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[Tetsuyuki Ochiai](https://orcid.org/0000-0003-2933-0014)

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[Degenerate spontaneous parametric down-conversion in nonlinear metasurfaces](https://mdr.nims.go.jp/datasets/7f8075a3-7300-4e9b-bff4-530e654eed93)

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Degenerate spontaneous parametric down-conversion in nonlinear metasurfacesResearch Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11065Degenerate spontaneous parametricdown-conversion in nonlinear metasurfacesTETSUYUKI OCHIAI*Research Center for Electronic and Optical Materials, National Institute for Materials Science (NIMS),Tsukuba 305-0044, Japan*OCHIAI.Tetsuyuki@nims.go.jpAbstract: 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 aband crossing between even- and odd-parity quasi-guided mode bands inside the light cone (abovethe light line) and a selection rule in the conversion efficiency of the SPDC. The efficiency canbe 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 noncentrosymmetricspheres and confirm that the scenario works well to enhance the SPDC.© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement1. IntroductionThe spontaneous parametric down-conversion (SPDC) is a quantum phenomenon of generatingtwo photons with angular frequency ω1 and ω2 by a pump light of angular frequency ω1 + ω2[1]. It becomes a source of entangled photon pairs [2], so that the SPDC can be used in quantuminformation, such as the Bell inequality test experiment [3,4].Compared with other entangled photon sources, such as the cascade emission from three-levelatoms or biexcitons in quantum dots [5–7], the SPDC has its advantages and disadvantages. Oneadvantage is it’s relatively easy to set up. The disadvantages include (in principle) low generationefficiency, lack of determinism, and need for filtering. These properties are tied with the SPDCbeing a nonlinear optical process.As a second-order nonlinear optical process, the SPDC is observed typically in uniaxial andnoncentrosymmetric media [8]. There, the phase matching via the birefringe is crucial. Forinstance, the phase matching of type II gives the polarization entanglement between the ordinaryand extraordinary waves with orthogonal polarizations.Optical media that are uniaxial and noncentrosymmetric are limited. Relaxing these twoproperties 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 artificialspatial periodicity in the system [9–12]. There, we can be free from the birefringe of the bulkmaterials. Another important direction is to employ surface modes in, even centrosymmetricmedia [13,14]. Since any material breaks the centrosymmetry at the surface, the system canhave χ(2) there. Combining possible surface modes with surface (or bulk) χ(2), we can expect asignificant enhancement of the second-order optical nonlinearity even in optically thin specimens.Recently, metasurfaces or photonic crystal slabs including nonlinear materials have attractedgrowing interest as a platform for enhanced nonlinear optical processes [15]. This trend is inthe latter direction. In particular, the so-called bound states in the continuum (BICs) [16–20]or, more generally, quasi-guided modes [21,22] are vital items there. These modes have high(ideally infinite) quality factors so that the light-matter interaction involving them becomes verystrong [23]. So far, various functions such as lasing [24–26], second-harmonic generation (SHG)[27–37], and Kerr effect [38–40], have been demonstrated via nonlinear metasurfaces. However,the study of SPDC in nonlinear metasurfaces is still limited [41–45].#514969 https://doi.org/10.1364/OE.514969Journal © 2024 Received 4 Dec 2023; revised 20 Feb 2024; accepted 21 Feb 2024; published 12 Mar 2024https://orcid.org/0000-0003-2933-0014https://doi.org/10.1364/OA_License_v2#VOR-OAResearch Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11066For instance, previous studies on the SPDC in two-dimensional (2D) metasurfaces rely ondeformations (in geometry) of symmetry-protected BICs at the Γ point so that the emission anglesof the signal and idler photons are close to the normal direction of the metasurfaces [43,44].There is a broad tunability of relevant quality factors in such cases. The tunability is reduced inone-dimensional (1D) metasurfaces [42,45].Instead of such a deformation, we here consider plain 2D structures and a conventionalphenomenon of band crossings at a generic k point on a mirror axis. This setting allows usto imitate the conventional SPDC scheme of the type II phase matching even in isotropic butnoncentrosymmetric media. The 2D structures are generally preferable to the 1D structures.The 2D structures are flexible in symmetry, allowing multiple mirror axes. As the BICs emergenormally on mirror axes, there are more opportunities to enhance the SPDC. Enriched symmetriesin 2D further control the possible combinations of pump, signal and idler photons via a selectionrule.In this paper, we present a theoretical analysis of the SPDC, taking account of metasurfacestructures and their spatial symmetries in a first-principles manner. We present a simple scenarioof the degenerate SPDC and polarization entanglement in metasurfaces with a certain photonicband structure. The essential quantity is the conversion efficiency from the pump light to biphotonstates. The spatial symmetries of the metasurface and the symmetry of χ(2) yields a selectionrule in the conversion efficiency factor. It restricts possible combinations of the biphoton statesand pump-light polarization. Moreover, the efficiency factor can be strongly enhanced by theexcitation of quasi-guided modes in the metasurface. We demonstrate these features in termsof a classical electromagnetic calculation of the reverse process, namely, the sum-frequencygeneration (SFG) [46–49] or the noncollinear SHG.This paper is organized as follows. In Sec. 2, we summarize several formulas of the SPDCin terms of an eigenmode expansion of the quantized radiation field. Section 3 is devotedto presenting the reverse process, namely, the SFG, in a fully classical approach. In Sec. 4,we present symmetry properties of the conversion efficiency factor in the SPDC and resultingpolarization entanglement in nonlinear metasurfaces assuming a mirror symmetry. In Sec. 5, wepresent a simulation of the conversion efficiency in a monolayer of noncentrosymmetric spheresas a toy model. Finally, in Sec. 6, summary and discussion are given.2. Spontaneous parametric down conversionWe first consider a generic photonic system with noncentrosymmetric media. The HamiltonianH of the radiation field including the second-order optical nonlinearity becomes [50]H = H0 +H1, (1)H0 =∫d3x(︃µ02H2 +12ϵ0D←→η (1)D)︃, (2)H1 =∫d3x13ϵ20η(2)ijk DiDjDk, (3)←→η (1) = (←→ϵ (1))−1, ←→ϵ (1) =←→1 +←→χ (1), (4)η(2)ijk = −χ(2)lmnη(1)il η(1)mj η(1)nk , (5)where D and H are the electric displacement and magnetic fields, respectively, µ0 and ϵ0 are thevacuum permeability and permittivity, respectively, and χ(1) and χ(2) are space-dependent linearResearch Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11067and second-order electric susceptibilities, respectively, of the media. That is,Pi = ϵ0(χ(1)ij Ej + χ(2)ijk EjEk), (6)being P and E the electric polarization and electric field, respectively, with constitutive relationD = ϵ0E + P.We introduce the dual vector potential Λ with the "Coulomb" gauge (∇ · Λ = 0) asD = ∇ × Λ, H = ∂Λ∂t. (7)Using the eigenmodes of Λ of the linear Maxwell equation, namely,∇ ×(︂←→η (1)(x)∇ × Λα(x))︂=ω2αc2 Λα(x) (8)∫d3xΛ∗α(x) · Λβ(x) = δαβ , (9)the radiation field can be expanded asD(x, t) =∑︂α√︄ℏ2µ0ωα(︂aα(t)∇ × Λα(x) + a†α(t)∇ × Λ∗α(x))︂, (10)H(x, t) = −i∑︂α√︄ℏωα2µ0(︂aα(t)Λα(x) − a†α(t)Λ∗α(x))︂, (11)where index α represents an eigenstate, ωα is its eigenfrequency, and Λα is its normalizedeigenfunction. In uniform media, the index α corresponds to a wave number and polarization.The unperturbed HamiltonianH0 then becomesH0 =∑︂αℏωα2(︂a†α(t)aα(t) + aα(t)a†α(t))︂, (12)[aα(t), a†β(t)] = δαβ , (13)assuming the time-reversal symmetry:Λ∗α = eiλαΛ−α, ωα = ω−α. (14)In the interaction picture, we haveaα(t) = aαe−iωα t. (15)The state vector is time-developed as|ψ(t)⟩ = T e−iℏ∫ t−∞ dt′H1(t′) |ψ(−∞)⟩ (16)where T represents the time-ordering product.Let us consider a strong continuous wave (CW) pump light of an eigenstate α = p (pump) isincident on the system. Its fluctuation can be safely neglected so that the interaction HamiltonianH1 relevant to the SPDC isH1 →∫d3x1ϵ20η(2)ijk Dpi Dfj Dfk, (17)Dp(x, t) =√︄ℏ2µ0ωp(︂βpe−iωpt∇ × Λp(x) + c.c.)︂, (18)where Dp is the classical-pump field, Df is the quantum-fluctuation expressed by Eq. (10), andc.c. represents the complex conjugate. Here, we assume the permutation symmetry concerningthe indices of η(2)ijk holds, assuming a nondispersive χ(2).Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11068In the first-order perturbation, if we start from the vacuum state |0⟩ at time t = −∞, we thenobtain the biphoton states after a long time:|ψ(∞)⟩ ≃ |0⟩ + 1iℏ(︃ℏ2µ0)︃ 32βp∑︂α1,α2|︁|︁1α1 ⊗ 1α2⟩︁ Fpα1α2√ωpωα1ωα22πδ(︁ωp − ωα1 − ωα2)︁(19)Fpα1α2 =∫d3xη(2)ijk (∇ × Λp)i(∇ × Λ∗α1 )j(∇ × Λ∗α2 )k, (20)where |1α⟩ represents the one photon state with eigenstate α. The second term is thus asuperposition of entangled biphoton states between eigenstates α1 and α2. The conversionefficiency from the pump light to the biphoton state is represented by factor Fpα1α2 .3. Sum-frequency generationTo evaluate the SPDC, we need the factor Fpα1α2 . The direct calculation of it according to Eq. (20)requires a detailed analysis of the eigenmode profiles. Instead, we consider the reverse process ofthe SPDC, namely, the SFG, in which the same factor emerges as we will see. Since the SFG doesnot need a quantum approach, we can access the factor fully classically in a perturbation scheme.Suppose that a incident light of eigenstates α1 and α2 is impinging to the system:D(1)(x, t) = Re[︄√︄2ℏµ0ωα1β1∇ × Λα1 (x)e−iωα1 t +√︄2ℏµ0ωα2β2∇ × Λα2 (x)e−iωα2 t]︄, (21)H(1)(x, t) = Re⎡⎢⎢⎢⎢⎣−i√︄2ℏωα1µ0β1Λα1 (x)e−iωα1 t − i√︄2ℏωα2µ0β2Λα2 (x)e−iωα2 t⎤⎥⎥⎥⎥⎦, (22)In the perturbative viewpoint, this field forms the nonlinear polarization P(NL) through Eq. (6) asP(NL)i (x, t) = ϵ0 χ(2)ijk (x)E(1)j (x, t)E(1)k (x, t), (23)E(1)(x, t) = 1ϵ0←→η (1)(x)D(1)(x, t). (24)The nonlinear polarization has several frequency components. The component of angularfrequency ω = ωα1 + ωα2 is given byP(NL)(x, t) → Re[P(NL)ω (x)e−iωt], (25)(P(NL)ω (x))i = −2ℏc2 β1β2√ωα1ωα2ϵ(1)ij (x)η(2)jkl (x)(∇ × Λα1 (x))k(∇ × Λα2 (x))l. (26)This polarization becomes the source of the SFG.Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11069The dual vector potential induced by the source satisfies∇×(←→η (1)∇×Λ(2)ω ) −ω2c2 Λ(2)ω = ∇×(←→η (1)P(NL)ω ). (27)By using the eigenmode expansion, the equation is solved asΛ(2)ω (x) =∑︂αc(2)α Λα(x), (28)c(2)α = 2ℏc4β1β2F∗αα1α2(ω2 − ω2α)√ωα1ωα2. (29)The radiation flux of the SFG is given by∫dS · 12Re[E(2)∗ω ×H(2)ω ]=πℏ2c6ϵ0|β1β2 |2∑︂αδ(ωα − ωα1 − ωα2 )|Fαα1α2 |2ωα1ωα2.(30)If the frequency-matched eigenstate is unique, we can access factor |Fαα1α2 |2 by fully classicallycalculating the radiation flux. This assumption is generally valid even if the relevant eigenstate isdegenerate. The condition of the frequency conservation and nonzero Fαα1α2 selects a uniqueeigenstate of particular momentum and polarization. A doubly degenerate mode consists oftwo modes with opposite parities, so that either one of the two modes or their particular linearcombination is selected.4. Selection rule in metasurfacesIn what follows, we consider a nonlinear metasurface that is periodic in the xy plane and is finitein the z direction. We do not care about the details of the system, but we impose that the systemhas a mirror symmetry on a particular axis of the system and the time-reversal symmetry. Wefurther assume the mirror axis bisects the unit cell (UC).According to the Bloch theorem, the system is characterized by a 2D Bloch momentum k. Fork on the mirror axis, say the x axis, the eigenmodes are classified according to the parity (of the yinversion) concerning the axis. Then, we have−σ̂yΛα̃k(σ̂yx) = ±Λα̃k(x), (31)Λα̃k(x) = ei(kxx+kyy)uα̃k(x), (32)uα̃k(x + X) = uα̃k(x), (33)where σ̂y is the y inversion operator, α = (α̃k), and X is a 2D real lattice vector. Thus, we haveFpα1α2 =∑︂Xei(kp−kα1−kα2 )·XF̃pα1α2 , (34)F̃pα1α2 =∫UCd3xη(2)ijk (∇×upkp )i(∇×u∗α̃1kα1)j(∇×u∗α̃2kα2)k, (35)The 2D phase matching requires kp = kα1 + kα2 (modulo 2D reciprocal lattice vector g).Suppose that the pump light is incident to the metasurface in the normal direction (kp = 0).It is then possible to produce the biphoton state having respective Bloch momenta of k and −kboth on the mirror axis. In this case, the pump, signal, and idler states are classified according toResearch Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11070the 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 P-polarized in the far field. Theodd 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 oddparities intersect at ±kc as shown in Fig. 1. The eigenfrequency there is denoted as ωc. Weassume the pump frequency ωp is twice the eigenfrequency, ωp = 2ωc. Then, both the even andodd parity eigenstates of ωc can be generated.kc-kcωckωeven (P-pol.)odd (S-pol.)Fig. 1. Schematic band structure for the SPDC. On a mirror axis of the system, thephotonic band structure is classified into even- and odd-parity bands, which are P- andS-polarized, respectively, in the far field below the diffraction threshold. At the crossingpoints, the selection rule in the conversion-efficiency factor of the SPDC results in apolarization entanglement between signal and idler photons for a pump light of thenormal incidence. The signal and idler photons are degenerate with angular frequency𝜔𝑐 , and the momentum ±𝒌𝑐 . The gray area is the region outside the light cone (belowthe light line).for (signal,idler) being (P,P) or (S,S) polarized and146(𝑖 𝑗 𝑘) = (𝑥𝑥𝑦), (𝑥𝑦𝑥), (𝑥𝑦𝑧), (𝑥𝑧𝑦) (37)for (signal,idler) being (P,S) or (S,P) polarized below the diffraction threshold. If the pump light147is 𝑦-polarized and off-resonant, and if the signal and idler photons are on the mirror axis of the 𝑥148direction, the nonvanishing 𝜂(2)𝑖 𝑗𝑘components are149(𝑖 𝑗 𝑘) = (𝑦𝑥𝑥), (𝑦𝑥𝑧), (𝑦𝑧𝑥), (𝑦𝑧𝑧), (𝑦𝑦𝑦) (38)for (signal,idler)=(P,P) or (S,S), and150(𝑖 𝑗 𝑘) = (𝑦𝑥𝑦), (𝑦𝑦𝑥), (𝑦𝑦𝑧), (𝑦𝑧𝑦) (39)for (signal,idler)=(P,S) or (S,P) below the diffraction threshold. A similar selection rule takes151place for signal and idler photons on other mirror axes.152In a simple noncentrosymmetric material whose crystal symmetry is 𝑇𝑑 , the allowed nonzero153component of 𝜒 (2) is154(𝑖 𝑗 𝑘) = (𝑥𝑦𝑧), (𝑥𝑧𝑦), (𝑦𝑧𝑥), (𝑦𝑥𝑧), (𝑧𝑥𝑦), (𝑧𝑦𝑥). (40)All the components are equal in 𝜒 (2) [8]. The linear susceptibility 𝜒(1)𝑖 𝑗is scalar in this point group.155According to Eq. (5), 𝜂 (2)𝑖 𝑗𝑘has the same nonzero components as 𝜒(2)𝑖 𝑗𝑘. Some semiconductors156such as GaAs, GaP, and ZnTe are in this category, having very large 𝜒 (2) compared with widely157used nonlinear-optical media such as KDP and BBO.158In this case, the 𝑥-polarized off-resonant pump gives the polarization entanglement as159𝑐PS |P𝒌𝑐⟩|S−𝒌𝑐⟩ + 𝑐SP |S𝒌𝑐⟩|P−𝒌𝑐⟩, (41)𝑐SP = ei(𝜆S𝒌𝑐+𝜆P(−𝒌𝑐 )−𝜆𝑝0 )𝑐∗PS, (42)Fig. 1. Schematic band structure for the SPDC. On a mirror axis of the system, the photonicband structure is classified into even- and odd-parity bands, which are P- and S-polarized,respectively, in the far field below the diffraction threshold. At the crossing points, theselection rule in the conversion-efficiency factor of the SPDC results in a polarizationentanglement 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 (below the light line).If the pump light is x-polarized and off-resonant, and if the signal and idler photons are on themirror axis of the x direction, the nonvanishing η(2)ijk components that contribute to Fpα1α2 is(ijk) = (xxx), (xxz), (xzx), (xzz), (xyy) (36)for (signal,idler) being (P,P) or (S,S) polarized and(ijk) = (xxy), (xyx), (xyz), (xzy) (37)for (signal,idler) being (P,S) or (S,P) polarized below the diffraction threshold. If the pump lightis y-polarized and off-resonant, and if the signal and idler photons are on the mirror axis of the xdirection, the nonvanishing η(2)ijk components are(ijk) = (yxx), (yxz), (yzx), (yzz), (yyy) (38)for (signal,idler)=(P,P) or (S,S), and(ijk) = (yxy), (yyx), (yyz), (yzy) (39)for (signal,idler)=(P,S) or (S,P) below the diffraction threshold. A similar selection rule takesplace for signal and idler photons on other mirror axes.Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11071In a simple noncentrosymmetric material whose crystal symmetry is Td, the allowed nonzerocomponent of χ(2) is(ijk) = (xyz), (xzy), (yzx), (yxz), (zxy), (zyx). (40)All the components are equal in χ(2) [8]. The linear susceptibility χ(1)ij is scalar in this point group.According to Eq. (5), η(2)ijk has the same nonzero components as χ(2)ijk . Some semiconductors suchas GaAs, GaP, and ZnTe are in this category, having very large χ(2) compared with widely usednonlinear-optical media such as KDP and BBO.In this case, the x-polarized off-resonant pump gives the polarization entanglement ascPS |Pkc⟩|S−kc⟩ + cSP |Skc⟩|P−kc⟩, (41)cSP = ei(λSkc+λP(−kc)−λp0)c∗PS, (42)and the y-polarized off-resonant pump givescPP |Pkc⟩|P−kc⟩ + cSS |Skc⟩|S−kc⟩ (43)under the time-reversal symmetry, namely,u∗α̃k(x) = eiλα̃kuα̃(−k)(x), λα̃k = λα̃(−k). (44)The polarization state is maximally entangled in the former case, whereas the latter is not. Theabsolute values of complex coefficients cPP, cPS, cSP, cSS are available classically via Eq. (30).Determining their phases is essential in entanglement manipulation, but requires a detailedanalysis of the eigenmodes. Here, we focus on the absolute values that relate to the conversionefficiency of the SPDC.5. Case of a monolayer of dielectric spheresAs a model calculation, let us consider the square lattice of dielectric spheres made of anoncentrosymmetric material with the Td point group. A schematic illustration of the systemunder study is shown in Fig. 2. The two incident plane-wave lights with the same frequencycome into the monolayer, and the noncollinear SHG is induced. The radiation flux of the SHGgives the conversion efficiency of the SPDC through Eq. (30).χ(2)(2ω,0)(ω,k) (ω,-k)ax,yzFig. 2. Schematic illustration of the toy model system under study. Two plane-wavelights are incident on a monolayer of spheres arranged in the square lattice withlattice constant 𝑎, and the noncollinear SHG takes place. The spheres are made of anoncentrosymmetric material with the second-order electric susceptibility 𝜒 (2) .and the 𝑦-polarized off-resonant pump gives160𝑐PP |P𝒌𝑐⟩|P−𝒌𝑐⟩ + 𝑐SS |S𝒌𝑐⟩|S−𝒌𝑐⟩ (43)under the time-reversal symmetry, namely,161𝒖∗𝛼̃𝒌 (𝒙) = ei𝜆𝛼̃𝒌𝒖 𝛼̃(−𝒌 ) (𝒙), 𝜆 𝛼̃𝒌 = 𝜆 𝛼̃(−𝒌 ) . (44)The polarization state is maximally entangled in the former case, whereas the latter is not.162The absolute values of complex coefficients 𝑐PP, 𝑐PS, 𝑐SP, 𝑐SS are available classically via Eq.163(30). Determining their phases is essential in entanglement manipulation, but requires a detailed164analysis of the eigenmodes. Here, we focus on the absolute values that relate to the conversion165efficiency of the SPDC.1665. Case of a monolayer of dielectric spheres167As a model calculation, let us consider the square lattice of dielectric spheres made of a168noncentrosymmetric material with the 𝑇𝑑 point group. A schematic illustration of the system169under study is shown in Fig. 2. The two incident plane-wave lights with the same frequency170come into the monolayer, and the noncollinear SHG is induced. The radiation flux of the SHG171gives the conversion efficiency of the SPDC through Eq. (30).172Figure 3 shows the photonic band structure of the true-guided and quasi-guided modes along173the ΓX direction of the square lattice. The band structure was calculated via the photonic layer174Korringa-Kohn-Rostoker (KKR) method [51–53]. Inside the light cone (above the light line), the175band structure is obtained by fitting the scattering phase shift 𝛿 [54] to the Breit-Wigner form as176e2i𝛿 = e2i𝛿0𝜔 − 𝜔0 − i𝛾𝜔 − 𝜔0 + i𝛾. (45)where 𝛿0 is the background phase shift, 𝜔0 is the resonance frequency, and 𝛾 is its width (or177inverse lifetime). Here, we plot 𝜔0 as a function of Bloch momentum 𝒌. The band structure is178classified according to the parities in the 𝑦 and 𝑧 directions. It have several band crossing points179between 𝑦-even and 𝑦-odd bands. At these crossing points (𝒌𝑐, 𝜔𝑐), the degenerate SPDC can180be enhanced.181Figure 4 shows a close-up view of the band structure together with the resonance width 𝛾.182Here, we focus on the band-crossing point between "ee" and "oo" bands. They have relatively183small 𝛾, so that the light-matter interaction is enhanced through these two modes.184Next, we consider the conversion efficiency of the SPDC via the classical calculation of the185noncollinear SHG. We assume two incident plane waves with the same angular frequency 𝜔/2186and opposite incident angles 𝜃𝑐 = ± sin−1 (𝑐𝑘𝑐/𝜔𝑐), which is adjusted for a particular crossing187Fig. 2. Schematic illustration of the toy model system under study. Two plane-wave lightsare incident on a monolayer of spheres arranged in the square lattice with lattice constanta, and the noncollinear SHG takes place. The spheres are made of a noncentrosymmetricmaterial with the second-order electric susceptibility χ(2).Figure 3 shows the photonic band structure of the true-guided and quasi-guided modes alongthe ΓX direction of the square lattice. The band structure was calculated via the photonic layerKorringa-Kohn-Rostoker (KKR) method [51–53]. Inside the light cone (above the light line), theband structure is obtained by fitting the scattering phase shift δ [54] to the Breit-Wigner form ase2iδ = e2iδ0ω − ω0 − iγω − ω0 + iγ. (45)where δ0 is the background phase shift, ω0 is the resonance frequency, and γ is its width (orinverse lifetime). Here, we plot ω0 as a function of Bloch momentum k. The band structure isResearch Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11072classified according to the parities in the y and z directions. It have several band crossing pointsbetween y-even and y-odd bands. At these crossing points (kc,ωc), the degenerate SPDC can beenhanced. 0.3 0.4 0.5 0.6 0.7 0.8 0  0.1  0.2  0.3  0.4  0.5ωa/2πckxa/2πee eo oe ooFig. 3. Photonic band structure along the ΓX (𝑘𝑥) direction in the square lattice ofdielectric spheres. The spheres have dielectric constant 12 and radius 0.4𝑎 where𝑎 is the lattice constant. The photonic band modes are classified according to theparities relevant to 𝑦 and 𝑧 directions. Symbols "ee","eo", "oe", and "oo" stand for(𝑦-even,𝑧-even), (𝑦-even,𝑧-odd), (𝑦-odd,𝑧-even), and (𝑦-odd,𝑧-odd), respectively. Solid,dashed, and dotted lines represent the light cone (light line), the diffraction threshold,and the line of constant incident angle 𝜃 = 15.26◦, respectively.point. The incident light induces the noncollinear SHG of angular frequency 𝜔. Then, the188resulting radiation field of the SHG in the far field is expressed as189𝑬 (2)± (𝒙) =open∑︁𝒈ei𝑲±𝒈 ·𝒙 𝒕 (2)±𝒈 , (46)𝑲±𝒈 = 𝒈 ± 𝑧Γ𝒈 , Γ𝒈 =√︂(𝜔𝑐)2− 𝒈2. (47)where superscript ± refers to the sign of 𝑧 (the monolayer center is taken to be 𝑧 = 0). The190plane-wave-expansion coefficient 𝒕 (2)±𝒈 can also be calculated via the photonic layer KKR method.191The radiation flux of the SHG per unit area is given by192𝐹 (2) =open∑︁𝒈Γ𝒈2𝜇0𝜔(| 𝒕 (2)+𝒈 |2 + | 𝒕 (2)−𝒈 |2). (48)This flux should be identified with Eq. (30) divided by the planar area of the system.193Under a non-resonant pumping, the eigenmode relevant to the pump light is like a plane wave.194Therefore, the electric field of the pump light has a little 𝑧 dependence inside the monolayer,195provided the monolayer thickness is in the subwavelength regime. In this case together with the196Fig. 3. Photonic band structure along the ΓX (kx) direction in the square lattice of dielectricspheres. The spheres have dielectric constant 12 and radius 0.4a where a is the latticeconstant. The photonic band modes are classified according to the parities relevant to y andz directions. Symbols "ee","eo", "oe", and "oo" stand for (y-even, z-even), (y-even, z-odd),(y-odd, z-even), and (y-odd, z-odd), respectively. Solid, dashed, and dotted lines representthe light cone (light line), the diffraction threshold, and the line of constant incident angleθ = 15.26◦, respectively.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 relativelysmall 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 thenoncollinear SHG. We assume two incident plane waves with the same angular frequency ω/2and opposite incident angles θc = ± sin−1(ckc/ωc), which is adjusted for a particular crossingpoint. The incident light induces the noncollinear SHG of angular frequency ω. Then, theresulting radiation field of the SHG in the far field is expressed asE(2)±(x) =open∑︂geiK±g ·xt(2)±g , (46)K±g=g ± ẑΓg, Γg =√︃(︂ωc)︂2− g2. (47)where superscript ± refers to the sign of z (the monolayer center is taken to be z = 0). Theplane-wave-expansion coefficient t(2)±g can also be calculated via the photonic layer KKR method.Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11073 0.64 0.642 0.644 0.646 0.648 0.65 0  0.1  0.2  0.3  0.4  0.5ωa/2πckxa/2π 0 0.0002 0.0004 0.0006 0.0008 0.001γa/2πcee eo oe ooFig. 4. Close-up view of Fig. 3 (lower panel) together with the imaginary photonicband structure (upper panel). The arrow indicated the band-crossing point we focus onin this paper.point group 𝑇𝑑 , the signal and idler lights are better to have opposite parities in the 𝑧 direction.197Otherwise, factor 𝐹𝑝𝛼1𝛼2 becomes small.198Figure 5 shows the radiation flux of the noncollinear SHG for the two incident waves of either199P or S-polarization. The incident angle 𝜃𝑐 is fixed to excite the modes at a crossing point [𝜃𝑐 =200± sin−1 (𝑐𝑘𝑐/𝜔𝑐)] and angular frequency is scanned. The SHG flux is normalized by the factor201|𝜒 (2)𝑡 𝑒20 |2/(2𝜇0𝑐), where 𝜒(2)𝑡 is the nonzero 𝜒 (2) component assuming a noncentrosymmetric202material with the𝑇𝑑 crystal symmetry, and 𝑒0 is the amplitude of the FH waves. This normalization203is used to extract the enhancement of the metasurface structure origin, independent of the actual204𝜒 (2) value and the incident-flux intensity. The physical SHG flux value in the theory is obtained205by multiplying the normalized value of Fig. 5 with206|𝜒 (2)𝑡 𝑒20 |22𝜇0𝑐= 0.7535 × 10−17𝐹02 ( 𝜒̃ (2)𝑡 )2 [W/(cm)2], (49)where 𝐹̃0 is the flux density 𝐹0 = |𝑒0 |2/(2𝜇0𝑐) of the FH waves in units of [W/(cm)2] and 𝜒̃(2)𝑡207is 𝜒(2)𝑡 in units of [pm/V].208The SHG flux is strongly enhanced at various first-harmonic (FH) frequencies. These peaks209represent the excitement of P- or S-polarized quasi-guided modes at FH frequencies. Or, the210excitement of the quasi-guided modes at the Γ point of second-harmonic frequencies. However,211all of these peaks are not directly related to the SPDC of interest, because the calculated flux212includes those of the diffraction channels other than the non-diffractive (𝒈 = 0) one of the normal213direction. The normal direction is supposed to be the pumping direction of the SPDC.214Instead, if we plot solely the flux contribution in the normal direction, the result changes215Fig. 4. Close-up view of Fig. 3 (lower panel) together with the imaginary photonic bandstructure (upper panel). The arrow indicated the band-crossing point we focus on in thispaper.The radiation flux of the SHG per unit area is given byF(2) =open∑︂gΓg2µ0ω(︂|t(2)+g |2 + |t(2)−g |2)︂. (48)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 thepoint group Td, the signal and idler lights are better to have opposite parities in the z direction.Otherwise, factor Fpα1α2 becomes small.Figure 5 shows the radiation flux of the noncollinear SHG for the two incident waves ofeither P or S-polarization. The incident angle θc is fixed to excite the modes at a crossing point[θc = ± sin−1(ckc/ωc)] and angular frequency is scanned. The SHG flux is normalized by thefactor |χ(2)t e20 |2/(2µ0c), where χ(2)t is the nonzero χ(2) component assuming a noncentrosymmetricmaterial with the Td crystal symmetry, and e0 is the amplitude of the FH waves. This normalizationis used to extract the enhancement of the metasurface structure origin, independent of the actualχ(2) value and the incident-flux intensity. The physical SHG flux value in the theory is obtainedby multiplying the normalized value of Fig. 5 with|χ(2)t e20 |22µ0c= 0.7535 × 10−17F̃02( χ̃(2)t )2[W/(cm)2], (49)where F̃0 is the flux density F0 = |e0 |2/(2µ0c) of the FH waves in units of [W/(cm)2] and χ̃(2)t isχ(2)t in units of [pm/V].Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 1107410-310-210-1100101102103104 0.6  0.62  0.64  0.66  0.68  0.7Normalized SHG fluxωa/2πc of FHPPPS,SPSSFig. 5. Normalized radiation flux of the noncollinear SHG from the monolayer of thespheres. The spheres are made of a noncentrosymmetric material with the𝑇𝑑 point group,which has 𝜒(2)𝑖 𝑗𝑘= 𝜒(2)𝑡 (≠ 0) for (𝑖 𝑗 𝑘) = (𝑥𝑦𝑧), (𝑥𝑧𝑦), (𝑦𝑧𝑥), (𝑦𝑥𝑧), (𝑧𝑥𝑦), (𝑧𝑦𝑥) andzero otherwise. The incident angle of the first-harmonic (FH) waves is fixed to be𝜃𝑐 = ±15.26◦ to excite the quasi-guided modes at the crossing point of Fig. 4. The twoFH waves are either P- or 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. Normalizedradiation flux is defined by 2𝑐𝜇0𝐹(2)/|𝜒 (2)𝑡 𝑒20 |2 where 𝑒0 is the electric field amplitudeof the FH light.drastically from Fig. 5, as shown in Fig. 6. There is a remarkable peak only for (signal,idler)=(P,S)216and (S,P) at 𝜔𝑎/2𝜋𝑐 = 0.646, where the P- and S-polarized bands cross as in Fig. 4 and the217SHG light is dominantly 𝑥-polarized as expected. The 𝑦-polarized SHG component is negligible218in the entire frequency range of Fig. 6. However, if we closely look at the small peak of the219crossing point concerned, we can observe that the 𝑦-polarized SHG component is enhanced only220for (signal,idler)=(P,P) or (S,S), as expected from the selection rule in Sec. 4. In this case, most221power is transmitted to the four diffraction channels with 𝒈 = ±𝑥,±𝑦̂ (×2𝜋/𝑎). Therefore, the222polarization entanglement is limited only if we send the pump light from the four different angles223that correspond to the above diffraction channels. Thus, it is better to design the resonance such224that 2𝜔𝑐 is below the diffraction threshold 2𝜋𝑐/𝑎.225For comparison, we show in Fig. 7 the radiation flux of the noncollinear SHG from a uniform226slab made of the same material as in Fig. 5. The thickness of the slab is taken to be the same227as the diameter of the spheres. There is no marked signal other than the three broad peaks for228(signal,idler)=(P,S) and (S,P). They correspond to the Fabri-Perot resonance of the slab. Moreover,229the normalized SHG flux is much smaller than in Figs. 5 and 6. As for (signal,idler)=(P,P) and230(S,S), the SHG flux is identically zero by symmetry.231Fig. 5. Normalized radiation flux of the noncollinear SHG from the monolayer of thespheres. The spheres are made of a noncentrosymmetric material with the Td point group,which has χ(2)ijk = χ(2)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 excitethe quasi-guided modes at the crossing point of Fig. 4. The two FH waves are either P-or 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 by2cµ0F(2)/|χ(2)t e20 |2 where e0 is the electric field amplitude of the FH light.The SHG flux is strongly enhanced at various first-harmonic (FH) frequencies. These peaksrepresent the excitement of P- or S-polarized quasi-guided modes at FH frequencies. Or, theexcitement of the quasi-guided modes at the Γ point of second-harmonic frequencies. However,all of these peaks are not directly related to the SPDC of interest, because the calculated fluxincludes those of the diffraction channels other than the non-diffractive (g = 0) one of the normaldirection. 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 changesdrastically 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 theSHG light is dominantly x-polarized as expected. The y-polarized SHG component is negligiblein the entire frequency range of Fig. 6. However, if we closely look at the small peak of thecrossing point concerned, we can observe that the y-polarized SHG component is enhanced onlyfor (signal,idler)=(P,P) or (S,S), as expected from the selection rule in Sec. 4. In this case, mostpower is transmitted to the four diffraction channels with g = ±x̂,±ŷ (×2π/a). Therefore, thepolarization entanglement is limited only if we send the pump light from the four different anglesthat correspond to the above diffraction channels. Thus, it is better to design the resonance suchthat 2ωc is below the diffraction threshold 2πc/a.Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11075Fig. 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. Normalized radiation flux of the noncollinear SHG from the uniform slab with thesame material as in Fig. 5. The setting of the incident wave is also the same. The slabthickness is taken to be 0.8a. The gray region corresponds to the frequency interval ofFigs. 5 and 6. The SHG flux for (signal,idler)=(P,P) and (S,S) vanishes.Research Article Vol. 32, No. 7 / 25 Mar 2024 / Optics Express 11076For comparison, we show in Fig. 7 the radiation flux of the noncollinear SHG from a uniformslab made of the same material as in Fig. 5. The thickness of the slab is taken to be the sameas 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.6. Summary and discussionIn summary, we have presented a theoretical analysis of the degenerate SPDC in nonlinearmetasurfaces. A band crossing between even- and odd-parity quasi-guided mode bands on amirror axis yields a boosted SPDC with a polarization-entangled biphoton state. The conversionefficiency 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 thesquare lattice.There are many issues that remain to be investigated. One is to determine and design therelative phase between |P⟩|S⟩ and |S⟩|P⟩ (or |P⟩|P⟩ and |S⟩|S⟩) mentioned in the text. It requiresa detailed analysis of the relevant eigenmodes and time-reversal symmetry.Another critical issue is an on-resonant pumping. To make the system simple enough, wehave assumed the off-resonant pumping. There, the pump light is plane-wave like even insidethe metasurfaces. The on-resonant pump implies the excitation of a quasi-guided mode at the Γpoint. The local field of the quasi-guided mode is no longer plane-wave like, so that the selectionrule becomes complicated, mixing all the four combinations of |P⟩|P⟩, |P⟩|S⟩, |S⟩|P⟩, and |S⟩|S⟩.At the same time, the on-resonant pump further boosts the conversion efficiency of the SPDC.It is also essential to implement the present scheme of the SPDC in experimentally moreaccessible platforms, e.g., dielectric slabs with a periodic array of air holes. Since the presenttheory 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 [55], to deal with the noncollinear SHG.We hope this paper stimulates further investigation of the SPDC in nonlinear metasurfaces.Funding. Japan Society for the Promotion of Science (22K03488).Disclosures. 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