Confining light in all-dielectric anisotropic metamaterial particles for nano-scale nonlinear optics
Saman Jahani, Joong Hwan Bahng, Arkadev Roy, Nicholas Kotov, Alireza Marandi
CConfining light in all-dielectric anisotropic metamaterial particles for nano-scalenonlinear optics
Saman Jahani, Joong Hwan Bahng, Arkadev Roy, Nicholas Kotov, and Alireza Marandi ∗ Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA. Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109, USA. (Dated: February 26, 2021)High-index dielectrics can confine light into nano-scale leading to enhanced nonlinear response.However, increased momentum in these media can deteriorate the overlap between different har-monics which hinders efficient nonlinear interaction in wavelength-scale resonators in the absence ofmomentum matching. Here, we propose an alternative approach for light confinement in anisotropicparticles. The extra degree of freedom in anisotropic media allows us to control the evanescent wavesnear the center and the radial momentum away from the center, independently. This can lead to astrong light confinement as well as an excellent field overlap between different harmonics which isideal for nonlinear wavelength conversion. Controlling the evanescent fields can also help to surpassthe constrains on the radiation bandwidth of isotropic dielectric antennas. This can improve thelight coupling into these particles, which is crucial for nano-scale nonlinear optics. We estimate thesecond-harmonic generation efficiency as well as optical parametric oscillation threshold in theseparticles to show the strong nonlinear response in these particles even away from the center ofresonances. Our approach is promising to be realized experimentally and can be used for manyapplications, such as large-scale parallel sensing and computing.
Efficient nonlinear light generation requires long rangenonlinear interaction and/or strong field enhancement[1]. For massive computing and sensing in mid-infrared,it is desirable to miniaturize nonlinear systems to nano-scale [2–4]. However, miniaturization of photonic devicesto nano-scale not only reduces the interaction length,but also deteriorates the light confinement because of thediffraction limit of light. Plasmonic and epsilon-near-zerostructures can enhance light confinement at nanoscaleleading to strong nonlinear response with limitations dueto the optical loss of metals [5–8].Recently, light confinement in all-dielectric high-indexnano-structures has emerged as a low loss alternative toenhance the nonlinear response at nano-scale [9–18]. Thehigh-Q Mie resonances in high-index particles with sub-wavelength sizes can help to confine energy inside theparticles which can be beneficial for nonlinear wavelengthconversion [19–21]. However, in isotropic media, the mo-mentum increases with increasing the refractive index.This constrains the field overlap especially for higherorder high-Q modes. As a result, exploiting higher or-der modes without a proper momentum matching doesnot necessarily improve the nonlinear response [22, 23].Besides, because of inefficient radiation of high-Q nano-antennas [24–29], in/out-coupling in high-index dielectricnano-antennas is weak which degrades the nonlinear con-version efficiency in these particles.Light can be confined using low-index particles basedon the multi-mode interaction in which due to the low-Qand small momentum of light, multiple modes can spa-tially and spectrally overlap and form a bright hot-spotwhich is known as ‘photonic nanojet’ [30, 31]. However,the intensity of the hot-spot in simple configurations isdirectly proportional to the size of the particle, which hinders miniaturization. Besides, the hot-spot is usu-ally formed outside the particle. Hence, it is difficultto construct an overlap between the optical mode and anonlinear material.Here, we propose a paradigm shift in light confinementapproaches and nonlinear optics at nano-scale using low-index particles surrounded by all-dielectric anisotropicmetamaterial shell ( ε r (cid:29) ε ⊥ = ε θ = ε ϕ ≈
1, and µ ij = 1) with second-order nonlinearity (Fig. 1(a)). Theanisotropy of the shell gives us two degrees of freedomto independently control the radial momentum of lightand the penetration of large angular momentum statestoward the center. Hence, we can enhance the field in-tensity without a significant change in the field profile(Fig. 1(b)). This allows us to achieve field confinementas well as strong overlap between harmonics (Fig. 1(c))leading to enhanced wavelength conversion in these par-ticles. We show that the radial anisotropy can also helpto convert reactive energy near the center into propagat-ing waves, and as a result, the radiation properties ofthese particles are remarkably improved. As a result, theefficiency of the coupling of input light and collection ofthe output light is enhanced. We estimate the second-harmonic generation (SHG) efficiency as well as the op-tical parametric oscillation threshold in these particles.We discuss that the nonlinear response in the proposedparticles can be orders of magnitude higher comparedto a high-index isotropic particle with similar nonlinearcoefficient and Q factors.Natural low loss dielectrics have limited anisotropy[19]. However, strong anisotropy can be achieved inhigh contrast dielectric nanostructures [32–38]. Re-cently, we have experimentally demonstrated particleswith anisotropic metamaterial shells [39]. The metama- a r X i v : . [ phy s i c s . op ti c s ] F e b ω2ω Confined andoverlapped (a) (b)
Metamaterial shell (c)
Metamaterial shellCore z y z Figure 1.
Confining light in anisotropic metamaterial particles to enhance nonlinear interaction . (a) Schematicrepresentation of a low-index ( √ ε <
2) particle with metamaterial shell. The radial anisotropy of the shell, with an opticalaxis in the r direction, offers an extra degree of freedom to engineer the electric Mie modes of the particle. ε θ and ε ϕ controlthe momentum while ε r can control the order of spherical waves. (b) The electric field distribution of the 5 th electric mode atresonance as a function of ε r in the shell while ε ⊥ = ε θ = ε ϕ = 1. The core is isotropic ε = 2 . R = 0 . µ m,and the shell radius is R = 1 . µ m. The shell has a graded-index profile such that ε r ( R ) = 1. Increasing the anisotropyenhances the field at the core/shell interface. Similar effect can be seen for other electric modes as well. (c) The anisotropy ofthe shell can enhance the fields at second harmonic at the core/shell interface as well. The field enhancement and strong overlapcan significantly enhance the nonlinear interaction for efficient second-harmonic generation and optical parametric oscillationprocesses. terial shell is composed of dielectric nanowires arrangedin spherical form [40]. Due to the sub-wavelength fea-ture size of the unit-cell in the shell, Maxwell-Garnetteffective medium theory can be applied to model the ef-fective response of the shell [41]. Since the nanowiresare mostly oriented in the radial direction, the nanowiresdemonstrate an effective spherical anisotropy. Also, asthe distance from the center is reduced, the nanowiresfilling factor reduces while the width of the nanowires isfixed. Hence, the effective response displays a graded-index profile as well. The analytical calculation of thefield distribution using a modified Mie theory is in goodagreement with the full-wave simulation except near thenanowires which is due to the inhomogeneity of the realstructure [42]. The anisotropy that we have achievedwith zinc oxide nanowires in our recent experiment [39]is limited, but it can be enhanced by using higher indexnanowires [43] or doping the nanowires [44].To understand the light confinement mechanism inthese particles, we first look at the wave equations inmedia with spherical anisotropy. Since, the magneticmodes (TE modes) are not affected by the non-magneticanisotropy, we only focus on the electric modes (TMmodes) here. The wave equation in uniaxial media withoptical axis in the r direction can be written as [42]: − ε ⊥ r ∂∂r (cid:18) r ∂∂r ( rE r ) (cid:19) + 1 ε r r (cid:126)LLL ( rE r ) = k ( rE r ) , (1)where (cid:126) (cid:126)LLL = (cid:126) i ( (cid:126)r × (cid:126) ∇ ) is the angular momentum operatorwith an eigenvalue of (cid:126) (cid:112) n ( n + 1) and n is an integer de-scribing the angular momentum mode number [45]. Thefirst term on the left-hand side of Eq. 1 corresponds tothe radial momentum with an eigenvalue of (cid:126) k r which can be expressed as [42]: k r ε ⊥ + n ( n + 1) ε r r = k . (2)The radial component of the electric field in a homoge-neous media with spherical anisotropy excited by a planewave can be written as a superposition of orthogonalmodes [42]: E r ( r, θ, ϕ ) = 1( k r ) ∞ (cid:88) n =1 c n z n e ( k √ ε ⊥ r ) P (1) n (cos θ ) e ± iϕ , (3) n e = (cid:114) ε ⊥ ε r n ( n + 1) + 14 − , where P (1) n is the associated Legendre polynomial of thefirst order, z n is one of the Ricatti-Bessel functions ortheir superposition [46, 47], c n is the amplitude of the n th -mode, k = ω/c is the momentum in free-space, ω is the angular frequency, and c is the speed of light invacuum.Figure 2 displays the electric field in media with andwithout spherical anisotropy. We have plotted only thefirst and the fifth modes. Without the loss of general-ity, the same arguments can be applied to other electricmodes as well. By increasing the angular momentummode number, as seen in Eq. 2, the radial momentumreduces, and at some point, it becomes imaginary. Thiscauses the field decays faster when it approaches towardthe center, which hampers light concentration with largeangular momentum in sub-wavelength regime in isotropicstructures. This also causes a weak radiation of gener-ated light in the sub-wavelength regime [48]. -101 n=1 k r -101 ( k r) E r k r -101 n=5 (a)(b)(c) ( k r) E r ( k r) E r Figure 2.
Field distribution in an infinite homogeneousmedia with radial anisotropy . Normalized electric fielddistribution for the first (left) and the fifth (right) electricmodes as a function of permittivity. (a) Isotropic media. (b)Anisotropic media with ε r = 1. (c) Anisotropic media with ε ⊥ = ε θ = ε ϕ = 1. ε ⊥ controls the momentum of spheri-cal waves while ε ⊥ /ε r changes the order of spherical Besselwaves. By increasing ε r while ε ⊥ is fixed, we can reducethe order without increasing the momentum. This results inan enhanced field intensity, especially in the sub-wavelengthregime ( k r (cid:28) Increasing the refractive index in isotropic media cancompress modes in the radial direction which results inincreasing the radial momentum as well as enhancingthe penetration of evanescent waves toward the center(Fig. 2(a)).Although the far-field momentum is independent of ε r as seen in Eq. 2, increasing ε ⊥ alone does not en-hance the field near the center (Fig. 2(b)). This is dueto the suppression of evanescent waves [41]. This typeof anisotropic media can be utilized to control the totalinternal reflection and to confine evanescent waves insidean isotropic core [35, 38, 41, 49–51].On the other hand, if we increase the anisotropy inthe opposite direction, as shown in Fig. 2(c), near-fieldevanescent waves can be enhanced without a significantchange in the momentum away from the center. Thefield enhancement using this approach in subwavelengthregime is more substantial than increasing the permit-tivity in isotropic media (see the Supplementary Mate-rials [42]) even though the averaged permittivity in theanisotropic media is lower. This can lead to a strongconversion of reactive (evanescent) fields near the cen-ter into propagating electromagnetic waves even withoutusing hyperbolic structures [48]. As a result, beside thefield enhancement, it is expected that the radiation froma particle composed of a material with radial anisotropy k R Q Anisotropic, coreAnisotropic, core/shellIsotropic, TMIsotropic, TE
Figure 3.
Comparing the Q value versus the size in dif-ferent types of spherical all-dielectric particles ( ε ij > and µ = 1 ) . Each point represents the resonant frequencyand the Q of the lowest order mode for a given value ofanisotropy with a fixed total size ( R ). The core in core/shellstructure is isotropic with ε = 2 . to outperform an isotropic dielectric nanoantenna.To describe the radiation properties of an anisotropicnano-antenna, we have calculated the Q values inanisotropic spherical particles compared to the isotropiccase (Fig. 3). The Q of an antenna is defined by the powerradiated by the antenna and the reactive energy storedin it ( Q = ωW stored /P radiated ), and it specifies the inher-ent limitation of the physical size of an antenna on itsperformance has been explored in the classical works byChu, Wheeler, and others [24–29]. Although increasingthe Q is desirable for field enhancement and increasinglight-matter interaction in a resonator [52], it causes anincrease in reactive power resulting inefficient couplingof light from and into the far-field. In bulk Fabry-Perotor whispering-gallery-mode resonators, efficient couplingis still achievable by evanescent coupling or impedancematching of the input port. However, in nano-scale res-onators in which multipolar modes can only be excitedfrom the far field, the radiation properties of the res-onator play significant roles for light-matter interactions.Figure 3 displays the Q factor of the first electric and thefirst magnetic modes in isotropic and anisotropic parti-cles. The Q factor in core/shell anisotropic structurescan be reduced and approach the Chu limit of dielectricantennas [53]. The same approach can also be used toimprove the radiation of dielectric resonant antennas inthe microwave regime where strong anisotropy is moreaccessible [54]. A similar argument can be applied to the Wavelength (nm) E l e c t r i c m ode s a m p li t ude
700 725 750 775 800 th mode 13 th mode x (c)(a) (b) | E | x ( λ = 774.88 nm) Signal y z | E | ( λ = 1549.8 nm) Pump
Wavelength (nm) -4 S H G e ff i c i en cy ( W - ) -5 -3 (d) Figure 4.
Second-harmonic generation in particles with anisotropic metamaterial shell . (a) Linear response of themodes at the fundamental and second-harmonic frequencies for a low-index particle with anisotropic metamaterial shell. Theparameters for the particle are the same as those in Fig. 1(b) with ε r ( R ) = 12. The normalized scattering amplitude of the 5 th (red) and 13 th electric modes of the particle. The Q factor for the modes are 25 and 1.6e4, respectively. The second harmonic ofthe 5 th mode coincides with the 13 th mode. The contributions of other modes on SHG are negligible because of weak scatteringresponse at the operating wavelengths. (b) The electric field distribution at the pump wavelength ( λ = 1549 . x -polarized plane-wave propagating in the z direction. The electric field amplitude is normalized to theamplitude of the plane-wave. (c) The electric field distribution of the 13 th electric mode which resonates at the second-harmonicof the pump excitation. Due to the anisotropy of the shell, the field is enhanced at the interface between the core and the shell.(d) Second-harmonic generation efficiency as a function of the pump wavelength. All the contributing modes at the pumpand the signal wavelengths are taken into account. The efficiency boosts as the second-harmonic wavelength approaches theresonance of the the 13 th electric mode. higher order electric modes.A particle composed of a low-index core and ananisotropic shell (Fig. 1(a)) can enhance and confine lightat the core/shell interface. The evanescent field enhance-ment because of the anisotropy of the shell as well asthe field enhancement in low-index core because of thecontinuity of the normal component of the displacementcurrent lead to generation of a hot-spot at the boundaryfor electric modes ( Fig. 1(b)).Since all the excited electric modes are confined atthe core/shell interface, there is a strong spatial over-lap between different harmonics at the hot-spot. Thiscan lead to enhanced nonlinear wavelength conversionin these particles. We consider an extreme anisotropyfor the shell (inset of Fig. 4(a)) to emphasize the role ofanisotropy for light confinement and wavelength conver-sion. The scattering coefficients for the electric and mag-netic modes are displayed in the Supplemental Material[42]. As expected, the magnetic modes are not altered bythe shell since they are TE modes. However, the electricmodes are significantly affected by the anisotropic shellleading to a field enhancement.We choose the fundamental harmonic to resonate atthe 5 th electric mode. The second harmonic spectrallyoverlaps with the 13 th electric mode. The scattering co- efficient for these modes are plotted in Fig. 4(a). Thescattering coefficient of other modes are illustrated inthe Supplementary Materials [42]. There is a good spec-tral overlap between the second-harmonic of the 5 th withthe 13 th modes. Hence, they can be employed for theSHG and optical parametric oscillation processes. Wefirst look at the SHG process in these particles. We haveassumed that the core has no nonlinearity and the shellhas a second-order nonlinearity with χ (2) = 200 pm/V.We excite the particle with a plane wave which excitesmultiple modes of the particle (Fig. 4(b)) at fundamentalharmonic. At second-harmonic, multiple modes can res-onate as well. However, since the 13 th electric mode hasthe highest Q around the second harmonic (Fig. 4(c)),most of the pump power is converted to this mode ifthe detuning from the resonant frequency is negligible[23, 39]. The calculated SHG efficiency considering allthe contributing modes is plotted in Fig. 4(d). The SHGefficiency can reach up to 2 × − W − near the res-onance. The highest measured SHG efficiency in singledielectric particles is ∼ − W − [15, 17]. It is notewor-thy that using higher order modes in isotropic high-indexdielectrics does not considerably improve the SHG effi-ciency without leveraging the phase matching [22].We have recently proposed the possibility of paramet- (a)(b)
770 772 774 776 778 780
Wavelength (nm) -0.500.5 ∆ ω / ω T h r e s ho l d ( W ) SignalIdler
Figure 5.
Optical parametric oscillation in particleswith anisotropic metamaterial shell . The structure isthe same as shown in Fig.1. (a) Oscillation threshold and (b)signal and idler separation as a function of the pump wave-length. All the contributing modes at the pump and the signalwavelengths are taken into account. The oscillation thresh-old drops remarkably as the pump wavelength approaches theresonance of the the 13 th electric mode. Because of the de-tuning of the resonant frequency of signal/idler modes fromthe fundamental harmonic and nonlinear interactions betweenmultiple modes a phase-transition from degenerate to non-degenerate case can happen. ric oscillation in wavelength-scale resonators [23]. Opti-cal parametric oscillators (OPOs) can generate entangledphoton pairs and squeezed vacuum states below the os-cillation threshold [55–57], while above the threshold atwhich the gain exceeds loss, they can generate mid-IRfrequency combs which can be used for many applica-tions, such as metrology, spectroscopy, and computationat degeneracy [58–60]. As we miniaturize a conventionalresonator, the nonlinear gain is reduced and field over-lap deteriorates if there is no phase matching. As a re-sult, it becomes extremely difficult to surpass the thresh-old. Since the SHG efficiency is strikingly high in theanisotropic particles introduced here, it is expected toachieve a low oscillation threshold in these particles aswell.Figure 5(a) displays the OPO threshold of the first os-cillating mode. The minimum threshold is around 0.37W which happens when the pump overlaps with the 13 th electric mode. This threshold is one order of magnitudelower than an isotropic particle with similar values for Qand nonlinearity [23]. This improvement is due to thefield enhancement and localization which is not achiev-able in isotropic particles. Due to the detuning of the res-onant frequency of the signal/idler from the fundamentalharmonic, the signal and idler separation is large. How-ever, the nonlinear interactions between them can lead to a phase transition from non-degenerate to degeneratecase [23, 61]. By engineering the resonant frequency ofthe modes and reducing the detuning, the OPO thresholdcan be reduced further.It is noteworthy that even away from the center of theresonance of the 13 th electric mode, the nonlinear re-sponse is still significant compared to an isotropic parti-cle [23]. Especially for OPO case, if we are in the low-Qregime, we can compress the pump into an ultra shortpulse which can lead to a considerable reduction in thethreshold.In summary, we have proposed a light confinementstrategy using particles with spherical anisotropic shell.We showed that in media with spherical anisotropy, theevanescent fields can be enhanced in the sub-wavelengthregime without a significant change in the field pro-file. This field enhancement in sub-wavelength regime, iseven stronger than the field enhancement in high-indexisotropic media. This allows to confine light in particleswith a low-index core and an anisotropic metamaterialshell and localize modes at the core/shell interface forall the electric modes. Controlling the evanescent wavesin the sub-wavelength regime can also improve the ra-diation properties of the nanoantennas which is essentialfor the efficient excitation and the collection of generatedlight. Our approach also suggests a strong field overlapbetween different harmonics. We have shown that if theshell is composed of a material with second-order nonlin-earity, we can enhance the SHG efficiency and reduce thethreshold of OPOs. Particles with anisotropic shell areachievable at optical frequencies [35, 36, 39], and theycan open opportunities for exploring nonlinear optics atnano-scale. Even though we have focused on sphericalparticles, the same concept can be applied to cylindricalMie resonators which are more amenable to fabricationon a chip.S. Jahani acknowledges Zubin Jacob for discussions. Confining light in all-dielectric anisotropic metamaterial particles fornano-scale nonlinear optics: Supplementary Materials
Saman Jahani , Joong Hwan Bahng , Arkadev Roy , Nicholas Kotov , and Alireza Marandi Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA. Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109, USA.
In this supplementary material, we report the wave equations in the spherical coordinate with radial anisotropy. We derivethe analytical solutions to the electric and magnetic fields. We demonstrate the scattering by an anisotropic sphere excitedby a plane-wave. We also compare full-wave simulation of a practical structure with our analytical calculations to confirm thevalidity of our model.
HELMHOLTZ EQUATIONS
In a homogeneous medium with spherical uniaxial anisotopic permittivity with the optical axis in the r direction(¯¯ ε = [ ε r , ε ⊥ , ε ⊥ ], where ε θ = ε ϕ = ε ⊥ ), any arbitrary electromagnetic field in spherical coordinate can be constructedas a superposition of TM ( H r = 0) and TE ( E r = 0) modes. We can write the scalar Helmholtz equation for E r and H r , and then derive the electric and magnetic fields in the θ and ϕ directions from the fields in the r direction. Forthe TE modes, the Helmhotrz equation can be written as:( ∇ × ∇ × (cid:126)H ) r = k ε ⊥ H r . (S1)Since ∇ . (cid:126)H = 0, the above equation is simplified to the familiar form of the wave equation [62]: ∇ ( rH r ) + k ε ⊥ ( rH r ) = 0 . (S2)For the TM modes, we can write the scalar Helmhotz equation for E r as:( ∇ × ∇ × (cid:126)E ) r = k ε r E r . (S3)However, since ∇ . (cid:126)E is not zero in anisotropic media, Eq. S3 is not simplified to the conventional form. Here, weshow how we can write the Helmholtz equation for E r for the anisotropic case. The left hand side of the Eq. S3 canbe written as:( ∇ × ∇ × (cid:126)E ) r = 1 r sin θ (cid:20) ∂∂θ (cid:16) ( ∇ × (cid:126)E ) ϕ sin θ (cid:17) − ∂∂ϕ (cid:16) ( ∇ × (cid:126)E ) θ (cid:17)(cid:21) (S4)= 1 r sin θ (cid:20) ∂∂θ (cid:18) r (cid:18) ∂∂r ( rE θ ) − ∂∂θ E r (cid:19) sin θ (cid:19) − ∂∂ϕ (cid:18) r (cid:18) θ ∂∂ϕ E r − ∂∂r ( rE ϕ ) (cid:19)(cid:19)(cid:21) = − r sin θ ∂∂θ (cid:18) sin θ ∂E r ∂θ (cid:19) − r sin θ ∂ E r ∂ϕ + 1 r sin θ ∂∂θ (cid:18) sin θ ∂∂r ( E θ ) (cid:19) + 1 r sin θ ∂∂ϕ ∂∂r ( rE ϕ )= −∇ ⊥ E r + 1 r sin θ ∂∂θ (cid:18) sin θ ∂∂r ( rE θ ) (cid:19) + 1 r sin θ ∂∂ϕ ∂∂r ( rE ϕ ) , where ∇ ⊥ is the transverse component of the Laplacian in the spherical coordinate. We can further simplify Eq. S4by adding and subtracting the radial component of the Laplacian which is multiplied by ε r / ( rε ⊥ ):( ∇ × ∇ × (cid:126)E ) r = − r ∇ ⊥ ( rE r ) − ε r rε ⊥ r ∂∂r (cid:18) r ∂∂r ( rE r ) (cid:19) (S5)+ (cid:20) ε r rε ⊥ r ∂∂r (cid:18) r ∂∂r ( rE r ) (cid:19) + 1 r sin θ ∂∂θ (cid:18) sin θ ∂∂r ( rE θ ) (cid:19) + 1 r sin θ ∂∂ϕ ∂∂r ( rE ϕ ) (cid:21) . After some algebra, it is easy to show that the last term on the right side of Eq. S5 can be written as the divergenceof the displacement current:( ∇ × ∇ × (cid:126)E ) r = 1 r (cid:20) −∇ ⊥ ( rE r ) − ε r ε ⊥ r ∂∂r (cid:18) r ∂∂r ( rE r ) (cid:19) + 1 ε ε ⊥ ∇ . (cid:126)D + 1 ε ε ⊥ ∂∂r (cid:16) r ∇ . (cid:126)D (cid:17)(cid:21) (S6)(S7) n=5 -7 n=1 k r k r (a)(b) ( k r) E r ( k r) E r ( k r) E r ( k r) E r Figure S1.
Field distribution in an infinite homogeneous media with radial anisotropy . Normalized electric fielddistribution for the first (left) and the fifth (right) electric modes as a function of permittivity. (a) Isotropic media. (b)Anisotropic media with ε ⊥ = ε θ = ε ϕ = 1. ε ⊥ controls the momentum of spherical waves while ε ⊥ /ε r changes the order ofspherical Bessel waves. By increasing ε r while ε ⊥ is fixed, we can reduce the order without increasing the momentum. Thisresults in an enhanced field intensity, especially in the sub-wavelength regime ( k r (cid:28) As ∇ . (cid:126)D = 0, the Helmholtz equation for E r can be written as: ε r ε ⊥ r ∂∂r (cid:18) r ∂∂r ( rE r ) (cid:19) + ∇ ⊥ ( rE r ) + k ε r ( rE r ) = 0 , (S8)or it can be expressed as: 1 ε ⊥ ppp r ( rE r ) + 1 ε r r (cid:126)LLL ( rE r ) = k ( rE r ) , (S9)where (cid:126) ppp r = (cid:126) i (ˆ r.(cid:126) ∇ ) and (cid:126) (cid:126)LLL = (cid:126) i ( (cid:126)r × (cid:126) ∇ ) are the radial momentum and the angular momentum operators witheigenvalues of (cid:126) k r and (cid:126) L = (cid:126) (cid:112) n ( n + 1), respectively. Hence, the eigenvalue problem can be simplified to: k r ε ⊥ + n ( n + 1) ε r r = k . (S10)By increase the angular momentum mode number, the second term on the left hand side of Eq. S10 exceeds theterm on the right hand side, especially when we are closer to the center. As a result, the radial momentum becomesimaginary which decays evanescently when we approach the center. This causes a weak excitation of higher ordermodes in the sub-wavelength regime and a weak coupling of these modes to the far-field radiating modes [48].If we rearrange the momentum as: k r = (cid:114) ε ⊥ ε r (cid:114) k ε r − n ( n + 1) r , (S11)it is seen that by controlling the anisotropy, we can control the evanescent fields near the center [41]. Especially, ifwe increase the ratio, ε ⊥ /ε r , the evanescent fields and as a result, the field is enhanced in the sub-wavelength regime( k r (cid:28) SOLUTION TO THE HELMHOLTZ EQUATIONSNon-magnetic anisotropic particle
We start with the simplest particle with non-magnetic anisotropy. We can use the approach of separating thevariables to find the solutions of E r and H r . Eq. S2 has the standard solution of [62]: rH r ( r, θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n (cid:2) c hn j n ( k √ ε ⊥ r ) + d hn n n ( k √ ε ⊥ r ) (cid:3) P ( m ) n (cos θ ) (cid:26) sin ( mϕ )cos ( mϕ ) (cid:27) , (S12)where P ( m ) n are the Legendre Polynomials. j n and n n are the spherical Bessel and Neumann functions defined as: j n ( x ) = (cid:16) π x (cid:17) J n + ( x ) (S13) n n ( x ) = (cid:16) π x (cid:17) N n + ( x ) , where J n ( x ) and N n ( x ) are the n th order Bessel and Neumann functions. Sometimes, it is more convenient to writethe solution as Ricatti-Bessel functions defined as [46, 63]: ψ n ( x ) = xj n ( x ) = (cid:16) πx (cid:17) J n + ( x ) (S14) χ n ( x ) = − xn n ( x ) = − (cid:16) πx (cid:17) N n + ( x ) , or as spherical Hankel function of the first kind and second kind for outward and inward radiations, respectively: h (1) n ( x ) = ξ n ( x ) /x = j n ( x ) + in n ( x ) (S15) h (2) n ( x ) = ζ n ( x ) /x = j n ( x ) − in n ( x )The angular part of the solution of Eq. S8 is the same as that in Eq. S12. However, the radial part is a bit morecomplicated than the standard form shown in Eq. S12: rE r ( r, θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n [ c en j n e ( k √ ε ⊥ r ) + d en n n e ( k √ ε ⊥ r )] P ( m ) n (cos θ ) (cid:26) sin ( mϕ )cos ( mϕ ) (cid:27) , (S16)where n e = (cid:113) ε ⊥ ε r n ( n + 1) + − . Note that if the medium is isotropic, the solution is simplified to the standardsolution as shown in Eq. S12.The tangential component of the electric and magnetic fields in the spherical anisotropic medium are expressed as: iωµ H θ = 1 r (cid:18) θ ∂∂ϕ E r − ∂∂r ( rE ϕ ) (cid:19) (S17) iωµ H ϕ = 1 r (cid:18) ∂∂r ( rE θ ) − ∂∂θ E r (cid:19) , and − iωε ε ⊥ E θ = 1 r (cid:18) θ ∂∂ϕ H r − ∂∂r ( rH ϕ ) (cid:19) (S18) − iωε ε ⊥ E ϕ = 1 r (cid:18) ∂∂r ( rH θ ) − ∂∂θ H r (cid:19) . TE modes
For TE modes, E r = 0, so Eq. S17 is simplified to: iωµ H TE θ = − r ∂∂r ( rE TE ϕ ) (S19) iωµ H TE ϕ = 1 r ∂∂r ( rE TE θ ) . By replacing Eq. S19 into Eq. S18 and multiplying the both sides by iωµ r , we obtain: ∂ ∂r ( rE TE θ ) + k ε ⊥ rE TE θ = iωµ sin θ ∂∂ϕ ( rH r ) (S20) ∂ ∂r ( rE TE ϕ ) + k ε ⊥ rE TE ϕ = − iωµ ∂∂θ ( rH r ) . Since the radial part of the right-hand side of Eq. S20 is a spherical Bessel function, the radial part of the left-handside must be a spherical function too. Using the recurrence relation for spherical Bessel functions: ∂ ∂r ( rz n ( kr )) + (cid:18) k − n ( n + 1) r (cid:19) rz n ( kr ) = 0 , (S21)where z n ( kr ) is a spherical Bessel, Neumann, or Hankel function, Eq. S20 is simplified to: E TE θ = iωµ n ( n + 1) sin θ ∂∂ϕ ( rH r ) (S22) E TE ϕ = − iωµ n ( n + 1) ∂∂θ ( rH r ) . Now if we insert Eq. S22 into Eq. S19, we can obtain the tangential component of the magnetic field: H TE θ = − iωµ r ∂∂r ( rE TE ϕ ) = 1 n ( n + 1) 1 r ∂ ∂θ∂r (cid:0) r H r (cid:1) (S23) H TE ϕ = 1 iωµ r ∂∂r ( rE TE θ ) = 1 n ( n + 1) 1 r sin θ ∂ ∂ϕ∂r (cid:0) r H r (cid:1) . TM modes
For TM modes, H r = 0. If we follow the same procedure that we used for TE modes, the tangential electric andmagnetic fields can be expressed as: H TM θ = − iωε ε ⊥ n e ( n e + 1) sin θ ∂∂ϕ ( rE r ) (S24) H TM ϕ = iωε ε ⊥ n e ( n e + 1) ∂∂θ ( rE r ) , and E TM θ = 1 iωε ε ⊥ r ∂∂r ( rH TM ϕ ) = 1 n e ( n e + 1) 1 r ∂ ∂θ∂r (cid:0) r E r (cid:1) (S25) E TM ϕ = − iωε ε ⊥ r ∂∂r ( rH TM θ ) = 1 n e ( n e + 1) 1 r sin θ ∂ ∂ϕ∂r (cid:0) r E r (cid:1) . General anisotropic particle
For a particle with both electric and magnetic anisotropy (¯¯ ε = [ ε r , ε ⊥ , ε ⊥ ] and ¯¯ µ = [ µ r , µ ⊥ , µ ⊥ ], where ε θ = ε ϕ = ε ⊥ and µ θ = µ ϕ = µ ⊥ ), the solution for both TE and TM modes are affected by the anisotropy [64–66]. The radialcomponent of the electromagnetic fields are written as: rE r ( r, θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n [ c en j n e ( k √ ε ⊥ µ ⊥ r ) + d en n n e ( k √ ε ⊥ µ ⊥ r )] P ( m ) n (cos θ ) (cid:26) sin ( mϕ )cos ( mϕ ) (cid:27) , (S26) rH r ( r, θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n (cid:2) c hn j n h ( k √ ε ⊥ µ ⊥ r ) + d hn n n h ( k √ ε ⊥ µ ⊥ r ) (cid:3) P ( m ) n (cos θ ) (cid:26) sin ( mϕ )cos ( mϕ ) (cid:27) , where n e = (cid:113) ε ⊥ ε r n ( n + 1) + − and n h = (cid:113) µ ⊥ µ r n ( n + 1) + − .The tangential components of the electric and magnetic fields for TE modes can be written as: E TE θ = iωµ µ ⊥ n h ( n h + 1) sin θ ∂∂ϕ ( rH r ) (S27) E TE ϕ = − iωµ µ ⊥ n h ( n h + 1) ∂∂θ ( rH r ) , and H TE θ = − iωµ µ ⊥ r ∂∂r ( rE TE ϕ ) = 1 n h ( n h + 1) 1 r ∂ ∂θ∂r (cid:0) r H r (cid:1) (S28) H TE ϕ = 1 iωµ µ ⊥ r ∂∂r ( rE TE θ ) = 1 n h ( n h + 1) 1 r sin θ ∂ ∂ϕ∂r (cid:0) r H r (cid:1) . For the TM modes, the the tangential components are: H TM θ = − iωε ε ⊥ n e ( n e + 1) sin θ ∂∂ϕ ( rE r ) (S29) H TM ϕ = iωε ε ⊥ n e ( n e + 1) ∂∂θ ( rE r ) , and E TM θ = 1 iωε ε ⊥ r ∂∂r ( rH TM ϕ ) = 1 n e ( n e + 1) 1 r ∂ ∂θ∂r (cid:0) r E r (cid:1) (S30) E TM ϕ = − iωε ε ⊥ r ∂∂r ( rH TM θ ) = 1 n e ( n e + 1) 1 r sin θ ∂ ∂ϕ∂r (cid:0) r E r (cid:1) . SCATTERING BY AN ANISOTROPIC SPHERE
Assuming the incident wave is a x polarized plane wave travelling in the z direction: (cid:126)E i = ˆ xE e ik z = ˆ xE e ik r cos θ , (S31)the incident electric and magnetic fields in the spherical coordinate can be written as [67]: E ir = cos ϕ sin θE ix = E k r cos ϕ (cid:88) n =1 i ( n +1) (2 n + 1) ψ n ( k r ) P (1) n (cos θ ) (S32) H ir = sin ϕ sin θ E ix η = E ηk r sin ϕ (cid:88) n =1 i ( n +1) (2 n + 1) ψ n ( k r ) P (1) n (cos θ ) , M agne t i c m ode s a m p li t ude Wavelength (nm) E l e c t r i c m ode s a m p li t ude M ode nu m be r Figure S2. Scattering amplitude for electric ( | a n | ) and magnetic modes ( | b n | ) for a low-index particle with anisotropicmetamaterial shell. The parameters for the particle are the same as those in Fig. 4 in the main text. Wavelength (nm) E l e c t r i c m ode s a m p li t ude M agne t i c m ode s a m p li t ude M ode nu m be r Figure S3. Scattering amplitude for electric ( | a n | ) and magnetic modes ( | b n | ) for a the structure shown in Fig. S2 withoutthe metamaterial shell. | E | (a)(c) (b)(d) Figure S4.
Practical realization of particles with anisotropic metamaterial shell. (a) A schematic representation ofa particle with metamaterial shell. The core is composed of glass with a radius of 500 µ m. The outer radius of the shell is 2.2 µ m. The nanowires are composed of zinc oxide with a width of 200 nm with a filling factor of 0.4 at the core/shell interface.(b) The electric field distribution in the particle when it is excited by a plane wave propagating (wavelength: 1550 nm) tothe right. (c) Effective medium modeling of the shell with anisotropic metamaterial. (d) The electric field distribution in theparticle with effective medium modeling. It is seen that there is a good agreement between the full-wave numerical simulationof the real structure and the homogenized one. where E is the incident electric field amplitude and η is the free-space impedance. Because of the interactionbetween the incident field and the particle, light is scattered. Since the scattered fields have to vanish in the infinitythe scattered light is expressed as: E sr = − E k r cos ϕ (cid:88) n =1 i ( n +1) (2 n + 1) a n ξ n ( k r ) P (1) n (cos θ ) (S33) H sr = − E ηk r sin ϕ (cid:88) n =1 i ( n +1) (2 n + 1) b n ξ n ( k r ) P (1) n (cos θ ) . The fields inside the sphere have to vanish at the origin. Hence they can be expressed as: E rr = E k r cos ϕ (cid:88) n =1 i ( n +1) (2 n + 1) c n ψ n e ( k √ ε ⊥ µ ⊥ r ) P (1) n (cos θ ) (S34) H rr = E ηk r sin ϕ (cid:88) n =1 i ( n +1) (2 n + 1) d n ψ n h ( k √ ε ⊥ µ ⊥ r ) P (1) n (cos θ ) . By applying the boundary conditions at the particle interfaces: E rθ ( k √ ε ⊥ µ ⊥ R ) = E iθ ( k R ) + E sθ ( k R ) (S35) H rθ ( k √ ε ⊥ µ ⊥ R ) = H iθ ( k R ) + H sθ ( k R ) , we can find a n and b n , which are electric and magnetic Mie scattering coefficients, respectively. The total scatteringand extinction cross-sections can be expressed as [63]: C sca = 2 πk ∞ (cid:88) n =1 (2 n + 1) (cid:0) | a n | + | b n | (cid:1) , (S36) C ext = 2 πk ∞ (cid:88) n =1 (2 n + 1)Re { a n + b n } . PRACTICAL REALIZATION
In the last few years, several approaches have been proposed to experimentally realize particles with radialanisotropic metamaterial shell [35, 36]. We have recently proposed and demonstrated particles with metamate-rial shell in colloidal platform (Fig. S4(a)) [39]. They are composed of a low-index nanoparticle covered by a shell ofnanowires with ability to engineer and tune their optical properties. Since the feature size of the nanowires is in sub-wavelength regime, effective medium theory can be applied to homogenize the shell. We have used Maxwell-Garnetteffective medium theory [41] to model the nanowires with an all-dielectric metamaterial representing radial anisotropy.Since the the filling factor reduces as the radius increases, the metamaterial shell also demonstrates a graded-indexprofile [39]. 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