagonally poled in the (x,z) plane orthogonal to the crystal optical axis (the crystal optical axis corresponds to the yaxis of the reference frame used in this work, shown in Fig. 1(a)). We consider type 0 phase-matching e,e, where both the pump and the down-converted field are extraordinarily polarized along the optical axis, and propagate mainly in the (x,z) plane, forming small angles with a mean propagation direction (the z-axis in the figure). In particular the pump beam consists of two waves slightly and symmetrically tilted with respect to the zaxis, giving rise to a spatial transverse modulation of the pumping of the medium. When the transverse modulation of the pump matches the one of the nonlinear pattern (Fig. 1(b)), a condition of spatial resonance is achieved. In such conditions, quasi phase-matching at the degenerate wavelength 2λp is achieved when the – poling period of the two-dimensional pattern is Λ=2πGz=7.782μm at a temperature of 85°C, according to the Sellmeier relations in [24]. These are the conditions chosen for the numerical simulations of the system that will be shown in the following.

The two-dimensional hexagonal pattern of the nonlinearity is described by keeping only the leading order terms of the Fourier expansion of the nonlinear-susceptibility d(x,z) [1]. Denoting with G1=GxexGzez and G2=+GxexGzez the two fundamental vectors of the reciprocal lattice which provide quasi-phase matching, we have accordingly

d(x,z)eiGzz[d01eiGxx+d10eiGxx]=2d01eiGzzcos (Gxx)

 figure: Fig. 1

Fig. 1 a) Hexagonally poled χ(2) crystal pumped by two waves, symmetrically tilted with respect to the z-axis. b) A spatial resonance is achieved by matching the pump transverse wave-vectors±qp with the transverse components ±Gxex of the lattice vectors G1 and G2.

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Considering e.g. inversion domains shaped as discs of radius R=0.28Λ (Fig. 2(a)), one has d01=d10=0.32deff [22], deff denoting the effective nonlinear susceptibility of the χ(2) material.

In the degenerate type 0 case, the signal and idler modes belong to the same wave-packet of central frequency ωs=ωp/2. The signal and pump envelope operators, denoted by A^s and A^p respectively, satisfy the following coupled propagation equations in the Fourier domain (see [14] and [25, 26])

zA^s(ws,z)=χd3wp(2π)32A^p(wp,z)[A^s(wpwsGx,z)eiD(ws,wpwsGx)z+A^s(wpws+Gx,z)eiD(ws,wpws+Gx)z]
zA^p(wp,z)=χ2d3ws(2π)32A^s(ws,z)[A^s(wpwsGx,z)eiD(ws,wpwsGx)z+A^s(wpws+Gx,z)eiD(ws,wpws+Gx)z]
where wj=(q,Ω), j=s,p, denotes the coordinate in the 3D Fourier space of the jth field, with q=(qx,qy) being the transverse component of the wave-vector, and Ω the offset frequency from the corresponding carrier frequencies ωs and ωp=2ωs. Gx=(Gx,0,0) is a short-hand notation for the x-component of the reciprocal lattice vector in the 3D Fourier space, and χd01ωpωs28ϵ0c3npns2. The two terms at r.h.s of Eq. (2b) describe all the possible three-photon processes wpws,wi=wpws±Gx mediated by the lattice vectors G1 and G2 respectively. Notice that for a given pump mode wp=(qp,Ωp), the energy and transverse momentum conservation imply that ws+wi=wp±Gx, i.e. Ωs+Ωi=Ωp, and qs+qi=qp±Gxex, the latter condition deriving from the simplifying assumption that the crystal is indefinitely extended in the transverse plane. On the other hand, since we are considering a crystal of finite length lc along the z-direction, the longitudinal momentum conservation is less stringent, and is expressed by the phase-matching functions appearing at the r.h.s. of Eqs. (2a) and (2b)
D(ws,wpws±Gx)=[ksz(ws)+ksz(wpws±Gx)kpz(wp)+Gz],
where kjz(wj)=kj2(wj)q2 is the z-components of the wave-vectors associated to mode wj, the wave number kj(wj)=ωj+Ωcnj(wj) being determined by the linear dispersion relation of the j-th wave in the medium. Notice that since all the fields are extraordinarily polarized, their refractive index in principle depends on the propagation direction through qy: nj(wj)=ne(ωj+Ωj,qy). However, spatial walk-off is negligible since the fields propagate nearly at π/2 from the optical axis, and more in general Lithium Tantalate is characterized by a very small birefringence [27]. Thus we shall neglect the crystal anisotropy both in analytical calculations and in the numerical simulations, letting nj(wj)=ne(ωj+Ω,qy=0), where ne is the crystal extraordinary refractive index in the lattice plane.

3. Dual plane-wave pump

In view of obtaining analytical results, we consider the parametric limit where the pump field in Eqs. (2) is treated as a classical coherent field undergoing negligible depletion during propagation. With respect to the pure phase modulation discussed in [14, 15], where a spatial resonance was realized by tilting a single pump wave, we assume here that the pump consists of two monochromatic plane-waves of frequency ωp, symmetrically tilted from the z-axis in the lattice plane, as shown in Fig. 1. Thus, our analysis includes the possibility of an intensity modulation of the pump. Denoting as q0p=±q0pex the pump transverse wave-vectors, in the direct space Ap(x,t)=α1eiq0px+α2eiq0px, while in the Fourier domain

Ap(q,Ωp)=dt2πdx2πeiΩtiqxAp(x,t)=(2π)3/2δ(Ω)δ(qy)[α1δ(qxq0p)+α2δ(qx+q0p)]

Quasi-phasematching is achieved for signal modes belonging to one of the four distinct QPM surfaces in the ws-space:

Σ11:D(ws,w0pws+Gx)=0[qs+qi=(q0p+Gx)ex]
Σ12:D(ws,w0pwsGx)=0[qs+qi=(q0pGx)ex]
Σ21:D(ws,w0pws+Gx)=0[qs+qi=(q0pGx)ex]
Σ22:D(ws,w0pwsGx)=0[qs+qi=(q0p+Gx)ex]
where ±w0p=(±q0p,qy=0,Ω=0) denote the 3D Fourier components of the two pump modes and the relations within the graph parentheses give the transverse momentum conservation associated to each QPM surface. We focus here on the condition of spatial resonance, obtained by matching the pump transverse modulation to that of the nonlinear lattice, i.e. setting
q0p=GxAp(z,x,t)=α1eiGxx+α2eiGxx

In this case Σ12 and Σ21 merge into the single surface

Σ12,Σ21Σ0:D(ws,ws)=0

It is important to stress that photon pairs belonging to Σ0 originate simultaneously from both pump modes and are quasi-phase matched by both vectors G1 and G2 of the grating of the nonlinearity. On the other hand, the two other QPM surfaces

Σ11:D(ws,ws+2Gx)=0,
Σ22:D(ws,ws2Gx)=0
are populated only by down-conversion either from pump 1 with the contribution of the lattice vector G1 (photon pairs appearing on Σ11), or from pump 2 mediated by the lattice vector G2 (photon pairs appearing on Σ22).

Substituting Eq. (4) into Eq. (2a), one obtains the following propagation equation

A^sz(ws)=(g1+g2)A^s(ws)eiD(ws,ws)z+g1A^s(ws+2Gx)eiD(ws,ws+2Gx)z+g2A^s(ws2Gx)eiD(ws,ws2Gx)z
where the complex parameters g1=χα1 and g2=χα2 represent the parametric gain per unit length associated to each pump mode. The first term at r.h.s. of Eq. (10), proportional to the sum of the two pump amplitudes α1+α2, account for PDC processes taking place on the QPM surface Σ0, while the last two terms are associated to the QPM surfaces Σ11 and Σ22 respectively. Equation (10) for a given mode ws has to be considered together with the propagation equations for the three coupled modes A^s(ws) and A^s(ws±2Gx) appearing at its right hand side, obtaining in principle an infinite chain of coupled equations for modes of all harmonic orders ±ws+nGx (see [14] for details). However, by inspecting which modes are effectively phase-matched to the considered signal mode ws, the chain can be truncated to finite sets of equations, obtaining thereby coupled equations for either 2-mode or 4-mode processes that will be discussed in the next sections.

 figure: Fig. 2

Fig. 2 Quasi phase-matching in a hexagonally poled LiTaO3 crystal pumped at 532nm by a dual pump at spatial resonance with the lattice: q0pex=Gxex, with Gx=2π/(3Λ)=0.466μm1. (a) QPM surfaces in the 3D Fourier space, and (b) its section at qy = 0.

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Figure 2 shows an example of the three QPM surfaces in the 3D Fourier space of the signal modes, obtained by numerically solving Eqs. (7)(9) by means of the Sellmeier formula reported in [24]. The period of the poling pattern is chosen to achieve QPM at degeneracy for the shared signal modes emitted at qx=±Gx,qy=0, as explained in App. A [see Eq. (32)]. In the same appendix we derive general expressions for these QPM surfaces valid within the paraxial approximation and not too far from degeneracy. In particular, for the chosen crystal parameters, Σ11 and Σ22 are well approximated by the two biconical surfaces given in Eqs. (33) and (34), having their vertexes at ±Gxex. Their projections onto the plane (λ,qx) (Fig. 2(b)), are two X-shaped curves symmetrically displaced along the qx axis, describing parametric emission collinear with each of the two tilted pumps. Within the same approximations, Σ0 has equation |qs|(Gx2+ksks Ωs2)1/2, corresponding approximately to a wide tube, centered at q=0, describing noncollinear PDC emission at an angle θGx/ks around the zaxis. Notice that the Σ0 surface includes the vertexes of the two cones.

3.1. Two-mode processes

Let us consider signal modes for which only one of the three QPM conditions described by Eqs. (7)-(9) is satisfied. These represent the vast majority of modes, with the remarkable exception of modes lying at the intersections of two distinct QPM surfaces, which will be analysed in the next section. Equation (10) reduces then to standard two-mode parametric equations of the form:

A^sz(ws)=γg¯eiϕ1A^s(wi)eiD(ws,wi)z
A^sz(wi)=γg¯eiϕ1A^s(ws)eiD(ws,wi)z
where wi=ws (for modes on the surface Σ0) or wi=ws±2Gx (on the surfaces Σ11,Σ22), represents the twin idler mode coupled to ws. Simple inspection of the different terms at the r.h.s. of Eq. (10) allows to determine how the gain of these 2-modes processes compares to the gain from a single pump mode with the same overall intensity |α|2=|α1|2+|α2|2. To this end, we set
g¯=|g1|2+|g2|2
representing the gain one would have by concentrating all the energy in a single pump. According to the standard solution of the parametric Eqs. (11) the number of photons of the phase-matched modes on each QPM surface grows as Nsinh2(|γ|g¯z) where, by introducing the complex parameter r=g2g1 (without loosing generality we assume that pump 1 is always present, and that |g2/g1|1),
γ={g1+g2g¯=1+r1+|r|2Σ0g1g¯=11+|r|2Σ11g2g¯=r1+|r|2Σ22

 figure: Fig. 3

Fig. 3 Photon-number distribution in the (qx,qy)-plane (top) and in the (λ,qx)-planes (bottom) for a single pump (a,d), two symmetric pumps (b,e) and two antisymmetric pumps (c,f), from numerical simulations of Eqs. (1), in the same NPC of Fig. 2. The pumps are plane-waves, g¯=0.4 mm 1, and results are shown after 7mm of propagation (g¯z=2.8). In (d,e) the scale was truncated to 1% of the peak value. For a dual symmetric pump the Σ0 branch is significantly more intense than for a single pump of equal energy, while it is absent for antisymmetric pumping. Lines of hot spots at qx=±Gx are clearly visible in panels (a) and (b).

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A striking feature of the spatial resonance (6) is that the parametric gain of modes belonging to the two-fold degenerate surface Σ0 can be controlled by varying the relative intensities and phases of the two tilted pumps. In particular:

  1. Maximum gain on Σ0 is achieved when the two pumps have the same intensity and phase (r = 1 in Figs. 3(b) and 3(e)): this symmetric configuration provides a gain enhancement by a factor γ=2 with respect to the use of a single pump of equivalent intensity. In the stimulated regime g¯z>1, this leads to a huge increase of the intensities of the Σ0 modes, as clearly shown by the results in Fig. 4: e.g. in Fig. 4(a) for g¯z=4 the Σ0 modes are roughly 30 times more intense when a dual pump is used instead of a single pump. (notice also the different scales of Fig. 3(a) and 3(b)). We stress that this gain enhancement affects also the PDC emission in the spontaneous regime g¯z1, in which splitting the available energy into a dual pump doubles the intensity of Σ0. This occurs because the resonant structure of the pump coherently couples the processes arising from the vectors G1 and G2 of the nonlinear pattern. Thus the use of a dual pump at resonance brings a net increase of the efficiency of parametric generation in photonic crystals, over a whole surface of QPM modes. This is quite different from typical configurations of PDC in nonlinear photonic crystals involving a single pump wave, where shared modes appear as isolated hot spots at the geometrical intersection of different QPM branches [15, 16]. Conversely, the gain on the side branches Σ11 and Σ22 is reduced by a factor 2.
  2. The Σ0-modes can be switched off by taking two antisymmetric pumps, with the same intensity and a π phase difference (case r = −1 shown in Figs. 3(c) and 3(f)). Notice that in this case the overall efficiency of quasi-phase-matching with the first order vector of the nonlinear grating is greatly reduced, so that one should consider also the contribution from higher order harmonics of the grating. The relative weight of those secondary PDC processes is however strongly dependent on the motif characterizing the poling grating and will not be discussed in this work.
  3. For a single pump, corresponding to a pure phase modulation, the Σ0 and Σ11 modes grow with the same gain g¯ and have the same intensity (r = 0 case in Figs. 3(a) and 3(d)). Clearly the Σ22 modes are in the vacuum state because the second pump wave is absent. This configuration is similar to that analysed in [14] and experimentally demonstrated in [15].

The numerical results shown in Figs. 3 and 4 fully confirm the analysis from the parametric model. They are obtained from 3D+1 stochastic simulations of the nonlinear propagation Eqs. (2). Numerical integration was performed using a pseudo-spectral (split-step) method in the framework of the Wigner representation, where the field operators are replaced by c-number fields (see e.g. [28]). The input signal field, in the vacuum state, is simulated by Gaussian white noise, while the pump beam is a stationary coherent field with the transverse spatial modulation required by the resonance condition Eq. (6). The two waves forming the dual pump are either plane-waves or have an elliptical Gaussian profile with waists 500μm x 200μm along the x and y axis. The numerical grid size, 512 × 256 × 256 along the x, y and temporal axis in the plane-wave pump case, was doubled along the x dimension for the more demanding Gaussian pump case. Figure 3 shows the intensity distributions of the down-converted field after 7 mm of propagation in the NPC, comparing the cases r = 0, r = 1 and r = −1. The bright lines of hot spots at qx=±Gx, originate from the 4-mode processes that will be discussed in the next section. Incidentally, despite the brightness of the hot-spots, for the chosen gain g¯=0.4mm1 we verified that the pump remains almost undepleted. Figure 4 plots the logarithm of the mean number of photons sampled on the QPM branches Σ0 and Σ11 as a function of g¯z along the crystal, comparing the use of a single pump versus a dual symmetric pump with the same total energy. The gain enhancement factors γ are estimated by fitting the numerical data, averaged over portions of the corresponding QPM branches far away from the hot-spots, with the prediction of the 2-mode parametric model (3.1) according to which N=sinh2(γg¯z). Precisely, they are obtained from linear fits of the approximated relation log (N)2γg¯z valid for g¯z1 (black lines). We verified that a nonlinear fit with the function N=sinh2(γg¯z) provides similar results. For plane-wave pumps, the values of γ inferred in this way are in good agreement with the analytical predictions (13). For Gaussian pumps, the effective gain is significantly reduced in all the cases, but the ratio of the gains for single and dual pumps are roughly preserved. In particular, a dual Gaussian pump provides an enhancement of the gain of the Σ0 modes by a factor 1.38 compared to a single-pump with the same energy, very close to the 2 factor predicted for plane-wave pumps. The number of generated photons on Σ0 at the crystal output, for g¯z=4, correspondingly becomes about two orders of magnitude larger.

 figure: Fig. 4

Fig. 4 Comparison between the use of a dual symmetric pump (red triangles) and a single pump (blue squares) with the same energy. The gain enhancement factor γ is evaluated from numerical simulations of Eqs. (1), The results for plane-wave pumps in (a,b) are very close to the γth predicted by the parametric model [Eq. (13)]. The lower panels (c,d) are obtained for Gaussian pumps, of waists 500μm and 200μm along the x and y axis. Other parameters as in Fig. 3.

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3.2. Shared mode processes

Modes belonging to the interceptions of two QPM surfaces play a special role, since two nonlinear processes concur to the generation of photons, leading to a local enhancement of the gain and to hot spots in the PDC emission [15–18]. Here the complexity is further increased by the dual pumping and by the spatial resonance between the pump and the nonlinear pattern. Similarly to what happens for a pure phase modulation of the pump [14, 15], the spatial resonance produces a 4-mode coupling among shared modes. As we shall see in the following, an intensity modulation of the pump can lead to a boost of the hot-spot intensities, and in general permit to tailor the coupling among the 4 modes.

With a dual pump at spatial resonance with the nonlinear grating [Eq. (6)], the shared modes are determined by the following equations

D(ws,ws+2Gx)=D(ws,ws)=0Σ0Σ11
D(ws,ws2Gx)=D(ws,ws)=0Σ0Σ22
D(ws,ws+2Gx)=D(ws,ws2Gx)=0Σ22Σ11

We focus here on the intersections between the central surface Σ0 and the two side branches Σ11, Σ22. As shown in Appendix A, the first equality in Eqs. (14) and (15) requires that qsx=+Gx and qsx=Gx, respectively. Requiring in addition that QPM is satisfied on these planes, one obtains two continuous lines of shared modes (see the red lines in Fig. 2(a)), that for the chosen phase-matching conditions are characterized by qsy=q¯sy(Ωs)±ksks Ωs, as can be inferred from Eqs. (33) and (34). Thus, these shared modes exist also at the degenerate frequency Ωs=0, where they coincide with the vertexes of the two conical surfaces Σ11 and Σ22. Conversely, the shared modes defined by Eq. (16), located at qx = 0, exist only away from degeneracy and are much weaker (barely visible in Fig. 3(c)), we shall not discuss them here.

By inspection of Eq. (10), we see that a given shared signal mode ws=(+Gx,qsy,Ωs) couples with (Gx,qsy,Ωs) through both pumps and with (Gx,qsy,Ωs) through pump 1 only. The common phase mismatch function vanishes for qsy=q¯sy(Ω). The third term at qx=3Gx, proportional to g2, is not phase-matched and can be discarded. A similar reasoning for the other shared mode (Gx,qsy,Ωs) leads us to conclude that there is a total of four interacting modes, for which we shall use the short hand notation

b^s:=A^s(+Gx,qsy,Ωs)b^i:=A^s(+Gx,qsy,Ωs)modesatqx=+Gx
c^s:=A^s(Gx,qsy,Ωs)c^i:=A^s(Gx,qsy,Ωs)modesatqx=Gx

An example of such a quadruplet of coupled modes is shown in Fig. 5.

 figure: Fig. 5

Fig. 5 Example of four-mode coupling process among shared modes at two conjugate frequencies ωp2±Ωs

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They satisfy the following coupled equations

db^sdz=[g1b^i+(g1+g2)c^i]eiD¯z
dc^sdz=[(g1+g2)b^i+g2c^i]eiD¯z
db^idz=[g1*b^s+(g1+g2)*c^s]eiD¯z
dc^idz=[(g1+g2)*b^s+g2*c^s]eiD¯z
where D¯ is the common phase-mismatch of the processes. For D¯=0, the eigenvalues of the 4x4 linear system (5) can be readily found as the four real eigenvalues ±Λ+,±Λ, where
Λ±=[2|g1+g2|+|g1|+|g2|2±12(|g1|2|g2|2]2+4|g1+g2|4+4[2Im(g1g2*)]2]12g¯1+r2|1+r2±125(1+r2)+6r|=g¯×{5±12r=032,12r=112,12r=1forr=g2g1ϵ
where the second line can be obtained with a little algebra in the simplest case where r=g2g1 is real, i.e. when the two pumps are either in phase or out of phase by π. These eigenvalues are plotted in Fig. 6, as a function of of the ratio |g2g1| of the two complex amplitudes of the pumps and of their the phase difference ϕ2ϕ1. For a single pump at resonance (r = 0) we thus retrieve the results of [14, 15], namely the eigenvalues are g¯Φ and g¯/Φ, where Φ=1+52=1.61803.. is the Golden ratio. When a second pump with nonzero amplitude is injected in the crystal, the eigenvalues become modulated with the phase difference between the pumps (Figs. 6(b) and 6(c)). The bigger eigenvalue Λ+ reaches its maximum and minimum values, 3g¯/2 and g¯/2, respectively, when the two pumps have equal intensities, either in phase or out of phase by π.

 figure: Fig. 6

Fig. 6 Eigenvalues Λ+ (upper surface) and Λ (lower surface) of the 4-mode propagation Eqs. (5), normalized to g¯=|g1|2+|g2|2 as a function of the ratio |r|=|g2g1| of the amplitudes of the two pumps and of their phase difference ϕ2ϕ1.

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In the high gain regime g¯z1, the number of generated photons is mainly determined by the big eigenvalue, which dominates the exponential rate of growth of the shared modes along the propagation distance. In this regime, a+/g¯ can thus be identified with the gain enhancement γ of the hot-spots involved in 4-mode processes compared to standard parametric processes from a single pump with the same energy. Figure 7 provides instead an estimation of the gain enhancement from numerical simulations of the nonlinear propagation Eqs. (1), comparing the use of single and dual pumps, for a given total energy.

Again, the values obtained for plane-wave pumps in Fig. 7(a) turn very close to the analytical predictions in Eq. (20) from the simplified 4-mode model, i.e. γ=Φ1.618 for the single pump (r = 0) and γ=3/22.121 for the dual pump (r = 1). When considering a Gaussian pump profile as in Fig. 7(b), the effective gains are reduced by about 20%, but the ratio between the dual pump and the single pump values is 1.27, which is close to the plane-wave pump prediction 32Φ1.31. This last result proves that the enhanced conversion efficiency achieved by using a dual pump with a spatial modulation resonant with the NPC grating is a robust feature which affects not only the whole central branch Σ0 but also the hot spots at qsx=±Gx.

 figure: Fig. 7

Fig. 7 Evaluation of the gain enhancement factor γ in the hot-spots at qx=±Gx, from numerical simulations of Eqs. (1). Comparison between the single pump (blue square) and the dual symmetric pump (red triangles), for (a) plane-wave pumps, (b) Gaussian pumps. The crystal and pump parameters are the same as in Figs. 3 and 4.

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 figure: Fig. 8

Fig. 8 (a) For =g2/g1ϵ, the 4-mode process (19) is equivalent to two independent standard parametric processes of gains Λ+ and Λ mixed on a beam splitter. Panel (b) and (c) show the eigenvalues Λ± and the ratio between the transmission and reflection coefficients of the beam-splitter as a function of r respectively.

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In the simplest case rϵ, where the eigenvalues are given by Eq. (20), the four-mode Eqs. (5) can be decoupled by means of a unitary transformation which involve b^s,c^s and b^i,c^i separately

(δ^jσ^j)=(cos Θsin Θsin Θcos Θ)(b^jc^j)j=s,i
with cos Θ=(121r25(1+r2)+6r)1/2, sin Θ=(12+1r25(1+r2)+6r)1/2. The new mode operators σ^j satisfy standard 2-mode parametric equations with a gain per unit length Λ+
dσ^sdz=Λ+σ^ieiD¯z
dσ^idz=Λ+σ^seiD¯z
while the δ^j mode operators satisfy similar equations with a comparatively lower gain Λ
dδ^sdz=Λδ^ieiD¯z
dδ^idz=Λδ^seiD¯z

The four-mode coupling process (19) is thus equivalent to the outcome of two independent 2-mode parametric processes of gains Λ+ and Λ, followed by an unbalanced beam splitter that mixes the two pairs of twin output modes according to the inverse transformation b^j=(cos Θ)σ^j+(sin Θ)δ^j, cj=(sin Θ)σ^j(sin Θ)δ^j, j=i,s, as shown in Fig. 8(a). Both the gains of the two processes and the reflection and transmission coefficients of the beam splitter can be controlled by varying the relative amplitude and phases of the two pumps, as shown in Figs. 8(b) and 8(c). For a single pump r=0, we have sin Θcos Θ=Φ, and the beam splitter performs a Golden ratio partition of the two pairs of twin beams of gains g¯Φ and g¯/Φ, in agreement with the result obtained in [14]. For a dual symmetical pump cos Θ=sin Θ=1/2: in this case two twin modes with gains 3g¯/2 and g¯/2 are generated and then a mixed on a 50:50 beam splitter. For an antisymmetric pump, cos Θ=0, sin Θ=1: indeed in this case g1+g2=0 and the four-mode process (19) uncouples trivially into two standard parametric processes with equal gain g¯/2, which populate only the side branches Σ11 and Σ22.

4. Conclusions

This work investigated parametric down-conversion in a hexagonally poled nonlinear photonic crystal pumped by two tilted pumps, forming a spatial transverse pattern. When the pump pattern matches the periodicity of the nonlinear grating, we have shownthe existence of a two-fold degenerate QPM branch of spatio-temporal modes (a surface Σ0 in the 3D Fourier space), where photon pairs are generated from both pump modes, and are quasi-phase-matched by both fundamental vectors of the 2D nonlinear grating.

In contrast, quasi-phasematching in nonlinear photonic crystals in the standard single pump configuration generally involves only one lattice vector at a time, except for restricted families of modes of lower dimensionality, the shared modes, which lie at the interceptions of different QPM branches. The two-fold degeneracy of the Σ0 branch is thus a distinctive feature of the spatial resonance of the dual pump with the nonlinear grating. As a consequence, its gain per unit propagation length isproportional to the sum |α1+α2| of the complex amplitudes of the two pumps, and can be controlled by varying their relative amplitudes and phases. For two symmetric pumps (α1=α2), the down-conversion process is strongly enforced by the pump transverse modulation cos (Gxx) which oscillates in phase with the nonlinear lattice. Conversely, down-conversion is inhibited for two anti-symmetric pumps α2=α1, because the pump modulation sin (Gxx) is in quadrature of phase with the nonlinear lattice.

Specifically, when comparing the use of a dual symmetric pump and a single pump with the same total energy, the conversion efficiency on the Σ0 branch simply doubles in the purely spontaneous PDC regime. However in the stimulated regime, where the number of generated photon pairs grows exponentially along the crystal, the 2 increase of the gain per unit length leads to a huge increase of the efficiency.

We also analysed the four-mode coupling characterizing the resonant condition, and we showed that is equivalent to two independent parametric processes of different gain generating each a pair of twin beams, followed by an unbalanced beam splitter that mixes the two outcomes with transmission and reflection coefficients related to the pump amplitudes. The present work is mainly devoted to the classical aspects; the quantum aspects will be discussed in a related work [23], showing how the control of intensities and phases of the dual pumps may be used to tailor the quadri-partite entanglement of the shared modes.

Appendix A: QPM surfaces and shared modes

In this appendix we provide analytical expressions for the QPM surfaces and for the shared modes at the interceptions of these surfaces. We consider the paraxial approximation with the longitudinal components of the signal wave-vector given by

ksz(ws)ks(Ωs)qs22ks(Ωs),

Noticing that the phase-matching functions which determine the various QPM surfaces (5a)-(5d) are distinguished only by the resultant of the pump and lattice transverse wave-vectors: R11=(q0p+Gx)ex; R12=(q0pGx)ex; R21=(q0p+Gx)ex; R22=(q0pGx)ex, we can write such functions as

D(ws,ws+Rlm)=ksz(ws)+ksz(ws+Rlm)kpz(w0p)+Gz
ks(Ω)+ks(Ωs)kpz+Gz|Rlm|22(ks(Ωs)+ks(Ωs))ks(Ωs)+ks(Ωs)2ks(Ωs)ks(Ω)|qks(Ωs)ks(Ωs)+ks(Ωs)Rlm|2,(l,m=1,2)
where kpz is the common z-component of the wavevectors of the two monochromatic pumps of wavelength λp and transverse wavevector ±q0pex. According to this result, at a given signal frequency Ωs, the down-converted field is generated provided that
D0(Ωs)=ks(Ω)+ks(Ωs)kpz+Gz|Rlm|22(ks(Ωs)+ks(Ωs))2kskpz+Gz|Rlm|24ks+ks Ωs20
where in the second line we made a Taylor series expansion up to second order in Ωs, valid not too far from degeneracy, and ks =d2ksdΩs2|Ωs=0. Provided that the inequality (29) is satisfied, then the QPM signal modes lie on circumferences in the (qx,qy) plane, whose centers
qlm=ks(Ω)ks(Ω)+ks(Ω)RlmRlm2
are distributed along the qx-axis, and of radial apertures
Qlm=[2ks(Ω)ks(Ω)ks(Ω)+ks(Ω)D0(Ωs)]12ksD0(Ωs)

In general, in the nonresonant case when qopGx, one has four distinct QPM branches, corresponding to the four possible resultants of the pump and lattice transverse wavevectors. Figure 9 shows an example of such four QPM branches, from a numerical simulation of the evolution equations, for a dual symmetric pump with qop=1.2Gx.

 figure: Fig. 9

Fig. 9 Results of simulations away from resonance, for two symmetric pumps with q0p=1.2Gx. (a) Intensity distribution in the (qx,qy) plane, showing four lines of hot spots at qx=±q0p and qx=±Gx. (b) Intensity distribution in the (λ,qx) plane (qy = 0). Other parameters as in Fig. 3.

Download Full Size | PPT Slide | PDF

In the resonant case the two surfaces Σ12 and Σ21

degenerate into a single surface Σ0 corresponding to a transverse resultant R12=R21=0. The other two QPM branches are characterized by the resultants R11=R22=2Gxex. We have chosen the poling period in such a way that

2kskpz+Gz|2Gx|24ks=0

Taking into account Eqs. (29)-(31), Σ11 and Σ22 are then well approximated by two bi- conical surfaces having their vertexes at qx=±Gx,qy=0,Ω=0:

Σ11:|qsGxex|ksks |Ωs|,
Σ22:|qs+Gxex|ksks |Ωs|,
which in the plane (Ωs,qx) appear as two X’s, vertically displaced along qx of an amount 2Gx (See Fig. 2(b)). Within the same approximation, Σ0 is given by
Σ0:|qs|(Gx2+ksks Ωs2)1/2,
which appears like a widely opened tube centered at q=0, intersecting the vertexes ±Gxex of the two conical surfaces as shown in Fig. 2(a).

Let us now determine the interceptions of the QPM surfaces, corresponding to the shared modes. In general, the QPM modes shared by two surfaces Σlm and Σpq lie on a curve defined by:

ΣlmΣpq:D(ws,ws+Rlm)=D(ws,ws+Rpq)=0,

Taking into account Eq. (27), the first equality in Eq. (36) implies that

ksz(ws+Rlm)=ksz(ws+Rpq)qsx=Rlm+Rpq2

Then for example the QPM modes shared by Σ11 and Σ12 are characterized by the x-component of the wave vector qsx=qop, while those shared by Σ11 and Σ21 are located at qsx=Gx. For reasons of symmetry the modes shared by Σ22 and Σ21 are located at qsx=qop, while those shared by Σ22 and Σ12 have qsx=Gx. In the plane (qx,qy) the shared modes form four straight lines at qx=±q0p, and qx=±Gx, as shown by the numerical simulations in Fig. 9(a). The corresponding hot spots have however a much lower intensity than in the resonant case(see Fig. 3(b)). At spatial resonance q0p=±Gx, the four lines of shared modes merge into two ones at qsx=±Gx. In the 3D Fourier space, they merge into the two curves of equations qsx=±Gx;qsy=q¯sy(Ω)=±ksks |Ωs| (red lines in Fig. 2(a)).

The remaining shared modes belonging to Σ11Σ22 and Σ12Σ21 are located at qsx=0 plane, generally on different curves. We shall not take these modes into consideration as they lie far from degeneracy.

Funding

Ministero dell’ Istruzione, dell’Università e della Ricerca (MIUR).

References

1. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998). [CrossRef]  

2. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000). [CrossRef]   [PubMed]  

3. S. Saltiel and Y. S. Kivshar, “Phase matching in nonlinear χ(2) photonic crystals,” Opt. Lett. 25, 1204–1206 (2000). [CrossRef]  

4. K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008). [CrossRef]   [PubMed]  

5. K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5. [CrossRef]  

6. M. Lazoul, A. Boudrioua, L.-M. Simohamed, and L.-H. Peng, “Multi-resonant optical parametric oscillator based on 2D-PPLT nonlinear photonic crystal,” Opt. Lett. 40, 1861–1864 (2015). [CrossRef]   [PubMed]  

7. Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017). [CrossRef]   [PubMed]  

8. H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017). [CrossRef]  

9. H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017). [CrossRef]  

10. H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013). [CrossRef]   [PubMed]  

11. Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012). [CrossRef]  

12. E. Megidish, A. Halevy, H. S. Eisenberg, A. Ganany-Padowicz, N. Habshoosh, and A. Arie, “Compact 2D nonlinear photonic crystal source of beamlike path entangled photons,” Opt. Express 21, 6689–6696 (2013). [CrossRef]   [PubMed]  

13. Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016). [CrossRef]   [PubMed]  

14. A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018). [CrossRef]  

15. O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018). [CrossRef]   [PubMed]  

16. H.-C. Liu and A. H. Kung, “Substantial gain enhancement for optical parametric amplification and oscillation in two-dimensional χ(2) nonlinear photonic crystals,” Opt. Express 16, 9714–9725 (2008). [CrossRef]   [PubMed]  

17. M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012). [CrossRef]  

18. L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014). [CrossRef]   [PubMed]  

19. M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014). [CrossRef]   [PubMed]  

20. R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas, and G. P. Banfi, “Self-diffraction through cascaded second-order frequency-mixing effects in β-barium borate,” Opt. Lett. 18, 574–576 (1993). [CrossRef]  

21. M. Tiihonen and V. Pasiskevicius, “Two-dimensional quasi-phase-matched multiple-cascaded four-wave mixing in periodically poled ktiopo4,” Opt. Lett. 31, 3324–3326 (2006). [CrossRef]   [PubMed]  

22. A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007). [CrossRef]  

23. A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

24. H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013). [CrossRef]  

25. A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003). [CrossRef]  

26. E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012). [CrossRef]  

27. K. Moutzouris, G. Hloupis, I. Stavrakas, D. Triantis, and M.-H. Chou, “Temperature-dependent visible to near-infrared optical properties of 8 mol% Mg-doped lithium tantalate,” Opt. Mater. Express 1, 458–465 (2011). [CrossRef]  

28. M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997). [CrossRef]  

References

  • View by:

  1. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
    [Crossref]
  2. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
    [Crossref] [PubMed]
  3. S. Saltiel and Y. S. Kivshar, “Phase matching in nonlinear χ(2) photonic crystals,” Opt. Lett. 25, 1204–1206 (2000).
    [Crossref]
  4. K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
    [Crossref] [PubMed]
  5. K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
    [Crossref]
  6. M. Lazoul, A. Boudrioua, L.-M. Simohamed, and L.-H. Peng, “Multi-resonant optical parametric oscillator based on 2D-PPLT nonlinear photonic crystal,” Opt. Lett. 40, 1861–1864 (2015).
    [Crossref] [PubMed]
  7. Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
    [Crossref] [PubMed]
  8. H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
    [Crossref]
  9. H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
    [Crossref]
  10. H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
    [Crossref] [PubMed]
  11. Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
    [Crossref]
  12. E. Megidish, A. Halevy, H. S. Eisenberg, A. Ganany-Padowicz, N. Habshoosh, and A. Arie, “Compact 2D nonlinear photonic crystal source of beamlike path entangled photons,” Opt. Express 21, 6689–6696 (2013).
    [Crossref] [PubMed]
  13. Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
    [Crossref] [PubMed]
  14. A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
    [Crossref]
  15. O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
    [Crossref] [PubMed]
  16. H.-C. Liu and A. H. Kung, “Substantial gain enhancement for optical parametric amplification and oscillation in two-dimensional χ(2) nonlinear photonic crystals,” Opt. Express 16, 9714–9725 (2008).
    [Crossref] [PubMed]
  17. M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
    [Crossref]
  18. L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
    [Crossref] [PubMed]
  19. M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014).
    [Crossref] [PubMed]
  20. R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas, and G. P. Banfi, “Self-diffraction through cascaded second-order frequency-mixing effects in β-barium borate,” Opt. Lett. 18, 574–576 (1993).
    [Crossref]
  21. M. Tiihonen and V. Pasiskevicius, “Two-dimensional quasi-phase-matched multiple-cascaded four-wave mixing in periodically poled ktiopo4,” Opt. Lett. 31, 3324–3326 (2006).
    [Crossref] [PubMed]
  22. A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
    [Crossref]
  23. A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.
  24. H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
    [Crossref]
  25. A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
    [Crossref]
  26. E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
    [Crossref]
  27. K. Moutzouris, G. Hloupis, I. Stavrakas, D. Triantis, and M.-H. Chou, “Temperature-dependent visible to near-infrared optical properties of 8 mol% Mg-doped lithium tantalate,” Opt. Mater. Express 1, 458–465 (2011).
    [Crossref]
  28. M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
    [Crossref]

2018 (2)

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

2017 (3)

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

2016 (1)

2015 (1)

2014 (2)

2013 (3)

E. Megidish, A. Halevy, H. S. Eisenberg, A. Ganany-Padowicz, N. Habshoosh, and A. Arie, “Compact 2D nonlinear photonic crystal source of beamlike path entangled photons,” Opt. Express 21, 6689–6696 (2013).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

2012 (3)

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

2011 (1)

2008 (2)

H.-C. Liu and A. H. Kung, “Substantial gain enhancement for optical parametric amplification and oscillation in two-dimensional χ(2) nonlinear photonic crystals,” Opt. Express 16, 9714–9725 (2008).
[Crossref] [PubMed]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

2007 (1)

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

2006 (1)

2003 (1)

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

2000 (2)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

S. Saltiel and Y. S. Kivshar, “Phase matching in nonlinear χ(2) photonic crystals,” Opt. Lett. 25, 1204–1206 (2000).
[Crossref]

1998 (1)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

1997 (1)

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

1993 (1)

Arie, A.

Assanto, G.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

Bahabad, A.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

Bai, Y. F.

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
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Baronio, F.

Beghoul, M. R.

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V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
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Björk, G.

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

Boudrioua, A.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

M. Lazoul, A. Boudrioua, L.-M. Simohamed, and L.-H. Peng, “Multi-resonant optical parametric oscillator based on 2D-PPLT nonlinear photonic crystal,” Opt. Lett. 40, 1861–1864 (2015).
[Crossref] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Chang, K.-H.

Chen, L.

Chikh-Touami, H.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Chou, M.-H.

Conforti, M.

Dai, M.

Danielius, R.

Drummond, P. D.

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

Dubietis, A.

Eisenberg, H. S.

Gallo, K.

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014).
[Crossref] [PubMed]

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

Ganany-Padowicz, A.

Gatti, A.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

Gong, Y. X.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Gong, Y.-X.

Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Habshoosh, N.

Halevy, A.

Hanna, D. C.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Hloupis, G.

Jedrkiewicz, O.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

Jin, H.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Katagai, T.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Kivshar, Y. S.

Kremer, R.

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Kung, A. H.

Kurimura, S.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Lazoul, M.

Lee, H.-J.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Lee, M.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Lee, M. W.

Leng, H. Y.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Levenius, M.

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014).
[Crossref] [PubMed]

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

Lim, H. H.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Liu, H.-C.

Lu, M. H.

Lugiato, L. A.

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

Luo, X. W.

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Megidish, E.

Moutzouris, K.

Offerhaus, H. L.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Pasiskevicius, V.

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

M. Tiihonen and V. Pasiskevicius, “Two-dimensional quasi-phase-matched multiple-cascaded four-wave mixing in periodically poled ktiopo4,” Opt. Lett. 31, 3324–3326 (2006).
[Crossref] [PubMed]

Pasquazi, A.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

Peng, L.-H.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

M. Lazoul, A. Boudrioua, L.-M. Simohamed, and L.-H. Peng, “Multi-resonant optical parametric oscillator based on 2D-PPLT nonlinear photonic crystal,” Opt. Lett. 40, 1861–1864 (2015).
[Crossref] [PubMed]

Piskarskas, A.

Podenas, D.

Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Ross, G. W.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Saltiel, S.

San Miguel, M.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

Shoji, I.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Simohamed, L.-M.

Stavrakas, I.

Stensson, K.

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

Stivala, S.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

Tamosauskas, G.

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

Tiihonen, M.

Trapani, P. Di

Triantis, D.

Werner, M. J.

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

Xie, Z. D.

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Xu, P.

Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
[Crossref] [PubMed]

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Yang, J.

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Yellas, Z.

Yu, W. J.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Zambrini, R.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

Zhang, S.

Zhao, G.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Zhong, M. L.

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Zhu, S. N.

Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
[Crossref] [PubMed]

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Appl. Phys. B: Lasers Opt. (1)

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Appl. Phys. Lett. (1)

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

J. Opt. (1)

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Jpn. J. Appl. Phys. (1)

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Opt. Mater. Express (1)

Opt. Quant. Electron. (1)

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

Phys. Rev. A (5)

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Phys. Rev. Lett. (4)

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Sci. Rep. (1)

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

Other (2)

A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 a) Hexagonally poled χ ( 2 ) crystal pumped by two waves, symmetrically tilted with respect to the z-axis. b) A spatial resonance is achieved by matching the pump transverse wave-vectors ± q p with the transverse components ± G x e x of the lattice vectors G 1 and G 2.
Fig. 2
Fig. 2 Quasi phase-matching in a hexagonally poled LiTaO3 crystal pumped at 532nm by a dual pump at spatial resonance with the lattice: q 0 p e x = G x e x, with G x = 2 π / ( 3 Λ ) = 0.466 μ m 1. (a) QPM surfaces in the 3D Fourier space, and (b) its section at qy = 0.
Fig. 3
Fig. 3 Photon-number distribution in the ( q x , q y )-plane (top) and in the ( λ , q x )-planes (bottom) for a single pump (a,d), two symmetric pumps (b,e) and two antisymmetric pumps (c,f), from numerical simulations of Eqs. (1), in the same NPC of Fig. 2. The pumps are plane-waves, g ¯ = 0.4 mm   1, and results are shown after 7mm of propagation ( g ¯ z = 2.8). In (d,e) the scale was truncated to 1% of the peak value. For a dual symmetric pump the Σ0 branch is significantly more intense than for a single pump of equal energy, while it is absent for antisymmetric pumping. Lines of hot spots at q x = ± G x are clearly visible in panels (a) and (b).
Fig. 4
Fig. 4 Comparison between the use of a dual symmetric pump (red triangles) and a single pump (blue squares) with the same energy. The gain enhancement factor γ is evaluated from numerical simulations of Eqs. (1), The results for plane-wave pumps in (a,b) are very close to the γ t h predicted by the parametric model [Eq. (13)]. The lower panels (c,d) are obtained for Gaussian pumps, of waists 500 μm and 200 μm along the x and y axis. Other parameters as in Fig. 3.
Fig. 5
Fig. 5 Example of four-mode coupling process among shared modes at two conjugate frequencies ω p 2 ± Ω s
Fig. 6
Fig. 6 Eigenvalues Λ + (upper surface) and Λ (lower surface) of the 4-mode propagation Eqs. (5), normalized to g ¯ = | g 1 | 2 + | g 2 | 2 as a function of the ratio | r | = | g 2 g 1 | of the amplitudes of the two pumps and of their phase difference ϕ 2 ϕ 1.
Fig. 7
Fig. 7 Evaluation of the gain enhancement factor γ in the hot-spots at q x = ± G x, from numerical simulations of Eqs. (1). Comparison between the single pump (blue square) and the dual symmetric pump (red triangles), for (a) plane-wave pumps, (b) Gaussian pumps. The crystal and pump parameters are the same as in Figs. 3 and 4.
Fig. 8
Fig. 8 (a) For = g 2 / g 1 ϵ , the 4-mode process (19) is equivalent to two independent standard parametric processes of gains Λ + and Λ mixed on a beam splitter. Panel (b) and (c) show the eigenvalues Λ ± and the ratio between the transmission and reflection coefficients of the beam-splitter as a function of r respectively.
Fig. 9
Fig. 9 Results of simulations away from resonance, for two symmetric pumps with q 0 p = 1.2 G x. (a) Intensity distribution in the ( q x , q y ) plane, showing four lines of hot spots at q x = ± q 0 p and q x = ± G x. (b) Intensity distribution in the ( λ , q x ) plane (qy = 0). Other parameters as in Fig. 3.

Equations (45)

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d ( x , z ) e i G z z [ d 01 e i G x x + d 10 e i G x x ] = 2 d 01 e i G z z cos  ( G x x )
z A ^ s ( w s , z ) = χ d 3 w p ( 2 π ) 3 2 A ^ p ( w p , z ) [ A ^ s ( w p w s G x , z ) e i D ( w s , w p w s G x ) z + A ^ s ( w p w s + G x , z ) e i D ( w s , w p w s + G x ) z ]
z A ^ p ( w p , z ) = χ 2 d 3 w s ( 2 π ) 3 2 A ^ s ( w s , z ) [ A ^ s ( w p w s G x , z ) e i D ( w s , w p w s G x ) z + A ^ s ( w p w s + G x , z ) e i D ( w s , w p w s + G x ) z ]
D ( w s , w p w s ± G x ) = [ k s z ( w s ) + k s z ( w p w s ± G x ) k p z ( w p ) + G z ] ,
A p ( q , Ω p ) = d t 2 π d x 2 π e i Ω t i q x A p ( x , t ) = ( 2 π ) 3 / 2 δ ( Ω ) δ ( q y ) [ α 1 δ ( q x q 0 p ) + α 2 δ ( q x + q 0 p ) ]
Σ 11 : D ( w s , w 0 p w s + G x ) = 0 [ q s + q i = ( q 0 p + G x ) e x ]
Σ 12 : D ( w s , w 0 p w s G x ) = 0 [ q s + q i = ( q 0 p G x ) e x ]
Σ 21 : D ( w s , w 0 p w s + G x ) = 0 [ q s + q i = ( q 0 p G x ) e x ]
Σ 22 : D ( w s , w 0 p w s G x ) = 0 [ q s + q i = ( q 0 p + G x ) e x ]
q 0 p = G x A p ( z , x , t ) = α 1 e i G x x + α 2 e i G x x
Σ 12 , Σ 21 Σ 0 : D ( w s , w s ) = 0
Σ 11 : D ( w s , w s + 2 G x ) = 0 ,
Σ 22 : D ( w s , w s 2 G x ) = 0
A ^ s z ( w s ) = ( g 1 + g 2 ) A ^ s ( w s ) e i D ( w s , w s ) z + g 1 A ^ s ( w s + 2 G x ) e i D ( w s , w s + 2 G x ) z + g 2 A ^ s ( w s 2 G x ) e i D ( w s , w s 2 G x ) z
A ^ s z ( w s ) = γ g ¯ e i ϕ 1 A ^ s ( w i ) e i D ( w s , w i ) z
A ^ s z ( w i ) = γ g ¯ e i ϕ 1 A ^ s ( w s ) e i D ( w s , w i ) z
g ¯ = | g 1 | 2 + | g 2 | 2
γ = { g 1 + g 2 g ¯ = 1 + r 1 + | r | 2 Σ 0 g 1 g ¯ = 1 1 + | r | 2 Σ 11 g 2 g ¯ = r 1 + | r | 2 Σ 22
D ( w s , w s + 2 G x ) = D ( w s , w s ) = 0 Σ 0 Σ 11
D ( w s , w s 2 G x ) = D ( w s , w s ) = 0 Σ 0 Σ 22
D ( w s , w s + 2 G x ) = D ( w s , w s 2 G x ) = 0 Σ 22 Σ 11
b ^ s : = A ^ s ( + G x , q s y , Ω s ) b ^ i : = A ^ s ( + G x , q s y , Ω s ) modes at q x = + G x
c ^ s : = A ^ s ( G x , q s y , Ω s ) c ^ i : = A ^ s ( G x , q s y , Ω s ) modes at q x = G x
d b ^ s d z = [ g 1 b ^ i + ( g 1 + g 2 ) c ^ i ] e i D ¯ z
d c ^ s d z = [ ( g 1 + g 2 ) b ^ i + g 2 c ^ i ] e i D ¯ z
d b ^ i d z = [ g 1 * b ^ s + ( g 1 + g 2 ) * c ^ s ] e i D ¯ z
d c ^ i d z = [ ( g 1 + g 2 ) * b ^ s + g 2 * c ^ s ] e i D ¯ z
Λ ± = [ 2 | g 1 + g 2 | + | g 1 | + | g 2 | 2 ± 1 2 ( | g 1 | 2 | g 2 | 2 ] 2 + 4 | g 1 + g 2 | 4 + 4 [ 2 I m ( g 1 g 2 * ) ] 2 ] 1 2 g ¯ 1 + r 2 | 1 + r 2 ± 1 2 5 ( 1 + r 2 ) + 6 r | = g ¯ × { 5 ± 1 2 r = 0 3 2 , 1 2 r = 1 1 2 , 1 2 r = 1 for r = g 2 g 1 ϵ
( δ ^ j σ ^ j ) = ( cos Θ sin Θ sin Θ cos Θ ) ( b ^ j c ^ j ) j = s , i
d σ ^ s d z = Λ + σ ^ i e i D ¯ z
d σ ^ i d z = Λ + σ ^ s e i D ¯ z
d δ ^ s d z = Λ δ ^ i e i D ¯ z
d δ ^ i d z = Λ δ ^ s e i D ¯ z
k s z ( w s ) k s ( Ω s ) q s 2 2 k s ( Ω s ) ,
D ( w s , w s + R l m ) = k s z ( w s ) + k s z ( w s + R l m ) k p z ( w 0 p ) + G z
k s ( Ω ) + k s ( Ω s ) k p z + G z | R l m | 2 2 ( k s ( Ω s ) + k s ( Ω s ) ) k s ( Ω s ) + k s ( Ω s ) 2 k s ( Ω s ) k s ( Ω ) | q k s ( Ω s ) k s ( Ω s ) + k s ( Ω s ) R l m | 2 , ( l , m = 1 , 2 )
D 0 ( Ω s ) = k s ( Ω ) + k s ( Ω s ) k p z + G z | R l m | 2 2 ( k s ( Ω s ) + k s ( Ω s ) ) 2 k s k p z + G z | R l m | 2 4 k s + k s   Ω s 2 0
q l m = k s ( Ω ) k s ( Ω ) + k s ( Ω ) R l m R l m 2
Q l m = [ 2 k s ( Ω ) k s ( Ω ) k s ( Ω ) + k s ( Ω ) D 0 ( Ω s ) ] 1 2 k s D 0 ( Ω s )
2 k s k p z + G z | 2 G x | 2 4 k s = 0
Σ 11 : | q s G x e x | k s k s   | Ω s | ,
Σ 22 : | q s + G x e x | k s k s   | Ω s | ,
Σ 0 : | q s | ( G x 2 + k s k s   Ω s 2 ) 1 / 2 ,
Σ l m Σ p q : D ( w s , w s + R l m ) = D ( w s , w s + R p q ) = 0 ,
k s z ( w s + R l m ) = k s z ( w s + R p q ) q s x = R l m + R p q 2

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