Degenerate spontaneous parametric down-conversion in nonlinear metasurfaces

We propose a simple scheme of degenerate spontaneous parametric down-conversion (SPDC) in nonlinear metasurfaces or photonic crystal slabs with quasi-guided modes. It employs a band crossing between P- and S-polarized quasi-guided mode bands inside the light cone and a selection rule in the conversion efficiency of the SPDC. The efficiency can be evaluated fully classically via the inverse process of noncollinear second-harmonic generation (SHG). As a toy model, we study the SPDC and SHG in a monolayer of noncentrosymmetric spheres and confirm that the scenario works well to enhance the SPDC.


I. INTRODUCTION
The spontaneous parametric down-conversion (SPDC) is a quantum phenomenon of generating two photons with angular frequency ω 1 and ω 2 by a pump light of angular frequency ω 1 + ω 2 [1].It becomes a source of entangled photon pairs [2], so that the SPDC can be used in quantum information, such as the Bell inequality test experiment [3,4].
Compared with other entangled photon sources, such as the cascade emission from three-level atoms or biexcitons in quantum dots [5][6][7], the SPDC has its advantages and disadvantages.One advantage is it's relatively easy to set up.The disadvantages include (in principle) low generation efficiency, lack of determinism, and need for filtering.These properties are tied with the SPDC being a nonlinear optical process.
As a second-order nonlinear optical process, the SPDC is observed typically in uniaxial and noncentrosymmetric media [8].There, the phase matching via the birefringe is crucial.For instance, the phase matching of type II gives the polarization entanglement between the ordinary and extraordinary waves with orthogonal polarizations.
Optical media that are uniaxial and noncentrosymmetric are limited.Relaxing these two properties are essential for practical applications involving the second-order optical nonlinearity.One crucial direction here is to employ the quasi-phase matching by introducing an artificial spatial periodicity in the system [9][10][11][12].There, we can be free from the birefringe of the bulk materials.Another important direction is to employ surface modes in, even centrosymmetric media [13,14].Since any material breaks the centrosymmetry at the surface, the system can have χ (2) there.Combining possible surface modes with surface (or bulk) χ (2) , we can expect a significant enhancement of the second-order optical nonlinearity even in optically thin specimens.
For instance, previous studies on the SPDC in twodimensional (2D) metasurfaces rely on deformations (in geometry) of symmetry-protected BICs at the Γ point so that the emission angles of the signal and idler photons are close to the normal direction of the metasurfaces [36,37].There is a broad tunability of relevant quality factors in such cases.The tunability is reduced in onedimensional (1D) metasurfaces [35,38].
Instead of such a deformation, we here consider plain 2D structures and a conventional phenomenon of band crossings at a generic k point on a mirror axis.This setting allows us to imitate the conventional SPDC scheme of the type II phase matching even in isotropic but noncentrosymmetric media.
In this paper, we present a theoretical analysis of the SPDC, taking account of metasurface structures and their spatial symmetries in a first-principles manner.We present a simple scenario of the degenerate SPDC and polarization entanglement in metasurfaces with a certain photonic band structure.The essential quantity is the conversion efficiency from the pump light to biphoton states.The spatial symmetries of the metasurface and the symmetry of χ (2) yields a selection rule in the conversion efficiency factor.It restricts possible combinations of the biphoton states and pump-light polarization.Moreover, the efficiency factor can be strongly enhanced by the excitation of quasi-guided modes in the metasurface.We demonstrate these features in terms of a classical electromagnetic calculation of the reverse process, namely, the sum-frequency generation (SFG) [39][40][41] or the noncollinear SHG.
This paper is organized as follows.In Sec.II, we summarize several formulas of the SPDC in terms of an eigenmode expansion of the quantized radiation field.Section III is devoted to presenting the reverse process, namely, the SFG, in a fully classical approach.In Sec.IV, we present symmetry properties of the conversion efficiency factor in the SPDC and resulting polarization entanglement in nonlinear metasurfaces assuming a mirror symmetry.In Sec.V, we present a simulation of the conversion efficiency in a monolayer of noncentrosymmetric spheres as a toy model.Finally, in Sec.VI, summary and discussion are given.

II. SPONTANEOUS PARAMETRIC DOWN CONVERSION
We first consider a generic photonic system with noncentrosymmetric media.The Hamiltonian H of the radiation field including the second-order optical nonlinearity becomes [42] il η (1) where D and H are the electric displacement and magnetic fields, respectively, µ 0 and ϵ 0 are the vacuum permeability and permittivity, respectively, and χ (1) and χ (2) are space-dependent linear and second-order electric susceptibilities, respectively, of the media.That is, ) being P and E the electric polarization and electric field, respectively, with constitutive relation D = ϵ 0 E + P .
We introduce the dual vector potential Λ with the "Coulomb" gauge (∇ • Λ = 0) as Using the eigenmodes of Λ of the linear Maxwell equation, namely, the radiation field can be expanded as where index α represents an eigenstate, ω α is its eigenfrequency, and Λ α is its normalized eigenfunction.The unperturbed Hamiltonian H 0 then becomes assuming the time-reversal symmetry: In the interaction picture, we have The state vector is time-developed as where T represents the time-ordering product.
Let us consider a strong continuous wave (CW) pump light of an eigenstate α = p is incident on the system.Its fluctuation can be safely neglected so that the interaction Hamiltonian H 1 relevant to the SPDC is where D p is the classical-pump field, D f is the quantumfluctuation expressed by Eq. ( 10), and c.c. represents the complex conjugate.Here, we assume the permutation symmetry concerning the indices of η ijk holds, assuming a nondispersive χ (2) .
In the first-order perturbation, if we start from the vacuum state |0⟩ at time t = −∞, we then obtain the biphoton states after a long time: where |1 α ⟩ represents the one photon state with eigenstate α.The second term is thus a superposition of entangled biphoton states between eigenstates α 1 and α 2 .The conversion efficiency from the pump light to the biphoton state is represented by factor F pα1α2 .

III. SUM-FREQUENCY GENERATION
To evaluate the SPDC, we need the factor F pα1α2 .The direct calculation of it according to Eq. ( 20) requires a detailed analysis of the eigenmode profiles.Instead, we consider the reverse process of the SPDC, namely, the SFG, in which the same factor emerges as we will see.Since the SFG does not need a quantum approach, we can access the factor fully classically in a perturbation scheme.
Suppose that a incident light of eigenstates α 1 and α 2 is impinging to the system: In the perturbative viewpoint, this field forms the nonlinear polarization P (NL) through Eq. ( 6) as The nonlinear polarization has several frequency components.The component of angular frequency ω = ω α1 + ω α2 is given by This polarization becomes the source of the SFG.
The dual vector potential induced by the source satisfies By using the eigenmode expansion, the equation is solved as The radiation flux of the SFG is given by dS If the frequency-matched eigenstate is unique, we can access factor |F αα1α2 | 2 by fully classically calculating the radiation flux.

IV. SELECTION RULE IN METASURFACES
In what follows, we consider a nonlinear metasurface that is periodic in the xy plane and is finite in the z direction.We do not care about the details of the system, but we impose that the system has a mirror symmetry on a particular axis of the system and the time-reversal symmetry.We further assume the mirror axis bisects the unit cell (UC).
According to the Bloch theorem, the system is characterized by a 2D Bloch momentum k.For k on the mirror axis, say the x axis, the eigenmodes are classified according to the parity (of the y inversion) concerning the axis.Then, we have where σy is the y inversion operator, α = ( αk), and X is a 2D real lattice vector.Thus, we have The 2D phase matching requires k p = k α1 + k α2 .Suppose that the pump light is incident to the metasurface in the normal direction (k p = 0).It is then possible to produce the biphoton state having respective Bloch momenta of k and −k both on the mirror axis.In this case, the pump, signal, and idler states are classified according to the parity on the mirror axis.This property constrains the biphoton states regarding the parity.Moreover, the parity is directly related to the polarization of the biphoton state in the far field.Namely, if the signal (idler) state has an even parity, then it is Ppolarized in the far field.The odd parity state becomes S-polarized in the far field, provided no open diffraction channel exists.
We here consider the case that two quasi-guided photonic bands of respective even and odd parities intersect 1.Schematic band structure for the SPDC.On a mirror axis of the system, the photonic band structure is classified into P-and S-polarized bands in the far field.At the crossing points, the selection rule in the conversion-efficiency factor of the SPDC results in a polarization entanglement between signal and idler photons for a pump light of the normal incidence.The signal and idler photons are degenerate with angular frequency ωc, and the momentum ±kc.The gray area is the region outside the light cone.
at ±k c as shown in Fig. 1.The eigenfrequency there is denoted as ω c .We assume the pump frequency ω p is twice the eigenfrequency, ω p = 2ω c .Then, both the even and odd parity eigenstates of ω c can be generated.
widely used nonlinear-optical media such as KDP and BBO.
In this case, the x-polarized off-resonant pump gives the polarization entanglement as and the y-polarized off-resonant pump gives under the time-reversal symmetry, namely, The polarization state is maximally entangled in the former case, whereas the latter is not.The absolute values of complex coefficients c PP , c PS , c SP , c SS are available classically via Eq.(30).Determining their phases is essential in entanglement manipulation, but requires a detailed analysis of the eigenmodes.Here, we focus on the absolute values that relate to the conversion efficiency of the SPDC.

V. CASE OF A MONOLAYER OF DIELECTRIC SPHERES
As a model calculation, let us consider the square lattice of dielectric spheres made of a noncentrosymmetric material with the T d point group.A schematic illustration of the system under study is shown in Fig. 2. The two incident plane-wave lights with the same frequency come into the monolayer, and the noncollinear SHG is induced.The radiation flux of the SHG gives the conversion efficiency of the SPDC through Eq. (30).
Figure 3 shows the photonic band structure of the trueguided and quasi-guided modes along the ΓX direction of the square lattice.The band structure was calculated via the photonic layer Korringa-Kohn-Rostoker (KKR) method [43][44][45].Inside the light cone, the band structure is obtained by fitting the scattering phase shift δ [46] to the Breit-Wigner form as  3. Photonic band structure along the ΓX (kx) direction in the square lattice of dielectric spheres.The spheres have dielectric constant 12 and radius 0.4a where a is the lattice constant.The photonic band modes are classified according to the parities relevant to y and z directions.Symbols "ee","eo", "oe", and "oo" stand for (y-even,z-even), (yeven,z-odd), (y-odd,z-even), and (y-odd,z-odd), respectively.Solid, dashed, and dotted lines represent the light cone, the diffraction threshold, and the line of constant incident angle θ = 15.26 • , respectively.
where δ 0 is the background phase shift, ω 0 is the resonance frequency, and γ is its width (or inverse lifetime).
Here, we plot ω 0 as a function of Bloch momentum k.The band structure is classified according to the parities in the y and z directions.It have several band crossing points between y-even and y-odd bands.At these crossing points (k c , ω c ), the degenerate SPDC can be enhanced.
Figure 4 shows a close-up view of the band structure together with the resonance width γ.Here, we focus on the band-crossing point between "ee" and "oo" bands.They have relatively small γ, so that the light-matter interaction is enhanced through these two modes.
Next, we consider the conversion efficiency of the SPDC via the classical calculation of the noncollinear SHG.We assume two incident plane waves with the same angular frequency ω/2 and opposite incident angles θ c = ± sin −1 (ck c /ω c ), which is adjusted for a particular crossing point.The incident light induces the noncollinear SHG of angular frequency ω.Then, the resulting radiation field of the SHG in the far field is expressed as where g is a 2D reciprocal lattice vector and superscript ± refers to the sign of z (the monolayer center is taken to be z = 0).The plane-wave coefficient t (2)± g can also be calculated via the photonic layer KKR method.
The radiation flux of the SHG per unit area is given by This flux should be identified with Eq. ( 30) divided by the planar area of the system.Under a non-resonant pumping, the eigenmode relevant to the pump light is like a plane wave.Therefore, the electric field of the pump light has a little z dependence inside the monolayer, provided the monolayer thickness is in the subwavelength regime.In this case together with the point group T d , the signal and idler lights are better to have opposite parities in the z direction.Otherwise, factor F pα1α2 becomes small.
Figure 5 shows the radiation flux of the noncollinear SHG for the two incident waves of either P or Spolarization.The incident angle θ c is fixed to excite the  t (̸ = 0) for (ijk) = (xyz), (xzy), (yzx), (yxz), (zxy), (zyx) and zero otherwise.The incident angle of the first-harmonic (FH) waves is fixed to be θc = ±15.26• to excite the quasi-guided modes at the crossing point of Fig. 4. The two FH waves are either Por S-polarized.Symbols "PP",PS","SP", and "SS" represent the (signal,idler) lights are (P,P), (P,S), (S,P), and (S,S) polarized, respectively.Normalized radiation flux is defined by 2cµ0F (2) /|χ modes at a crossing point [θ c = ± sin −1 (ck c /ω c )] and angular frequency is scanned.The SHG flux is strongly enhanced at various first-harmonic (FH) frequencies.These peaks represent the excitement of P-or S-polarized quasiguided modes at FH frequencies.Or, the excitement of the quasi-guided modes at the Γ point of secondharmonic frequencies.However, all of these peaks are not directly related to the SPDC of interest, because the calculated flux includes those of the diffraction channels other than the non-diffractive (g = 0) one of the normal direction.The normal direction is supposed to be the pumping direction of the SPDC.
Instead, if we plot solely the flux contribution in the normal direction, the result changes drastically from Fig. 5, as shown in Fig. 6.There is a remarkable peak only for (signal,idler)=(P,S) and (S,P) at ωa/2πc = 0.646, where the P-and S-polarized bands cross as in Fig. 4 and the SHG light is dominantly x-polarized as expected.The y-polarized SHG component is negligible in the entire frequency range of Fig. 6.However, if we closely look at the small peak of the crossing point concerned, we can observe that the y-polarized SHG component is enhanced only for (signal,idler)=(P,P) or (S,S), as expected from the selection rule in Sec.IV.In this case, most power is transmitted to the four diffraction channels with g = ±x, ±ŷ (×2π/a).Therefore, the polarization entanglement is limited only if we send the pump light from the four different angles that correspond to the above diffraction channels.It is better to design the resonance such that 2ω c is below the diffraction threshold 2πc/a.
For comparison, we show in Fig. 7 the radiation flux of the noncollinear SHG from a uniform slab made of the same material as in Fig. 5.The thickness of the slab is taken to be the same as the diameter of the spheres.There is no marked signal other than the three broad peaks for (signal,idler)=(P,S) and (S,P).They correspond to the Fabri-Perot resonance of the slab.Moreover, the normalized SHG flux is much smaller than in Figs. 5 and 6.As for (signal,idler)=(P,P) and (S,S), the SHG flux is identically zero by symmetry.

VI. SUMMARY AND DISCUSSION
In summary, we have presented a theoretical analysis of the degenerate SPDC in nonlinear metasurfaces.A band crossing between P-and S-polarized quasi-guided modes on a mirror axis yields a boosted SPDC with a polarization-entangled biphoton state.The conversion efficiency in the SPDC is available classically via the inverse process of the noncollinear SHG.We demonstrate these features in a monolayer of noncentrosymmetric spheres arranged in the square lattice.
There are many issues that remain to be investigated.The setting of the incident wave is also the same.The slab thickness is taken to be 0.8a.The gray region corresponds to the frequency interval of Figs. 5 and 6.The SHG flux for (signal,idler)=(P,P) and (S,S) vanishes.
One is to determine and design the relative phase between |P⟩|S⟩ and |S⟩|P⟩ (or |P⟩|P⟩ and |S⟩|S⟩) mentioned in the text.It requires a detailed analysis of the relevant eigenmodes and time-reversal symmetry.
Another critical issue is an on-resonant pumping.To make the system simple enough, we have assumed the off-resonant pumping.There, the pump light is planewave like even inside the metasurfaces.The on-resonant pump implies the excitation of a quasi-guided mode at the Γ point.The local field of the quasi-guided mode is no longer plane-wave like, so that the selection rule becomes complicated, mixing all the four combinations of |P⟩|P⟩, |P⟩|S⟩, |S⟩|P⟩, and |S⟩|S⟩.At the same time, the onresonant pump further boosts the conversion efficiency of the SPDC.
It is also essential to implement the present scheme of the SPDC in experimentally more accessible platforms, e.g., dielectric slabs with a periodic array of air holes.Since the present theory assumes a mirror symmetry and the time-reversal symmetry as a minimal requirement, there is no obstacle to applying the scheme.In the theory side, we need to extend, for instance, the rigorous coupled wave analysis [47], to deal with the noncollinear SHG.
We hope this paper stimulates further investigation of the SPDC in nonlinear metasurfaces.

FIG. 4 .
FIG. 4. Close-up view of Fig. 3 (lower panel) together with the imaginary photonic band structure (upper panel).The arrow indicated the band-crossing point we focus on in this paper.

FIG. 5 .
FIG. 5.Normalized radiation flux of the noncollinear SHG from the monolayer of the spheres.The spheres are made of a noncentrosymmetric material with the T d point group, which has χ (2) ijk = χ

2
where e0 is the electric field amplitude of the FH light.

FIG. 6 .
FIG. 6.Same as in Fig.5.However, the radiation flux is limited to the contribution of (t(2)+ g=0 )x.The fluxes of the incident PP and SS polarizations of the FH wave are very tiny (but nonzero) and are not visible in the graph scale.

FIG. 7 .
FIG.7.Normalized radiation flux of the noncollinear SHG from the uniform slab with the same material as in Fig.5.The setting of the incident wave is also the same.The slab thickness is taken to be 0.8a.The gray region corresponds to the frequency interval of Figs.5 and 6.The SHG flux for (signal,idler)=(P,P) and (S,S) vanishes.