Polar dielectrics are being actively investigated as a promising platform for mid-infrared nanophotonics, and it indicates dramatic nonlocal phenomena at the nanoscale. Based on full-wave nonlocal extended Mie theory, we analytically derive the equivalent permittivity of nonlocal polar dielectric nanospheres. Then, we establish the conditions for surface phonon amplification by stimulated emission of radiation of the nonlocal polar dielectric core-gain shell nanoparticle laser based on phonon polaritons. The results show that the nonlocality dramatically affects the selection of the gain medium for lasing condition, which also indicates a new degree of freedom in the modulation of the gain medium. The validity of the derived equivalent permittivity is demonstrated by comparing the obtained extinction spectra and the choice of the threshold gain with those under nonlocal extended Mie theory. Our research may provide a practical tool for designing phonon polariton nanoparticle lasers.

## I. INTRODUCTION

With the rapid progress of nanofabrication technology and optical metamaterials, nanophotonics in the infrared light range has attracted increasing attention.^{1–3} Recently, significant research was conducted on polar dielectrics and their potential applications in nanophotonics, including the development of sensing,^{4} nonlinear optics,^{5,6} and thermal radiation control.^{7} Similar to metal-excited surface plasmons, surface phonon polaritons (SPhPs) can be excited in polar dielectric systems, which are quasiparticles resulting from the coupling of phonons and photons. The vibration wavelength of polaritons is generally much smaller than that of exciting light, allowing for the modulation of electromagnetic waves at subwavelength scales. Phonon–polariton lasers are based on matter–light hybrid modes formed via strong coupling between excitons and phonons,^{8–11} which can provide a unique solution for ultralow power consumption photonics and quantum photonics due to their low lasing threshold density, large nonlinearity, and matter–light hybrid nature. Meanwhile, it has been demonstrated that phonon–polariton lasers have a wide range of practical working wavelengths from ultraviolet to terahertz.^{9} These lasers can be controlled via both phonon and exciton components, providing greater flexibility in tuning and controlling the mode properties.

In the theoretical scenario of investigating the electromagnetic characteristics of polar dielectrics, the local solution of the traditional Maxwell equation can no longer describe the electromagnetic properties when the size of polar dielectric nanostructures is down to a few nanometers, where the coupling between the longitudinal optical phonon and SPhPs in the polar dielectric has to be considered.^{12} These intrinsic quantum properties of polar dielectrics could be taken into account by the nonlocal nature of the dielectric response in the framework of the semiclassical phenomenological model.^{12–16} It has been widely demonstrated that neglecting this nonlocality can result in erroneous predictions of modal frequencies and significant overestimation of achievable field confinement.^{17,18} Therefore, the properties of phonon–polariton lasers at the nanoscale are expected to be influenced by the nonlocality. In addition, the nonlocal response has important implications for the propagation and manipulation of light in polar dielectrics.

In this paper, we establish the nonlocal equivalent medium theory for polar dielectric crystals with a mesoscopic nonlocal theory to analyze the nonlocality in polar dielectric nanostructures. Based on electromagnetic scattering theory, we derive the equivalent permittivity *ɛ*_{eq} of the polar dielectric nanosphere with spatially dispersive permittivity *ɛ*(*k*, *ω*) by considering the scattering problem of the nonlocal polar dielectric sphere embedded in an equivalent medium. Moreover, we take one step forward to design nonlocal phononic core–shell lasers composed of nonlocal polar dielectrics and obtain “sphaser” (short for surface phonon amplification by stimulated emission of radiation) generation conditions, whose structures are similar to those based on plasmonic metamaterials.^{19–21} We show that the required gain threshold and gain medium are dramatically influenced by the nonlocality parameter. The relevant phenomena can be well explained by the proposed equivalent permittivity. Our investigation may provide a new path for designing phonon polariton scattering lasers in the mid-infrared frequency region.

## II. MODEL AND FORMULATIONS

### A. Equivalent permittivity for the nonlocal polar dielectric nanosphere

*ɛ*(

*k*,

*ω*) and radius

*a*embedded in an equivalent medium with permittivity

*ɛ*

_{eq}and permeability

*μ*

_{eq}. For the nonlocal longitudinal permittivity of the polar dielectrics, it reads

^{12,22}

*ɛ*

_{∞}is the high-frequency dielectric constant,

*ω*

_{L}and

*ω*

_{T}represent the longitudinal optic (LO) phonon frequency and transverse optic (TO) phonon frequency, respectively,

*γ*is the damping rate, and

*β*is the phase velocity depicting the dispersion of LO phonons. The phase velocity can be extracted from experimental measurements of polar lattice phonon dispersion by fitting the quadratic expansion of the low-wave vector region.

^{23}In contrast, the transverse permittivity is described by

Note that both longitudinal and transverse waves can propagate in the polar dielectric system, and the wave vector of the longitudinal phonon wave is determined by the condition *ɛ*_{L}(*ω*, *k*_{L}) = 0; however, the transverse electromagnetic modes satisfy the dispersion relation $kT2(\omega )=(\omega /c)2\epsilon T(\omega )$.

*E*

^{I}, scattering electric field

*E*

^{R}, transverse electric field

*E*

^{T}, and longitudinal electric field

*E*

^{L}inside the nonlocal polar dielectric nanosphere can be written as

*j*

_{l}(

*x*) [or

*h*

_{l}(

*x*)] represents the spherical Bessel (or Hankel) functions of the first kind and

*k*

_{eq}satisfies the dispersion relation $keq2(\omega )=(\omega /c)2\epsilon eq(\omega )\mu eq(\omega )$. The corresponding magnetic fields can be obtained by Maxwell's equations, and no magnetic field corresponds to the longitudinal mode of the electric field. The tangential components of electric and magnetic fields satisfy the following boundary continuity conditions at the interface (

*r*=

*a*):

*r*=

*a*due to the excitation of the longitudinal mode as follows:

^{12}

*a*

^{R},

*b*

^{R},

*a*

^{T},

*b*

^{T}, and

*a*

^{L}can be determined by the above-mentioned boundary conditions, and we derive the scattering coefficients

*a*

^{R}and

*b*

^{R}of polar dielectric nanoparticles as

*k*

_{eq}

*a*≪ 1), the scattering cross section is dominated by the dipolar mode (

*l*= 1), i.e., $alR$ and $blR$. The electromagnetic property of the dipolar nanosphere can be regarded as the same as the background medium when the scattering cross section is 0. In this connection, the equivalent permittivity

*ɛ*

_{eq}and the equivalent permeability

*μ*

_{eq}can be regarded as the equivalent constitutive parameters of the nonlocal dipolar dielectrics. When

*x*=

*k*

_{eq}

*a*≪ 1, the Bessel and Hankel functions can be reduced to $j1(x)\u2245x3,h1(x)\u2245x3\u2212i1x2$, and when substituted into $a1R=0,b1R=0$,

^{24–26}the equivalent parameters can be obtained as

*μ*

_{eq}is close to 1 and that its variation is quite small that we will not conduct a detailed analysis on it in the subsequent discussion. The imaginary part of

*k*

_{L}of the propagating longitudinal mode will become infinite when the nonlocality is not taken into account. In such a situation, the excitation of the LO phonon mode is no longer supported within the polar dielectric nanosphere. By substituting Im(

*k*

_{L}) →

*∞*and the quasistatic approximation condition

*k*

_{T}

*a*≪ 1 into Eq. (7), the equivalent permittivity

*ɛ*

_{eq}will be reduced to the local permittivity

*ɛ*

_{T}.

### B. Nanosphere lasers containing nonlocal polar dielectric core and gain-assistant shell

In this subsection, we study the surface phonon amplification by stimulated emission of radiation based on nonlocal core–shell nanolasers composed of a nonlocal polar dielectric core and gain shell, as shown in Fig. 1, where polar dielectric nanoparticles with an inner radius of *r*_{c} are surrounded by a gain shell layer with an outer radius of *r*_{s} and permittivity *ɛ*_{G}. The background medium surrounding them has a relative permittivity of *ɛ*_{b}.

*E*

^{I}, scattering wave

*E*

^{R}, transverse wave $EcT$, and longitudinal wave $EcL$ in the polar dielectric core and the gain shell $EsT$ are written as

*y*

_{l}(

*x*) represents the spherical Bessel function of the second kind and $kb=(\omega /c)\epsilon b$ is the wave vector of the electric field in the surrounding background medium. After some tedious derivations, the scattering coefficient related to electric field polarization can be obtained, and the denominator of the scattering coefficient can be written as

^{27}i.e.,

*n*

_{G}) of the gain shell and the threshold gain

*G*

_{th}(

*G*

_{th}= −4

*π*Im(

*n*

_{G})/

*λ*), with $nG=\epsilon G$, to achieve the singularity of the field intensity can be numerically determined. Incidentally, if we neglect the nonlocality, i.e.,

*Q*= 0, Eq. (11) is reduced to the local case,

## III. RESULTS AND DISCUSSION

We are now in a position to present some numerical results. For numerical calculations, we choose 3C–SiC as the nonlocal polar dielectric material, and its relevant parameters are shown as follows:^{1,28} *ɛ*_{∞} = 6.52, *ω*_{T} = 796.1 cm^{−1}, *ω*_{L} = 973 cm^{−1}, *γ* = 4 cm^{−1}, and *β* = 15.39 × 10^{5 }cm⋅s^{−1}.

We first investigate the equivalent permittivity *ɛ*_{eq} as a function of the incident frequency for nonlocal polar dielectric materials with different radii, as shown in Fig. 2. For the local case *β* = 0, *ɛ*_{eq} is reduced to *ɛ*_{T}, and one observes the maximal absorption at *ω*_{T} = 796.1 cm^{−1} [see Fig. 2(b)], around which Re(*ɛ*_{T}) exhibits a ripple-like line shape and changes from maximum positive to maximum negative. When the nonlocality is taken into account, the magnitude of the maximal absorption is decreased, and the corresponding frequency slightly blueshifts or redshifts. In addition, a series of small peaks generated by the LO phonon modes arise in both the real part Re(*ɛ*_{eq}) and the imaginary part Im(*ɛ*_{eq}), with their amplitudes increasing as the particle size is decreased.

To verify the accuracy of the proposed equivalent permittivity *ɛ*_{eq}, we compare the extinction efficiency obtained by full wave nonlocal Mie theory^{12} with that obtained by local Mie scattering theory, adopting the equivalent permittivity. In Fig. 3, we plot the extinction efficiency *σ*_{ext} for 3C–SiC spheres with radii of 5, 15, and 150 nm. We find a dipolar Fröhlich resonance of the transverse mode near *ω*_{F} = 934 cm^{−1} and numerous additional closely spaced peaks in the frequency region *ω* < *ω*_{L}_{.} These are the result of the screening charge induced at the particle surface due to the excitation of the longitudinal modes. It is demonstrated that the scattering feature predicted by nonlocal full-wave Mie theory could be well observed by local theory involving our equivalent permittivity on a small particle scale (*a* = 5 nm and *a* = 15 nm). Actually, even for a larger size (a = 150 nm), the local theory with our equivalent permittivity agrees well with the nonlocal theory, except for a slight difference in the magnitude of magnetic-dipole modes near *ω*_{T}. The reason is that the coupling between the transverse and longitudinal modes occurs at the interface and that the screening charge generated on the surface cannot propagate to the whole particle. In addition, as the size is increased, the absorption in the nonlocal polar dielectric sphere significantly increases; hence, the longitudinal modes are greatly suppressed.

Next, we perform a numerical study on the sphaser generation of nonlocal polar core-gain shell nanoparticle lasers with a fixed outer radius *r*_{s} = 10 nm. In Figs. 4(a) and 4(b), the dependence of the refractive index Re(*n*_{G}) and the threshold intensity G_{th} on the incident frequency and the inner radius *r*_{c} is shown for the local situation. In the spectral region between longitudinal optical (LO) and transverse optical (TO) phonon frequencies (*ω*_{T} < *ω* < *ω*_{L}), polar dielectrics exhibit negative permittivity, and surface phonon polaritons (SPhPs) exist. This spectral region, known as the Reststrahlen band of the material, can serve as a low-loss, negative permittivity frequency band. Actually, *ω*_{T} and *ω*_{L} of the polar dielectrics can be tuned by adjusting the temperature.^{29} Within the range of *ω* < *ω*_{T} and *ω* > *ω*_{L}, the polar dielectric barely supports SPhPs so that is not considered. When *ω* is slightly larger than *ω*_{T}, to compensate for the large loss due to the imaginary part of the polar dielectric materials, G_{th} exhibits the maximal value, which increases with increasing core radius *r*_{c}. Actually, the increasing *r*_{c} will lead to much more absorptive polar dielectric materials and less gain materials. As a consequence, a higher gain threshold is needed to offset the loss of polar dielectrics. According to Fig. 4(c), a low and relatively stable region of G_{th} is observed in the frequency range of 900 cm^{−1} < *ω* < 950 cm^{−1}. Near the resonance frequency of SPhPs, *ω*_{F} = 934 cm^{−1}, an extremely low G_{th} (∼10^{2} cm^{−1}) can be obtained by reducing the core radius *r*_{c}. In addition, Re(*n*_{G}) is relatively small near this frequency, as shown in Fig. 4(d), which makes it more feasible to achieve nanolasers in the mid-infrared frequency region.

In reality, the size of polar dielectric spheres can be in the range of a few nanometers, and it is natural to expect the existence of nonlocality or spatial dispersion. Considering the nonlocal effect, the relationships between the gain threshold G_{th} and the real part of the complex refractive index Re(*n*_{G}) with respect to the incident frequency and inner radius *r*_{c} are shown in Fig. 5. When the incident frequency is in the Reststrahlen band, due to the resonant coupling between the SPhP mode and LO phonon mode in the nonlocal polar dielectrics, longitudinal modes are excited and propagate inside the polar dielectric nanosphere, resulting in a series of oscillations in G_{th} and Re(*n*_{G}) with respect to *r*_{c}. Moreover, it is evident that the required gain threshold and the gain refractive index generally become larger when nonlocality is introduced. Even for this, we still find a broad region of low gain threshold less than 10^{4} cm^{−1} and much realistic Re(*n*_{G}) ∈ (1.5, 4) when *r*_{c} is in the range between 2 and 8 nm and the frequency is in the range (840, 900 cm^{−1}) simultaneously.

To provide a more visual representation, Fig. 6 illustrates the variations in the threshold intensity G_{th} and the refractive index Re(*n*_{G}) as a function of *r*_{c} for both local and nonlocal cases at different frequencies. For the local case, with increasing *r*_{c}, both G_{th} and Re(*n*_{G}) increase monotonically. However, for the nonlocal case, one observes that G_{th} is always larger than the one for the local case and Re(*n*_{G}) fluctuates around the value obtained in the local case. Both of them exhibit periodic oscillations due to the excitation of longitudinal resonant modes. Such a rich nonlocal phenomenology is qualitatively different from that of plasmonic systems. Therefore, for a given frequency, there is an optimal *r*_{c} for one to achieve the minimal gain threshold and the suitable real part of the complex refractive index. Note that as the size of the polar dielectric core increases, the effect of nonlocality weakens; hence, the results for the nonlocal case approach those of the local case.

To further analyze the reasons for the oscillations in the Reststrahlen band, curves of the real part and the imaginary part of the equivalent permittivity and the refractive index of the gain medium varying with the inner radius are drawn for *ω* = 875 cm^{−1}. It is evident that an appropriately lower gain threshold value, as well as lower Re(*n*_{G}), can be achieved in core–shell spasers at *ω* = 875 cm^{−1} [see Figs. 6(b) and 6(e)]. It can be clearly seen from Fig. 7 that the oscillation phenomenon results from the excitation of longitudinal resonant modes and can be well characterized by the intrinsic behavior of the equivalent permittivity *ɛ*_{eq}. For a nonlocal polar dielectric core with the given *r*_{c}, when the damping is higher (or lower), more (or less) gain threshold is required to compensate for the losses.

## IV. CONCLUSION

In conclusion, we establish the nonlocal effective medium theory and derive the equivalent permittivity for the nonlocal polar dielectric nanosphere. We present the condition of nonlocal surface phonon amplification for nonlocal polar dielectric core–shell nanolasers in the mid-infrared frequency region. When the nonlocality is taken into account, both the threshold intensity and the required refractive index exhibit periodic oscillations due to the excitation of longitudinal resonant modes. As far as their magnitudes are concerned, the threshold intensity is generally enhanced, while the required refractive index fluctuates around the value of the local case. The validity of the equivalent permittivity is demonstrated from both the calculations of the extinction efficiency and the explanation of the oscillations in the threshold intensity and the refractive index.

Here are some appropriate comments. The inherently high optical losses in metal-based plasmonic components restrict their applications in spasers. For comparison, loss in present dielectric materials depends on the scattering time of optical phonons. The longer scattering times of optical phonons in polar dielectrics than of electrons in metals result in negative permittivity materials with lower loss. As a consequence, G_{th} obtained from our calculations for polar dielectrics at small sizes is also relatively smaller than that of metal-based nanolasers. Metal materials require a higher gain threshold and a higher refractive index of the gain medium in the nonlocal case than in the local case.^{20,21} However, nonlocal effects in certain frequency ranges can cause the gain threshold and refractive index of the gain medium to remain essentially unchanged for the polar dielectric case. Therefore, we can control the impact of nonlocal effects by selecting specific incident frequencies and core–shell ratios, thereby reducing the gain threshold. Our method provides theoretical support for the design of ultrasmall nanoparticle lasers in the mid-infrared frequency region. The study of polar dielectrics has led to significant advances in our understanding of the interaction between light and matter and has opened up new avenues for the design of advanced optical devices with unprecedented performance. On the other hand, many works on the calculation of spontaneous emission nearby spherical structures have been proposed, and the change in spontaneous emission of radiation near a dielectric sphere could result in metal enhanced fluorescence or surface enhanced fluorescence. It is of interest to take into account the spontaneous emission in our model.^{30–32}

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 92050104 and 12274314) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20221240). This study was also funded by the Jiangsu Key Laboratory of Thin Films, Soochow University.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Naifu Yu**: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (lead); Writing – original draft (lead); Writing – review & editing (supporting). **Lei Gao**: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (lead). **Yang Huang**: Conceptualization (supporting); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.