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Photonic integrated circuits with bound states in the continuum: comment

Jiří Čtyroký and Jiří Petráček

Open Access Open Access

Abstract

We point out that the electro-optic modulation of the bound state in the continuum in a low-index dielectric-loaded slab ${{\rm LiNbO}_3}$ waveguide reported by Yu et al., [Optica 6, 1342 (2019) [CrossRef]  ] is treated incorrectly. According to our analysis based on both the perturbation approach and numerical simulation, the electro-optic modulation is more than an order of magnitude less efficient than the authors claim. It thus remains unclear how the experimental results described in the paper were obtained.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Corrections

22 June 2022: A correction was made to the funding block.

In [1], the authors describe an interesting problem of a bound mode in the continuum (BIC) created by loading of a lithium niobate on insulator (LNOI) planar waveguide with a low-index (polymer) stripe. Although nothing is said in this paper about the crystallographic orientation of the ${{\rm LiNbO}_3}$ used in the LNOI structure, it can be deduced from the waveguide parameters that the device was built on the Z-cut. This is explicitly stated only in Section 3.6, Electro-optic modulators, in Supplement 1 of the paper. In Fig. S17 in Supplement 1, the cross section of the waveguide is shown, together with laterally placed electrodes intended for electro-optic modulation. The spatial distribution of the applied electric field is shown in Fig. S17, too. As expected, the dominant electric field component of the applied field is directed along the $y$ coordinate in the co-ordinate system used by the authors. The authors then describe a theory of electro-optic modulation, on the basis of which they conclude that the variation of the extraordinary refractive index (predominantly “seen” by the quasi-TM BIC mode) is given by Eq. (S8) in Supplement 1, which is reproduced here:

$$\Delta {n_e} = - \frac{1}{2}n_e^3{\gamma _{51}}{E_y}.$$

In this equation, ${\gamma _{51}}$ is the pertinent electro-optic coefficient of the ${{\rm LiNbO}_3}$ crystal.

However, this conclusion is not correct. The (small) electro-optic variation of the optical permittivity tensor due to the applied electric field intensity ${\textbf E}$ is generally given by the equation

$$\Delta \bar \varepsilon = - \bar \varepsilon \cdot \tilde \gamma \cdot {\textbf E} \cdot \bar \varepsilon ,$$
where $\bar \varepsilon = {\rm diag}(n_o^2,n_o^2,n_e^2{)}$ and $\tilde \gamma$ are the (second rank) optical permittivity tensor and the (third-rank) electro-optic tensor, respectively. For the applied electric field in the $y$ direction and the electro-optic tensor of the ${{\rm LiNbO}_3}$ crystal [2], Eq. (2) results in the following expression for the variation of the relative permittivity tensor:
$$\Delta \bar \varepsilon = - \left({\begin{array}{*{20}{c}}{- n_o^4{\gamma _{22}}}&0&0\\0&{n_o^4{\gamma _{22}}}&{n_o^2n_e^2{\gamma _{51}}}\\0&{n_o^2n_e^2{\gamma _{51}}}&0\end{array}} \right){E_y}.$$

The variation of the relative permittivity component that influences the dominant electric field component of the TM polarized BIC mode of the waveguide oriented in the $z$ direction is apparently $\Delta {\varepsilon _{33}}$; however, this component is equal to zero. The electrode configuration shown in Fig. S17 cannot thus ensure an efficient electro-optic modulation of the TM (BIC) mode.

We investigated this situation in more detail using the perturbation approach as well as the numerical solution with the help of COMSOL Multiphysics software. The electro-optic variation of the effective refractive index induced by the applied electric field ${E_y}$ can be approximately calculated as [3]

$$\Delta {{N}_{\rm eff}}\approx -\frac{{{Y}_{0}}}{2}\frac{\iint\nolimits_{S}{\left[ n_{o}^{4}{{\gamma }_{22}}\left( e_{x}^{2}+e_{y}^{2} \right)+2n_{o}^{2}n_{e}^{2}{{\gamma }_{51}}{{e}_{y}}{{e}_{z}} \right]}{{E}_{y}}{\rm d}S}{\iint\nolimits_{S}{\left( {{e}_{y}}{{h}_{z}}-{{e}_{z}}{{h}_{y}} \right){\rm d}S}},$$
where ${Y_0}$ is the vacuum admittance, and ${e_x},{e_y},{e_z}$ and ${h_y},{h_z}$ are the components of the electric and magnetic fields of the pertinent waveguide mode, respectively. For calculation of these fields, we also used COMSOL Multiphysics, and the waveguide parameters were taken from [1] and the corresponding Supplement 1. For comparison, we calculated the variations of the effective refractive indices of both the BIC TM mode and the standard (non-BIC) TE mode. The calculated results are shown in the graphs in Fig. 1.
 figure: Fig. 1.

Fig. 1. Dependence of the effective index of the (BIC) TM mode (a) and the standard TE mode (b) on the applied electric field ${E_y}$.

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While the results of both methods are in excellent agreement for the TE mode, the agreement is not as good for the TM mode. The probable reason for this is that the influence of the electro-optic effect on mode field distribution is stronger for TM polarization; in the first-order perturbation method [Eq. (4)], this effect is neglected, and the electro-optic variation of the effective refractive index of the (BIC) TM mode is very weak. According to the results shown in Fig. 1(a), the low limit of the half-wave voltage of the Mach–Zehnder modulator described in the paper can be estimated as 260 V, which is in marked disagreement with the measured value of 16 V declared on p. 1347 of the paper [1]. It thus remains unclear how the authors obtained experimental results based on electro-optic modulation reported in the paper.

Funding

Grantová Agentura České Republiky (1900062S).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. Z. J. Yu, X. Xi, J. W. Ma, H. K. Tsang, C. L. Zou, and X. K. Sun, “Photonic integrated circuits with bound states in the continuum,” Optica 6, 1342–1348 (2019). [CrossRef]  

2. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).

3. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics. Topics in Applied Physics, 2nd ed., T. Tamir, ed. (Springer, 1979).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (1)
Fig. 1.
Fig. 1. Dependence of the effective index of the (BIC) TM mode (a) and the standard TE mode (b) on the applied electric field ${E_y}$.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

Δ n e = − 1 2 n e 3 γ 51 E y .
Δ ε ¯ = − ε ¯ ⋅ γ ~ ⋅ E ⋅ ε ¯ ,
Δ ε ¯ = − ( − n o 4 γ 22 0 0 0 n o 4 γ 22 n o 2 n e 2 γ 51 0 n o 2 n e 2 γ 51 0 ) E y .
Δ N e f f ≈ − Y 0 2 ∬ S [ n o 4 γ 22 ( e x 2 + e y 2 ) + 2 n o 2 n e 2 γ 51 e y e z ] E y d S ∬ S ( e y h z − e z h y ) d S ,
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