Abstract

Second harmonic generation from the two-layer structure where a transition-metal dichalcogenide monolayer is put on a one-dimensional grating has been studied. This grating supports bound states in the continuum which have no leakage lying within the continuum of radiation modes, we can enhance the second harmonic generation from the transition-metal dichalcogenide monolayer by more than four orders of magnitude based on the critical field enhancement near the bound states in the continuum. In order to complete this calculation, the scattering matrix theory has been extended to include the nonlinear effect and the scattering matrix of a two-dimensional material including nonlinear terms; furthermore, two methods to observe the bound states in the continuum are considered, where one is tuning the thickness of the grating and the other is changing the incident angle of the electromagnetic wave. We have also discussed various modulation of the second harmonic generation enhancement by adjusting the azimuthal angle of the transition-metal dichalcogenide monolayer.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, the optical properties of transition-metal dichalcogenide (TMDC) monolayers have been the subject of studies because of the recent advances in the applications and physical understanding of two-dimensional materials learned from graphene [1–4]. As a centrosymmetric material, graphene exhibits large third-order nonlinear effects such as third-harmonic generation [5], strong optical Kerr nonlinearity [6], however the even-order nonlinear processes of graphene are inhibited. As a complementary the TMDC monolayers have noncentrosymmetric atomic lattice and hence allow even-order nonlinear optical processes [7–10]. These properties allow one to employ TMDC monolayers in active photonic devices with improved functionality, the strength of interaction between TMDC monolayers and electromagnetic (EM) waves plays a central role in the application. In general, such an interaction is very weak due to the single atom thickness of TMDC monolayers. The weak interactions block efficient generation of nonlinear effects and potential applications of TMDC monolayers in optoelectronic devices. Thus, how to improve the interaction between TMDC monolayers and EM waves and realize the efficient generation of nonlinear effects has become an important research topic.

Various methods to improve the interaction between TMDC monolayers and EM waves have been proposed. For example, the third-harmonic generation may be dramatically enhanced in graphene by inserting it in a properly designed one-dimensional photonic crystals (PCs) [11–13], using ultrastrong localized surface plasmon resonances [14–16] or combing with waveguide [17]. In our work, we attempt to explore whether or not this interaction can be realized by utilizing light bound states in the continuum (BIC) which are embedded trapped modes corresponding to discrete eigenvalues coexisting with extended modes of a continuous spectrum [18–20]. It is found that many structures support BIC such as the dielectric grating [18], waveguide structures [19, 21–23], surface of the object [20, 24], photonic crystal slabs [25, 26], they may be implemented more easily than the electron system in artificially designing the potential. The effect of the bound states in the continuum on the second harmonic (SH) generation from TMDC monolayer has not been investigated before, in our work we attempt to explore whether or not the large enhancement can be realized utilizing the field confinement.

To study this problem, a two layer structure where a TMDC monolayer is put on a grating is proposed, BICs which are realized in this structure can be view as a mode with an infinite quality factor Q which has no radiation leakage, however, the Fano resonances close to these BICs support radiation modes with ultrahigh Q factors [27, 28]. We find that the SH generation from the TMDC monolayer can be enhanced dramatically by using the Fano resonance near BICs. In order to complete this study, the scattering matrix method to study the layered structured has been developed to include the nonlinear effect. The physical origin for this phenomenon has also been analyzed. The rest of the paper is organized as follows: we will describe the theory and method in Sec. II, present the results and discussions in Sec. III, and give a summary in Sec. IV.

2. Theory and method

We consider a two layer structure where one TMDC monolayer is deposited on the one dimensional (1D) grating as shown in Fig. 1. Now suppose the grating is periodic along the x direction and the lattice constant is a. The width, thickness and relative permittivity of the grating are represented by w, d and ε, respectively, the grating is immersed in the air. The relative permittivity and effective thickness of the TMDC monolayer are denoted by εt and dt, meanwhile, the EM properties of two dimensional material can alternatively be described by the conductivity σ, which is related to εt and dt through the following relation: σ=iε0ωdtχt, where ε0 represents the permittivity of the vacuum and χt=εt1 is the linear polarizability, the angular frequency of EM wave is denoted by ω. All these material are nonmagnetic. The EM can be well confined in the grating due to the formation of a bound states in a continuum (BICs) of radiation modes, near BICs the EM can be dramatically enhanced at the TMDC monolayer location, utilizing this property, we study the SH generation from TMDC monolayer near BICs.

 figure: Fig. 1

Fig. 1 Diagram of the TMDC monolayer-grating structure and the SH generation process (oblique view shown in (a) and side view presented in (b)). The lattice constant, width, thickness and relative permittivity of the grating are marked by a, w, d and ε, respectively, the grating is immersed in the air. When a plane wave with angular frequency ω is incident on the structure at the incident angle θi, the transmitted and reflected SH fields are generated due to the second-order nonlinear effect of TMDC monolayer.

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The relative permittivity of TMDC monolayers which is equal to a superposition of N Lorentzian functions is given by εTMDC=1+k=1Nfk/(ωk2ω2iωγk) [29, 30], where fk,ωk and γk represent the oscillator strength, resonance frequency and spectral width of the kth oscillator, respectively, the values of these model parameters are provided in [30]. In contrast to graphene, TMDC monolayers are noncentrosymmetric and therefore the second order nonlinear effect is permitted. Based on the symmetry properties of their space group D3h, it can be shown that the structure of their quadratically susceptibility tensor χ(2) yields only one independent and nonvanishing component [7,8]: χ(2)=χyyy(2)=χyxx(2)=χxyx(2)=χxxy(2), where we set the zigzag and armchair directions of the monolayer as the x and y directions. Numerical values for dt of the TMDC monolayers are also taken from [30]. In a similar manner to the linear property, the second-order optical property can be alternatively described by the corresponding nonlinear surface current J(2)(2ω)=idt2ωP(2)(2ω), where 2ω and P(2)(2ω) are the angular frequency of the SH EM wave and second-order polarization density. After giving the optical parameters of the two layer structure, in the following we describe detailedly the calculation of the EM fields at fundamental and SH frequencies in this structure via the surface current description of the TMDC monolayer.

When a plane wave E0+(r,ω) at fundamental frequency is incident upon the two layer structure at the incident angle θi as shown in Fig. 1(a), the transmitted and reflected SH fields (E2+(r,2ω) and E0(r,2ω)) are generated due to the second-order nonlinear effect of TMDC monolayer. Here we assume that each layer is surrounded by an infinitely thin film of air in both hand sides and the number of the subscript of the electric field represents the corresponding air region as shown in Fig. 1(b). In our work, the EM waves Ek(x,y,ω) (k=0,1,2) at infinitely thin films of air are expanded by plane waves due to the periodicity of this structure

Ek(x,y,ω)=mn(Ekmn+(ω)+Ekmn(ω))ei(kmnx(ω)x+kmny(ω)y),
here the wave vector is (kmnx(ω),kmny(ω))=(k0x(ω),k0y(ω))+(Gmnx,Gmny) and (Gmnx,Gmny)=mb1+nb2, b1 and b2 represent the reciprocal lattice vector, k0x(ω) and k0y(ω) represent the components of the wave vector in the first Brillouin zone, the superscript + (-) denotes a wave propagating to the downside (upside). In a similar way, the corresponding EM waves at SH frequency at infinitely thin films of air are given by

Ek(x,y,2ω)=mn(Ekmn+(2ω)+Ekmn(2ω))ei(kmnx(2ω)x+kmny(2ω)y).

The EM field at fundamental frequency can be calculated by the scattering matrix theory, of which the details of the calculation process are described in [31, 32]. In the first place, we calculate every scattering matrix of these two layers at fundamental frequency [33]. elaborates the method to get the scattering matrix of the graphene via its conductivity, it is very easy to extend this method to include the TMDC monolayer, the scattering matrix of the photonic crystal slab can be calculated using plane wave expansion method in [34]. In the next place, the total scattering matrix of the two layer structure can be obtained by combing every scattering matrix of each layer. At last, the transmitted and reflected fields are calculated out via total scattering matrix, then the fields at every infinite thin film are obtained through every scattering matrix of each layer.

In the following, we describe the calculation of the SH electric field in great detail. The previous scattering matrix method is used to calculate the linear optical property, in our work, this method is extended enormously to include the nonlinear effect. Based on the calculated fundamental electric field at the TMDC monolayer location and the quadratically susceptibility tensor χ(2), the nonlinear surface current J(2)(2ω) is obtained and can be also expanded in plane waves

J(2)(r,2ω)=idt2ωε0χ(2):E1(r,ω)E1(r,ω).
Substitute the plane wave expansion at fundamental frequency Eq. (1) into Eq. (3) and the nonlinear surface current can be expressed as
J(2)(r,2ω)=idt2ωε0χ(2):mn(pqE1pq(ω)E1mp,nq(ω))ei(kmnx(2ω)x+kmny(2ω)y).
The scattering matrix of the TMDC monolayer which is calculated in the appendix can be written as in the matrix form
(E1+(2ω)E0(2ω))=(Q1I(2ω)Q1II(2ω)Q1III(2ω)Q1IV(2ω))(E0+(2ω)E1(2ω))+(C1I(2ω)C1II(2ω)),
where the electric fields up and down the TMDC monolayer are written as in the matrix form E0±(2ω)=(,E0mnx±(2ω),E0mny±(2ω),)T and E1±(2ω)=(,E1mnx±(2ω),E1mny±(2ω),)T, the superscript T represents matrix transposition, the subscript of the scattering matrix stands for the layer number. In the same way, the electric field down the grating can be also expressed in the matrix form E2±(2ω)=(,E2mnx±(2ω),E2mny±(2ω),)T. The columns C1I(2ω) and C1II(2ω) represent the nonlinear terms. The relation between the electric fields up and down the grating
(E2+(2ω)E1(2ω))=(Q2I(2ω)Q2II(2ω)Q2III(2ω)Q2IV(2ω))(E1+(2ω)E2(2ω))+(C2I(2ω)C2II(2ω)),
here the columns C2I(2ω) and C2II(2ω) appear due to the second-order effect of the grating, there are already a lot of good work about SH generation in 1D periodic structures [35–37], however, in our work the material of the grating we choose is only linear because we only want to study the SH generation from TMDC monolayer, that is to say C2I(2ω)=0 and C2II(2ω)=0. In order to achieve a complete theory, these nonlinear terms are not ignored theoretically.

Based on the scattering matrixes Q1η(2ω) (η=I,II,III,IV), C1κ(2ω) (κ=I,II) of the TMDC monolayer and Q2η(2ω) (η=I,II,III,IV), C2κ(2ω) (κ=I,II) of the grating, the total scattering matrixes Q1,2η(2ω) (η=I,II,III,IV), C1,2κ(2ω) (κ=I,II) of the two layer structure can be obtained

(E2+(2ω)E0(2ω))=(Q1,2I(2ω)Q1,2II(2ω)Q1,2III(2ω)Q1,2IV(2ω))(E0+(2ω)E2(2ω))+(C1,2I(2ω)C1,2II(2ω)),
here E0+(2ω)=0 and E2(2ω)=0 because the SH fields are radiated from the two layer structure. The nonlinear terms C1,2I(2ω) and C1,2II(2ω) are calculated by us for the first time
C1,2I(2ω)=Q2I(2ω)[1Q1II(2ω)Q2III(2ω)]1[Q1II(2ω)C2II(2ω)+C1I(2ω)]+C2I(2ω),
C1,2II(2ω)=Q1IV(2ω)[1Q2III(2ω)Q1II(2ω)]1[Q2III(2ω)C1I(2ω)+C2II(2ω)]+C1II(2ω).
The method to calculate the matrixes Q1,2I(2ω), Q1,2II(2ω), Q1,2III(2ω) and Q1,2IV(2ω) is same as the corresponding linear case which is detailed in the [31].

Utilizing the total scattering matrix shown in Eq. (7), we can obtain the generated SH electric fields. Then the electric field intensities of the transmitted (It(2ω)) and reflected (Ir(2ω)) SH fields can be expressed as

It(2ω)=12ε0mn,i[E3mni+(2ω)][E3mni+(2ω)]*,
Ir(2ω)=12ε0mn,i[E0mni(2ω)][E0mni(2ω)]*,
here and in the following i=x,y,z. The electric field intensity of the incident field at fundamental frequency can be also calculated based on the known incident field
Ii(ω)=12ε0mn,i[E0mni+(ω)][E0mni+(ω)]*.
In the calculating the sums of Eq. (9) waves with imaginary z components of wave vectors (decaying waves) are dropped, as they do not transfer energy along the z direction.

In order to compare the SH generation from the two layer structure with those from the freestanding TMDC monolayer, the enhancements of the transmitted and reflected SH field (ET and ER) are introduced as

ET=It(2ω)I0t(2ω),ER=Ir(2ω)I0r(2ω)
where I0t(2ω) and I0r(2ω) represent the electric field intensities of the transmitted and reflected SH fields generated from the freestanding TMDC monolayer. Based on the above relations, the enhancements for SH fields can be obtained easily by numerical calculations.

3. Results and discussions

In this section, we present numerical results for the SH generation from the two layer structure. The relative permittivity of the grating is taken as ε=15.21, which corresponds to Si0.7Ge0.3 [38, 39]. As a typical material of TMDC monolayers, we take WS2 monolayer as the object to give the numerical results, similar phenomena can recur for the other TMDC monolayers. The effective thickness of WS2 monolayer is dt=0.618nm and its relative permittivity can also be obtained [30]. The second-order nonlinear susceptibility of the monolayer WS2 is chosen to be χ(2)=100pm/V which is compatible with the results in [7] and the second-order effect of the grating can be neglected.

We plot the transmission and absorption as a function of the wavelength λ of the EM wave and the thickness d of the grating under normal incident EM wave in Figs. 2(a) and 2(b). The lattice constant and width of the grating are selected as a=460nm and w=0.6a, the EM wave is incident normally and polarized along the y direction. It is obviously seen that the transmission and absorption spectrums exhibit many sharp resonance features with high Q factor, there are many embedded states of which the linewidth and absorption become zero, these are called BICs [40] indicated by the arrows and square in Figs. 2(a) and 2(b). These BICs are symmetry-protected bound states at zero in-plane wave vector, at these BICs the reason why leakage radiation to the surface normal direction and absorption are forbidden is that the symmetry of the EM field in the grating is incompatible with the external radiation. In order to give a further analysis, the square absorption region shown in Fig. 2(b) are magnified in Fig. 2(c) and three absorption spectrums indicated by the dashed lines in Fig. 2(c) for three different thicknesses d=312.8nm, d=321.71434nm, and d=331.2nm are plotted in Figs. 2(d), 2(e) and 2(f), respectively.

 figure: Fig. 2

Fig. 2 Transmission (a) and absorption (b) for the two layer structure as a function of the wavelength λ of the EM wave and thickness d of the grating. (c) Magnified absorption map for the square region in (b). (d)-(f) Corresponding absorption spectrums for different thicknesses of the grating d=312.8nm (d), d=321.71434nm (e) and d=331.2nm (f). The parameters are assumed as follows: a=460nm and w=0.6a.

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Exact BICs with no radiation leakage do not interact with the continuum states, hence the absorption at BICs are zero. A BIC is formed in Fig. 2(c) that can be seen clearly by the asymptotic behavior of the absorption spectrums shown in Figs. 2(d)-2(f), at d=321.71434nm the absorption shown in Fig. 2(e) become nearly zero and the BIC appear, near this BIC sharp absorption resonances at d=312.8nm and d=331.2nm are shown in Figs. 2(d) and 2(f). We actually measure the BICs from the experiment in an asymptotic method that the transmission, reflection or reflection of the leaky modes asymptotically approach a singularity in the corresponding spectrums. As shown in Fig. 2(c) when the configuration approach the BIC, the absorption increases gradually and then decreases to nearly zero at the BIC, meanwhile, the linewidth of the absorption narrows gradually and becomes nearly zero at the BIC.

We are interested in the slow light effect near BICs which are very benefit for the design of optical devices. A critical point is the maximal electric field enhancement at the surface of the grating which is related to absorption in our work, this field enhancement is defined as (|E/Ei|dS/dS), where the relative field E are integrated over the period surface of the grating and Ei represents the incident field. Figures 3(a) and 3(b) illustrate the maximal absorption and maximal field enhancement as a function of the thickness d of the grating, respectively. One maximal point is gotten by selecting the maximum from the absorption or field enhancement spectrum at different wavelength for a known thickness of the grating. It can be found that the maximal absorption and maximal field strength exhibit similar phenomena, in order to disclose this, we plot the position, the wavelength λ of the EM wave and thickness d of the grating, of the maximal absorption and maximal field enhancement in Fig. 3(c) and find that these two positions are coincide with each other, that is to say the increase or decrease of the absorption is originate from the electric field enhancement at the WS2 monolayer location. The valley points in Figs. 3(a) and 3(b) correspond to the BIC which exhibits lowest absorption and field enhancement, as the configuration approach to the BIC the maximal absorption and maximal field enhancement at the resonance increases gradually and then decrease to minimal value at the BIC, two peaks at d=317.75972nm and d=325.87182nm are formed near the BIC for each of these two figures, the corresponding peaks are 0.437 and 0.574 in Fig. 3(a) and are 18.966 and 20.615 in Fig. 3(b). We plot the absorption spectrum at the second peak in Fig. 3(d) and find that a very sharp resonance appears, the sharpness can be described by the relative line width Δλ/λm, where λm is the center wavelength of the peak and Δλ represents full width at half maximum of the peak, in this figure the relative linewidth is 0.0167% which is much less than that in other works [41]. The topic we are interested in is the effect of the critical field enhancement on the SH generation from the two dimensional material WS2 monolayer.

 figure: Fig. 3

Fig. 3 (a) Maximal absorption near BIC as a function of the thickness of the grating. (b) Maximal field enhancement near BIC as a function of the thickness d of the grating. (c) The position spectrum of the maximal absorption (black line with black symbol) and maximal field enhancement (red line with red symbol) indicated by the wavelength λ of the EM wave and the thickness d of the grating. (d) Absorption spectrum at the maximum point in (a) d=325.87182nm. The other parameters are identical to those in Fig. 2.

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In order to study the effect of the resonant states near the BIC on the SH generation from the WS2 monolayer, we plot in Fig. 4 the SH generation enhancement ET of the transmitted field and ER of the reflected field from the two layer structure compared with that from the freestanding WS2 monolayer for different thicknesses of the grating. It is clearly seen that the resonant enhancements ET and ER appear and are redshifted that is in agreement with the resonant features in Figs. 2(c)-2(f) and 3(d). The enhancements ET and ER simultaneously reach 6.000 and 3.3326 at λ=1015.076nm and d=321.71434nm which is corresponding to the BIC, due to no radiation leakage and interaction with the continuum states at the BIC, the enhancements ET and ER are so small that these can be neglected. At λ=1017.821nm and d=325.87182nm which is corresponding to the maximum of maximal field enhancement in Fig. 3 ET and ER can reach as much as 19880.9 and 14912.7 shown in Fig. 4(c). For comparison we also provide the ET and ER spectrum for ordinary Fano resonances, specially, ET and ER reach 2637.22 and 887.492 at λ=1008.362nm and d=312.8nm in Fig. 4(a), 7458.30 and 9188.38 at λ=1021.120nm and d=331.2nm in Fig. 4(d). The relative linewidths of the ET spectrums are 0.04166%, 0.01101% and 0.02939% for Figs. 4(a), 4(c) and 4(d), respectively, hence the resonant peak is narrowed when the configuration approaches to the BIC.

 figure: Fig. 4

Fig. 4 SH generation enhancement ET of the transmitted field (a)-(d) and ER of the reflected field (e)-(h) as a function of the wavelength λ of the incident field for different thicknesses of the grating. The thicknesses of the grating are taken as follows: d=312.8nm for (a) and (e), d=321.71434nm for (b) and (f), d=325.87182nm for (c) and (g), and d=331.2nm for (d) and (h). The other parameters are identical to those in Fig. 2.

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In addition to the resonance, the SH generation enhancement ET and ER can also be tuned by the azimuthal angle ϕ of the WS2 monolayer. This is because its anisotropic second-order nonlinear susceptibility depends on ϕ, if the WS2 monolayer is rotated ϕ clockwise, the corresponding second-order nonlinear polarization can be expressed as

Px=ε02χ(2)sinϕcosϕExEx+ε02χ(2)(sin2ϕcos2ϕ)ExEy+ε02χ(2)sinϕcosϕEyEy,Py=ε0χ(2)(sin2ϕcos2ϕ)ExEx+ε0χ(2)4sinϕcosϕExEy+ε0χ(2)(cos2ϕsin2ϕ)EyEy,
where Ex and Ey are electric plane wave on the WS2 monolayer. A periodic modulation of the SH generation enhancement with period 90° can be observed clearly, ET reaches maximum 19886.7 and simultaneously ER reaches minimum 14917.1 at ϕ=0°, ET reaches minimum 4865.38 and simultaneously ER reaches maximum 45313.4 at ϕ=45°. This periodic modulation can be understood easily from the physical rotation and utilizing this property we can tune the SH generation in a large range by changing the azimuthal angle ϕ of the WS2 monolayer (see Fig. 5).

 figure: Fig. 5

Fig. 5 SH generation enhancement ET of the transmitted field (solid line) and ER of the reflected field (dashed line) as a function of the azimuthal angle of the WS2 monolayer. The wavelength of the EM wave and thicknesses of the grating are taken as λ=1017.821nm and d=325.87182nm corresponding to the maximum point in Fig. 4(c).

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So far the BICs are observed via tuning the thickness of the grating, in fact the BICs can be observed by changing the incident angle θi of the EM wave. Figure 6(a) shows the absorption as a function of the wavelength λ and incident angle θi of the EM wave. The incident wave is TE (transverse electric) polarized along the y direction. Furthermore, the corresponding maximal field enhancement and maximal absorption as a function of the incident angle θi are shown in Figs. 6(b) and 6(c), respectively, it can be observed clearly that there are three BICs where the values of absorption are nearly zero and there are also peaks of the maximal field enhancement and maximal absorption around these three BICs that is similar to the previous study. In Fig. 6(d), we plot the absorption spectrum at θi=16.1723° which corresponds to the maximum point in Fig. 6(b) and find a very sharp resonant peak with relative linewidth 0.0184%.

 figure: Fig. 6

Fig. 6 (a) Absorption for two layer structure as a function of the wavelength λ and incident angle θi of the EM wave. Maximal field enhancement (b) and maximal absorption (c) near BIC as a function of the incident angle θi of the EM wave. (d) Absorption spectrum at the maximum point in (b) θi=16.1723°. The parameters are assumed as follows: a=460nm, w=0.6a and d=1.0a.

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Figures 7(a) and 7(b) display the SH generation enhancement ET of the transmitted field and ER of the reflected field from the two layer structure compared with the freestanding WS2 monolayer at θi=16.1723° which corresponds to the peak of maximal field enhancement in Fig. 6(b). Like the case in Fig. 4, sharp resonant peaks appear for the transmitted and reflected fields, ET and ER reach 353.5 and 1893.1 at λ=1030.075nm, respectively, and the relative linewidth of the ET peak is 0.01651%.

 figure: Fig. 7

Fig. 7 SH generation enhancement ET of the transmitted field (a) and ER of the reflected field (b) as a function of the wavelength λ of the incident field at θi=16.1723°. The other parameters are identical to those in Fig. 6.

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4. Conclusion

We have designed TMDC monolayer-grating structure to realize the large SH generation enhancement based on the BICs and proposed the extended scattering matrix theory including the nonlinear effect of TMDC monolayer. Two methods to observe the optical BICs in this structure are considered, one is tuning the thickness of the grating, the SH generation can be enhanced by more than four orders of magnitude, the other is changing the incident angle of the EM wave, the enhancement can reach three orders of magnitude. The enhancements in the two cases also exhibit outstanding monochromaticity due to the small linewidth about 0.01% and can be tuned in a large range by adjusting the azimuthal angle of the TMDC monolayer. We believe these phenomena are very beneficial for the design of optical devices.

Appendix Scattering matrix of TMDC monolayer at second harmonic frequency

1 Transmission and reflection of TMDC monolayer including second-order nonlinear effect

In order to obtain the transmission and reflection of the TMDC monolayer including second-order nonlinear effect [42], we assume an incident plane wave Ei at second harmonic angular frequency ωSH=2ω is incident from the x-z plane shown in Fig. 8, the transmitted and reflected plane waves (Et and Er) are generated. Due to the ultrathin thickness the TMDC monolayer is regarded as an interface, the electric and magnetic fields at the interface can be related to the following boundary conditions:

{ExtExiExr=0,HxtHxiHxr=σEy+Jy(2),EytEyiEyr=0,HytHyiHyr=σExJx(2),
here Hi, Ht and Hr represent the incident, transmitted and reflected magnetic field, Jx(2) and Jy(2) denote the nonlinear surface current originating from the second-order nonlinear effect of the TMDC monolayer.

 figure: Fig. 8

Fig. 8 Transmission and reflection of a plane wave from a TMDC monolayer at the SH frequency. The incident, transmitted and reflected waves are denoted by Ei, Et and Er, the mediums up and down the interface are marked by the number 1 and 2.

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The magnetic field can be expressed by the electric field with the Maxwell equations, then after careful calculation the relation between the electric fields up and down the interface is obtained

{Ext=TxExi+Ctx,Eyt=TyEyi+Cty,Exr=RxExi+Crx,Eyr=RyEyi+Cry.
The transmission and reflection coefficients Tx, Ty, Rx and Ry are given by
{Tx=2q2z/ε2q1z/ε1+q2z/ε2+q1zq2zσ/(ωε0ε1ε2),Ty=2q1zq1z+q2z+ωμ0σ,Rx=q2z/ε2q1z/ε1q1zq2zσ/(ωε0ε1ε2)q1z/ε1+q2z/ε2+q1zq2zσ/(ωε0ε1ε2),Ry=q1zq2zωμ0σq1z+q2z+ωμ0σ,
and the nonlinear terms Ctx, Cty, Crx and Cry are written as
{Ctx=Jx(2)q1zq2z/(ωε0ε1ε2)q1z/ε1+q2z/ε2+q1zq2zσ/(ωε0ε1ε2),Cty=Jy(2)ωμ0q1z+q2z+ωμ0σ,Crx=Jx(2)q1zq2z/(ωε0ε1ε2)q1z/ε1+q2z/ε2+q1zq2zσ/(ωε0ε1ε2),Cry=Jy(2)ωμ0q1z+q2z+ωμ0σ,
here q1z (q2z) is the z component of wave vector in the upper medium 1 (down medium 2), ε0 and μ0 are the permittivity and permeability of the vacuum respectively, ε1 and ε2 represent the relative permittivity of the upper medium 1 (down medium 2). Due to the consideration of the second-order nonlinear effect of TMDC monolayer, four terms (Ctx, Cty, Crx and Cry) emerge in the equation. This method can be extended to calculate the other nonlinear effect of other two dimensional materials.

This method in the x-z incident plane is required to extended to the arbitrary plane, we can construct a new axis x', y' and z', the azimuthal angle between the x'-z' plane and the x-z plane is represented by ϕ, then the electric fields in the new axis are related by

(Ex'tEy't)=(cos2ϕTx+sin2ϕTysinϕcosϕ(TxTy)sinϕcosϕ(TxTy)sin2ϕTx+cos2ϕTy)(Ex'iEy'i)+(cosϕCtxsinϕCtysinϕCtxcosϕCty),
where (Ex't,Ey't)T and (Ex'i,Ey'i)T represent the transmitted and incident electric fields in the new axis, a similar relation for the reflected electric fields can be obtained easily. Thus the transmission and reflection of the TMDC monolayer including second-order nonlinear effect in the arbitrary axis are calculated.

2 Scattering matrix of TMDC monolayer including second-order nonlinear effect

In the following we calculate the scattering matrix of the TMDC monolayer at SH frequency including second-order nonlinear effect based on the transmission and reflection derived in the former part. In Fig. 9 two electric fields E1+ and E2 are incident on the TMDC monolayer and two electric fields E2+ and E1 are generated.

 figure: Fig. 9

Fig. 9 Scattering of two waves from a TMDC monolayer. These two waves indicated by E1+ and E2 are incident from the up and down mediums, two waves E2+ and E1 are generated.

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The electric fields in the up and down interface are expressed as E1x(y)=mn[E1mnx(y)+eiβmnz+E1mnx(y)eiβmnz]eikmnxx+ikmnyy and E2x(y)=mn[E2mnx(y)+eiβmnz+E2mnx(y)eiβmnz]eikmnxx+ikmnyy, here (kmnx,kmny) is the Bragg wave vector, the superscript + (-) of the electric field denotes that the wave vector along z axis is positive (negative). These electric fields are related by the scattering matrixes

(E2+E1)=(T(1,2)R(2,1)R(1,2)T(2,1))(E1+E2)+(C1C2),
where E1±=(E1mnx±,E1mny±)T and E2±=(E2mnx±,E2mny±)T are amplitude columns, C1 and C2 represent the columns resulting from the second-order nonlinear effect of TMDC monolayer. T(1,2) or R(1,2) (T(2,1) or R(2,1)) represent the matrixes relating the transmitted or reflected electric field with the incident EM wave from medium 1 (2). Based on the transmission and reflection derived in former part, the elements of these scattering matrixes are given by
Tmn,pq(i,j)=δmn,pq(cos2ϕmnTmnx(i,j)+sin2ϕmnTmny(i,j)sinϕmncosϕmn(Tmnx(i,j)Tmny(i,j))sinϕmncosϕmn(Tmnx(i,j)Tmny(i,j))sin2ϕmnTmnx(i,j)+cos2ϕmnTmny(i,j)),
Rmn,pq(i,j)=δmn,pq(cos2ϕmnRmnx(i,j)+sin2ϕmnRmny(i,j)sinϕmncosϕmn(Rmnx(i,j)Rmny(i,j))sinϕmncosϕmn(Rmnx(i,j)Rmny(i,j))sin2ϕmnRmnx(i,j)+cos2ϕmnRmny(i,j)),
C1mn=(cosϕmnCmntx(1,2)sinϕmnCmnty(1,2)+cosϕmnCmnrx(2,1)sinϕmnCmnry(2,1)sinϕmnCmntx(1,2)+cosϕmnCmnty(1,2)+sinϕmnCmnrx(2,1)+cosϕmnCmnry(2,1)),
C2mn=(cosϕmnCmnrx(1,2)sinϕmnCmnry(1,2)+cosϕmnCmntx(2,1)sinϕmnCmnty(2,1)sinϕmnCmnrx(1,2)+cosϕmnCmnry(1,2)+sinϕmnCmntx(2,1)+cosϕmnCmnty(2,1)),
here the matrixes Tmn,pq(i,j), Rmn,pq(i,j) are the transmission and reflection matrixes derived in the former part when the wave vector is taken as (kmnx,kmny,βmn) and (i,j) represents (1,2) or (2,1), C1mn and C2mn are the corresponding nonlinear terms.

Funding

National Natural Science Foundation of China (NSFC) (Grants No.11574400 and No. 11204379).

References and links

1. M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013). [CrossRef]   [PubMed]  

2. A. H. Castro Neto, “Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides,” Phys. Rev. Lett. 86(19), 4382–4385 (2001). [CrossRef]   [PubMed]  

3. F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]  

4. S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013). [CrossRef]   [PubMed]  

5. S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood Jr., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013). [CrossRef]  

6. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012). [CrossRef]   [PubMed]  

7. C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014). [CrossRef]   [PubMed]  

8. L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013). [CrossRef]  

9. Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013). [CrossRef]   [PubMed]  

10. K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015). [CrossRef]   [PubMed]  

11. M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014). [CrossRef]  

12. S. Zhang and X. Zhang, “Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 33(3), 452–460 (2016). [CrossRef]  

13. Y. Zeng, Y. Fu, X. Chen, W. Lu, and H. Ågren, “Highly efficient generation of entangled photon pair by spontaneous parametric down conversion in defective photonic crystal,” J. Opt. Soc. Am. B 24(6), 1365–1368 (2007). [CrossRef]  

14. J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33(4), 615–621 (2016). [CrossRef]  

15. A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013). [CrossRef]  

16. G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017). [CrossRef]  

17. H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017). [CrossRef]  

18. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108(7), 070401 (2012). [CrossRef]   [PubMed]  

19. J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012). [CrossRef]   [PubMed]  

20. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). [CrossRef]   [PubMed]  

21. C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013). [CrossRef]  

22. F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014). [CrossRef]  

23. Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014). [CrossRef]   [PubMed]  

24. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014). [CrossRef]   [PubMed]  

25. E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008). [CrossRef]  

26. Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011). [CrossRef]   [PubMed]  

27. J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015). [CrossRef]   [PubMed]  

28. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010). [CrossRef]  

29. Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014). [CrossRef]  

30. M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third- harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016). [CrossRef]  

31. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998). [CrossRef]  

32. T. Wang and X. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photon. Res. 5(6), 629–639 (2017). [CrossRef]  

33. M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015). [CrossRef]   [PubMed]  

34. Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003). [CrossRef]   [PubMed]  

35. C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18(3), 348–351 (2001). [CrossRef]  

36. Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002). [CrossRef]   [PubMed]  

37. M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007). [CrossRef]   [PubMed]  

38. E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).

39. J. Humlícek, A. Röseler, T. Zettler, M. G. Kekoua, and E. V. Khoutsishvili, “Infrared refractive index of germanium-silicon alloy crystals,” Appl. Opt. 31(1), 90–94 (1992). [CrossRef]   [PubMed]  

40. C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016). [CrossRef]  

41. A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017). [CrossRef]  

42. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4(4), 481–489 (1987). [CrossRef]  

References

  • View by:

  1. M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
    [Crossref] [PubMed]
  2. A. H. Castro Neto, “Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides,” Phys. Rev. Lett. 86(19), 4382–4385 (2001).
    [Crossref] [PubMed]
  3. F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
    [Crossref]
  4. S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
    [Crossref] [PubMed]
  5. S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
    [Crossref]
  6. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
    [Crossref] [PubMed]
  7. C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
    [Crossref] [PubMed]
  8. L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
    [Crossref]
  9. Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
    [Crossref] [PubMed]
  10. K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
    [Crossref] [PubMed]
  11. M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
    [Crossref]
  12. S. Zhang and X. Zhang, “Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 33(3), 452–460 (2016).
    [Crossref]
  13. Y. Zeng, Y. Fu, X. Chen, W. Lu, and H. Ågren, “Highly efficient generation of entangled photon pair by spontaneous parametric down conversion in defective photonic crystal,” J. Opt. Soc. Am. B 24(6), 1365–1368 (2007).
    [Crossref]
  14. J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33(4), 615–621 (2016).
    [Crossref]
  15. A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013).
    [Crossref]
  16. G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
    [Crossref]
  17. H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
    [Crossref]
  18. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108(7), 070401 (2012).
    [Crossref] [PubMed]
  19. J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
    [Crossref] [PubMed]
  20. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
    [Crossref] [PubMed]
  21. C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
    [Crossref]
  22. F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
    [Crossref]
  23. Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
    [Crossref] [PubMed]
  24. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
    [Crossref] [PubMed]
  25. E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
    [Crossref]
  26. Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
    [Crossref] [PubMed]
  27. J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015).
    [Crossref] [PubMed]
  28. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
    [Crossref]
  29. Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
    [Crossref]
  30. M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third- harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016).
    [Crossref]
  31. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998).
    [Crossref]
  32. T. Wang and X. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photon. Res. 5(6), 629–639 (2017).
    [Crossref]
  33. M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015).
    [Crossref] [PubMed]
  34. Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003).
    [Crossref] [PubMed]
  35. C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18(3), 348–351 (2001).
    [Crossref]
  36. Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
    [Crossref] [PubMed]
  37. M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
    [Crossref] [PubMed]
  38. E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).
  39. J. Humlícek, A. Röseler, T. Zettler, M. G. Kekoua, and E. V. Khoutsishvili, “Infrared refractive index of germanium-silicon alloy crystals,” Appl. Opt. 31(1), 90–94 (1992).
    [Crossref] [PubMed]
  40. C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
    [Crossref]
  41. A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
    [Crossref]
  42. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4(4), 481–489 (1987).
    [Crossref]

2017 (4)

G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
[Crossref]

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

T. Wang and X. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photon. Res. 5(6), 629–639 (2017).
[Crossref]

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

2016 (4)

C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
[Crossref]

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third- harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016).
[Crossref]

S. Zhang and X. Zhang, “Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 33(3), 452–460 (2016).
[Crossref]

J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33(4), 615–621 (2016).
[Crossref]

2015 (3)

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
[Crossref] [PubMed]

J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015).
[Crossref] [PubMed]

M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015).
[Crossref] [PubMed]

2014 (6)

F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
[Crossref]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

2013 (8)

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
[Crossref] [PubMed]

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
[Crossref] [PubMed]

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

2012 (3)

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108(7), 070401 (2012).
[Crossref] [PubMed]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

2011 (1)

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

2010 (2)

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

2008 (1)

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
[Crossref]

2007 (2)

Y. Zeng, Y. Fu, X. Chen, W. Lu, and H. Ågren, “Highly efficient generation of entangled photon pair by spontaneous parametric down conversion in defective photonic crystal,” J. Opt. Soc. Am. B 24(6), 1365–1368 (2007).
[Crossref]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

2003 (1)

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003).
[Crossref] [PubMed]

2002 (1)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

2001 (2)

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18(3), 348–351 (2001).
[Crossref]

A. H. Castro Neto, “Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides,” Phys. Rev. Lett. 86(19), 4382–4385 (2001).
[Crossref] [PubMed]

1998 (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998).
[Crossref]

1992 (1)

1987 (1)

Abram, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

Adamashvili, G. T.

G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
[Crossref]

Ågren, H.

Aitchison, J. S.

Alencar, T. V.

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Alù, A.

F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
[Crossref]

Ana Paula Barboza, M.

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Andreani, L. C.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

Aryshev, A.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Bao, Q.

Blanchard, C.

C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
[Crossref]

Butler, S. Z.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Cao, L.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Castro Neto, A. H.

A. H. Castro Neto, “Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides,” Phys. Rev. Lett. 86(19), 4382–4385 (2001).
[Crossref] [PubMed]

Chen, H.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
[Crossref] [PubMed]

Chen, X.

Chenet, D. A.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Chernikov, A.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Choi, D.-Y.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

Chua, S. L.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

Corboliou, V.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

Crespi, V.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Cui, Y.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

D’Orazio, A.

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

Dadap, J. I.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

de Angelis, C.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18(3), 348–351 (2001).
[Crossref]

de Ceglia, D.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

de Paula, A. M.

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Dean, C. R.

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
[Crossref] [PubMed]

Dreisow, F.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

Dumeige, Y.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

Elías, A. L.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Ferrari, A.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Fu, Y.

Godbout, N.

Goldberger, J. E.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Gong, P.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
[Crossref] [PubMed]

Gorbach, A. V.

A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013).
[Crossref]

Grande, M.

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

Gringoli, F.

Gupta, J. A.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Gutiérrez, H. R.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Heinrich, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

Heinz, T. F.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
[Crossref] [PubMed]

Hill, H. M.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Hollen, S. M.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Honda, Y.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Hone, J.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

Hong, S. S.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
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C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
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Ismach, A. F.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
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C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
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C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
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Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
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Mehta, N.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
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Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
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R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
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Noda, S.

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S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
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C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
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Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
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Shan, J.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Shapira, O.

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

Shevelev, M.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Shi, L.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Shi, M.

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
[Crossref] [PubMed]

Shih, E.-M.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Sipe, J. E.

Soljacic, M.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

Solntsev, A. S.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

Song, S. H.

J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015).
[Crossref] [PubMed]

Spencer, M. G.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Stefanou, N.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998).
[Crossref]

Stone, A. D.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

Sukhikh, L.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Szameit, A.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

Terrones, M.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Terunuma, N.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Urakawa, J.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

van der Zande, A. M.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Vidakovic, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

Vincenti, M. A.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

Virally, S.

Wang, S.

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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Wang, T.

Wang, Y.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Weismann, M.

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third- harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016).
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Windl, W.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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Wu, S.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
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Xu, M.

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
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Xu, X.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
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Yan, J.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
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Yang, Y.

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
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N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998).
[Crossref]

Yao, W.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
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Yeh, P.-C.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
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Yoon, J. W.

J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015).
[Crossref] [PubMed]

You, Y.

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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Zeng, Y.

Zettler, T.

Zhang, H.

Zhang, M.

M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015).
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Zhang, S.

Zhang, X.

T. Wang and X. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photon. Res. 5(6), 629–639 (2017).
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S. Zhang and X. Zhang, “Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 33(3), 452–460 (2016).
[Crossref]

M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015).
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Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Zhen, B.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

ACS Nano (1)

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Appl. Opt. (1)

Chem. Rev. (1)

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
[Crossref] [PubMed]

Comput. Phys. Commun. (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998).
[Crossref]

J. Math. Phys. (1)

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
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J. Opt. Soc. Am. B (5)

Light Sci. Appl. (2)

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

Nano Lett. (1)

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
[Crossref] [PubMed]

Nat. Photonics (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Nature (1)

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

Opt. Lett. (1)

Photon. Res. (1)

Phys. Rev. A (2)

A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013).
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G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
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Phys. Rev. Accel. Beams (1)

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
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Phys. Rev. B (6)

C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
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M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
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Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third- harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
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Phys. Rev. E (1)

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003).
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Phys. Rev. Lett. (9)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
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Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
[Crossref]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
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J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

Phys. Rev. X (1)

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

Sci. Rep. (3)

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015).
[Crossref] [PubMed]

M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 Diagram of the TMDC monolayer-grating structure and the SH generation process (oblique view shown in (a) and side view presented in (b)). The lattice constant, width, thickness and relative permittivity of the grating are marked by a, w, d and ε, respectively, the grating is immersed in the air. When a plane wave with angular frequency ω is incident on the structure at the incident angle θ i , the transmitted and reflected SH fields are generated due to the second-order nonlinear effect of TMDC monolayer.
Fig. 2
Fig. 2 Transmission (a) and absorption (b) for the two layer structure as a function of the wavelength λ of the EM wave and thickness d of the grating. (c) Magnified absorption map for the square region in (b). (d)-(f) Corresponding absorption spectrums for different thicknesses of the grating d=312.8nm (d), d=321.71434nm (e) and d=331.2nm (f). The parameters are assumed as follows: a=460nm and w=0.6a.
Fig. 3
Fig. 3 (a) Maximal absorption near BIC as a function of the thickness of the grating. (b) Maximal field enhancement near BIC as a function of the thickness d of the grating. (c) The position spectrum of the maximal absorption (black line with black symbol) and maximal field enhancement (red line with red symbol) indicated by the wavelength λ of the EM wave and the thickness d of the grating. (d) Absorption spectrum at the maximum point in (a) d=325.87182nm. The other parameters are identical to those in Fig. 2.
Fig. 4
Fig. 4 SH generation enhancement ET of the transmitted field (a)-(d) and ER of the reflected field (e)-(h) as a function of the wavelength λ of the incident field for different thicknesses of the grating. The thicknesses of the grating are taken as follows: d=312.8nm for (a) and (e), d=321.71434nm for (b) and (f), d=325.87182nm for (c) and (g), and d=331.2nm for (d) and (h). The other parameters are identical to those in Fig. 2.
Fig. 5
Fig. 5 SH generation enhancement ET of the transmitted field (solid line) and ER of the reflected field (dashed line) as a function of the azimuthal angle of the WS 2 monolayer. The wavelength of the EM wave and thicknesses of the grating are taken as λ=1017.821nm and d=325.87182nm corresponding to the maximum point in Fig. 4(c).
Fig. 6
Fig. 6 (a) Absorption for two layer structure as a function of the wavelength λ and incident angle θ i of the EM wave. Maximal field enhancement (b) and maximal absorption (c) near BIC as a function of the incident angle θ i of the EM wave. (d) Absorption spectrum at the maximum point in (b) θ i =16.1723°. The parameters are assumed as follows: a=460nm, w=0.6a and d=1.0a.
Fig. 7
Fig. 7 SH generation enhancement ET of the transmitted field (a) and ER of the reflected field (b) as a function of the wavelength λ of the incident field at θ i =16.1723°. The other parameters are identical to those in Fig. 6.
Fig. 8
Fig. 8 Transmission and reflection of a plane wave from a TMDC monolayer at the SH frequency. The incident, transmitted and reflected waves are denoted by E i , E t and E r , the mediums up and down the interface are marked by the number 1 and 2.
Fig. 9
Fig. 9 Scattering of two waves from a TMDC monolayer. These two waves indicated by E 1 + and E 2 are incident from the up and down mediums, two waves E 2 + and E 1 are generated.

Equations (24)

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E k ( x,y,ω )= mn ( E kmn + ( ω )+ E kmn ( ω ) ) e i( k mnx ( ω )x+ k mny ( ω )y ) ,
E k ( x,y,2ω )= mn ( E kmn + ( 2ω )+ E kmn ( 2ω ) ) e i( k mnx ( 2ω )x+ k mny ( 2ω )y ) .
J ( 2 ) ( r ,2ω )=i d t 2ω ε 0 χ ( 2 ) : E 1 ( r ,ω ) E 1 ( r ,ω ).
J ( 2 ) ( r ,2ω )=i d t 2ω ε 0 χ ( 2 ) : mn ( pq E 1pq ( ω ) E 1mp,nq ( ω ) ) e i( k mnx ( 2ω )x+ k mny ( 2ω )y ) .
( E 1 + ( 2ω ) E 0 ( 2ω ) )=( Q 1 I ( 2ω ) Q 1 II ( 2ω ) Q 1 III ( 2ω ) Q 1 IV ( 2ω ) )( E 0 + ( 2ω ) E 1 ( 2ω ) )+( C 1 I ( 2ω ) C 1 II ( 2ω ) ),
( E 2 + ( 2ω ) E 1 ( 2ω ) )=( Q 2 I ( 2ω ) Q 2 II ( 2ω ) Q 2 III ( 2ω ) Q 2 IV ( 2ω ) )( E 1 + ( 2ω ) E 2 ( 2ω ) )+( C 2 I ( 2ω ) C 2 II ( 2ω ) ),
( E 2 + ( 2ω ) E 0 ( 2ω ) )=( Q 1,2 I ( 2ω ) Q 1,2 II ( 2ω ) Q 1,2 III ( 2ω ) Q 1,2 IV ( 2ω ) )( E 0 + ( 2ω ) E 2 ( 2ω ) )+( C 1,2 I ( 2ω ) C 1,2 II ( 2ω ) ),
C 1,2 I ( 2ω )= Q 2 I ( 2ω ) [ 1 Q 1 II ( 2ω ) Q 2 III ( 2ω ) ] 1 [ Q 1 II ( 2ω ) C 2 II ( 2ω )+ C 1 I ( 2ω ) ]+ C 2 I ( 2ω ),
C 1,2 II ( 2ω )= Q 1 IV ( 2ω ) [ 1 Q 2 III ( 2ω ) Q 1 II ( 2ω ) ] 1 [ Q 2 III ( 2ω ) C 1 I ( 2ω )+ C 2 II ( 2ω ) ]+ C 1 II ( 2ω ).
I t ( 2ω )= 1 2 ε 0 mn,i [ E 3mni + ( 2ω ) ] [ E 3mni + ( 2ω ) ] * ,
I r ( 2ω )= 1 2 ε 0 mn,i [ E 0mni ( 2ω ) ] [ E 0mni ( 2ω ) ] * ,
I i ( ω )= 1 2 ε 0 mn,i [ E 0mni + ( ω ) ] [ E 0mni + ( ω ) ] * .
ET= I t ( 2ω ) I 0t ( 2ω ) , ER= I r ( 2ω ) I 0r ( 2ω )
P x = ε 0 2 χ ( 2 ) sinϕcosϕ E x E x + ε 0 2 χ ( 2 ) ( sin 2 ϕ cos 2 ϕ ) E x E y + ε 0 2 χ ( 2 ) sinϕcosϕ E y E y , P y = ε 0 χ ( 2 ) ( sin 2 ϕ cos 2 ϕ ) E x E x + ε 0 χ ( 2 ) 4sinϕcosϕ E x E y + ε 0 χ ( 2 ) ( cos 2 ϕ sin 2 ϕ ) E y E y ,
{ E x t E x i E x r =0, H x t H x i H x r =σ E y + J y ( 2 ) , E y t E y i E y r =0, H y t H y i H y r =σ E x J x ( 2 ) ,
{ E x t = T x E x i + C tx , E y t = T y E y i + C ty , E x r = R x E x i + C rx , E y r = R y E y i + C ry .
{ T x = 2 q 2z / ε 2 q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , T y = 2 q 1z q 1z + q 2z +ω μ 0 σ , R x = q 2z / ε 2 q 1z / ε 1 q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , R y = q 1z q 2z ω μ 0 σ q 1z + q 2z +ω μ 0 σ ,
{ C tx = J x ( 2 ) q 1z q 2z / ( ω ε 0 ε 1 ε 2 ) q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , C ty = J y ( 2 ) ω μ 0 q 1z + q 2z +ω μ 0 σ , C rx = J x ( 2 ) q 1z q 2z / ( ω ε 0 ε 1 ε 2 ) q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , C ry = J y ( 2 ) ω μ 0 q 1z + q 2z +ω μ 0 σ ,
( E x' t E y' t )=( cos 2 ϕ T x + sin 2 ϕ T y sinϕcosϕ( T x T y ) sinϕcosϕ( T x T y ) sin 2 ϕ T x + cos 2 ϕ T y )( E x' i E y' i )+( cosϕ C tx sinϕ C ty sinϕ C tx cosϕ C ty ),
( E 2 + E 1 )=( T ( 1,2 ) R ( 2,1 ) R ( 1,2 ) T ( 2,1 ) )( E 1 + E 2 )+( C 1 C 2 ),
T mn,pq ( i,j ) = δ mn,pq ( cos 2 ϕ mn T mnx ( i,j ) + sin 2 ϕ mn T mny ( i,j ) sin ϕ mn cos ϕ mn ( T mnx ( i,j ) T mny ( i,j ) ) sin ϕ mn cos ϕ mn ( T mnx ( i,j ) T mny ( i,j ) ) sin 2 ϕ mn T mnx ( i,j ) + cos 2 ϕ mn T mny ( i,j ) ),
R mn,pq ( i,j ) = δ mn,pq ( cos 2 ϕ mn R mnx ( i,j ) + sin 2 ϕ mn R mny ( i,j ) sin ϕ mn cos ϕ mn ( R mnx ( i,j ) R mny ( i,j ) ) sin ϕ mn cos ϕ mn ( R mnx ( i,j ) R mny ( i,j ) ) sin 2 ϕ mn R mnx ( i,j ) + cos 2 ϕ mn R mny ( i,j ) ),
C 1mn =( cos ϕ mn C mntx ( 1,2 ) sin ϕ mn C mnty ( 1,2 ) +cos ϕ mn C mnrx ( 2,1 ) sin ϕ mn C mnry ( 2,1 ) sin ϕ mn C mntx ( 1,2 ) +cos ϕ mn C mnty ( 1,2 ) +sin ϕ mn C mnrx ( 2,1 ) +cos ϕ mn C mnry ( 2,1 ) ),
C 2mn =( cos ϕ mn C mnrx ( 1,2 ) sin ϕ mn C mnry ( 1,2 ) +cos ϕ mn C mntx ( 2,1 ) sin ϕ mn C mnty ( 2,1 ) sin ϕ mn C mnrx ( 1,2 ) +cos ϕ mn C mnry ( 1,2 ) +sin ϕ mn C mntx ( 2,1 ) +cos ϕ mn C mnty ( 2,1 ) ),

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