Directional photon pairs generation by dielectric nanoparticles Anna - PowerPoint PPT Presentation

Directional photon pairs generation by dielectric nanoparticles Anna Nikolaeva 1 , Kristina Frizyuk 1 , Nikita Olekhno 1 , Alexander Solntsev 2 and Mihail Petrov 1 1 ITMO University, St. Petersburg 197101, Russia 2 University of Technology Sydney, 15 Broadway, Ultimo NSW 2007, Australia Introduction Theoretical approach / SPDC-SHG correspondence Nonlinear Kerker effect Polarization correlations / Conclusion Introduction Mie resonances and Kerker’s-type scattering According to the Mie theory [1], expression for the scattered electric field 𝑭 𝒕 and Fig. 1. Geometry of the w z G a A s Creation of correlated photon pairs is one of the key the field inside nanoparticle 𝑭 𝒒 at a fundamental frequency 𝜕 ! can be expressed as considered problem. a = 1 1 0 n m topics in contemporary quantum optics. Here, we Spherical particle of wurtzite theoretically describe the generation of photon pairs in GaAs with a radius 𝑏 = the process of spontaneous parametric down-conversion 110 𝑜𝑛 , the incident plane in a resonant spherical nanoparticle made of a dielectric wave propagates along z- material with bulk 𝜓 (%) nonlinearity. We pick the axis, the electric field ε2 ( ω ) ε1 nanoparticle size that satisfies the condition of resonant z oscillates along x-axis, the eigenmodes described by Mie theory. We reveal that k figure also shows the highly directional photon-pair generation can be H 0 orientation of the crystal Firstly, we need to understand linear Kerker-type scattering that was done in [2]. y x observed utilizing the nonlinear Kerker-type effect, and lattice relative to the The scattering intensity is described by Poynting vector S , which in the dipole E 0 that this regime provides useful polarization incident pane wave. approximation in the forward ( 𝜄 = 0 ) and backward ( 𝜄 = 𝜌 ) direction has the form correlations. 10 Spontaneous parametric Scattering cross section (arb. u.) (a) (b) (c) Total downconversion (SPDC) is an Fig. 2. Scattering cross-section ED 8 important nonlinear process, MD included different multipoles during this process one pump EQ (green - total, red - electric photon at frequency 𝜕 ! is 6 MQ dipole ED, blue-magnetic absorbed and two photons, dipole MD, orange - electric the idler and the signal, are 4 quadrupole EQ, purple - generated at frequencies 𝜕 " Fig. 3. Scalar products of vector spherical harmonics: (a,b) terms giving magnetic quadrupole MQ) and 𝜕 # . Energy conservation the same contribution in the Poynting vector in the forward and 2 depending on fundamental law is satisfied: ℏ𝜕 ! = ℏ𝜕 " + backward direction (c) term responsible for the directionality, is wavelength 𝜇 ! . included with a different sign in the forward and backward direction. ℏ𝜕 # , we consider degenerate 0 process assuming 𝜕 " = 𝜕 # = 400 600 800 1000 1200 1400 1600 [1] C. F. Bohren and D. R. Huffman. "Absorption and scattering of light by small particles” (1983) 𝜕 ! /2. Wavelength, nm [2] Fu, Y. H. et al. Directional visible light scattering by silicon nanoparticles. Nat. Commun. 4:1527 (2013)

Anna Nikolaeva, K. Frizyuk, N. Olekhno, A. Solntsev and M. Petrov Directional photon pairs generation by dielectric nanoparticles Introduction Theoretical approach / SPDC-SHG correspondence Nonlinear Kerker effect Polarization correlations / Conclusion Theoretical approach SPDC-SHG correspondence . Let’s consider collinear decay, when signal and idler photons emitted in the same direction. Two-photon amplitude can be expand in vector spherical harmonics Fig. 4. Schematic SPDC process in collinear decay geometry (a), signal and idler photons are being detected at where 𝑿 𝑲 is vector spherical harmonic (VSH) and 𝐸 𝑲 𝒒 →𝑲 𝒋 ,𝑲 𝒕 is coefficient describing decaying channels, it contain VSH 𝑿 𝑲 𝒒 from the same point in the the pump field 𝑭 𝒒 ( 𝒔 𝟏 ) and harmonics from dyadic green functions 1 𝑯 ( 𝒔, 𝒔 𝟏 , 𝜕/2 ). The selection rules for the reverse process - far-field 𝒔 𝒋 = 𝒔 𝒋 = 𝒔 , and second harmonic generation are determined by similar overlapping integrals [4]. Thus, we have a correspondence between in non-collinear decay spontaneous parametric downconversion (SPDC) and second harmonic generation (SHG) in the Mie configuration. geometry (b) Using the approach developed in the work [3], we describe correlations between photons of (b) (a) the produced pair via the two-photon amplitude 1 N x M y M y N x N z N z 1 0.8 D (arb. u) 0.6 M y N y M x N x M y M y 0.4 0.2 0.5 where are dipole moments of the idler and the signal detectors N x N x N x N x M x M x 0 is dyadic Green's function of the generating system N z N y is the generation matrix N x N x N z N x N y N y M z M z N y M z N x M y M z is the second-order nonlinear susceptibility tensor M x M y M x 0 is the pump field which causes the nonlinear generation Fig. 5. (a) D-coefficients normalized to the maximum for all possible dipole decays at a wavelength 𝜇 ! =720 nm. (b) Possible decay channels in considering geometry, grey or yellow needed to directivity: solid - decay to the crossed dipoles, dashed - Two-photon counting rate: decay to the same dipoles. [3] A. N. Poddubny et al., Phys. Rev. Lett. 117 , 123901 (2016) [4] K. Frizyuk, I. Volkovskaya, D. Smirnova, A. Poddubny, and M. Petrov, Phys. Rev. B 99, 075425 (2019).

Anna Nikolaeva, K. Frizyuk, N. Olekhno, A. Solntsev and M. Petrov Directional photon pairs generation by dielectric nanoparticles Introduction Theoretical approach / SPDC-SHG correspondence Nonlinear Kerker effect Polarization correlations / Conclusion Nonlinear Kerker effect We consider the difference of unpolarized counting rates in the forward and backward directions in Fig. 6. The far-field patterns of collinear geometry and derive the condition for observing directivity. It can be assumed that the phase collinear two-photon (%) coefficients in the decay is (%) and magnetic dipole 𝑏 4 generation for different difference between the electric dipole 𝑐 4 approximately zero 𝜒 5 % (') - 𝜒 6 % (') ~0. wavelengths 𝜇 𝑞 =660 nm, 720 nm, 786 nm, 1086 nm. (b) Forward and backward counting rate and their ratio depending on pump wavelength 𝜇 𝑞 . (c) Amplitudes of coefficients |𝑑 1 | and |𝑒 1 | in decomposition of pump field Necessary decays to observe directivity in emission: inside nanoparticle 𝐹 𝑞 . (d) 1) Crossed magnetic dipole MD and electric dipole ED modes: Phases of this coefficients and → 𝑁 2 , 𝑂 3 ( 𝛽 ≠ 𝛾 ) their phase difference 𝜒 𝑑 1 - 𝜒 𝑒 1 . 2) Identical magnetic or electric dipole modes: → 𝑁 2 , 𝑁 2 or → 𝑂 3 , 𝑂 3 Decay on crossed Groups of Directivity Decay on co-aligned dipoles Non-zero χ(2) components dipoles symmetry Table 1. The first column - N z M y M y N z M y along X xxx xxx xyy xyy xzz xzz xyz xxz xxz xxy Or directionality along one of C 1 C 1 h the x, y, z axes; the second - yxx yyy yzz yyz yyz yxz yxy yxy M z N y N y M z M z C 3 N y decay into crossed dipoles, Or zxx zyy zxx zyy zzz zzz zyz zxz zxz zxy magnetic and electric; the third is decay into identical N z N z N z M x M x M x xxx xyy xxx xyy xzz xzz xyz xyz xxz xxy along Y dipoles; fourth – non-zero Or C 1 components of the second- C 3 yxx yyy yzz yyz yxz yxz yxy yxy M z M z order nonlinear susceptibility M z D 3 N x Or N x N x zxx zyy zzz zyz zxz zxz zxy zxy tensor 𝜓 (%) ; last column- crystal symmetry groups C 1 M y N x M y M y N x xxx xxx xyy xyy xzz xzz xyz xxz xxy required to observe N x Or C 1 h along Z directivity. yxx yyy yzz yyz yxz yxy yxy C 3 N y M x M x N y N y M x D 3 Or zxx zyy zzz zyz zxz zxz zxy C 3 h