Abstract

A generalized formalism is developed to model second-order nonlinear processes in finite-difference time-domain (FDTD) simulations. The method is capable of modeling frequency-conversion from all 18 elements of the second-order nonlinear tensor, where dispersion of the tensor elements is included at both the pump and generated frequencies. The model is validated by considering frequency-conversion in a LiNbO3 crystal, which has highly dispersive second-order nonlinear susceptibilities near the phonon resonances. The developed nonlinear formalism is able to model any arbitrary excitation polarization state and can be applied to investigate second-order nonlinear processes in type I or type II phase-matching. This generalized second-order nonlinear formalism represents an advancement for the FDTD computational technique and can provide more realistic modeling of second-order nonlinear interactions in nanoscale devices and waveguiding structures.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The finite-difference time-domain (FDTD) technique has evolved to become one of the most powerful numerical methods in modeling electromagnetic phenomena and optical devices. In part, this is due to the simplicity of the numerical algorithm in calculating the fields’ evolution in real time. With recent advances in integrated nonlinear photonics, the FDTD method has proven to be ideal in modeling on-chip devices based on second or third order optical nonlinearities. However, such FDTD approaches have been restricted to a basic representation of the nonlinear material properties, such that intricate and complex nonlinear interactions have not been fully explored by these models. In particular, this is very important when dealing with a second-order nonlinear optical material having highly-dispersive nonlinearities and multiple non-vanishing nonlinear tensor components. Naturally, the next major advancement for FDTD modeling of second-order nonlinear effects [e.g. second harmonic generation, sum frequency generation, difference frequency generation (DFG), and optical rectification] is to incorporate realistic models for the nonlinear material.

Second-order nonlinear optical materials, especially those that are rich in optical resonances, exhibit high nonlinear dispersion across a wide spectral range. Second-order nonlinear susceptibility, χ(2), dispersion is prominent near the phonon resonance of a material [1], where crystals such as LiNbO3, LiTaO3, GaAs, GaP, InP, and CdTe exhibit enhancement of their χ(2) tensor coefficients [1–5]. Another situation where χ(2) dispersion occurs is near the band edge of a material. Here, χ(2) can vary by as much as ~100 pm/V for 4¯3m point group materials (e.g. GaAs, GaP, ZnTe, ZnSe, ZnS, InAs, and InSb) [6–8]. Likewise, χ(2) dispersion is crucial for organic molecules [9] and 2D materials [10], as their χ(2) coefficients differ by >100% across the near-infrared and visible spectral ranges, respectively. Since a high-degree of complexity arises when including all the dispersive χ(2) elements into FDTD formalisms, current FDTD methods implement a maximum of three dispersive tensor elements. Ignoring the remaining 15 dispersive χ(2) elements lead to critical restrictions, as it precludes realistic modeling of second-order nonlinear effects for numerous material classes. In 3m point group materials, frequency-conversion can occur via eight nonlinear elements (i.e. χ15(2), χ16(2), χ21(2), χ22(2), χ24(2), χ31(2), χ32(2), and χ33(2)) [11]. Even more involved are organic crystals, where 10 of the 18 nonlinear elements are non-zero for 2-methyl-4-nitroaniline [12].

Here, we present a generalized FDTD method for modeling dispersive second-order nonlinear effects. The model is based on Miller’s rule, which allows dispersion to be included for all 18 elements in the second order-nonlinear tensor. To validate the developed formalism, frequency-conversion is investigated in a LiNbO3 crystal, which has highly dispersive χ(2) elements near its phonon modes. The developed method is able to model any arbitrary excitation polarization state and can be applied to investigate nonlinear χ(2) processes in type I or type II phase-matched configurations. This generalized second-order nonlinear formalism represents a major advancement for the FDTD computation technique and is envisioned to have a strong impact in integrated optics.

2. Derivation of the second-order nonlinear current density

Light interaction with a non-centrosymmetric material can be described by the current density, J=J(1)+J(2), where J(1) and J(2) are the linear (i.e. first-order) and the second-order current densities, respectively. Although J(1) and its implementation into an FDTD formalism has been well-established [13], a generalized FDTD formalism needs to be developed for nonlinear light interactions involving J(2). When a non-centrosymmetric material is excited using optical electric fields having the frequencies of ω and Ω-ω, a nonlinear optical current density, J(2), is induced at the frequency Ω. The second-order nonlinear susceptibility tensor, utilizing contracted notation for the tensor elements, can be written as,

χ(2)¯¯(Ω,ω,Ωω)=[χ11(2)(Ω,ω,Ωω)χ12(2)(Ω,ω,Ωω)χ13(2)(Ω,ω,Ωω)χ21(2)(Ω,ω,Ωω)χ22(2)(Ω,ω,Ωω)χ23(2)(Ω,ω,Ωω)χ31(2)(Ω,ω,Ωω)χ32(2)(Ω,ω,Ωω)χ33(2)(Ω,ω,Ωω)χ14(2)(Ω,ω,Ωω)χ15(2)(Ω,ω,Ωω)χ16(2)(Ω,ω,Ωω)χ24(2)(Ω,ω,Ωω)χ25(2)(Ω,ω,Ωω)χ26(2)(Ω,ω,Ωω)χ34(2)(Ω,ω,Ωω)χ35(2)(Ω,ω,Ωω)χ36(2)(Ω,ω,Ωω)].
However, in the following derivations, it is more intuitive to implement the uncontracted tensor element representation,
χ(2)¯¯(Ω,ω,Ωω)=[χxxx(2)(Ω,ω,Ωω)χxyy(2)(Ω,ω,Ωω)χxzz(2)(Ω,ω,Ωω)χyxx(2)(Ω,ω,Ωω)χyyy(2)(Ω,ω,Ωω)χyzz(2)(Ω,ω,Ωω)χzxx(2)(Ω,ω,Ωω)χzyy(2)(Ω,ω,Ωω)χzzz(2)(Ω,ω,Ωω)χxyz(2)(Ω,ω,Ωω)χxxz(2)(Ω,ω,Ωω)χxxy(2)(Ω,ω,Ωω)χyyz(2)(Ω,ω,Ωω)χyxz(2)(Ω,ω,Ωω)χyxy(2)(Ω,ω,Ωω)χzyz(2)(Ω,ω,Ωω)χzxz(2)(Ω,ω,Ωω)χzxy(2)(Ω,ω,Ωω)],
where χhjk(2) is the second-order nonlinear susceptibility representing generation along the h coordinate due to excitations along the j and k coordinates. Notably, the subscripts h, j, and k can be either x, y or z. Each individual element in the nonlinear tensor produces a second-order nonlinear polarization according to the following equation,
Ph:j,k(2)(Ω)=ε0χhjk(2)(Ω,ω,Ωω)Ej(ω)Ek(Ωω)dω,
where ε0 is the permittivity of free space and Ej,k are the optical excitation electric fields having polarizations along the j and k coordinates, respectively. Moreover, the subscript h:j,k signifies that excitation frequencies oriented along the j and k coordinates produce a second-order nonlinear polarization along the h coordinate. To incorporate second-order nonlinear dispersive effects, the FDTD model implements χhjk(2) coefficients according to Miller’s rule [1],
χhjk(2)(Ω,ω,Ωω)=δhjkχh(Ω)χj(ω)χk(Ωω),
where δhjk is Miller’s proportionality constant and χh,j,k are the linear susceptibilities along the h, j, and k coordinates, respectively. It should be noted that Miller’s formalism inherently links the nonlinear dispersion of an optical material to the linear material dispersion, at both the excitation frequencies (i.e. ω and Ω-ω) and the generation frequency (i.e. Ω). Using the definition in Eq. (4) for χhjk(2), Eq. (3) becomes,
Ph:j,k(2)(Ω)=ε0δhjkχh(Ω)Sj(ω)Sk(Ωω)dω,
where,
Sj,k(ω)=χj,k(ω)Ej,k(ω).
The integral in Eq. (5) is the convolution operation, such that it can be recasted in a more compact form,
Ph:j,k(2)(Ω)=ε0δhjkχh(Ω){SjSk}(Ω).
By transforming Ph:j,k(2) to the time-domain, we obtain,
Ph:j,k(2)(t)=ε0δhjk{χh(SjSk)}(t),
where t is time. Using Eq. (8), the complete set of second-order nonlinear polarizations along the x, y, and z directions are [14],
Px(2)(t)=Px:x,x(2)(t)+Px:y,y(2)(t)+Px:z,z(2)(t)+2Px:y,z(2)(t)+2Px:x,z(2)(t)+2Px:x,y(2)(t),
Py(2)(t)=Py:x,x(2)(t)+Py:y,y(2)(t)+Py:z,z(2)(t)+2Py:y,z(2)(t)+2Py:x,z(2)(t)+2Py:x,y(2)(t),
Pz(2)(t)=Pz:x,x(2)(t)+Pz:y,y(2)(t)+Pz:z,z(2)(t)+2Pz:y,z(2)(t)+2Pz:x,z(2)(t)+2Pz:x,y(2)(t).
The second-order nonlinear polarizations are related to the second-order nonlinear current densities through,
Jx(2)(t)=dPx(2)(t)dt,
Jy(2)(t)=dPy(2)(t)dt,
Jz(2)(t)=dPz(2)(t)dt.
Equations (12-14) allow the full, dispersive second-order nonlinear susceptibility tensor to be implemented in the FDTD algorithm. Nonetheless, an accurate representation of the linear susceptibility is required. Since most physical processes permit the linear susceptibility (i.e. χh,j,k) to be represented by a frequency-independent term and a summation of Lorentzian oscillators, the linear susceptibilities along the x, y, and z directions are,
χx,y,z(ω)=χsx,y,z+m=1Nx,y,z(ωmx,y,z)2χmx,y,z(ωmx,y,z)2+iγmx,y,zωω2,
where χsx,y,z are the frequency-independent linear susceptibilities, χmx,y,z are the Lorentz susceptibilities of the mth Lorentzian oscillator, ωmx,y,z are the resonant frequencies of the mth Lorentzian oscillator, γmx,y,z are the damping factors of the mth Lorentzian oscillator, and Nx,y,z are the number of Lorentzian oscillators used to describe the linear susceptibilities.

3. Discretization of the second-order nonlinear current density

To discretize the Jh(2) terms in Eqs. (12-14), we must first discretize the Sj,k and Ph:j,k(2) terms in Eq. (6) and Eq. (7), respectively. Using Eq. (15), Sj,k can be written as,

Sj,k(ω)=Gsj,k(ω)+m=1Nj,kGmj,k(ω),
where,
Gsj,k(ω)=χsj,kEj,k(ω),
and
Gmj,k(ω)=(ωmj,k)2χmj,k(ωmj,k)2+iγmj,kωω2Ej,k(ω)form=1,2,...Nj,k.
To solve the term in Eq. (17), it is first transformed to the time-domain,
Gsj,k(t)=χsj,kEj,k(t).
Using temporal averaging of the Gsj,k term, Eq. (19) is then discretized for the time iteration of n to obtain,
Gsj,k(n+1)=2χsj,kEj,k(n)Gsj,k(n1),
where n is related to the time step, Δt, via the relationship t = nΔt. To solve Eq. (18), the terms are first rearranged to,
(ωmj,k)2Gmj,k(ω)+iγmj,kωGmj,k(ω)ω2Gmj,k(ω)=(ωmj,k)2χmj,kEj,k(ω)form=1,2,...Nj,k.
Next, using the fact that iωGmj,k(ω)dGmj,k(t)dt and ω2Gmj,k(ω)d2Gmj,k(t)dt2, Eq. (21) can be transformed to its time-domain form,
(ωmj,k)2Gmj.k(t)+γmj,kdGmj,k(t)dt+d2Gmj,k(t)dt2=(ωmj,k)2χmj,kEj,k(t)form=1,2,...Nj,k.
Using central-differencing techniques, this equation is discretized for the time iteration n to obtain,
Gmj,k(n+1)=2Δt2(ωmj,k)2χmj,kγmj,kΔt+2Ej,k(n)+42Δt2(ωmj,k)2γmj,kΔt+2Gmj,k(n)+γmj,kΔt2γmj,kΔt+2Gmj,k(n1)form=1,2,...Nj,k.
Now that both Gsj,k and Gmj,k have been retrieved in their discretized form, a solution for Sj,k is obtained by converting Eq. (16) to the time-domain and discretizing it for the time iteration of n + 1,
Sj,k(n+1)=Gsj,k(n+1)+m=1Nj,kGmj,k(n+1).
Importantly, Sj,k(n) must also be obtained which leads to the following equations,
Gsj,k(n)=χsj,kEj,k(n),
Gmj,k(n)=2Δt2(ωmj,k)2χmj,kγmj,kΔt+2Ej,k(n1)+42Δt2(ωmj,k)2γmj,kΔt+2Gmj,k(n1)+γmj,kΔt2γmj,kΔt+2Gmj,k(n2)form=1,2,...Nj,k,
Sj,k(n)=Gsj,k(n)+m=1Nj,kGmj,k(n).
Alternatively, Gsj,k(n) and Gmj,k(n) can instead be obtained from Eq. (20) and (23), which is achieved by using the results from the current time iteration at the next time iteration. Next, using Eq. (15), Ph:j,k(2) in Eq. (7) can be recasted as,
Ph:j,k(2)(Ω)=Kshjk(Ω)+m=1NhKmhjk(Ω),
where,
Kshjk(Ω)=ε0δhjkχsh{SjSk}(Ω),
and
Kmhjk(Ω)=ε0δhjk(ωmh)2χmh(ωmh)2+iγmhΩΩ2{SjSk}(Ω)form=1,2,...Nh.
Equation (29) is solved by first converting it to the time-domain,
Kshjk(t)=ε0δhjkχshSj(t)Sk(t),
and then discretizing it for the time iteration of n + 1 to obtain,
Kshjk(n+1)=ε0δhjkχshSj(n+1)Sk(n+1).
Clearly, this depends on the solution to Sj,k, which is presented in Eq. (24). Equation (30) is solved by first rearranged it into the form,
(ωmh)2Kmhjk(Ω)+iγmhΩKmhjk(Ω)Ω2Kmhjk(Ω)=ε0δhjk(ωmh)2χmh{SjSk}(Ω)form=1,2,...Nh.
Utilizing iΩKmhjk(Ω)dKmhjk(t)dt and Ω2Kmhjk(Ω)d2Kmhjk(t)dt2 with Eq. (33), we obtain the time-domain equation,
(ωmh)2Kmhjk(t)+γmhdKmhjk(t)dt+d2Kmhjk(t)dt2=ε0δhjk(ωmh)2χmhSj(t)Sk(t)form=1,2,...Nh.
Using central-differencing techniques, Eq. (34) is discretized for the time iteration of n,
Kmhjk(n+1)=ε0δhjk2Δt2(ωmh)2χmhγmhΔt+2Sj(n)Sk(n)+42Δt2(ωmh)2γmhΔt+2Kmhjk(n)+γmhΔt2γmhΔt+2Kmhjk(n1)form=1,2,...Nh,
where Eq. (35) depends on the Sj,k solution in Eq. (27). Now that Kshjk and Kmhjk are obtained in their discretized form, a discretized solution can be attained for Ph:j,k(2) by converting Eq. (28) to the time-domain and discretizing it for the time iteration of n + 1,
Ph:j,k(2)(n+1)=Kshjk(n+1)+m=1NhKmhjk(n+1).
The above equation is the final, discretized form for Ph:j,k(2), where Kshjk and Kmhjk are given in Eqs. (32) and (35), respectively. Finally, the discretization of Ph(2) [Eqs. (9-11)] for the time iteration of n + 1 is straightforward and results in,
Px(2)(n+1)=Px:x,x(2)(n+1)+Px:y,y(2)(n+1)+Px:z,z(2)(n+1)+2Px:y,z(2)(n+1)+2Px:x,z(2)(n+1)+2Px:x,y(2)(n+1),
Py(2)(n+1)=Py:x,x(2)(n+1)+Py:y,y(2)(n+1)+Py:z,z(2)(n+1)+2Py:y,z(2)(n+1)+2Py:x,z(2)(n+1)+2Py:x,y(2)(n+1),
Pz(2)(n+1)=Pz:x,x(2)(n+1)+Pz:y,y(2)(n+1)+Pz:z,z(2)(n+1)+2Pz:y,z(2)(n+1)+2Pz:x,z(2)(n+1)+2Pz:x,y(2)(n+1),
whereas the discretization of Jh(2) [Eqs. (12-14)] for the time iteration of n + 1/2 provides,
Jx(2)(n+1/2)=Px(2)(n+1)Px(2)(n)Δt,
Jy(2)(n+1/2)=Py(2)(n+1)Py(2)(n)Δt,
Jz(2)(n+1/2)=Pz(2)(n+1)Pz(2)(n)Δt.
To include second-order nonlinear effects in the FDTD formalism, these Jh(2) terms are incorporated in the update equations,
Exp1/2,r,s(n+1)=Exp1/2,r,s(n)Δtε0εrxJ'x(1)p1/2,r,s(n+12)Δtε0εrxJx(2)p1/2,r,s(n+12)+Δtε0εrxΔy[Hzp1/2,r+1/2,s(n+12)Hzp1/2,r1/2,s(n+12)]Δtε0εrxΔz[Hyp1/2,r,s+1/2(n+12)Hyp1/2,r,s1/2(n+12)],
Eyp,r1/2,s(n+1)=Eyp,r1/2,s(n)Δtε0εryJ'y(1)p,r1/2,s(n+12)Δtε0εryJy(2)p,r1/2,s(n+12)+Δtε0εryΔz[Hxp,r1/2,s+1/2(n+12)Hxp,r1/2,s1/2(n+12)]Δtε0εryΔx[Hzp+1/2,r1/2,s(n+12)Hzp1/2,r1/2,s(n+12)],
Ezp,r,s1/2(n+1)=Ezp,r,s1/2(n)Δtε0εrzJ'z(1)p,r,s1/2(n+12)Δtε0εrzJz(2)p,r,s1/2(n+12)+Δtε0εrzΔx[Hyp+1/2,r,s1/2(n+12)Hyp1/2,r,s1/2(n+12)]Δtε0εrzΔy[Hxp,r+1/2,s1/2(n+12)Hxp,r1/2,s1/2(n+12)],
Hxp,r1/2,s1/2(n+12)=Hxp,r1/2,s1/2(n12)+Δtμ0Δz[Eyp,r1/2,s(n)Eyp,r1/2,s1(n)]Δtμ0Δy[Ezp,r,s1/2(n)Ezp,r1,s1/2(n)],
Hyp1/2,r,s1/2(n+12)=Hyp1/2,r,s1/2(n12)+Δtμ0Δx[Ezp,r,s1/2(n)Ezp1,r,s1/2(n)]Δtμ0Δz[Exp1/2,r,s(n)Exp1/2,r,s1(n)],
Hzp1/2,r1/2,s(n+12)=Hzp1/2,r1/2,s(n12)+Δtμ0Δy[Exp1/2,r,s(n)Exp1/2,r1,s(n)]Δtμ0Δx[Eyp,r1/2,s(n)Eyp1,r1/2,s(n)],
where μ0 is the permeability free space, p, r, and s are the spatial indices of the field components along the x, y, and z-directions, respectively, Δx, Δy, and Δz are the mesh steps along the x, y, and z-directions, respectively, Hx, Hy, and Hz are the magnetic field components along the x, y, and z-directions, respectively, εrx, εry, and εrz are the relative permittivities along the x, y, and z-directions, respectively, and J'x(1), J'y(1), and J'z(1) are the dispersive parts of the linear components of the current densities along the x, y, and z-directions, respectively.

4. Frequency-conversion in a LiNbO3 crystal

The derived FDTD method is evaluated by simulating the representative effects of optical rectification and DFG. These nonlinear effects are investigated using a LiNbO3 crystal layer having a length l, which is illustrated in Fig. 1. LiNbO3 serves as a prime material to evaluate the generalized second-order nonlinear method, since it has strong nonlinear dispersion in the terahertz (THz) frequency regime, contains a nonlinear tensor with numerous non-vanishing elements, and has sufficient nonlinear experimental data available. The LiNbO3 layer is excited using an electric field pulse having a central-wavelength of 1550 nm, a duration of 80 fs, and a polarization angle of θ with respect to the c-axis of the crystal (see Fig. 1). Here, it is assumed that an index-matched layer and free space are positioned at the input and output faces of the LiNbO3 layer, respectively. The c-axis of the crystal is oriented along the z-axis, the [100] crystal direction is aligned with the x-axis, and the y-axis is defined as the direction of propagation. Since a bulk crystal is being simulated, periodic boundary conditions are implemented for the boundaries normal to the x and z-directions, whereas perfectly matched layers are used for boundaries normal to the propagation direction (i.e. y-direction). A mesh size of 40 nm is used, which is sufficiently small for both the excitation and generated frequency components.

 

Fig. 1 An illustration of the LiNbO3 crystal having a length l and an excitation electric field polarization at the angle θ with respect to the crystal c-axis.

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It is critical to determine the frequency dependence of the LiNbO3 refractive indices and extinction coefficients, along with its second-order nonlinear susceptibilities. The extraordinary, ne, and ordinary, no, refractive indices are shown in Fig. 2(a), 2(c), and 2(e) for the frequency ranges of interest. Furthermore, the extraordinary, кe, and ordinary, кo, extinction coefficients are shown in Fig. 2(b) and 2(d), respectively. The experimental data is obtained from references [15,16]. and the curve fits are achieved by using a superposition of Lorentzian oscillators. Clearly, the experimental data matches the fitted curves very well. ne and кe show a phonon resonance at 7.6 THz, corresponding to the lowest-frequency A1 mode of LiNbO3, whereas three E mode phonon resonances are observed in no and кo, which are located at 4.6, 7.9, and 9.7 THz [15]. Since nonlinear dispersion is directly related to linear dispersion via Miller’s rule, the χ(2) elements of the LiNbO3 crystal cannot be taken as constants. The LiNbO3 crystal belongs to the 3m point group symmetry class, such that its second-order nonlinear susceptibly tensor is written as [11],

χ(2)¯¯(Ω,ω,Ωω)=[000χ22(2)(Ω,ω,Ωω)χ22(2)(Ω,ω,Ωω)0χ31(2)(Ω,ω,Ωω)χ31(2)(Ω,ω,Ωω)χ33(2)(Ω,ω,Ωω)0χ15(2)(Ω,ω,Ωω)χ22(2)(Ω,ω,Ωω)χ15(2)(Ω,ω,Ωω)00000],
where χ15(2)(Ω,ω,Ωω), χ22(2)(Ω,ω,Ωω), χ31(2)(Ω,ω,Ωω)c, and  χ33(2)(Ω,ω,Ωω) are the non-vanishing coefficients. Notably, since the THz region exhibits significant dispersion, the Kleinman symmetry condition is invalid and the χ15(2)(Ω,ω,Ωω) element differs from the χ31(2)(Ω,ω,Ωω) element. From Eq. (3), (9), (11), (49), and the fact that propagation is along the y-direction, it can be determined that the second-order nonlinear polarizations occurring along the x and z directions are,
Px(2)(Ω)=2ε0χ15(2)(Ω,ω,Ωω)Ex(ω)Ez(Ωω)dω,
and,
Pz(2)=ε0χ31(2)(Ω,ω,Ωω)Ex(ω)Ex(Ωω)dω+ε0χ33(2)(Ω,ω,Ωω)Ez(ω)Ez(Ωω)dω,
such that it is necessary to consider dispersion for the χ15(2)(Ω,ω,Ωω), χ31(2)(Ω,ω,Ωω), and χ33(2)(Ω,ω,Ωω) tensor elements. Figure 3(a) illustrates the highly-dispersive χ33(2)(Ω,ω,Ωω)  element, which is calculated using Miller’s rule [Eq. (4)], a proportionality constant of δ = 1.3 pm/V, and the experimental data in [17,18]. Similarly, the χ31(2)(Ω,ω,Ωω) element [see Fig. 3(b)] is obtained by using a proportionality constant of δ = 0.21 pm/V and the experimental data in [19], whereas χ15(2)(Ω,ω,Ωω) [see Fig. 3(c)] is described by δ = 0.71 pm/V and the experimental data from [19]. Importantly, in comparison to the low-frequency nonlinear susceptibility values, all three of the nonlinear elements exhibit an enhancement of >7 times at their lowest-frequency phonon resonance.

 

Fig. 2 (a) Refractive index and (b) extinction coefficient for the extraordinary LiNbO3 crystal direction in the THz frequency regime. (c) Refractive index and (d) extinction coefficient for the ordinary LiNbO3 crystal direction in the THz frequency regime. (e) Extraordinary and ordinary refractive indices in the optical frequency regime. The experimental data is obtained from [15,16] and the LiNbO3 crystal is taken to be lossless in the optical regime.

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Fig. 3 Magnitude and phase of the second-order nonlinear susceptibility elements of LiNbO3 for (a) χ33(2)(Ω,ω,Ωω), (b) χ31(2)(Ω,ω,Ωω), and (c) χ15(2)(Ω,ω,Ωω) as a function of frequency. The curve fits are calculated using Miller’s rule with the experimental data from [17–19].

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To illustrate the implication of a dispersive second-order nonlinear susceptibility on the nonlinear frequency-conversion process, simulations are performed using l = 150 µm and θ = 0°, such that the 1550 nm excitation pulse is polarized along the c-axis. Here, the only contributing nonlinear element is χ33(2)(Ω,ω,Ωω). The simulations are conducted using the fully-dispersive χ33(2)(Ω,ω,Ωω) element [see Fig. 3(a)], as well as the frequency-independent χ33(2) = 348 pm/V. Figure 4(a) depicts the spectral power generated along the z-direction, ξz, where the phase-mismatching effects appear as dips (e.g. f = 0.8 THz, 1.5 THz, etc.) in the power spectrum, in agreement with those predicted by the theoretical formula,

fd=cql|ngoptnTHz(fd)|,
where c is the speed of light, q is a positive integer, ngopt is the extraordinary group refractive index at the wavelength of 1550 nm, and nTHz is the extraordinary phase refractive index in the THz regime. When comparing ξz obtained using the dispersive χ33(2)(Ω,ω,Ωω) and the frequency-independent χ33(2) = 348 pm/V, significant disagreement is evident, especially at frequencies near the phonon resonance of LiNbO3 at 7.6 THz. Figure 4(b) shows the relative spectral power, which is defined as ξz obtained using the dispersive χ33(2)(Ω,ω,Ωω) divided by ξz obtained using the frequency independent χ33(2) = 348 pm/V. While the discrepancy in the relative spectral power is moderate at frequencies <6 THz, it is more than 60 times higher at frequencies near 7.6 THz. Therefore, a dispersive nonlinear element is critical to accurately model frequency-conversion near the phonon resonance of a crystal.

 

Fig. 4 (a) The z-component of the spectral power obtained using the dispersive χ33(2)(Ω,ω,Ωω) and a frequency-independent χ33(2) = 348 pm/V. (b) Relative spectral power calculated when implementing the dispersive χ33(2)(Ω,ω,Ωω) and the frequency-independent χ33(2) = 348 pm/V.

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The nonlinear algorithm can be used to model dispersive type II phase-matching for l = 50-150 µm and θ = 45°. Herein, the THz radiation generation occurs along the x-direction through the process of DFG. In DFG, phase-matching can occur between the pump frequency, fp, the signal frequency, fs, and either the forward, fi+, or backward, fi, propagating idler frequencies. The forward propagation DFG coherence length, Lc+, is,

Lc+=c2|npfpnsfsnifi+|1,
and the backward propagation DFG coherence length, Lc, is,
Lc=c2|npfpnsfs+nifi|1,
where np, ns, and ni are the pump, signal, and idler refractive indices, respectively, and fi=|fpfs|. Since the excitation angle is at θ = 45°, the electric fields at fp, fs, and fi+ (or fi) are polarized along the ordinary, extraordinary, and ordinary crystal directions, respectively. When considering THz radiation generation in the forward direction [i.e. Equation (53)], phase-matching occurs at fi+ = 3 THz, as shown in Fig. 5(a). This agrees with the FDTD model, where the x-component of the spectral power, ξx, exhibits THz radiation generation at fi+ = 3 THz, as seen in Fig. 5(b). Similarly, Lc from Eq. (54) is displayed in Fig. 5(c), which shows that phase-matching occurs at the idler frequency of fi = 1.8 THz. By performing FDTD simulations and recording ξx near the input facet of the LiNbO3 layer, we confirm that fi = 1.8 THz [Fig. 5(d)]. Clearly, the versatility of this nonlinear FDTD approach arises from its ability in modeling excitation electric fields at arbitrary polarization states, along with the dispersive second-order nonlinear susceptibilities.

 

Fig. 5 (a) Coherence length for THz radiation generation in the forward direction. (b) The x-component of the spectral power recorded in free space. (c) Coherence length for THz radiation generation in the backward direction. (d) The x-component of the spectral power recorded near the input face of the LiNbO3 crystal.

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Next, by using l = 150 µm and θ = 0°-90°, we examine the two simultaneous nonlinear frequency-conversion processes of DFG and optical rectification in the THz regime. These two processes depend on the polarization state of the excitation electric field, and the generated THz radiation is polarized along different crystal directions. Here, the broadband THz radiation generated via optical rectification is polarized along the z-axis, whereas the narrowband THz radiation generated via DFG is polarized along the x-axis. Figure 6(a) depicts ξz, where the broadband THz radiation generation is highest when the excitation pulse is oriented along the c-axis (i.e. θ = 0°), since Pz(2) is maximum and the only contribution is from the χ33(2)(Ω,ω,Ωω) term [see Eq. (51)]. THz radiation generation decreases as θ increases, since the contribution from χ33(2)(Ω,ω,Ωω) is reduced; nonetheless, the weaker χ31(2)(Ω,ω,Ωω) coefficient now contributes to Pz(2). At θ = 90°, THz radiation generation is lowest, since Pz(2) is only influenced by χ31(2)(Ω,ω,Ωω). Figure 6(b) depicts ξx, where the THz radiation generation vanishes at both θ = 0° and 90°, since Px(2) = 0 [see Eq. (50)]. The power spectra are similar in magnitude at the intermediate angels of θ = 22.5° and 67.5°, whereas THz radiation generation is highest when θ = 45°. The presented generalized FDTD method offers more accurate modeling of dispersive nonlinear frequency-conversion processes that strongly depend on the excitation polarization state and the generated electric field polarization.

 

Fig. 6 (a) The z-component of the spectral power and (b) the x-component of the spectral power, at various electric field polarization angles.

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5. Summary

A generalized nonlinear FDTD formalism is developed that considers all 18 elements of the second-order nonlinear tensor and incorporates dispersion of the nonlinear elements. The versatility of the method was demonstrated by modeling optical rectification and DFG in a LiNbO3 crystal, where the tensor elements are highly dispersive at the phonon resonances and the enhancement of the nonlinear interaction is more pronounced. When implementing the generalized nonlinear formalism, THz radiation generation near the phonon resonance is found to be 60 times higher than that obtained using frequency-independent nonlinear methods. Furthermore, the nonlinear formalism permits frequency-conversion for an arbitrary electric field polarization state. Although the implantation was focused on nonlinear frequency-conversion in the LiNbO3 crystal, the nonlinear algorithm can be applied to model more complex structures, such as nano-dimensional waveguides that support a longitudinal electric field, plasmonic structures that enhance frequency-conversion [20], or the generation of radiation in bulk materials that are excited by a focused laser beam.

Funding

Natural Sciences and Engineering Research Council of Canada.

Acknowledgments

This work was performed under NSERC’s Engage grant program in collaboration with Optiwave Systems Inc. The authors are grateful for the support provided by Optiwave Systems Inc., especially Scott Newman, Ahmad Atieh, and Kevin Chu, and for the access to Optiwave's OptiFDTD simulation software platform.

References

1. D. H. Auston and M. C. Nuss, “Electrooptical generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]  

2. Y. J. Ding, “Efficient generation of high-frequency terahertz waves from highly lossy second-order nonlinear medium at polariton resonance under transverse-pumping geometry,” Opt. Lett. 35(2), 262–264 (2010). [CrossRef]   [PubMed]  

3. M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010). [CrossRef]  

4. A. Mayer and F. Keilmann, “Far-infrared nonlinear optics. I. χ (2) near ionic resonance,” Phys. Rev. B Condens. Matter 33(10), 6954–6961 (1986). [CrossRef]   [PubMed]  

5. D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974). [CrossRef]  

6. J. Seres, “Dispersion of second-order nonlinear optical coefficient,” Appl. Phys. B 73(7), 705–709 (2001). [CrossRef]  

7. H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998). [CrossRef]  

8. M. I. Bell, “Frequency Dependence of Miller’s Rule for Nonlinear Susceptibilities,” Phys. Rev. B 6(2), 516–521 (1972). [CrossRef]  

9. R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988). [CrossRef]  

10. M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017). [CrossRef]  

11. G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964). [CrossRef]  

12. R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000). [CrossRef]  

13. A. Taflove, Computational electrodynamics: the finite-difference time-domain method (Artech House Publishers, 1995).

14. M. C. Gupta and J. Ballato, The Handbook of Photonics (CRC, 2007).

15. S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016). [CrossRef]  

16. A. S. Barker Jr. and R. Loudon, “Dielectric Properties and Optical Phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967). [CrossRef]  

17. G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973). [CrossRef]  

18. G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971). [CrossRef]  

19. N. Courjal, M.-P. Bernal, A. Caspar, G. Ulliac, F. Bassignot, L. Gauthier-Manuel, and M. Suarez, “Lithium niobate optical waveguides and microwaveguides,” Emerging Waveguide Technology (IntechOpen, 2018).

20. A. Cala’ Lesina, L. Ramunno, and P. Berini, “Dual-polarization plasmonic metasurface for nonlinear optics,” Opt. Lett. 40(12), 2874–2877 (2015). [CrossRef]   [PubMed]  

References

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  1. D. H. Auston and M. C. Nuss, “Electrooptical generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988).
    [Crossref]
  2. Y. J. Ding, “Efficient generation of high-frequency terahertz waves from highly lossy second-order nonlinear medium at polariton resonance under transverse-pumping geometry,” Opt. Lett. 35(2), 262–264 (2010).
    [Crossref] [PubMed]
  3. M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
    [Crossref]
  4. A. Mayer and F. Keilmann, “Far-infrared nonlinear optics. I. χ (2) near ionic resonance,” Phys. Rev. B Condens. Matter 33(10), 6954–6961 (1986).
    [Crossref] [PubMed]
  5. D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
    [Crossref]
  6. J. Seres, “Dispersion of second-order nonlinear optical coefficient,” Appl. Phys. B 73(7), 705–709 (2001).
    [Crossref]
  7. H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
    [Crossref]
  8. M. I. Bell, “Frequency Dependence of Miller’s Rule for Nonlinear Susceptibilities,” Phys. Rev. B 6(2), 516–521 (1972).
    [Crossref]
  9. R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
    [Crossref]
  10. M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
    [Crossref]
  11. G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
    [Crossref]
  12. R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
    [Crossref]
  13. A. Taflove, Computational electrodynamics: the finite-difference time-domain method (Artech House Publishers, 1995).
  14. M. C. Gupta and J. Ballato, The Handbook of Photonics (CRC, 2007).
  15. S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
    [Crossref]
  16. A. S. Barker and R. Loudon, “Dielectric Properties and Optical Phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
    [Crossref]
  17. G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
    [Crossref]
  18. G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
    [Crossref]
  19. N. Courjal, M.-P. Bernal, A. Caspar, G. Ulliac, F. Bassignot, L. Gauthier-Manuel, and M. Suarez, “Lithium niobate optical waveguides and microwaveguides,” Emerging Waveguide Technology (IntechOpen, 2018).
  20. A. Cala’ Lesina, L. Ramunno, and P. Berini, “Dual-polarization plasmonic metasurface for nonlinear optics,” Opt. Lett. 40(12), 2874–2877 (2015).
    [Crossref] [PubMed]

2017 (1)

M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
[Crossref]

2016 (1)

S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
[Crossref]

2015 (1)

2010 (2)

Y. J. Ding, “Efficient generation of high-frequency terahertz waves from highly lossy second-order nonlinear medium at polariton resonance under transverse-pumping geometry,” Opt. Lett. 35(2), 262–264 (2010).
[Crossref] [PubMed]

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

2001 (1)

J. Seres, “Dispersion of second-order nonlinear optical coefficient,” Appl. Phys. B 73(7), 705–709 (2001).
[Crossref]

2000 (1)

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

1998 (1)

H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
[Crossref]

1988 (2)

D. H. Auston and M. C. Nuss, “Electrooptical generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988).
[Crossref]

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

1986 (1)

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics. I. χ (2) near ionic resonance,” Phys. Rev. B Condens. Matter 33(10), 6954–6961 (1986).
[Crossref] [PubMed]

1974 (1)

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[Crossref]

1973 (1)

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

1972 (1)

M. I. Bell, “Frequency Dependence of Miller’s Rule for Nonlinear Susceptibilities,” Phys. Rev. B 6(2), 516–521 (1972).
[Crossref]

1971 (1)

G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
[Crossref]

1967 (1)

A. S. Barker and R. Loudon, “Dielectric Properties and Optical Phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

1964 (1)

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

Auston, D. H.

D. H. Auston and M. C. Nuss, “Electrooptical generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988).
[Crossref]

Barker, A. S.

A. S. Barker and R. Loudon, “Dielectric Properties and Optical Phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Bell, M. I.

M. I. Bell, “Frequency Dependence of Miller’s Rule for Nonlinear Susceptibilities,” Phys. Rev. B 6(2), 516–521 (1972).
[Crossref]

Berini, P.

Bivona, S.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Bond, W. L.

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

Boyd, G. D.

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
[Crossref]

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

Bridges, T. J.

G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
[Crossref]

Busacca, A. C.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Cala’ Lesina, A.

Card, A.

M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
[Crossref]

Cherchi, M.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Cino, A. C.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Coleman, P. D.

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[Crossref]

Damman, P.

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

Ding, Y. J.

Dosiére, M.

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

Ganikhanov, F.

M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
[Crossref]

Hoshina, T.

S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
[Crossref]

Hvam, J. M.

H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
[Crossref]

Ito, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Kaneda, Y.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Kanehara, K.

S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
[Crossref]

Keilmann, F.

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics. I. χ (2) near ionic resonance,” Phys. Rev. B Condens. Matter 33(10), 6954–6961 (1986).
[Crossref] [PubMed]

Kojima, S.

S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
[Crossref]

Kondo, T.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Kühnelt, M.

H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
[Crossref]

Langbein, W.

H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
[Crossref]

Leone, C.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Loudon, R.

A. S. Barker and R. Loudon, “Dielectric Properties and Optical Phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Mayer, A.

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics. I. χ (2) near ionic resonance,” Phys. Rev. B Condens. Matter 33(10), 6954–6961 (1986).
[Crossref] [PubMed]

Miller, R. C.

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

Mokim, M.

M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
[Crossref]

Morita, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Nassau, K.

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

Nuss, M. C.

D. H. Auston and M. C. Nuss, “Electrooptical generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988).
[Crossref]

Ogasawara, N.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Oliveri, R. L.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Pollack, M. A.

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
[Crossref]

Ramunno, L.

Sah, B.

M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
[Crossref]

Sanseverino, S. R.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Savage, A.

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

Seres, J.

J. Seres, “Dispersion of second-order nonlinear optical coefficient,” Appl. Phys. B 73(7), 705–709 (2001).
[Crossref]

Stivala, S.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Sugihashi, A.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Taormina, A.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

Thompson, D. E.

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[Crossref]

Toussaere, E.

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

Tsurumi, T.

S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
[Crossref]

Turner, E. H.

G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
[Crossref]

Umegaki, S.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Vallée, R.

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

Wagner, H. P.

H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
[Crossref]

Zyss, J.

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

AIP Adv. (1)

M. Mokim, A. Card, B. Sah, and F. Ganikhanov, “Dispersion of the resonant second order nonlinearity in 2D semiconductors probed by femtosecond continuum pulses,” AIP Adv. 7(10), 105121 (2017).
[Crossref]

Appl. Phys. B (1)

J. Seres, “Dispersion of second-order nonlinear optical coefficient,” Appl. Phys. B 73(7), 705–709 (2001).
[Crossref]

Appl. Phys. Lett. (1)

G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, and A. Savage, “LiNbO3: An efficient phase matchable nonlinear optical material,” Appl. Phys. Lett. 5(11), 234–236 (1964).
[Crossref]

IEEE J. Quantum Electron. (2)

D. H. Auston and M. C. Nuss, “Electrooptical generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988).
[Crossref]

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron. 46(3), 368–376 (2010).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[Crossref]

J. Am. Chem. Soc. (1)

R. Vallée, P. Damman, M. Dosiére, E. Toussaere, and J. Zyss, “Nonlinear optical properties and crystalline orientation of 2-Methyl-4-nitroaniline layers grown on nanostructured poly(tetrafluoroethylene) substrates,” J. Am. Chem. Soc. 122(28), 6701–6709 (2000).
[Crossref]

Jpn. J. Appl. Phys. (2)

S. Kojima, K. Kanehara, T. Hoshina, and T. Tsurumi, “Optical phonons and polariton dispersions of congruent LiNbO3 studied by far-infrared spectroscopic ellipsometry and Raman scattering,” Jpn. J. Appl. Phys. 55(10S), 10TC02 (2016).
[Crossref]

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-Methyl-4-Nitroaniline (MNA),” Jpn. J. Appl. Phys. 27(6), L1131–L1133 (1988).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

A. S. Barker and R. Loudon, “Dielectric Properties and Optical Phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967).
[Crossref]

Phys. Rev. B (2)

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

M. I. Bell, “Frequency Dependence of Miller’s Rule for Nonlinear Susceptibilities,” Phys. Rev. B 6(2), 516–521 (1972).
[Crossref]

Phys. Rev. B Condens. Matter (1)

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics. I. χ (2) near ionic resonance,” Phys. Rev. B Condens. Matter 33(10), 6954–6961 (1986).
[Crossref] [PubMed]

Phys. Rev. B Condens. Matter Mater. Phys. (1)

H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B Condens. Matter Mater. Phys. 58(16), 10494–10501 (1998).
[Crossref]

Phys. Rev. Lett. (1)

G. D. Boyd, T. J. Bridges, M. A. Pollack, and E. H. Turner, “Microwave nonlinear susceptibilities due to electronic and ionic anharmonicities in acentric crystals,” Phys. Rev. Lett. 26(7), 387–390 (1971).
[Crossref]

Other (3)

N. Courjal, M.-P. Bernal, A. Caspar, G. Ulliac, F. Bassignot, L. Gauthier-Manuel, and M. Suarez, “Lithium niobate optical waveguides and microwaveguides,” Emerging Waveguide Technology (IntechOpen, 2018).

A. Taflove, Computational electrodynamics: the finite-difference time-domain method (Artech House Publishers, 1995).

M. C. Gupta and J. Ballato, The Handbook of Photonics (CRC, 2007).

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Figures (6)

Fig. 1
Fig. 1 An illustration of the LiNbO3 crystal having a length l and an excitation electric field polarization at the angle θ with respect to the crystal c-axis.
Fig. 2
Fig. 2 (a) Refractive index and (b) extinction coefficient for the extraordinary LiNbO3 crystal direction in the THz frequency regime. (c) Refractive index and (d) extinction coefficient for the ordinary LiNbO3 crystal direction in the THz frequency regime. (e) Extraordinary and ordinary refractive indices in the optical frequency regime. The experimental data is obtained from [15,16] and the LiNbO3 crystal is taken to be lossless in the optical regime.
Fig. 3
Fig. 3 Magnitude and phase of the second-order nonlinear susceptibility elements of LiNbO3 for (a) χ 33 (2) ( Ω,ω,Ωω ), (b) χ 31 (2) ( Ω,ω,Ωω ), and (c) χ 15 (2) ( Ω,ω,Ωω ) as a function of frequency. The curve fits are calculated using Miller’s rule with the experimental data from [17–19].
Fig. 4
Fig. 4 (a) The z-component of the spectral power obtained using the dispersive χ 33 (2) ( Ω,ω,Ωω ) and a frequency-independent χ 33 (2) = 348 pm/V. (b) Relative spectral power calculated when implementing the dispersive χ 33 (2) ( Ω,ω,Ωω ) and the frequency-independent χ 33 (2) = 348 pm/V.
Fig. 5
Fig. 5 (a) Coherence length for THz radiation generation in the forward direction. (b) The x-component of the spectral power recorded in free space. (c) Coherence length for THz radiation generation in the backward direction. (d) The x-component of the spectral power recorded near the input face of the LiNbO3 crystal.
Fig. 6
Fig. 6 (a) The z-component of the spectral power and (b) the x-component of the spectral power, at various electric field polarization angles.

Equations (54)

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χ (2) ¯ ¯ (Ω,ω,Ωω)=[ χ 11 (2) (Ω,ω,Ωω) χ 12 (2) (Ω,ω,Ωω) χ 13 (2) (Ω,ω,Ωω) χ 21 (2) (Ω,ω,Ωω) χ 22 (2) (Ω,ω,Ωω) χ 23 (2) (Ω,ω,Ωω) χ 31 (2) (Ω,ω,Ωω) χ 32 (2) (Ω,ω,Ωω) χ 33 (2) (Ω,ω,Ωω) χ 14 (2) (Ω,ω,Ωω) χ 15 (2) (Ω,ω,Ωω) χ 16 (2) (Ω,ω,Ωω) χ 24 (2) (Ω,ω,Ωω) χ 25 (2) (Ω,ω,Ωω) χ 26 (2) (Ω,ω,Ωω) χ 34 (2) (Ω,ω,Ωω) χ 35 (2) (Ω,ω,Ωω) χ 36 (2) (Ω,ω,Ωω) ].
χ (2) ¯ ¯ (Ω,ω,Ωω)=[ χ xxx (2) (Ω,ω,Ωω) χ xyy (2) (Ω,ω,Ωω) χ xzz (2) (Ω,ω,Ωω) χ yxx (2) (Ω,ω,Ωω) χ yyy (2) (Ω,ω,Ωω) χ yzz (2) (Ω,ω,Ωω) χ zxx (2) (Ω,ω,Ωω) χ zyy (2) (Ω,ω,Ωω) χ zzz (2) (Ω,ω,Ωω) χ xyz (2) (Ω,ω,Ωω) χ xxz (2) (Ω,ω,Ωω) χ xxy (2) (Ω,ω,Ωω) χ yyz (2) (Ω,ω,Ωω) χ yxz (2) (Ω,ω,Ωω) χ yxy (2) (Ω,ω,Ωω) χ zyz (2) (Ω,ω,Ωω) χ zxz (2) (Ω,ω,Ωω) χ zxy (2) (Ω,ω,Ωω) ],
P h:j,k (2) (Ω)= ε 0 χ hjk (2) (Ω,ω,Ωω) E j (ω) E k (Ωω)dω ,
χ hjk (2) (Ω,ω,Ωω)= δ hjk χ h (Ω) χ j (ω) χ k (Ωω),
P h:j,k (2) (Ω)= ε 0 δ hjk χ h (Ω) S j (ω) S k (Ωω)dω ,
S j,k (ω)= χ j,k (ω) E j,k (ω).
P h:j,k (2) (Ω)= ε 0 δ hjk χ h (Ω){ S j S k } (Ω).
P h:j,k (2) (t)= ε 0 δ hjk { χ h ( S j S k ) }(t),
P x (2) (t)= P x:x,x (2) (t)+ P x:y,y (2) (t)+ P x:z,z (2) (t)+2 P x:y,z (2) (t)+2 P x:x,z (2) (t)+2 P x:x,y (2) (t),
P y (2) (t)= P y:x,x (2) (t)+ P y:y,y (2) (t)+ P y:z,z (2) (t)+2 P y:y,z (2) (t)+2 P y:x,z (2) (t)+2 P y:x,y (2) (t),
P z (2) (t)= P z:x,x (2) (t)+ P z:y,y (2) (t)+ P z:z,z (2) (t)+2 P z:y,z (2) (t)+2 P z:x,z (2) (t)+2 P z:x,y (2) (t).
J x (2) ( t )= d P x (2) ( t ) dt ,
J y (2) ( t )= d P y (2) ( t ) dt ,
J z (2) ( t )= d P z (2) ( t ) dt .
χ x,y,z (ω)= χ s x,y,z + m=1 N x,y,z ( ω m x,y,z ) 2 χ m x,y,z ( ω m x,y,z ) 2 +i γ m x,y,z ω ω 2 ,
S j,k (ω)= G s j,k (ω)+ m=1 N j,k G m j,k (ω),
G s j,k (ω)= χ s j,k E j,k (ω),
G m j,k (ω)= ( ω m j,k ) 2 χ m j,k ( ω m j,k ) 2 +i γ m j,k ω ω 2 E j,k (ω)form=1,2,... N j,k .
G s j,k (t)= χ s j,k E j,k (t).
G s j,k (n+1)=2 χ s j,k E j,k (n) G s j,k (n1),
( ω m j,k ) 2 G m j,k (ω)+i γ m j,k ω G m j,k (ω) ω 2 G m j,k (ω) = ( ω m j,k ) 2 χ m j,k E j,k (ω)form=1,2,... N j,k .
( ω m j,k ) 2 G m j.k (t)+ γ m j,k d G m j,k (t) dt + d 2 G m j,k (t) d t 2 = ( ω m j,k ) 2 χ m j,k E j,k (t)form=1,2,... N j,k .
G m j,k (n+1)= 2Δ t 2 ( ω m j,k ) 2 χ m j,k γ m j,k Δt+2 E j,k (n)+ 42Δ t 2 ( ω m j,k ) 2 γ m j,k Δt+2 G m j,k (n) + γ m j,k Δt2 γ m j,k Δt+2 G m j,k (n1)form=1,2,... N j,k .
S j,k (n+1)= G s j,k (n+1)+ m=1 N j,k G m j,k (n+1).
G s j,k (n)= χ s j,k E j,k (n),
G m j,k (n)= 2Δ t 2 ( ω m j,k ) 2 χ m j,k γ m j,k Δt+2 E j,k (n1)+ 42Δ t 2 ( ω m j,k ) 2 γ m j,k Δt+2 G m j,k (n1) + γ m j,k Δt2 γ m j,k Δt+2 G m j,k (n2)form=1,2,... N j,k ,
S j,k (n)= G s j,k (n)+ m=1 N j,k G m j,k (n).
P h:j,k (2) (Ω)= K s hjk (Ω)+ m=1 N h K m hjk (Ω) ,
K s hjk (Ω)= ε 0 δ hjk χ s h { S j S k } (Ω),
K m hjk (Ω)= ε 0 δ hjk ( ω m h ) 2 χ m h ( ω m h ) 2 +i γ m h Ω Ω 2 { S j S k } (Ω)form=1,2,... N h .
K s hjk (t)= ε 0 δ hjk χ s h S j (t) S k (t),
K s hjk (n+1)= ε 0 δ hjk χ s h S j (n+1) S k (n+1).
( ω m h ) 2 K m hjk (Ω)+i γ m h Ω K m hjk (Ω) Ω 2 K m hjk (Ω) = ε 0 δ hjk ( ω m h ) 2 χ m h { S j S k } (Ω)form=1,2,... N h .
( ω m h ) 2 K m hjk (t)+ γ m h d K m hjk (t) dt + d 2 K m hjk (t) d t 2 = ε 0 δ hjk ( ω m h ) 2 χ m h S j (t) S k (t)form=1,2,... N h .
K m hjk (n+1)= ε 0 δ hjk 2Δ t 2 ( ω m h ) 2 χ m h γ m h Δt+2 S j (n) S k (n)+ 42Δ t 2 ( ω m h ) 2 γ m h Δt+2 K m hjk (n) + γ m h Δt2 γ m h Δt+2 K m hjk (n1)form=1,2,... N h ,
P h:j,k (2) (n+1)= K s hjk (n+1)+ m=1 N h K m hjk (n+1) .
P x (2) (n+1)= P x:x,x (2) (n+1)+ P x:y,y (2) (n+1)+ P x:z,z (2) (n+1) +2 P x:y,z (2) (n+1)+2 P x:x,z (2) (n+1)+2 P x:x,y (2) (n+1),
P y (2) (n+1)= P y:x,x (2) (n+1)+ P y:y,y (2) (n+1)+ P y:z,z (2) (n+1) +2 P y:y,z (2) (n+1)+2 P y:x,z (2) (n+1)+2 P y:x,y (2) (n+1),
P z (2) (n+1)= P z:x,x (2) (n+1)+ P z:y,y (2) (n+1)+ P z:z,z (2) (n+1) +2 P z:y,z (2) (n+1)+2 P z:x,z (2) (n+1)+2 P z:x,y (2) (n+1),
J x (2) ( n+1/2 )= P x (2) ( n+1 ) P x (2) ( n ) Δt ,
J y (2) ( n+1/2 )= P y (2) ( n+1 ) P y (2) ( n ) Δt ,
J z (2) ( n+1/2 )= P z (2) ( n+1 ) P z (2) ( n ) Δt .
E x p1/2,r,s ( n+1 )= E x p1/2,r,s ( n ) Δt ε 0 ε rx J ' x (1)p1/2,r,s ( n+ 1 2 ) Δt ε 0 ε rx J x (2)p1/2,r,s ( n+ 1 2 ) + Δt ε 0 ε rx Δy [ H z p1/2,r+1/2,s ( n+ 1 2 ) H z p1/2,r1/2,s ( n+ 1 2 ) ] Δt ε 0 ε rx Δz [ H y p1/2,r,s+1/2 ( n+ 1 2 ) H y p1/2,r,s1/2 ( n+ 1 2 ) ],
E y p,r1/2,s ( n+1 )= E y p,r1/2,s ( n ) Δt ε 0 ε ry J ' y (1)p,r1/2,s ( n+ 1 2 ) Δt ε 0 ε ry J y (2)p,r1/2,s ( n+ 1 2 ) + Δt ε 0 ε ry Δz [ H x p,r1/2,s+1/2 ( n+ 1 2 ) H x p,r1/2,s1/2 ( n+ 1 2 ) ] Δt ε 0 ε ry Δx [ H z p+1/2,r1/2,s ( n+ 1 2 ) H z p1/2,r1/2,s ( n+ 1 2 ) ],
E z p,r,s1/2 ( n+1 )= E z p,r,s1/2 ( n ) Δt ε 0 ε rz J ' z (1)p,r,s1/2 ( n+ 1 2 ) Δt ε 0 ε rz J z (2)p,r,s1/2 ( n+ 1 2 ) + Δt ε 0 ε rz Δx [ H y p+1/2,r,s1/2 ( n+ 1 2 ) H y p1/2,r,s1/2 ( n+ 1 2 ) ] Δt ε 0 ε rz Δy [ H x p,r+1/2,s1/2 ( n+ 1 2 ) H x p,r1/2,s1/2 ( n+ 1 2 ) ],
H x p,r1/2,s1/2 ( n+ 1 2 )= H x p,r1/2,s1/2 ( n 1 2 ) + Δt μ 0 Δz [ E y p,r1/2,s ( n ) E y p,r1/2,s1 ( n ) ] Δt μ 0 Δy [ E z p,r,s1/2 ( n ) E z p,r1,s1/2 ( n ) ],
H y p1/2,r,s1/2 ( n+ 1 2 )= H y p1/2,r,s1/2 ( n 1 2 ) + Δt μ 0 Δx [ E z p,r,s1/2 ( n ) E z p1,r,s1/2 ( n ) ] Δt μ 0 Δz [ E x p1/2,r,s ( n ) E x p1/2,r,s1 ( n ) ],
H z p1/2,r1/2,s ( n+ 1 2 )= H z p1/2,r1/2,s ( n 1 2 ) + Δt μ 0 Δy [ E x p1/2,r,s ( n ) E x p1/2,r1,s ( n ) ] Δt μ 0 Δx [ E y p,r1/2,s ( n ) E y p1,r1/2,s ( n ) ],
χ (2) ¯ ¯ (Ω,ω,Ωω)=[ 0 0 0 χ 22 (2) (Ω,ω,Ωω) χ 22 (2) (Ω,ω,Ωω) 0 χ 31 (2) (Ω,ω,Ωω) χ 31 (2) (Ω,ω,Ωω) χ 33 (2) (Ω,ω,Ωω) 0 χ 15 (2) (Ω,ω,Ωω) χ 22 (2) (Ω,ω,Ωω) χ 15 (2) (Ω,ω,Ωω) 0 0 0 0 0 ],
P x (2) (Ω)=2 ε 0 χ 15 (2) (Ω,ω,Ωω) E x (ω) E z (Ωω)dω ,
P z (2) = ε 0 χ 31 (2) (Ω,ω,Ωω) E x (ω) E x (Ωω)dω + ε 0 χ 33 (2) (Ω,ω,Ωω) E z (ω) E z (Ωω)dω ,
f d = cq l| n g opt n THz ( f d ) | ,
L c + = c 2 | n p f p n s f s n i f i + | 1 ,
L c = c 2 | n p f p n s f s + n i f i | 1 ,

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