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Impact of Kerr nonlinearity on the whispering gallery modes of a microsphere

Shovasis Kumar Biswas and Shiva Kumar

Open Access Open Access

Abstract

To describe the temporal evolution of the mode amplitude of a spherical microcavity, a nonlinear equation is developed by considering loss and Kerr nonlinear effect as perturbations. In order to study the impact of Kerr nonlinearity, the tensor components of χ(3) in spherical coordinates are calculated. To describe the impact of Kerr nonlinearity, effective mode volume and effective nonlinear coefficient are defined. We found that the resonant modes undergo a negative frequency shift proportional to the injected energy, consistant with the reported experimental observations.

© 2015 Optical Society of America

1. Introduction

The theory of the modes of a spherical dielectric material was developed by Mie at the beginning of the 19th century [1]. These modes are often called whispering gallery modes (WGM) [2] or morphology dependent resonances (MDR) [3]. At resonance, the light beam is trapped for a long time inside the surface of the sphere due to repeated total internal reflections. WGM of microcavity have drawn significant attention for different applications [4], which include cavity quantum electrodynamics (CQED) experiments [5,6], microlasers [7,8], third harmonic generation (THG) [9], sensing [8, 10–12], parametric oscillation [13] and frequency comb [14–17].

Typically, it is hard to separate the impact of thermal and Kerr nonlinearities on WGMs. When the excitation wavelength of the incident light is swept through the resonances of microsphere, Kerr and thermal effects contribute differently to the total nonlinear response depending on the speed of the wavelength scanning [18]. If the scanning time τscan is much larger than the resonance building time τmode, the power absorbed due to optical losses heats the microsphere leading to thermal nonlinear effect due to temperature dependence of the refractive index. The difference in response times of the thermal and Kerr nonlinearities can be used to separate their contributions to the nonlinear properties of microspheres [18]. Treussart et al. have discussed a scheme to separate Kerr and thermo-optical effects by placing a microsphere in the superfluid helium [19]. Carmon and Vahala have experimentally demonstrated third harmonic generation in the visible region from a silica microresonator using an infrared pump source and verified that the third harmonic (TH) emission scales cubically with pump power [9]. This confirms that the third order nonlinearity is the dominant nonlinear process in their experiment. Xiao et al. have theoretically investigated a quantum nondemolition (QND) measurement with optical Kerr effect in a micro-toroidal system and their results predict that the QND measurement scheme allows for detection of a few photons or even a single photon [20]. Kippenberg et al. have observed the Kerr nonlinearity induced parametric oscillation in a toroid microcavity at record low threshold levels more than 2 order of magnitude lower than for optical fiber based optical parametric oscillation. Del’Haye et al. have reported the generation of equally spaced frequency combs produced by the interaction between a continuous-wave pump laser of a known frequency with the modes of a monolithic ultra-high Q microresonator via the Kerr nonlinearity [14, 15].

Treussart et al. [19] have observed a bistable behaviour of WGM resonances due to the intrinsic Kerr nonlinearity of silica with a threshold power as small as 10 μW. In this paper, we develop a theoretical model to describe the Kerr nonlinear effects in spherical microcavity. We treat the loss and Kerr nonlinearity as perturbations on the resonant modes of the cavity and develop a nonlinear equation that describes the temporal evolution of the mode amplitude. Typically the components of nonlinear susceptibility tensor, χ(3) are provided in Cartesian coordinates. In this paper, we develop a framework to express the tensor components in spherical coordinates. In the absence of loss and nonlinear effects, the mode amplitude is a constant. However, in the presence of loss and Kerr nonlinearity, mode amplitude decays and acquires a phase shift as a function of time. To account for the impact of Kerr nonlinearity, we defined effective mode volume, Veff which is found to be always less than the physical volume of the microsphere. We note that the effective mode volume, Veff has been defined to describe the Raman interaction in microsphere [21–24]. However, to the best of our knowledge, Veff has not been defined previously to describe the self-phase modulation in microsphere due to Kerr nonlinearity. Our results show that the WGMs undergo a negative frequency shift proportional to injected energy, due to Kerr nonlinearity, consistant with the obsevation in [19]. In this paper, we focussed on the impact of Kerr nonlinearity on a single mode. However, the proposed approach can be easily generalized to describe the cross-phase modulation (XPM) and four-wave mixing (FWM) effects among two or more WGMs.

Effective mode volume (or effective mode area in the context of optical fibers) is specific to the type of nonlinearity under consideration. The effective mode volume discussed in [25], cannot be used to predict accurately the frequency shift of WGM due to the Kerr effect. The mode volume defined in [23] to describe the Raman process should not be used to describe the impact of Kerr nonlinearity on WGM. As pointed in [24], the definition of mode volume is not rigorous and it depends on the physical problem studied. In the context of optical fibers, the effective mode area defined to account for Kerr nonlinearity is found to predict inaccurate stimulated brillouin scattering (SBS) threshold [26, 27]. Effective mode area that takes into account the overlap of optical and acoustic modes is found to predict the SBS threshold in agreement with experiment [26]. If the WGM is approximated by a 3-D Gaussian mode profile given by exp(−x2/αx2+y2/αy2+z2/αz2), the mode volume is characterized by αxαyαz (based on the linear properties of the mode). However, such a definition does not yield the accurate expression for the frequency-shift of WGM due to Kerr nonlinearity.

WGMs are obtained by solving the vector wave equation. If the modes reflect with grazing incidence at the boundary between two materials (in other words, index contrast is very small), scalar wave approximation can be done [24,28]. Under the scalar approximation, tensor nature of χ(3) becomes less important and the results are not accurate. However, in this paper, we use the exact vector mode as the unperturbed WGM and treat nonlinearity and loss as perturbations. In this case, tensor nature of χ(3) can not be ignored. If the dielectric material is anisotropic, tensor nature of χ(3) becomes even more important.

2. Review of modes of spherical microcavity

The optical modes can be characterized by its polarization (TE or TM) and three integer orders (ν, l, m), where ν denotes the radial mode order, l indicates the angular momentum mode number and m refers the azimuthal mode number. The radial mode number ν indicates the number of peaks in the radial intensity distribution of the internal electric field, and the mode number l corresponds to the number of wavelengths around the circumference. For each angular momentum mode number l, the allowed azimuthal mode numbers in the range of −l ≤ m ≤ l. To gain a better understanding of the modes, it is necessary to solve Maxwells equations for a simple geometry of a sphere.

Consider a dielectric microsphere in the air. The radius of the microsphere is denoted by a, and the refractive index is denoted by n(r). In the region inside the sphere, r ≤ a, the refractive index has a constant value, n1 and in the region outside the sphere, r > a, n(r) = 1. The evolution of electric field in the dielectric structure is governed by the vector wave equation [3]

Δ×Δ×E→−k2n2(r)∂2E→∂t2=0.
The solution to vector wave equation can be computed by expanding the electric field in terms of the spherical vector wave functions [3]. The modes can be divided into two types: transverse electric (TE) modes and transverse magnetic (TM) modes. In this paper, we focus mainly on the TE modes, which can be written as [3]
E→(r→)=Eθθ→+Eϕϕ→,
where
Eθ=12[AqrXθ(θ,ϕ)exp(imϕ)Rl(r)exp(iωt)+c.c.],
Eϕ=12[AqrsinθXϕ(θ,ϕ)exp(imϕ)Rl(r)exp(iωt)+c.c.].
The angular functions Xθ (θ, ϕ) and Xϕ (θ, ϕ) are defined as
Xθ(θ,ϕ)=imsinθPlm(cosθ),
Xϕ(θ,ϕ)=−∂∂θPlm(cosθ).
The function Plm(cosθ) is the associated Legendre polynomial. In electromagnetic literature, the field components are typically denoted using subscripts such as Eθ. However, to facilitate the use of tensor calculus in Section 3, we denote the contra-variant vectors by superscripts and covariant vectors by subscripts. In Eqs.(3) and (4), |q|2 is proportional to energy and A is a normalization constant discussed in Appendix B.

In Eq. (3) and (4), Rl(r) is the radial distribution for the electric field, which can be determined from the following second order differential equation [3, 29]:

[d2dr2+k2n2(r)−l(l+1)r2]rRl(r)=0,
where k = ω/c. The solution of this differential equation is given by
Rl(r)={jl(k1r)for  r≤aBhl(1)(kr)for  r>a
where jl and hl(1) are the spherical Bessel function and spherical Hankel function of first kind respectively and k1 = kn1. At the boundary r = a, continuity of Rl(r) and dRl(r)/dr leads to two equations:
jl(k1a)=Bhl(1)(ka),
jl′(k1r)k1=Bhl′(1)(kr)k.
where a prime denotes the differentiation with respect to the argument of the function. Dividing Eq. (10) by Eq. (9), we get the characteristics equation
jl′(k1a)k1jl(k1a)=hl′(1)(ka)khl(1)(ka).
For a given angular momentum mode number l, there are many solutions of the characteristic equation. Each solution corresponds to a radial resonant mode. The characteristic equation does not have solution when k is real and hence, the modes are leaky. The real part of k corresponds to the center frequency of the resonance and the imaginary part is proportional to the width of the resonance.

3. Impact of nonlinearity on cavity modes

The electric field in microcavity is governed by the nonlinear wave equation [30]

∇2E→−1c2∂2E→∂t2=μ0∂2P→L∂t2+μ0∂2P→NL∂t2,
where the linear and nonlinear polarization components are related to the electric field E⃗(r⃗, t) by
P→L(r→,t)=ε0χ(1)E→(r→,t),
P→NL(r→,t)=ε0χ(3)⋮E→(r→,t)E→(r→,t)E→(r→,t).
respectively. Here, ε0 is the free space permittivity, μ0 is the free space permeability, χ(1) is the linear susceptibility and χ(3) is the nonlinear susceptibility tensor. Substituting Eqs. (13) and (14) in Eq. (12), we obtain
∇2E→−n2(r)c2∂2E→∂t2=1c2∂2P→NL∂t2,
where c is the speed of light in vacuum and n2(r) = 1 + χ(1)(r) is the refractive index profile. Let the electric field be
E→=Eθθ→+Eϕϕ→,
where Eθ and Eϕ are given by Eqs. (3) and (4), respectively. In the absence of loss and nonlinear effects, the mode amplitude q is a constant. In this section, we assume that loss and nonlinear effects are perturbations on the resonant modes and hence, q becomes a function of time t. Substituting Eq. (16) in the second term of Eq. (15) and under the slowly varying envelope approximation, we find
n2(r)c2∂2E→∂t2=n2(r)c212[(2∂q∂tiω−qω2)f→(r→)exp(iωt)+c.c.],
where
f→(r→)=fθ(r→)θ→+fϕ(r→)ϕ→.
The function fθ (r⃗) and fϕ (r⃗) are defined as
fθ(r→)=ARl(r)imrsinθPlm(cosθ)exp(imϕ),
fϕ(r→)=ARl(r)rsinθ[−∂∂θPlm(cosθ)]exp(imϕ).
f⃗(r⃗) are the eigen modes that satisfy the vector Helmholtz equation,
∇2E→+ω2c2nr2(r)E→=0,
where
nr2(r)=Re[n2(r)]=εr(r),
ni2(r)=Im[n2(r)]=εi(r),
ε = εr(r) + iεi(r) is the permittivity. Using Eqs. (17) and (21), the left hand side of Eq. (15) is simplified as
∇2E→−n2(r)c2∂2E→∂t2=[−ω2c2q2nr2(r)f→(r→)−n2(r)c212(2∂q∂tiω   −qω2)f→(r→)]exp(iωt)+c.c.,=[−n2(r)c2∂q∂tiω−q2iεi(r)c2ω2]f→(r→)exp(iωt)+c.c..
Using Eqs. (14) and (16), the right hand side of Eq. (15) is simplified as (see Appendix A)
∇2E→−n2(r)c2∂2E→∂t2=−ω2c2(PNLθθ→+PNLϕϕ→),
where
PNLθ=|q|2qχxxx(3)x{[38r2|fθ(r→)|2fθ(r→)+14r2sin2θ|fϕ(r→)|2fθ(r→)+18r2sin2θ(fϕ(r→))2fθ*(r→)]}exp(iωt)+c.c.,
PNLϕ=|q|2qχxxx(3)x{[38r2sin2θ|fϕ(r→)|2fϕ(r→)+14r2|fθ(r→)|2fϕ(r→)+18r2(fθ(r→))2fϕ*(r→)]}exp(iωt)+c.c.
Combining Eqs. (24) and (25), we find
i[∂q∂t+ωq2εi(r)n2(r)]f→(r→)=ω|q|2qχxxx(3)xn2(r)(PNL(1)θθ→+PNL(1)ϕϕ→),
where
PNL(1)θ=[38r2|fθ(r→)|2fθ(r→)+14r2sin2θ|fϕ(r→)|2fθ(r→)+18r2sin2θ(fϕ(r→))2fθ*(r→)]+c.c.,
PNL(1)ϕ=[38r2sin2θ|fϕ(r→)|2fϕ(r→)+14r2|fθ(r→)|2fϕ(r→)+18r2(fθ(r→))2fϕ*(r→)]+c.c..
Multiplying both sides of Eq. (28) by f⃗*(r⃗) and integrating over the volume, we find
i∂q∂t+iqω2∫[εi(r)f→(r→)⋅f→*(r→)]dv/n2(r)∫f→(r→).f→*(r→)dv=ω|q|2qχxxx(3)x∫[(PNL(1)θθ→+PNL(1)ϕϕ→).f→*(r→)]dv/n2(r)∫f→(r→).f→*(r→)dv
Eq. (31) may be written as
i(∂q∂t+q2τp)=γ|q|2q,
where τp is the lifetime, given by
τp=∫f→(r→).f→*(r→)dvω∫[εi(r)f→(r→).f→*(r→)]dv/n2(r).
γ is the effective nonlinear coefficient,
γ=χxxx(3)xω  cVeff,
where Veff is effective modal volume, given by
Veff=c∫f→(r→).f→*(r→)dv∫[(PNL(1)θθ→+PNL(1)ϕϕ→).f→*(r→)]dv/n2(r).
Using Eqs. (29) and (30), effective mode volume can be calculated as (see Appendix B)
Veff=c2Iθ+Iϕ,
where
Iθ=π4m2|A|4(3m2k1+2k2)∫r2|Rl(r)|4n2(r)dr,
Iϕ=π4|A|4(3k3+2m2k2)∫r2|Rl(r)|4n2(r)dr,
k1=∫−11[plm(x)]4(1−x2)2dx,
k2=∫−11[plm(x)]2[(1−x2)∂∂xplm(x)−xplm(x)]2(1−x2)2dx,
k3=∫−11[(1−x2)∂∂xplm(x)−xplm(x)]4(1−x2)2dx.
Eqs. (32), (36)–(38) are the main results of this paper. To solve Eq. (32), let
q(t)=X(t)eiθ(t).
where X(t) and θ(t) are amplitude and phase, respectively. Subtituting Eq. (42) in Eq. (32), we find
dXdt=−X2τp,
X(t)=X(0)exp(−t/2τp),
dθdt=−γX2(t)=−γexp(−t/τp)X2(0).
Integrating Eq. (45), we find
θ(t)=θ(0)−γ[1−exp(−t/τp)]τpX2(0).
Instantaneous frequency shift is
Δf(t)=12πdθdt=−γ2πEinjexp(−t/τp),
where Einj = |q(0)|2 is the injected energy. Using Eqs. (44) and (46), we find
q(t)=q(0)e−t/2τp−iγEinj[1−exp(−t/τp)]τp,
From Eq. (48), we see that the mode amplitude decays exponentially due to loss and it acquires a phase factor due to nonlinear effects which corresponds to the frequency shift of the cavity resonance. When τp is large, the Taylor expansion of the phase term yields,
1−exp(−t/τp)≅t/τp,
θ(t)≅−γEinjt,
Δf=12πdθdt=−γEinj/2π.
With γ = 1.22 × 1020 J−1s−1, Einj = 10−12 J, the resonant modes undergo a negative frequency shift of 19.4 MHz, consistent with the observation of [19]. Eq. (50) is the analog of Eq. (3.1.5) of [31] derived for the case of nonlinear Fabry-Perot resonators,
ϕNL=γPavLR,
where γ is the nonlinear coefficient (of a different dimension), Pav is the average power and LR is the roundtrip distance. To the best of our knowledge, for spherical microcavity, calculation of effective mode volume taking into account the tensor components of χ(3) in spherical coordinates and the corresponding nonlinear phase shift has not been done before. For a nonlinear Fabry-Perot interferrometer, Eq. (52) is combined with the Airy formula to explain bistable behaviour [31]. It may be possible to explain bistability in microsphere based on Eq. (48) which is a subject of future investigation.

4. Results and discussion

For a given angular momentum mode number l, there are many possible solutions according to the characteristic Eq. (11). Figure 1 shows the distributions of the electric field intensity for the first three radial mode numbers. The following parameters are used: angular momentum mode number l = 80, radius of microsphere a = 15 μm, refractive index of the silica microsphere n1 = 1.52. Solving Eq. (11) for l = 80 yields the values (real parts) λ1,80TE=1640.03 nm, λ2,80TE=1527.77  nm and λ3,80TE=1443.92 nm. It can be seen from Fig. 1, that the modes are well confined inside the cavity with a small fraction outside the cavity in the evanescent form.

 figure: Fig. 1

Fig. 1 Distribution of electric field intensity |Rl(r)|2 with radius 15 μm and angular momentum mode number l = 80 for first three mode numbers (ν = 1, 2, 3). The field shows slower decay for higher order radial numbers (ν).

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To calculate the effective mode volume, we use Eqs. (36)–(38). The integrations in Eqs. (37) and (38) are carried out using Simpson’s 1/3 rule. The following parameters are used for calculating the effective mode volume: angular momentum mode number l = 80, resonance wavelength 1640.03 nm, radius of microsphere a = 15 μm, refractive index n1 = 1.52 and azimuthal mode number m = 1. The calculated effective mode volume in the microsphere is 2.9561 × 10−16 m3 which is much smaller than the microsphere’s physical volume of 1.41×10−14 m3. Using a loss of 0.2 dB/km for silica, the calculated lifetime is 2.588 × 10−5 s.

Table 1 lists the effective mode volume and effective nonlinear coefficient for the first three radial mode numbers. As can be seen, the effective mode volume increases with the radial mode order, ν which is attributed to the weaker confinement of the higher order modes. Figures 2(a) and 2(b) show the phase θ(t) calculated using Eq. (48) for different modes and for differentinjected energy Einj, and Figs. 2(c) and 2(d) show the corresponding frequency shift Δf(t) calculated using Eq. (47). When the radial mode order is larger, effective nonlinear coefficient is lower and hence we get a lower frequency shift.

Tables Icon

Table 1. Effective mode volumes and effective nonlinear coefficients for the first three radial mode numbers.

View Table
 figure: Fig. 2

Fig. 2 Observation of phase change and frequency shift. Parameters: radius a = 15 μm and angular momentum mode number l = 80 for first three radial mode numbers (ν = 1, 2, 3) for different injected energy Einj in the microsphere.

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The theoretical framework presented here is only the first step to predict accurately the impact of third order nonlinear effects on WGMs. For the experimental work on third harmonic generation [9] or optical parametric oscillation [13], the energy conservation (or frequency matching) is important to increase the efficiency. Due to self phase modulation, the pump and TH light of [9] undergo different amounts of frequency shifts (as predicted by Eq. (47) of the manuscript) and hence, there is an additional source of frequency detuning, which should be considered in the analysis of the experimental work of [9]. The consequence of taking or not taking into account the effect of Kerr nonlinearity depends on the application. If the injected power is small, Kerr nonlinear effects can be ignored. However, for applications such as parametric oscillations, frequency comb and third harmonic generation, impact of Kerr nonlinearity should be accurately modeled.

The optical properties of a material are characterized by its linear susceptibility, χ(1), 2nd order nonlinear susceptibility, χ(2) and 3rd order nonlinear susceptibility χ(3), which are unique to the material. Typically, in sensing applications, the refractive index change (mainly due to χ(1)) translates into the frequency shift of WGMs, which enables the detection of the object. However, in highly nonlinear materials and/or when large power is injected into microspheres, the frequency shift due to χ(2) and χ(3) could be comparable to that due to χ(1). Since the power dependence of these nonlinear effects are different, by doing a power sweep, it may be possible to separate the frequency shifts due to χ(2) and χ(3). Since χ(j), j = 1, 2, 3 are unique to the object, it would be possible to identify the object. For example, for centro-symmetric materials, χ(2) is zero. Using the equation for effective mode volume Eqs. (35) and (36), it would be possible to estimate χ(3) of the material based on the experimentally observed frequency shifts.

5. Conclusions

We have investigated the impact of Kerr nonlinearity on the whispering gallery modes of a microsphere. By treating loss and Kerr nonlinear effect as perturbations, a nonlinear equation to describe the temporal evolution of the mode amplitude is developed. We defined effective mode volume taking into account the tensor components of χ(3) in spherical coordinates to describe the impact of Kerr nonlinearity. Our results showed that the resonant modes undergo a negative frequency shift proportional to the injected energy, consistent with the experimental observations in [19].

Appendix A: Calculation of PNLθ and PNLϕ

The third order nonlinear polarization is given by

PNLn=χklm(3)nEkElEm,            k,l,m,n=x,y,  or  z.
Here, the Einstein convention of summation over the repeated indices is assumed [32]. For an isotropic medium, there are 21 non-zero elements of the tensor χ(3) of which only 3 are independent [33]. They are
χyzz(3)y=χzyy(3)z=χzxx(3)z=χxzz(3)x=χxyy(3)x=χyxx(3)y,
χzyz(3)y=χxyx(3)y=χyzy(3)z=χxzx(3)z=χzxz(3)x=χyxy(3)x,
χzzy(3)y=χxxy(3)y=χyyz(3)z=χxxz(3)z=χxzz(3)x=χyyx(3)x,
and
χxxx(3)x=χyyy(3)y=χzzz(3)z=χxyy(3)x+χyxy(3)x+χyyx(3)x.
For any odd number of x, y or z, tensor components of χ(3) are zero.
χxxy(3)x=χxxz(3)x=χyyz(3)x=……=0.
The spherical coordinates (r, θ, ϕ) are related to the Cartesian coordinates (x, y, z) by
r=(x2+y2+z2)1/2,
θ=tan−1((x2+y2)1/2z),
ϕ=tan−1(yx).
Consider a transformation matrix
Λ=(∂r∂x∂r∂y∂r∂z∂θ∂x∂θ∂y∂θ∂z∂ϕ∂x∂ϕ∂y∂ϕ∂z)=(sinθcosϕsinθsinϕcosθcosθcosϕrcosθsinϕr−sinθr−sinϕrsinθcosϕrsinθ0).
The displacement vector Δ⃗r can be written in two coordinate systems as
Δ→r=Δrr→+Δθ  θ→+Δϕ  ϕ→,=Δxx→+Δyy→+Δzz→,
and
Δr=∂r∂xΔx+∂r∂yΔy+∂r∂zΔz,=Λij′(Δx)i,         j′=r.
where, Λij′ are the elements of the matrix Λ and Δx1 = Δx, Δx2 = Δy and Δx3 = Δz. The components of Δ⃗r in spherical coordinates can be written as
(Δ→r)j′=Λij′(Δx)i,     j′=r, θ  or   ϕ.
If Ai(i = x, y or z) is a component of the vector A⃗ in Cartesian coordinates, it is transformed to the spherical coordinates by the following rule
Aj′=Λij′A′,            i=x,y  or   z     and    j′=r, θ  or  ϕ.
Similarly, the component of tensor χ(3) in Cartesian coordinates are transformed to spherical coordinates by the following rule,
χk′l′m′(3)j′=Λjj′Λk′kΛl′lΛm′mχklm(3)j.
Using Eq. (67), we find
χθθθ(3)θ=ΛjθΛθkΛθlΛθmχklm(3)j           =ΛxθΛθxΛθxΛθxχxxx(3)x+ΛxθΛθxΛθyΛθyχxyy(3)x+ΛxθΛθxΛθzΛθzΛxzz(3)x                +ΛxθΛθyΛθxΛθyχyxy(3)x+ΛxθΛθyΛθyΛθxχyyx(3)x+ΛxθΛθzΛθxΛθzχzxz(3)x                +ΛxθΛθzΛθzΛθxχzzx(3)x+ΛyθΛθyΛθxΛθxχyxx(3)y+ΛyθΛθyΛθyΛθyχyyy(3)y                +ΛyθΛθyΛθzΛθzχyzz(3)y+ΛyθΛθxΛθxΛθyχxxy(3)y+ΛyθΛθxΛθyΛθxχxyx(3)y                +ΛyθΛθzΛθyΛθzχzyz(3)y+ΛyθΛθzΛθzΛθyχzzy(3)y+ΛzθΛθxΛθxΛθzχxxz(3)z                +ΛzθΛθxΛθzΛθxχxzx(3)z+ΛzθΛθyΛθyΛθzχyyz(3)z+ΛzθΛθyΛθzΛθyχyzy(3)z,                +ΛzθΛθzΛθxΛθxχzxx(3)z+ΛzθΛθzΛθyΛθyχzyy(3)z+ΛzθΛθzΛθzΛθzχzzz(3)z,=[cosθcosϕrr3cos3θcos3ϕ+cosθsinϕrr3cos3θsin3ϕ+(−sinθ)r(−r3sin3θ)+cosθcosϕrrcosθcosϕr2cos2θsin2ϕ+cosθsinϕrrcosθsinϕr2cos2θcos2ϕ+cosθcosϕrrcosθcosϕr2sin2θ+cosθsinϕrrcosθsinϕr2sin2θ+(−sinθ)r(−rsinθ)r2cos2θcos2ϕ+(−sinθ)r(−rsinθ)r2cos2θsin2θ]χxxx(3)x,−r2χxxx(3)x,
χϕϕϕ(3)ϕ=∑j∑k∑l∑mΛjϕΛϕkΛϕlΛϕmχklm(3)j,=[Λxϕ(Λϕx)3+Λyϕ(Λϕy)3+Λzϕ(Λϕz)3+(Λxϕ)(Λϕx)(Λϕy)2)+(Λyϕ)(Λϕy)(Λϕx)2    +(Λxϕ)(Λϕx)(Λϕz)2+(Λyϕ)(Λϕy)(Λϕz)2+(Λzϕ)(Λϕz)(Λϕx)2+(Λzϕ)(Λϕz)(Λϕy)2]χxxx(3)x,=[(−sinϕ)sinθ(−r3sin3θsin3ϕ+cosϕrsinθr3sin3θcos3ϕ+cosϕrsinθrsinθcosϕ)     r2sin2θsin2ϕ+sinϕrsinθ(−rsinθsinϕ)r2sin2θcos2ϕ]χxxx(3)x,=(r2sin2θsin4ϕ+r2sin2θcos4ϕ+r2cos2ϕsin2θsin2ϕ    +r2cos2ϕsin2θsin2ϕ)χxxx(3)x,=r2sin2θχxxx(3)x,
χθϕϕ(3)θ=∑j∑k∑l∑mΛjθΛθkΛϕlΛϕmχklm(3)j,=ΛxθΛθxΛϕxΛϕxχxxx(3)x+ΛxθΛθxΛϕyΛϕyχxyy(3)x+ΛxθΛθxΛϕzΛϕzχxzz(3)x+ΛxθΛθyΛϕxΛϕyχyxy(3)x    +ΛxθΛθyΛϕyΛϕxχyyx(3)x+ΛxθΛθzΛϕxΛϕzχzxz(3)x+ΛxθΛθzΛϕzΛϕxχzzx(3)x+ΛyθΛθyΛϕxΛϕxχyxx(3)x    +ΛyθΛθyΛϕyΛϕyχyyy(3)y+ΛyθΛθyΛϕzΛϕyχyzz(3)y+ΛyθΛθxΛϕxΛϕyχxxy(3)y+ΛyθΛθxΛϕyΛϕxχxyx(3)y    +ΛyθΛθzΛϕyΛϕzχzyz(3)y+ΛyθΛθzΛϕzΛϕyχzzy(3)y+ΛzθΛθxΛϕxΛϕzχxxz(3)z    +ΛzθΛθxΛϕzΛϕxχxzx(3)z+ΛzθΛθyΛϕyΛϕzχyyz(3)z+ΛzθΛθyΛϕzΛϕyχyzy(3)z   +ΛzθΛθzΛϕxΛϕxχzxx(3)z+ΛzθΛθyΛϕyΛϕzχzyy(3)z+ΛzθΛθzΛϕzΛϕzχzzz(3)z,=13r2sin2θχxxx(3)x.
Similarly
χϕϕθ(3)θ=13r2sin2θχxxx(3)x,
χϕθϕ(3)θ=13r2sin2θχxxx(3)x,
χϕθθ(3)ϕ=13r2χxxx(3)x,
χθϕθ(3)ϕ=13r2χxxx(3)x,
χθθϕ(3)ϕ=13r2χxxx(3)x,
χθθθ(3)ϕ=χϕϕϕ(3)θ=0.
In spherical coordinates, Eq. (53) may be rewritten as
PNLn=χklm(3)nEkElEm,             k,l,m,n=r,θ  or  ϕ.
For TE modes, Er = 0. So, from Eq. (77), we have
PNLθ=χθθθ(3)θ(Eθ)3+[χϕϕθ(3)θ+χϕθϕ(3)θ++χθϕϕ(3)θ](Eϕ)2Eθ,=r2[(Eθ)3+13sin2θ(Eϕ)2Eθ+13sin2θ(Eϕ)2Eθ    +13sin2θ(Eϕ)2Eθ]χxxx(3)x,=r2[(Eθ)3+sin2θ(Eϕ)2Eθ]χxxx(3)x,=[α1(Eθ)2+α2(Eϕ)2]Eθ,
where
α1=r2χxxx(3)x,
α2=r2sin2θχxxx(3)x,
Eθ=12[q(t)fθ(r→)exp(iωt)+c.c.],
Eϕ=12[q(t)fϕ(r→)exp(iωt)+c.c.].
Using Eq. (81), we find
(Eθ)3=18[qfθ(r→)exp(iωt)+q*fθ*(r→)exp(−iωt)][qfθ(r→)exp(iωt)     +q*fθ*(r→)exp(−iωt)][q  fθ(r→)exp(iωt)+q*fθ*(r→)exp(−iωt)],=18[q2(fθ(r→))2exp(2iωt)+2|q|2|fθ(r→)|2+q*fθ*(r→)q*fθ*(r→)exp(−2iωt)     [qfθ(r→)exp(iωt)+q*fθ*(r→)exp(−iωt)],=18[q3(fθ(r→))3exp(3iωt)+2|q|2q|fθ(r→)|2fθ(r→)exp(iωt)   +|q|2|fθ(r→)|2q*fθ*(r→)exp(−iωt)+|q|2|fθ(r→)|2qfθ(r→)exp(iωt)   +2|q|2|fθ(r→)|2q*fθ*(r→)exp(−iωt)+q*fθ*(r→)q*fθ*(r→)fθ*(r→)q*exp(−3iωt).
In Eq. (83), there are terms proportional to exp(3iωt) and exp(−3iωt) which are responsible for third harmonic generation. In the absence of phase matching, the efficiency of third harmonic generation is very small and hence these terms can be ignored [34]. Now, Eq. (83), may be rewritten as,
(Eθ)3=|q|2|q8[3|fθ(r→)|2fθ(r→)exp(iωt)+c.c.].
Similarly, we find
(Eϕ)2Eθ=|q|2|q8{[2|fϕ(r→)|2fθ(r→)+(fϕ*(r→))2q  fθ*(r→)]exp(iωt)+c.c.}.
Using Eqs. (84) and (85) in Eq. (78), we find
PNLθ=[α1(Eθ)2+α2(Eϕ)2]Eθ,=|q|2qχxxx(3)x[(38r2|fθ(r→)|2fθ(r→)+14r2sin2θ|fϕ(r→)|2fθ(r→)    +18r2sin2θ(fϕ(r→))2fθ*(r→))exp(iωt)+c.c.].
Similarly, using Eq. (77), we obtain
PNLϕ=χϕϕϕ(3)ϕ(Eϕ)3+[χϕθθ(3)ϕ+χθθϕ(3)ϕ+χθϕθ(3)ϕ](Eθ)2Eϕ,=r2[sin2θ(Eϕ)3+13(Eθ)2Eϕ+13(Eθ)2Eϕ+13(Eθ)2Eϕ]χxxx(3)x,=r2[sin2θ(Eϕ)3+(Eθ)2Eϕ]χxxx(3)x,=[α2(Eϕ)2+α1(Eθ)2]Eϕ,
Using Eqs. (81) and Eq. (82) and proceeding as before, we find
(Eϕ)3=|q|2|q8[3|fϕ(r→)|2fϕ(r→)exp(iωt)+c.c.],
and
(Eθ)2Eϕ=|q|2|q8[(2|fθ(r→)|2fϕ(r→)exp(iωt)+c.c.)+((fθ(r→))2fϕ*(r→)exp(iωt)+c.c.)].
Using Eqs. (88) and (89) in Eq. (87), we find
PNLϕ=[α2(Eϕ)2+α1(Eθ)2]Eϕ,=|q|2qχxxx(3)x[(38r2sin2θ|fϕ(r→)|2fϕ(r→)+14r2|fθ(r→)|2fϕ(r→)+18r2(fθ(r→))2fϕ*(r→))exp(iωt)+c.c.].

Appendix B: Calculation of effective mode volume

The metric coefficients are given by [32]

gθθ=θ→.θ→=r2,
gϕϕ=ϕ→.ϕ→=r2sin2θ,
gθϕ=gϕθ=0.
We like to express |q(0)|2 in the units of energy. In order to do that, the modes are so normalized that
1c∫(f→(r→).f→*(r→))dv=1c∫(|fθ(r→)|2gθθ+|fϕ(r→)|2gϕϕ)dv,=1,
where
fθ(r→)=A  Rl(r)imrsinθPlm(cosθ)exp(imϕ),
fϕ(r→)=A  Rl(r)rsinθ[−∂∂θPlm(cosθ)]exp(imϕ).
Substituting Eqs. (95) and (96) in Eq. (94) and simplifying, we obtain
|A|2=c∫[r2|Rl(r)|2m2sinθ[plm(cosθ)]2+r2|Rl(r)|2[−∂∂θplm(cosθ)]2sinθ]   drdθ  dϕ.
Eq. (35) may be rewritten as
Veff=c2Iθ+Iϕ,
where
Iθ=∫PNL(1)θfθ*(r→)gθθdvn2(r),
Iϕ=∫PNL(1)ϕfϕ*(r→)gϕϕdvn2(r).
Substituting Eq. (29) and Eqs. (95)–(97) in Eq. (99), we find
Iθ=∫[38r2|fθ(r→)|4n2(r)gθθ+14r2|fϕ(r→)|2|fθ(r→)|2n2(r)sin2θgθθ]dv,=38|A|4∫r2|Rl(r)|4n2(r)m4sin3θ[plm(cosθ)]4drdθdϕ    +14|A|4∫r2|Rl(r)|4n2(r)m2sinθ[plm(cosθ)]2[−∂∂θplm(cosθ)]2drdθdϕ,=π4m2|A|4(3m2k1+2k2)∫r2|Rl(r)|4n2(r)dr,
where
k1=∫−11[plm(x)]4(1−x2)2dx,
k2=∫−11[plm(x)]2[(1−x2)∂∂xplm(x)−xplm(x)]2(1−x2)2dx.
Substituting Eq. (30) and Eqs. (95)–(97) in Eq. (100), we find
Iϕ=∫[38r2|fϕ(r→)|4n2(r)sin2θgϕϕ+14r2|fϕ(r→)|2|fθ(r→)|2n2(r)sin2θgϕϕ]dv,=38|A|4∫r2|Rl(r)|4n2(r)[−∂∂θplm(cosθ)]4sinθdrdθdϕ    +14|A|4∫r2|Rl(r)|4n2(r)m2sinθ[plm(cosθ)]2[−∂∂θplm(cosθ)]2drdθdϕ,=π4|A|4(3k3+2m2k2)∫r2|Rl(r)|4n2(r)dr,
where
k3=∫−11[(1−x2)∂∂xplm(x)−xplm(x)]4(1−x2)2dx.

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Figures (2)
Fig. 1
Fig. 1 Distribution of electric field intensity |Rl(r)|2 with radius 15 μm and angular momentum mode number l = 80 for first three mode numbers (ν = 1, 2, 3). The field shows slower decay for higher order radial numbers (ν).

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Fig. 2
Fig. 2 Observation of phase change and frequency shift. Parameters: radius a = 15 μm and angular momentum mode number l = 80 for first three radial mode numbers (ν = 1, 2, 3) for different injected energy Einj in the microsphere.

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Tables (1)

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Table 1 Effective mode volumes and effective nonlinear coefficients for the first three radial mode numbers.

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Equations (105)

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Δ × Δ × E → − k 2 n 2 ( r ) ∂ 2 E → ∂ t 2 = 0 .
E → ( r → ) = E θ θ → + E ϕ ϕ → ,
E θ = 1 2 [ Aq r X θ ( θ , ϕ ) exp ( im ϕ ) R l ( r ) exp ( i ω t ) + c . c . ] ,
E ϕ = 1 2 [ Aq r sin θ X ϕ ( θ , ϕ ) exp ( im ϕ ) R l ( r ) exp ( i ω t ) + c . c . ] .
X θ ( θ , ϕ ) = im sin θ P l m ( cos θ ) ,
X ϕ ( θ , ϕ ) = − ∂ ∂ θ P l m ( cos θ ) .
[ d 2 d r 2 + k 2 n 2 ( r ) − l ( l + 1 ) r 2 ] r R l ( r ) = 0 ,
R l ( r ) = { j l ( k 1 r ) for     r ≤ a B h l ( 1 ) ( k r ) for     r > a
j l ( k 1 a ) = B h l ( 1 ) ( k a ) ,
j l ′ ( k 1 r ) k 1 = B h l ′ ( 1 ) ( k r ) k .
j l ′ ( k 1 a ) k 1 j l ( k 1 a ) = h l ′ ( 1 ) ( k a ) k h l ( 1 ) ( k a ) .
∇ 2 E → − 1 c 2 ∂ 2 E → ∂ t 2 = μ 0 ∂ 2 P → L ∂ t 2 + μ 0 ∂ 2 P → NL ∂ t 2 ,
P → L ( r → , t ) = ε 0 χ ( 1 ) E → ( r → , t ) ,
P → NL ( r → , t ) = ε 0 χ ( 3 ) ⋮ E → ( r → , t ) E → ( r → , t ) E → ( r → , t ) .
∇ 2 E → − n 2 ( r ) c 2 ∂ 2 E → ∂ t 2 = 1 c 2 ∂ 2 P → NL ∂ t 2 ,
E → = E θ θ → + E ϕ ϕ → ,
n 2 ( r ) c 2 ∂ 2 E → ∂ t 2 = n 2 ( r ) c 2 1 2 [ ( 2 ∂ q ∂ t i ω − q ω 2 ) f → ( r → ) exp ( i ω t ) + c . c . ] ,
f → ( r → ) = f θ ( r → ) θ → + f ϕ ( r → ) ϕ → .
f θ ( r → ) = A R l ( r ) im r sin θ P l m ( cos θ ) exp ( im ϕ ) ,
f ϕ ( r → ) = A R l ( r ) r sin θ [ − ∂ ∂ θ P l m ( cos θ ) ] exp ( im ϕ ) .
∇ 2 E → + ω 2 c 2 n r 2 ( r ) E → = 0 ,
n r 2 ( r ) = Re [ n 2 ( r ) ] = ε r ( r ) ,
n i 2 ( r ) = Im [ n 2 ( r ) ] = ε i ( r ) ,
∇ 2 E → − n 2 ( r ) c 2 ∂ 2 E → ∂ t 2 = [ − ω 2 c 2 q 2 n r 2 ( r ) f → ( r → ) − n 2 ( r ) c 2 1 2 ( 2 ∂ q ∂ t i ω       − q ω 2 ) f → ( r → ) ] exp ( i ω t ) + c . c . , = [ − n 2 ( r ) c 2 ∂ q ∂ t i ω − q 2 i ε i ( r ) c 2 ω 2 ] f → ( r → ) exp ( i ω t ) + c . c . .
∇ 2 E → − n 2 ( r ) c 2 ∂ 2 E → ∂ t 2 = − ω 2 c 2 ( P NL θ θ → + P NL ϕ ϕ → ) ,
P NL θ = | q | 2 q χ x x x ( 3 ) x { [ 3 8 r 2 | f θ ( r → ) | 2 f θ ( r → ) + 1 4 r 2 sin 2 θ | f ϕ ( r → ) | 2 f θ ( r → ) + 1 8 r 2 sin 2 θ ( f ϕ ( r → ) ) 2 f θ * ( r → ) ] } exp ( i ω t ) + c . c . ,
P NL ϕ = | q | 2 q χ x x x ( 3 ) x { [ 3 8 r 2 sin 2 θ | f ϕ ( r → ) | 2 f ϕ ( r → ) + 1 4 r 2 | f θ ( r → ) | 2 f ϕ ( r → ) + 1 8 r 2 ( f θ ( r → ) ) 2 f ϕ * ( r → ) ] } exp ( i ω t ) + c . c .
i [ ∂ q ∂ t + ω q 2 ε i ( r ) n 2 ( r ) ] f → ( r → ) = ω | q | 2 q χ x x x ( 3 ) x n 2 ( r ) ( P NL ( 1 ) θ θ → + P NL ( 1 ) ϕ ϕ → ) ,
P NL ( 1 ) θ = [ 3 8 r 2 | f θ ( r → ) | 2 f θ ( r → ) + 1 4 r 2 sin 2 θ | f ϕ ( r → ) | 2 f θ ( r → ) + 1 8 r 2 sin 2 θ ( f ϕ ( r → ) ) 2 f θ * ( r → ) ] + c . c . ,
P NL ( 1 ) ϕ = [ 3 8 r 2 sin 2 θ | f ϕ ( r → ) | 2 f ϕ ( r → ) + 1 4 r 2 | f θ ( r → ) | 2 f ϕ ( r → ) + 1 8 r 2 ( f θ ( r → ) ) 2 f ϕ * ( r → ) ] + c . c . .
i ∂ q ∂ t + i q ω 2 ∫ [ ε i ( r ) f → ( r → ) ⋅ f → * ( r → ) ] d v / n 2 ( r ) ∫ f → ( r → ) . f → * ( r → ) d v = ω | q | 2 q χ x x x ( 3 ) x ∫ [ ( P NL ( 1 ) θ θ → + P NL ( 1 ) ϕ ϕ → ) . f → * ( r → ) ] d v / n 2 ( r ) ∫ f → ( r → ) . f → * ( r → ) d v
i ( ∂ q ∂ t + q 2 τ p ) = γ | q | 2 q ,
τ p = ∫ f → ( r → ) . f → * ( r → ) d v ω ∫ [ ε i ( r ) f → ( r → ) . f → * ( r → ) ] d v / n 2 ( r ) .
γ = χ x x x ( 3 ) x ω     c V eff ,
V eff = c ∫ f → ( r → ) . f → * ( r → ) d v ∫ [ ( P NL ( 1 ) θ θ → + P NL ( 1 ) ϕ ϕ → ) . f → * ( r → ) ] d v / n 2 ( r ) .
V eff = c 2 I θ + I ϕ ,
I θ = π 4 m 2 | A | 4 ( 3 m 2 k 1 + 2 k 2 ) ∫ r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
I ϕ = π 4 | A | 4 ( 3 k 3 + 2 m 2 k 2 ) ∫ r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
k 1 = ∫ − 1 1 [ p l m ( x ) ] 4 ( 1 − x 2 ) 2 d x ,
k 2 = ∫ − 1 1 [ p l m ( x ) ] 2 [ ( 1 − x 2 ) ∂ ∂ x p l m ( x ) − x p l m ( x ) ] 2 ( 1 − x 2 ) 2 d x ,
k 3 = ∫ − 1 1 [ ( 1 − x 2 ) ∂ ∂ x p l m ( x ) − x p l m ( x ) ] 4 ( 1 − x 2 ) 2 d x .
q ( t ) = X ( t ) e i θ ( t ) .
d X d t = − X 2 τ p ,
X ( t ) = X ( 0 ) exp ( − t / 2 τ p ) ,
d θ d t = − γ X 2 ( t ) = − γ exp ( − t / τ p ) X 2 ( 0 ) .
θ ( t ) = θ ( 0 ) − γ [ 1 − exp ( − t / τ p ) ] τ p X 2 ( 0 ) .
Δ f ( t ) = 1 2 π d θ d t = − γ 2 π E inj exp ( − t / τ p ) ,
q ( t ) = q ( 0 ) e − t / 2 τ p − i γ E inj [ 1 − exp ( − t / τ p ) ] τ p ,
1 − exp ( − t / τ p ) ≅ t / τ p ,
θ ( t ) ≅ − γ E inj t ,
Δ f = 1 2 π d θ d t = − γ E inj / 2 π .
ϕ NL = γ P av L R ,
P NL n = χ k l m ( 3 ) n E k E l E m ,                         k , l , m , n = x , y ,     or     z .
χ y z z ( 3 ) y = χ z y y ( 3 ) z = χ z x x ( 3 ) z = χ x z z ( 3 ) x = χ x y y ( 3 ) x = χ y x x ( 3 ) y ,
χ z y z ( 3 ) y = χ x y x ( 3 ) y = χ y z y ( 3 ) z = χ x z x ( 3 ) z = χ z x z ( 3 ) x = χ y x y ( 3 ) x ,
χ z z y ( 3 ) y = χ x x y ( 3 ) y = χ y y z ( 3 ) z = χ x x z ( 3 ) z = χ x z z ( 3 ) x = χ y y x ( 3 ) x ,
χ x x x ( 3 ) x = χ y y y ( 3 ) y = χ z z z ( 3 ) z = χ x y y ( 3 ) x + χ y x y ( 3 ) x + χ y y x ( 3 ) x .
χ x x y ( 3 ) x = χ x x z ( 3 ) x = χ y y z ( 3 ) x = … … = 0 .
r = ( x 2 + y 2 + z 2 ) 1 / 2 ,
θ = tan − 1 ( ( x 2 + y 2 ) 1 / 2 z ) ,
ϕ = tan − 1 ( y x ) .
Λ = ( ∂ r ∂ x ∂ r ∂ y ∂ r ∂ z ∂ θ ∂ x ∂ θ ∂ y ∂ θ ∂ z ∂ ϕ ∂ x ∂ ϕ ∂ y ∂ ϕ ∂ z ) = ( sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ r cos θ sin ϕ r − sin θ r − sin ϕ r sin θ cos ϕ r sin θ 0 ) .
Δ → r = Δ r r → + Δ θ     θ → + Δ ϕ     ϕ → , = Δ x x → + Δ y y → + Δ z z → ,
Δ r = ∂ r ∂ x Δ x + ∂ r ∂ y Δ y + ∂ r ∂ z Δ z , = Λ i j ′ ( Δ x ) i ,                   j ′ = r .
( Δ → r ) j ′ = Λ i j ′ ( Δ x ) i ,           j ′ = r ,   θ     or       ϕ .
A j ′ = Λ i j ′ A ′ ,                         i = x , y     or       z           and         j ′ = r ,   θ     or     ϕ .
χ k ′ l ′ m ′ ( 3 ) j ′ = Λ j j ′ Λ k ′ k Λ l ′ l Λ m ′ m χ k l m ( 3 ) j .
χ θ θ θ ( 3 ) θ = Λ j θ Λ θ k Λ θ l Λ θ m χ k l m ( 3 ) j                       = Λ x θ Λ θ x Λ θ x Λ θ x χ x x x ( 3 ) x + Λ x θ Λ θ x Λ θ y Λ θ y χ x y y ( 3 ) x + Λ x θ Λ θ x Λ θ z Λ θ z Λ x z z ( 3 ) x                                 + Λ x θ Λ θ y Λ θ x Λ θ y χ y x y ( 3 ) x + Λ x θ Λ θ y Λ θ y Λ θ x χ y y x ( 3 ) x + Λ x θ Λ θ z Λ θ x Λ θ z χ z x z ( 3 ) x                                 + Λ x θ Λ θ z Λ θ z Λ θ x χ z z x ( 3 ) x + Λ y θ Λ θ y Λ θ x Λ θ x χ y x x ( 3 ) y + Λ y θ Λ θ y Λ θ y Λ θ y χ y y y ( 3 ) y                                 + Λ y θ Λ θ y Λ θ z Λ θ z χ y z z ( 3 ) y + Λ y θ Λ θ x Λ θ x Λ θ y χ x x y ( 3 ) y + Λ y θ Λ θ x Λ θ y Λ θ x χ x y x ( 3 ) y                                 + Λ y θ Λ θ z Λ θ y Λ θ z χ z y z ( 3 ) y + Λ y θ Λ θ z Λ θ z Λ θ y χ z z y ( 3 ) y + Λ z θ Λ θ x Λ θ x Λ θ z χ x x z ( 3 ) z                                 + Λ z θ Λ θ x Λ θ z Λ θ x χ x z x ( 3 ) z + Λ z θ Λ θ y Λ θ y Λ θ z χ y y z ( 3 ) z + Λ z θ Λ θ y Λ θ z Λ θ y χ y z y ( 3 ) z ,                                 + Λ z θ Λ θ z Λ θ x Λ θ x χ z x x ( 3 ) z + Λ z θ Λ θ z Λ θ y Λ θ y χ z y y ( 3 ) z + Λ z θ Λ θ z Λ θ z Λ θ z χ z z z ( 3 ) z , = [ cos θ cos ϕ r r 3 cos 3 θ cos 3 ϕ + cos θ sin ϕ r r 3 cos 3 θ sin 3 ϕ + ( − sin θ ) r ( − r 3 sin 3 θ ) + cos θ cos ϕ r r cos θ cos ϕ r 2 cos 2 θ sin 2 ϕ + cos θ sin ϕ r r cos θ sin ϕ r 2 cos 2 θ cos 2 ϕ + cos θ cos ϕ r r cos θ cos ϕ r 2 sin 2 θ + cos θ sin ϕ r r cos θ sin ϕ r 2 sin 2 θ + ( − sin θ ) r ( − r sin θ ) r 2 cos 2 θ cos 2 ϕ + ( − sin θ ) r ( − r sin θ ) r 2 cos 2 θ sin 2 θ ] χ x x x ( 3 ) x , − r 2 χ x x x ( 3 ) x ,
χ ϕ ϕ ϕ ( 3 ) ϕ = ∑ j ∑ k ∑ l ∑ m Λ j ϕ Λ ϕ k Λ ϕ l Λ ϕ m χ k l m ( 3 ) j , = [ Λ x ϕ ( Λ ϕ x ) 3 + Λ y ϕ ( Λ ϕ y ) 3 + Λ z ϕ ( Λ ϕ z ) 3 + ( Λ x ϕ ) ( Λ ϕ x ) ( Λ ϕ y ) 2 ) + ( Λ y ϕ ) ( Λ ϕ y ) ( Λ ϕ x ) 2         + ( Λ x ϕ ) ( Λ ϕ x ) ( Λ ϕ z ) 2 + ( Λ y ϕ ) ( Λ ϕ y ) ( Λ ϕ z ) 2 + ( Λ z ϕ ) ( Λ ϕ z ) ( Λ ϕ x ) 2 + ( Λ z ϕ ) ( Λ ϕ z ) ( Λ ϕ y ) 2 ] χ x x x ( 3 ) x , = [ ( − sin ϕ ) sin θ ( − r 3 sin 3 θ sin 3 ϕ + cos ϕ r sin θ r 3 sin 3 θ cos 3 ϕ + cos ϕ r sin θ r sin θ cos ϕ )           r 2 sin 2 θ sin 2 ϕ + sin ϕ r sin θ ( − r sin θ sin ϕ ) r 2 sin 2 θ cos 2 ϕ ] χ x x x ( 3 ) x , = ( r 2 sin 2 θ sin 4 ϕ + r 2 sin 2 θ cos 4 ϕ + r 2 cos 2 ϕ sin 2 θ sin 2 ϕ         + r 2 cos 2 ϕ sin 2 θ sin 2 ϕ ) χ x x x ( 3 ) x , = r 2 sin 2 θ χ x x x ( 3 ) x ,
χ θ ϕ ϕ ( 3 ) θ = ∑ j ∑ k ∑ l ∑ m Λ j θ Λ θ k Λ ϕ l Λ ϕ m χ k l m ( 3 ) j , = Λ x θ Λ θ x Λ ϕ x Λ ϕ x χ x x x ( 3 ) x + Λ x θ Λ θ x Λ ϕ y Λ ϕ y χ x y y ( 3 ) x + Λ x θ Λ θ x Λ ϕ z Λ ϕ z χ x z z ( 3 ) x + Λ x θ Λ θ y Λ ϕ x Λ ϕ y χ y x y ( 3 ) x         + Λ x θ Λ θ y Λ ϕ y Λ ϕ x χ y y x ( 3 ) x + Λ x θ Λ θ z Λ ϕ x Λ ϕ z χ z x z ( 3 ) x + Λ x θ Λ θ z Λ ϕ z Λ ϕ x χ z z x ( 3 ) x + Λ y θ Λ θ y Λ ϕ x Λ ϕ x χ y x x ( 3 ) x         + Λ y θ Λ θ y Λ ϕ y Λ ϕ y χ y y y ( 3 ) y + Λ y θ Λ θ y Λ ϕ z Λ ϕ y χ y z z ( 3 ) y + Λ y θ Λ θ x Λ ϕ x Λ ϕ y χ x x y ( 3 ) y + Λ y θ Λ θ x Λ ϕ y Λ ϕ x χ x y x ( 3 ) y         + Λ y θ Λ θ z Λ ϕ y Λ ϕ z χ z y z ( 3 ) y + Λ y θ Λ θ z Λ ϕ z Λ ϕ y χ z z y ( 3 ) y + Λ z θ Λ θ x Λ ϕ x Λ ϕ z χ x x z ( 3 ) z         + Λ z θ Λ θ x Λ ϕ z Λ ϕ x χ x z x ( 3 ) z + Λ z θ Λ θ y Λ ϕ y Λ ϕ z χ y y z ( 3 ) z + Λ z θ Λ θ y Λ ϕ z Λ ϕ y χ y z y ( 3 ) z       + Λ z θ Λ θ z Λ ϕ x Λ ϕ x χ z x x ( 3 ) z + Λ z θ Λ θ y Λ ϕ y Λ ϕ z χ z y y ( 3 ) z + Λ z θ Λ θ z Λ ϕ z Λ ϕ z χ z z z ( 3 ) z , = 1 3 r 2 sin 2 θ χ x x x ( 3 ) x .
χ ϕ ϕ θ ( 3 ) θ = 1 3 r 2 sin 2 θ χ x x x ( 3 ) x ,
χ ϕ θ ϕ ( 3 ) θ = 1 3 r 2 sin 2 θ χ x x x ( 3 ) x ,
χ ϕ θ θ ( 3 ) ϕ = 1 3 r 2 χ x x x ( 3 ) x ,
χ θ ϕ θ ( 3 ) ϕ = 1 3 r 2 χ x x x ( 3 ) x ,
χ θ θ ϕ ( 3 ) ϕ = 1 3 r 2 χ x x x ( 3 ) x ,
χ θ θ θ ( 3 ) ϕ = χ ϕ ϕ ϕ ( 3 ) θ = 0 .
P NL n = χ k l m ( 3 ) n E k E l E m ,                           k , l , m , n = r , θ     or     ϕ .
P NL θ = χ θ θ θ ( 3 ) θ ( E θ ) 3 + [ χ ϕ ϕ θ ( 3 ) θ + χ ϕ θ ϕ ( 3 ) θ + + χ θ ϕ ϕ ( 3 ) θ ] ( E ϕ ) 2 E θ , = r 2 [ ( E θ ) 3 + 1 3 sin 2 θ ( E ϕ ) 2 E θ + 1 3 sin 2 θ ( E ϕ ) 2 E θ         + 1 3 sin 2 θ ( E ϕ ) 2 E θ ] χ x x x ( 3 ) x , = r 2 [ ( E θ ) 3 + sin 2 θ ( E ϕ ) 2 E θ ] χ x x x ( 3 ) x , = [ α 1 ( E θ ) 2 + α 2 ( E ϕ ) 2 ] E θ ,
α 1 = r 2 χ x x x ( 3 ) x ,
α 2 = r 2 sin 2 θ χ x x x ( 3 ) x ,
E θ = 1 2 [ q ( t ) f θ ( r → ) exp ( i ω t ) + c . c . ] ,
E ϕ = 1 2 [ q ( t ) f ϕ ( r → ) exp ( i ω t ) + c . c . ] .
( E θ ) 3 = 1 8 [ q f θ ( r → ) exp ( i ω t ) + q * f θ * ( r → ) exp ( − i ω t ) ] [ q f θ ( r → ) exp ( i ω t )           + q * f θ * ( r → ) exp ( − i ω t ) ] [ q     f θ ( r → ) exp ( i ω t ) + q * f θ * ( r → ) exp ( − i ω t ) ] , = 1 8 [ q 2 ( f θ ( r → ) ) 2 exp ( 2 i ω t ) + 2 | q | 2 | f θ ( r → ) | 2 + q * f θ * ( r → ) q * f θ * ( r → ) exp ( − 2 i ω t )           [ q f θ ( r → ) exp ( i ω t ) + q * f θ * ( r → ) exp ( − i ω t ) ] , = 1 8 [ q 3 ( f θ ( r → ) ) 3 exp ( 3 i ω t ) + 2 | q | 2 q | f θ ( r → ) | 2 f θ ( r → ) exp ( i ω t )       + | q | 2 | f θ ( r → ) | 2 q * f θ * ( r → ) exp ( − i ω t ) + | q | 2 | f θ ( r → ) | 2 q f θ ( r → ) exp ( i ω t )       + 2 | q | 2 | f θ ( r → ) | 2 q * f θ * ( r → ) exp ( − i ω t ) + q * f θ * ( r → ) q * f θ * ( r → ) f θ * ( r → ) q * exp ( − 3 i ω t ) .
( E θ ) 3 = | q | 2 | q 8 [ 3 | f θ ( r → ) | 2 f θ ( r → ) exp ( i ω t ) + c . c . ] .
( E ϕ ) 2 E θ = | q | 2 | q 8 { [ 2 | f ϕ ( r → ) | 2 f θ ( r → ) + ( f ϕ * ( r → ) ) 2 q     f θ * ( r → ) ] exp ( i ω t ) + c . c . } .
P NL θ = [ α 1 ( E θ ) 2 + α 2 ( E ϕ ) 2 ] E θ , = | q | 2 q χ x x x ( 3 ) x [ ( 3 8 r 2 | f θ ( r → ) | 2 f θ ( r → ) + 1 4 r 2 sin 2 θ | f ϕ ( r → ) | 2 f θ ( r → )         + 1 8 r 2 sin 2 θ ( f ϕ ( r → ) ) 2 f θ * ( r → ) ) exp ( i ω t ) + c . c . ] .
P NL ϕ = χ ϕ ϕ ϕ ( 3 ) ϕ ( E ϕ ) 3 + [ χ ϕ θ θ ( 3 ) ϕ + χ θ θ ϕ ( 3 ) ϕ + χ θ ϕ θ ( 3 ) ϕ ] ( E θ ) 2 E ϕ , = r 2 [ sin 2 θ ( E ϕ ) 3 + 1 3 ( E θ ) 2 E ϕ + 1 3 ( E θ ) 2 E ϕ + 1 3 ( E θ ) 2 E ϕ ] χ x x x ( 3 ) x , = r 2 [ sin 2 θ ( E ϕ ) 3 + ( E θ ) 2 E ϕ ] χ x x x ( 3 ) x , = [ α 2 ( E ϕ ) 2 + α 1 ( E θ ) 2 ] E ϕ ,
( E ϕ ) 3 = | q | 2 | q 8 [ 3 | f ϕ ( r → ) | 2 f ϕ ( r → ) exp ( i ω t ) + c . c . ] ,
( E θ ) 2 E ϕ = | q | 2 | q 8 [ ( 2 | f θ ( r → ) | 2 f ϕ ( r → ) exp ( i ω t ) + c . c . ) + ( ( f θ ( r → ) ) 2 f ϕ * ( r → ) exp ( i ω t ) + c . c . ) ] .
P NL ϕ = [ α 2 ( E ϕ ) 2 + α 1 ( E θ ) 2 ] E ϕ , = | q | 2 q χ x x x ( 3 ) x [ ( 3 8 r 2 sin 2 θ | f ϕ ( r → ) | 2 f ϕ ( r → ) + 1 4 r 2 | f θ ( r → ) | 2 f ϕ ( r → ) + 1 8 r 2 ( f θ ( r → ) ) 2 f ϕ * ( r → ) ) exp ( i ω t ) + c . c . ] .
g θ θ = θ → . θ → = r 2 ,
g ϕ ϕ = ϕ → . ϕ → = r 2 sin 2 θ ,
g θ ϕ = g ϕ θ = 0 .
1 c ∫ ( f → ( r → ) . f → * ( r → ) ) d v = 1 c ∫ ( | f θ ( r → ) | 2 g θ θ + | f ϕ ( r → ) | 2 g ϕ ϕ ) d v , = 1 ,
f θ ( r → ) = A     R l ( r ) im r sin θ P l m ( cos θ ) exp ( im ϕ ) ,
f ϕ ( r → ) = A     R l ( r ) r sin θ [ − ∂ ∂ θ P l m ( cos θ ) ] exp ( im ϕ ) .
| A | 2 = c ∫ [ r 2 | R l ( r ) | 2 m 2 sin θ [ p l m ( cos θ ) ] 2 + r 2 | R l ( r ) | 2 [ − ∂ ∂ θ p l m ( cos θ ) ] 2 sin θ ]       d r d θ     d ϕ .
V eff = c 2 I θ + I ϕ ,
I θ = ∫ P NL ( 1 ) θ f θ * ( r → ) g θ θ d v n 2 ( r ) ,
I ϕ = ∫ P NL ( 1 ) ϕ f ϕ * ( r → ) g ϕ ϕ d v n 2 ( r ) .
I θ = ∫ [ 3 8 r 2 | f θ ( r → ) | 4 n 2 ( r ) g θ θ + 1 4 r 2 | f ϕ ( r → ) | 2 | f θ ( r → ) | 2 n 2 ( r ) sin 2 θ g θ θ ] d v , = 3 8 | A | 4 ∫ r 2 | R l ( r ) | 4 n 2 ( r ) m 4 sin 3 θ [ p l m ( cos θ ) ] 4 d r d θ d ϕ         + 1 4 | A | 4 ∫ r 2 | R l ( r ) | 4 n 2 ( r ) m 2 sin θ [ p l m ( cos θ ) ] 2 [ − ∂ ∂ θ p l m ( cos θ ) ] 2 d r d θ d ϕ , = π 4 m 2 | A | 4 ( 3 m 2 k 1 + 2 k 2 ) ∫ r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
k 1 = ∫ − 1 1 [ p l m ( x ) ] 4 ( 1 − x 2 ) 2 d x ,
k 2 = ∫ − 1 1 [ p l m ( x ) ] 2 [ ( 1 − x 2 ) ∂ ∂ x p l m ( x ) − x p l m ( x ) ] 2 ( 1 − x 2 ) 2 d x .
I ϕ = ∫ [ 3 8 r 2 | f ϕ ( r → ) | 4 n 2 ( r ) sin 2 θ g ϕ ϕ + 1 4 r 2 | f ϕ ( r → ) | 2 | f θ ( r → ) | 2 n 2 ( r ) sin 2 θ g ϕ ϕ ] d v , = 3 8 | A | 4 ∫ r 2 | R l ( r ) | 4 n 2 ( r ) [ − ∂ ∂ θ p l m ( cos θ ) ] 4 sin θ d r d θ d ϕ         + 1 4 | A | 4 ∫ r 2 | R l ( r ) | 4 n 2 ( r ) m 2 sin θ [ p l m ( cos θ ) ] 2 [ − ∂ ∂ θ p l m ( cos θ ) ] 2 d r d θ d ϕ , = π 4 | A | 4 ( 3 k 3 + 2 m 2 k 2 ) ∫ r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
k 3 = ∫ − 1 1 [ ( 1 − x 2 ) ∂ ∂ x p l m ( x ) − x p l m ( x ) ] 4 ( 1 − x 2 ) 2 d x .
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