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

Directional photon pairs generation by dielectric nanoparticles

Anna Nikolaeva1, Kristina Frizyuk1, Nikita Olekhno1, Alexander Solntsev2 and Mihail Petrov1

1ITMO University, St. Petersburg 197101, Russia 2University of Technology Sydney, 15 Broadway, Ultimo NSW 2007, Australia

Introduction Polarization correlations / Conclusion

Introduction Mie resonances and Kerker’s-type scattering

Nonlinear Kerker effect Theoretical approach / SPDC-SHG correspondence

Spontaneous parametric downconversion (SPDC) is an important nonlinear process, during this process one pump photon at frequency 𝜕! is absorbed and two photons, the idler and the signal, are generated at frequencies 𝜕" and 𝜕#. Energy conservation law is satisfied: ℏ𝜕! = ℏ𝜕" + ℏ𝜕#, we consider degenerate process assuming 𝜕" = 𝜕# = 𝜕!/2. Creation of correlated photon pairs is one of the key topics in contemporary quantum optics. Here, we theoretically describe the generation of photon pairs in the process of spontaneous parametric down-conversion in a resonant spherical nanoparticle made of a dielectric material with bulk 𝜓(%) nonlinearity. We pick the nanoparticle size that satisfies the condition of resonant eigenmodes described by Mie theory. We reveal that highly directional photon-pair generation can be

• bserved utilizing the nonlinear Kerker-type effect, and

that this regime provides useful polarization correlations. According to the Mie theory [1], expression for the scattered electric field 𝑭𝒕 and the field inside nanoparticle 𝑭𝒒 at a fundamental frequency 𝜕! can be expressed as Firstly, we need to understand linear Kerker-type scattering that was done in [2]. The scattering intensity is described by Poynting vector S, which in the dipole approximation in the forward (𝜄 = 0) and backward (𝜄 = 𝜌) direction has the form

a = 1 1 n m

ε2(ω) ε1 k E0 H0 y z x

w z G a A s

Fig. 1. Geometry

• f

the considered problem. Spherical particle of wurtzite GaAs with a radius 𝑏 = 110 𝑜𝑛, the incident plane wave propagates along z- axis, the electric field

• scillates along x-axis, the

figure also shows the

• rientation of the crystal

lattice relative to the incident pane wave.

• Fig. 2. Scattering cross-section

included different multipoles (green - total, red - electric dipole ED, blue-magnetic dipole MD, orange - electric quadrupole EQ, purple - magnetic quadrupole MQ) depending on fundamental wavelength 𝜇!.

400 600 800 1000 1200 1400 1600 Wavelength, nm 2 4 6 8 10 Scattering cross section (arb. u.) Total ED MD EQ MQ

• Fig. 3. Scalar products of vector spherical harmonics: (a,b) terms giving

the same contribution in the Poynting vector in the forward and backward direction (c) term responsible for the directionality, is included with a different sign in the forward and backward direction. (a) (b) (c)

[1] C. F. Bohren and D. R. Huffman. "Absorption and scattering of light by small particles” (1983) [2] Fu, Y. H. et al. Directional visible light scattering by silicon nanoparticles. Nat. Commun. 4:1527 (2013)

SPDC-SHG correspondence. Theoretical approach

Directional photon pairs generation by dielectric nanoparticles Anna Nikolaeva, K. Frizyuk, N. Olekhno, A. Solntsev and M. Petrov

• Fig. 4. Schematic SPDC

process in collinear decay geometry (a), signal and idler photons are being detected at the same point in the far-field 𝒔𝒋= 𝒔𝒋= 𝒔, and in non-collinear decay geometry (b) Using the approach developed in the work [3], we describe correlations between photons of the produced pair via the two-photon amplitude are dipole moments of the idler and the signal detectors is dyadic Green's function of the generating system is the generation matrix is the second-order nonlinear susceptibility tensor is the pump field which causes the nonlinear generation where

[3] A. N. Poddubny et al., Phys. Rev. Lett.117, 123901 (2016)

Two-photon counting rate:

Introduction Polarization correlations / Conclusion Nonlinear Kerker effect Theoretical approach / SPDC-SHG correspondence

My My Nx My Mx Ny Nx Nx Nx Nx Nz Nz Ny Ny Mx My Mz Mz Nx Nx Nx Nx My Mx

Mx My Mz

Nx Ny

Mx My Mz

Nx Ny Nz

1 0.8 0.6 0.4 0.2

1 0.5

Nz

D (arb. u)

• 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 - decay to the same dipoles. (a) (b)

[4] K. Frizyuk, I. Volkovskaya, D. Smirnova, A. Poddubny, and M. Petrov, Phys. Rev. B 99, 075425 (2019).

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 where 𝑿𝑲 is vector spherical harmonic (VSH) and 𝐸𝑲𝒒→𝑲𝒋 ,𝑲𝒕 is coefficient describing decaying channels, it contain VSH 𝑿𝑲𝒒 from the pump field 𝑭𝒒(𝒔𝟏) and harmonics from dyadic green functions 1 𝑯(𝒔, 𝒔𝟏,𝜕/2). The selection rules for the reverse process - second harmonic generation are determined by similar overlapping integrals [4]. Thus, we have a correspondence between spontaneous parametric downconversion (SPDC) and second harmonic generation (SHG) in the Mie configuration.

Nonlinear Kerker effect

Directional photon pairs generation by dielectric nanoparticles Anna Nikolaeva, K. Frizyuk, N. Olekhno, A. Solntsev and M. Petrov

• Fig. 6. The far-field patterns of

collinear two-photon generation for different wavelengths 𝜇𝑞=660 nm, 720 nm, 786 nm, 1086 nm. (b) Forward and backward counting rate and their ratio depending on pump wavelength 𝜇𝑞. (c) Amplitudes

• f coefficients |𝑑1| and |𝑒1| in

decomposition of pump field inside nanoparticle 𝐹𝑞. (d) Phases of this coefficients and their phase difference 𝜒𝑑1- 𝜒𝑒1.

Directivity

xxx xxx xyy xyy xzz xzz xyz xxy yxy yxy zxy yxx yyy yzz yyz yyz zyz zxx zxx zyy zyy zzz zzz zxz zxz yxz xxz xxz xxx xxx xyy xyy xzz xzz xyz xyz xxy yxy yxy zxy zxy yxx yyy yzz yyz zyz zxx zyy zzz zxz zxz yxz yxz xxz xxx xxx xyy xyy xzz xzz xyz xxy yxy yxy zxy yxx yyy yzz yyz zyz zxx zyy zzz zxz zxz yxz xxz

along X along Y along Z Non-zero χ(2) components

Ny Nz Nx Mx Mz Nz Mz Mx Nx My Ny My My Nz Ny Ny Mz Mz Mx Mx Nz Nz Nx Nx Mz Mz My My Nx Nx Ny Ny Mx Mx

Or Or Or Or Or Or Decay on crossed dipoles C1 C1h C3 D3 C3h C1 C3 D3 C1 C1h C3 Groups of symmetry Decay on co-aligned dipoles

My

Introduction Polarization correlations / Conclusion Nonlinear Kerker effect Theoretical approach / SPDC-SHG correspondence

Table 1. The first column - directionality along one of the x, y, z axes; the second - decay into crossed dipoles, magnetic and electric; the third is decay into identical dipoles; fourth – non-zero components of the second-

• rder nonlinear susceptibility

tensor 𝜓(%); last column- crystal symmetry groups required to observe directivity. Necessary decays to observe directivity in emission: 1) Crossed magnetic dipole MD and electric dipole ED modes: → 𝑁2 , 𝑂3 (𝛽 ≠ 𝛾) 2) Identical magnetic or electric dipole modes: → 𝑁2 , 𝑁2

• r → 𝑂

3 , 𝑂3

We consider the difference of unpolarized counting rates in the forward and backward directions in collinear geometry and derive the condition for observing directivity. It can be assumed that the phase difference between the electric dipole 𝑐4

(%)and magnetic dipole 𝑏4 (%) coefficients in the decay is

approximately zero 𝜒5%

(')- 𝜒6% (')~0.

Polarization correlations

Directional photon pairs generation by dielectric nanoparticles Anna Nikolaeva, K. Frizyuk, N. Olekhno, A. Solntsev and M. Petrov

• Fig. 7. (a) Configuration of detection in the area limited by the maximum angle

𝜄 = 60∘relative to the backward direction, radius of the sphere and all other parameters are similar as at previous slides (b, c) polarization correlations at 𝜇E=720 nm in linear basis (b) and in circular basis (c), R- right circular polarization, L- left circular polarization.

1

x x y y

Signal polarization Idler polarization 1 0.5 Idler polarization Signal polarization

L R R L

0.5 0.4 0.2 0.3 0.1 0.8 0.6 0.4 0.2

z x y k E0 H0

ωp

θ

di ds

ωp/2

(a) (b)

ωp/2

Introduction Polarization correlations / Conclusion Nonlinear Kerker effect Theoretical approach / SPDC-SHG correspondence

Conclusion

We have proposed the theory describing the generation of correlated photons through spontaneous down-conversion process in sub-wavelength dielectric resonator supporting lower Mie resonant modes. Using the two-photon amplitude approach, we have identified the mechanism of spontaneous photons decay inn terms of electromagnetic

• multipoles. As a result we have shown that by proper

designing the mode content for particular class crystalline materials one can achieve strongly directional emission of correlated photons. For the collinear geometry provided by idler and signal detectors positioned in the same place, we have formulated the conditions of strongly forward/backward photons generation which surprisingly appeared to be very similar to classical Kerker-effect conditions. You can see that the detected photons in non-collinear geometry have dominantly the same 𝑦-polarizations Fig. 7, which corresponds to the D- coefficients shown in the Fig. 5a, where it can be seen that the decays into two electric dipoles directed along x-axis and to the electric dipole along x- axis and magnetic dipole along y-axis are several times higher than all other possible decays.