Guided Modes in the Plane Array of Optical Waveguides

TL;DR

This paper demonstrates that an infinite array of dielectric waveguides can support guided modes below the cutoff frequency of an isolated waveguide, with the mode quality improving as the array size increases.

Contribution

It reveals the existence of sub-cutoff guided modes in periodic waveguide arrays and provides a quantitative relation between array size and mode quality factor.

Findings

Guided modes exist below the cutoff frequency in infinite arrays.

Mode quality factor scales as N^3 with the number of waveguides.

Numerical and analytical methods confirm the results.

Abstract

It is known that for an isolated dielectric cylinder waveguide there exists the cutoff frequency $\omega_\ast$ below which there is no guided mode. It is shown in the paper that the infinite plane periodic array of such waveguides possesses the guided modes in the frequency domain which is below the frequency $\omega_\ast$. In the case of a finite array, the modes in this frequency domain are weakly radiating ones, but their quality factor $Q$ increases with the number of waveguides $N$ as $Q(N)\sim N^3$. This dependence is obtained both numerically, using the multiple scattering formalism, and is justified with a simple analytical model.

Equations

$$\begin{array}[]{l}\tilde{\mathbf{E}}_{j}(t,\mathbf{r})=e^{-i\omega t+i\beta z}\,\sum\limits_{m=0,\pm 1...}e^{im\phi_{j}}\,\Bigl{(}c_{jm}\,\tilde{\mathbf{N}}_{\omega^{\prime}\beta m}(\rho_{j})-d_{jm}\,\tilde{\mathbf{M}}_{\omega^{\prime}\beta m}(\rho_{j})\Bigr{)},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \tilde{\mathbf{H}}_{j}(t,\mathbf{r})=e^{-i\omega t+i\beta z}\,n\sum\limits_{m=0,\pm 1...}e^{im\phi_{j}}\,\Bigl{(}c_{jm}\,\tilde{\mathbf{M}}_{\omega^{\prime}\beta m}(\rho_{j})+d_{jm}\,\tilde{\mathbf{N}}_{\omega^{\prime}\beta m}(\rho_{j})\Bigr{)},\end{array}$$

$${\tilde{\mathbf{N}}}_{{\omega }^{\prime }\beta m}({\rho }_j)={\mathbf{e}}_r\frac{i\beta }{\kappa }{J}_m^{\prime }(\kappa {\rho }_j)-{\mathbf{e}}_{\varphi }\frac{m\beta }{{\kappa }^2{\rho }_j}{J}_m(\kappa {\rho }_j)+{\mathbf{e}}_z{J}_m(\kappa {\rho }_j),$$

$${\tilde{\mathbf{M}}}_{{\omega }^{\prime }\beta m}({\rho }_j)={\mathbf{e}}_r\frac{m{\omega }^{\prime }}{{\kappa }^2{\rho }_j}{J}_m(\kappa {\rho }_j)+{\mathbf{e}}_{\varphi }\frac{i{\omega }^{\prime }}{\kappa }{J}_m^{\prime }(\kappa {\rho }_j),$$

$$\mathbf{E}(t,\mathbf{r})=\sum _{j=1}^N{\mathbf{E}}_j(t,\mathbf{r}),\mathbf{H}(t,\mathbf{r})=\sum _{j=1}^N{\mathbf{H}}_j(t,\mathbf{r}).$$

$$\begin{array}[]{l}\mathbf{E}_{j}(t,\mathbf{r})=e^{-i\omega t+i\beta z}\sum\limits_{m}e^{im\phi_{j}}\,\Bigl{(}a_{jm}\,\mathbf{N}_{\omega_{0}\beta m}(\rho_{j})-b_{jm}\,\mathbf{M}_{\omega_{0}\beta m}(\rho_{j})\Bigr{)},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \mathbf{H}_{j}(t,\mathbf{r})=e^{-i\omega t+i\beta z}n_{0}\sum\limits_{m}e^{im\phi_{j}}\,\Bigl{(}a_{jm}\,\mathbf{M}_{\omega_{0}\beta m}(\rho_{j})+b_{jm}\,\mathbf{N}_{\omega_{0}\beta m}(\rho_{j})\Bigr{)}.\end{array}$$

$${\mathbf{N}}_{{\omega }_0\beta m}({\rho }_j)={\mathbf{e}}_r\frac{i\beta }{{\kappa }_0}{H}_m^{\prime }({\kappa }_0{\rho }_j)-{\mathbf{e}}_{\varphi }\frac{m\beta }{{\kappa }_0^2{\rho }_j}{H}_m({\kappa }_0{\rho }_j)+{\mathbf{e}}_z{H}_m({\kappa }_0{\rho }_j),$$

$${\mathbf{M}}_{{\omega }_0\beta m}({\rho }_j)={\mathbf{e}}_r\frac{m{\omega }_0}{{\kappa }_0^2{\rho }_j}{H}_m({\kappa }_0{\rho }_j)+{\mathbf{e}}_{\varphi }\frac{i{\omega }_0}{{\kappa }_0}{H}_m^{\prime }({\kappa }_0{\rho }_j),$$

$${\tilde{\mathbf{E}}}_j(\mathbf{r})=c_j{\tilde{\mathbf{N}}}_{{\omega }^{\prime }\beta 0}({\rho }_j),{\tilde{\mathbf{H}}}_j(\mathbf{r})=c_{j}n{\tilde{\mathbf{M}}}_{{\omega }^{\prime }\beta 0}({\rho }_j),{\rho }_j<R$$

$${\mathbf{E}}_j(\mathbf{r})=a_j{\mathbf{N}}_{{\omega }_0\beta 0}({\rho }_j),{\mathbf{H}}_j(\mathbf{r})=a_j{n}_0{\mathbf{M}}_{{\omega }_0\beta 0}({\rho }_j),{\rho }_j>R$$

$$[\mathbf{E}({\mathbf{R}}_j){]}_z=[\widetilde{{\mathbf{E}}_j}({\mathbf{R}}_j){]}_z,[\mathbf{H}({\mathbf{R}}_j){]}_{\varphi }=[\widetilde{{\mathbf{H}}_j}({\mathbf{R}}_j){]}_{\varphi }.$$

$$\left[\begin{array}[]{c}\mathbf{N}_{\omega_{0}\beta n}(\rho_{l})\\ \mathbf{M}_{\omega_{0}\beta n}(\rho_{l})\end{array}\right]\,e^{in\phi_{l}}=\sum\limits_{m=0}^{+\infty}U_{nm}^{lj}(\omega,\beta)\,\left[\begin{array}[]{c}\tilde{\mathbf{N}}_{\omega_{0}\beta m}(\rho_{j})\\ \tilde{\mathbf{M}}_{\omega_{0}\beta m}(\rho_{j})\end{array}\right]\,e^{im\phi_{j}},$$

$$U_{nm}^{lj}(\omega ,\beta )=H_{n-m}({\kappa }_{0}a|l-j|)[\mathrm{sign}(j-l){]}^{n-m}.$$

$$\begin{array}[]{l}\mathbf{N}_{\omega_{0}\beta 0}(\rho_{l})\approx U_{l-j}(\omega,\beta)\,\tilde{\mathbf{N}}_{\omega_{0}\beta 0}(\rho_{j}),\\ \mathbf{M}_{\omega_{0}\beta 0}(\rho_{l})\approx U_{l-j}(\omega,\beta)\,\tilde{\mathbf{M}}_{\omega_{0}\beta 0}(\rho_{j}),\end{array}$$

$$\begin{array}[]{l}\mathbf{E}_{l}(\mathbf{r})=a_{l}\,U_{l-j}(\omega,\beta)\,\tilde{\mathbf{N}}_{\omega_{0}\beta 0}(\rho_{j}),\\ \mathbf{H}_{l}(\mathbf{r})=a_{l}\,n_{0}\,U_{l-j}(\omega,\beta)\,\tilde{\mathbf{M}}_{\omega_{0}\beta 0}(\rho_{j}).\end{array}$$

$$\begin{array}[]{l}\mathbf{E}(\mathbf{r})=a_{j}\,\mathbf{N}_{\omega_{0}\beta 0}(\rho_{j})+\sum\limits_{l\neq j}a_{l}\,U_{l-j}(\omega,\beta)\,\tilde{\mathbf{N}}_{\omega_{0}\beta 0}(\rho_{j}),\\ \mathbf{H}(\mathbf{r})=a_{j}\,n_{0}\,\mathbf{M}_{\omega_{0}\beta 0}(\rho_{j})+\sum\limits_{l\neq j}a_{l}\,n_{0}\,U_{l-j}(\omega,\beta)\,\tilde{\mathbf{M}}_{\omega_{0}\beta 0}(\rho_{j}).\end{array}$$

$$\begin{array}[]{l}a_{j}\,H_{0}(\kappa_{0}R)+\sum\limits_{l\neq j}a_{l}\,U_{l-j}(\omega,\beta)\,J_{0}(\kappa_{0}R)=c_{j}\,J_{0}(\kappa R),\\ a_{j}\,n_{0}\,\frac{i\omega_{0}}{\kappa_{0}}H_{0}^{\prime}(\kappa_{0}R)+\sum\limits_{l\neq j}a_{l}\,n_{0}\,U_{l-j}(\omega,\beta)\frac{i\omega_{0}}{\kappa_{0}}J_{0}^{\prime}(\kappa_{0}R)=c_{j}\,n\,\frac{i\omega^{\prime}}{\kappa}J_{0}^{\prime}(\kappa R).\end{array}$$

$$\frac{a_j}{\bar{a}(\omega ,\beta )}-\sum _{l\ne j}{U}_{l-j}(\omega ,\beta )a_l=0,$$

$$c_j=\bar{c}(\omega ,\beta )a_j,$$

$$\bar{a}(\omega ,\beta )=\frac{n^2{\kappa }_0{J}_0^{\prime }(\kappa R)J_0({\kappa }_{0}R)-n_0^2\kappa J_0(\kappa R)J_0^{\prime }({\kappa }_{0}R)}{n_0^2\kappa J_0(\kappa R)H_0^{\prime }({\kappa }_{0}R)-n^2{\kappa }_0{J}_0^{\prime }(\kappa R)H_0({\kappa }_{0}R)},$$

$$\bar{c}(\omega ,\beta )=\frac{n_0^2\kappa \{H_0({\kappa }_{0}R)J_0^{\prime }({\kappa }_{0}R)-H_0^{\prime }({\kappa }_{0}R)J_0({\kappa }_{0}R)\}}{n_0^2\kappa J_0^{\prime }({\kappa }_{0}R)J_0(\kappa R)-n^2\kappa J_0({\kappa }_{0}R)J_0^{\prime }(\kappa R)}.$$

$$\det \Vert \frac{{\delta }_{jl}}{\bar{a}(\omega ,\beta )}-U_{l-j}(\omega ,\beta )\Vert =0.$$

$$a_j=a_0{e}^{ikaj},-\pi /a<k\le \pi /a.$$

$$\frac{1}{\bar{a}(\omega ,\beta )}-U(\omega ,\beta ,k)=0,$$

$$U(\omega ,\beta ,k)=\sum _{l\ne 0}{U}_l(\omega ,\beta )e^{ikal}.$$

$$\beta <n_0\omega <\sqrt{{\beta }^2+k^2},$$

$$i\frac{dA_{j}}{dt}(t)+V\Bigl{(}A_{j-1}(t)+A_{j+1}(t)\Bigr{)}=0.$$

$$A_j(t)=A_0{e}^{-i\omega t+ikj},-\pi <k\le \pi .$$

$$\omega (k)=-2V\cos k.$$

$$i\frac{dA_{j}}{dt}(t)+V\,\Bigl{(}1-\delta_{j1}\Bigr{)}\,A_{j-1}(t)+V\,\Bigl{(}1-\delta_{jN}\Bigr{)}\,A_{j+1}(t)+i\gamma\,\Bigl{(}\delta_{j1}+\delta_{jN}\Bigr{)}\,A_{j}(t)=0.$$

$$\omega A_j+V(1-{\delta }_{j1})A_{j-1}+V(1-{\delta }_{jN})A_{j+1}+i\gamma ({\delta }_{j1}+{\delta }_{jN})A_j=0.$$

Full text

Optical waveguides, guided modes, waveguide array,

††preprint:

I Introduction

The optical waveguides are the inherent component of the optical and optoelectronic devices, indispensable for the optical signals transmission between different parts of the system. The interaction between the closely spaced waveguides usually results in the undesirable effects that distorts the transmitted signal. However, in some cases, the interaction between the waveguides can be exploited for practical purpose. In particular, this concerns the plane periodic arrays of waveguides, which is a special kind of low-dimensional photonic crystals. The main feature of such systems is a band structure of the optical spectrum which defines its peculiar properties Lourtioz ; Joannopoulos ; Busch ; Longhi . Low-dimensional photonic crystal composed of the parallel rods are of a special interest. For the first time, the band structure of the plane array composed of the semiconductor cylinders was investigated in Ref. maradudin1 . Then a similar metallic structure which took into account dissipation was computed in Ref. maradudin2 . Superconducting photonic crystals of such kind were considered in lozovik . Photonic crystals of such kind are useful for various applications vergeles1 . In particular, they reveal negative-angle refraction and reflection vergeles2 . On the other hand, the arrays of parallel interacting cylinders is an ideal system to simulate various physical phenomena inherent in condensed matter physics such as Anderson localization, Bloch oscillations, Bloch-Zener tunneling, etc. cite1 ; cite2 ; cite3 ; cite4 ; cite5 .

The electromagnetic filed which describes a guided mode in a waveguide is finite inside the waveguide, while it is vanishing at a large distance from it. It is well known that, for the isolated cylinder waveguide, the guided modes can exist only above the so-called cutoff frequency $\omega_{\ast}$ Marcuse . This is due to the fact that the conversion of the modes with the frequency below $\omega_{\ast}$ into a free photon is possible. This frequency depends on the material refractive index and the waveguide diameter. However, in various applied tasks, it may be necessary to have a guided mode below the frequency $\omega_{\ast}$ for the given waveguides. In particular, such problem may appear in the following case. Indeed, along with the radiation, there exist losses connected with the absorption by the material itself. If the frequency window is located below the cutoff frequency $\omega_{\ast}$, there arise a problem to shift the frequency $\omega_{\ast}$ into this window region.

In this paper, we investigate a possibility of a formation of high quality guided modes in the plane waveguide array which are below the cutoff frequency of the isolated waveguide $\omega_{\ast}$. Thus our aim is to suppress the radiation loss using the array of the waveguides. First, we consider the infinite plane periodical array of the waveguides an show that, taking into account the interaction between them, results in the appearance of the guided modes below $\omega_{\ast}$. These modes possess the infinite $Q-$factor. However, actually we deal with the array of a finite size and the modes become low-radiating ones. For this reason, we investigate how the $Q-$factor of the low-radiating modes depends on the number of the waveguides in the array $N$. Using the multiple scattering formalism (MSF), it is shown numerically that $Q(N)\sim N^{3}$. This formalism is based on the exact description of the electromagnetic waves which are scattered by the infinite cylinder VanDeHulst . Besides, we propose a simple model which qualitatively explains this cubic dependence.

Note that the effect of increasing the $Q-$factor for the radiative modes in the array of the interacting spherical particles with increasing the array size, was discovered in We-OptExpr2007 ; Burin-PhysRevE2006 ; deich ; wePRE2010 .

The paper is organized as follows. In Section II we describe the MSF giving a brief derivation of the principal relations. Based on the relations obtained, the numerical simulation for the infinite and the finite arrays of cylinder waveguides is given in Section III. A clear qualitative explanation of the results obtained numerically is given in Section IV.

II Multiple scattering formalism

Let us consider the array of $N$ parallel cylindrical dielectric waveguides (see figure 1). The axes of the waveguides are in the $xz$-plane and are parallel to the $z$-axis. The array is equidistant, $a$ being the distance between the axes of the nearest waveguides. All the waveguides are assumed to have the same radius $R$ and the same refractive index $n$. The refractive index of the environment is $n_{0}$. The system of units where the speed of light $c=1$ is used.

Let a guided mode with a frequency $\omega$ is excited. Because of the translation invariance in the $z$ direction, all the components of the electromagnetic field describing the guided mode depend on the coordinate $z$ as $e^{i\beta z}$, $\beta$ being a propagation constant. Thus, all the components of the electromagnetic field describing the guided mode are proportional to the factor $e^{-i\omega t+i\beta z}$. First, let us describe the electromagnetic field inside the waveguides. The field inside the $j$-th waveguide, being of a finite value, may be represented as follows

[TABLE]

where $\mathbf{r}=(x,y,z)=(\bm{\rho},z)$, $\rho_{j}=|\bm{\rho}-\mathbf{a}j|<R$, $\phi_{j}$ is the polar angle for the vector $\bm{\rho}-\mathbf{a}j$ (see figure 1), $\omega^{\prime}=n\omega$. The vector cylinder harmonics $\tilde{\mathbf{M}}_{\omega^{\prime}\beta m}(\rho_{j})$ and $\tilde{\mathbf{N}}_{\omega^{\prime}\beta m}(\rho_{j})$ are defined as follows

[TABLE]

[TABLE]

here $\kappa=\sqrt{n^{2}\omega^{2}-\beta^{2}}$, $J_{m}(\kappa\rho_{j})$ is the Bessel function, and the prime means the derivative with respect to the argument $\kappa\rho_{j}$. Thus, the guided mode inside the $j$-th waveguide is determined by the frequency $\omega$, by the propagation constant $\beta$ and by the partial amplitudes $c_{jm}$, $d_{jm}$.

Now let us turn to the electromagnetic field for the same guided mode outside of the array. This field is a sum of the contributions induced by all the waveguides:

[TABLE]

For the guided mode, the contribution of the $j$-th waveguide should vanish at $\rho_{j}\to\infty$. Therefore, one can write

[TABLE]

where $\rho_{j}>R$. In (5), another kind of the vector cylinder harmonics is introduced

[TABLE]

[TABLE]

where $H_{m}(\kappa_{0}\rho_{j})$ is the Hankel function of the first kind, $\omega_{0}=n_{0}\omega$, $\kappa_{0}=\sqrt{n_{0}^{2}\omega^{2}-\beta^{2}}$. Thus, the contribution of the $j$-th waveguide into the guided mode field outside of the array is determined by the frequency $\omega$, by the propagation constant $\beta$, by the partial amplitudes $a_{jm}$, $b_{jm}$. Note that, for $\beta=0$, expansions (1) and (5) transform into the corresponding expressions in VanDeHulst , however different notations are used there. Below, the factor $e^{-i\omega t+i\beta z}$ is omitted.

Below in this paper, for the purpose of illustration of the effect, we confine ourselves to the zero-harmonic approximation. This means that only the terms with $m=0$ are taken into account in (1) and (5). It is easy to convince ourselves that in this case the guided modes are either transverse magnetic modes (TM) or transverse electric (TE) ones. For the TM mode $b_{j0}=d_{j0}=0$, while for the TE mode $a_{j0}=c_{j0}=0$. As an example, let us consider the TM modes. Then, equations (1) and (5) take the form

[TABLE]

[TABLE]

Here the notations $a_{j}$, $c_{j}$ are used instead of $a_{j0}$, $c_{j0}$.

Let $\mathbf{R}_{j}$ be the radius-vector of a point on the surface of the $j$-th waveguide. Then the fields $\tilde{\mathbf{E}}_{j}(\mathbf{R}_{j})$, $\tilde{\mathbf{H}}_{j}(\mathbf{R}_{j})$ in Eq.(1) and the fields $\mathbf{E}(\mathbf{R}_{j})$ $\mathbf{H}(\mathbf{R}_{j})$ in Eq.(4) obey the boundary conditions on the surface of this waveguide. Generally, there are six boundary conditions. However, for the TM-modes only two of them are required.

[TABLE]

These equations determine the partial amplitude $a_{j}$ and $c_{j}$ for given $\omega$ and $\beta.$ To represent Eqs.(10) in a convenient form, one should express the fields $\mathbf{E}_{l}(\mathbf{R}_{j})$ for $l\neq j$ entering in Eq.(4) in terms of the functions $\tilde{\mathbf{N}}$ using the Graph formula

[TABLE]

here

[TABLE]

For the $m=0$ approximation one has

[TABLE]

where $U_{l-j}(\omega,\beta)=U_{00}^{lj}(\omega,\beta)$. Then, it follows from Eq.(9) that

[TABLE]

Thus,

[TABLE]

Substituting (15) and (8) into (10), one obtains

[TABLE]

Then the system of equations (16) is reduced to the form

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

The terms $U_{l-j}(\omega,\beta)$ in Eq.(17) desribe the interaction between the waveguides. If the terms $U_{l-j}(\omega,\beta)$ in Eq.(17) are neglected, the poles of $\bar{a}(\omega,\beta)$ or $\bar{c}(\omega,\beta)$ determines the guied modes for the isolated wavegide.

System of equations (17) possesses nontrivial solutions if

[TABLE]

This equation relates the frequency of the guided mode $\omega$ and its propagation constant $\beta$ implicitly.

For the infinite periodical array of identical waveguides, the solution of Eq.(17) reads

[TABLE]

In this case, the nontrivial solution exists if

[TABLE]

where

[TABLE]

Equation (23) determines the dispersion law $\omega(\beta,k)$. If the propagation constant $\beta$ is real, the corresponding mode frequencies may be either real or complex. If the frequency is real, the mode possesses an infinite $Q-$factor. Otherwise the mode has a finite lifetime and the imaginary part of the frequency determines the mode decay rate. However, if the corresponding quality factor is large, the mode may be considered as a guided one.

III Numerical simulation for the infinite and the finite arrays.

Let us consider the infinite the array of the waveguides with the geometric parameters and the refractive indices which are chosen to be close to those in Refs. cite5 . The specific values of the parameters are taken so that the illustration of the results looks quite representative. For this reason, one takes the waveguide radius $R=1.975\mathrm{\mu m}$, the refractive index of the waveguides $n=1.554$, and the refractive index of the environment $n_{0}=0.99n=1.538$. Using (19) one finds the cutoff frequency for the isolated waveguide $\omega_{\ast}=5.57\mathrm{\mu m}^{-1}.$ This value corresponds to the cutoff propagation constant $\beta_{\ast}=8.57\mathrm{\mu m}^{-1}$. (Let us remind that the speed of light $c=1$ and, therefore, the frequency has a dimension of the inverse length).

The modes we are interested in, appear due to the interaction between the waveguides. For this reason, we assume that the neighbor waveguides touch each other, since in this case the interaction reveals itself the most strongly. As an example, let us choose the propagation constant $\beta=8\mathrm{\mu m}^{-1}<\beta_{\ast}$. The dispersion curve $\omega(\beta,k),$which is a numerical solution to Eq. (23), is presented in Fig. 2 by the thick solid line.

One sees that the curve is completely located within the domain $\beta<n_{0}\omega(k)<\sqrt{\beta^{2}+k^{2}}.$ A physical explanation to this fact is given below. This numerical results completely supports the kinematic criterion for the mode to be a radiationless one. Thus the infinite periodical array of the waveguides may possess the guides modes with the frequencies below the cutoff frequency of the single waveguide.

The radiationless guided modes inherent in the periodical array found above (see Fig.2) possess the *infinite *quality factor $Q=2\mathrm{Re}\,\omega/|\mathrm{Im}\,\omega|$. This is due to the fact that the array is infinite. However, actually one deals with the arrays composed of a finite number of the waveguides $N$. On the other hand, it is evident that for $N\gg 1$ the array should manifest the features similar to the infinite array. In particular, the guided modes should possess a high quality. Let us investigate how the quality factor $Q$ depends on the number of the waveguides $N.$

Using Eq. (21) one can obtain numerically that, for a finite $N\gg 1,$ the highest quality factor is reached for the modes whose frequency is close to the upper edge of the Brillouin zone $k\approx\pi/a$ (such a feature is inherent also for the array of spherical particles We-OptExpr2007 ; Burin-PhysRevE2006 ; wePRE2010 ). The dependence of the quality factor on the number of the waveguides $N$ just for modes with the highest $Q-$factor is illustrated by two example: the waveguides are touching, $a=2R,$ and the waveguides are spatially separated, $a=3R$. The three values for the propagation constant smaller than $\beta_{\ast}$ are taken: $\beta_{1}=6\mathrm{\mu m}^{-1}$, $\beta_{2}=7\mathrm{\mu m}^{-1}$, $\beta_{3}=8\mathrm{\mu m}^{-1}<\beta_{\ast}$. The results of the numerical simulation are presented in Fig. 3. The analysis of the dependencies in this figure reveals a remarkable feature: for $N>10$ the dependence the quality factor $Q(N)\sim N^{3}$.

Let us retrieve, using Eq. (MSF_Main), the relation between the partial amplitudes $a_{j}.$ A typical dependence, obtained numerically, is presented, as an example, in Fig. 4 for the case $N=15.$

IV The interpretation of the numerical results

Knowing the dependence $\omega(\beta,k)$ allows us to determine the features of the guided modes. Let us pay attention, that the reason for the mode to possesses a finite lifetime is a conversion of it into a free photon. That is the mode is a radiative one. First, let us consider a single waveguide. Then the mode is described by a frequency $\omega$ and a propagation constant $\beta$. For the conversion into a free photon to takes place, the photon wave vector $\mathbf{q}$ should satisfy the two conditions: $|\mathbf{q}|=n_{0}\omega$ and $q_{z}=\beta$. Since $q_{z}<|\mathbf{q}|$, the photon can be emitted only if $\beta<n_{0}\omega$. In the opposite case $\beta>n_{0}\omega$, the mode is a radiationless one and it is a guided one with the infinite quality factor. Let us turn to the infinite plane periodic array. In this case, a mode is determined by a quasi-wave vector $k$, in addition to the frequency $\omega$ and the propagation constant $\beta$. Thus, the wave vector $\mathbf{q}$ of the emitted free photon satisfies three conditions: $|\mathbf{q}|=n_{0}\omega$, $q_{z}=\beta$ and $q_{x}=k$. It is obvious that $\sqrt{q_{x}^{2}+q_{z}^{2}}<|\mathbf{q}|$. So, the mode can be converted into a free photon only if $\sqrt{k^{2}+\beta^{2}}<n_{0}\omega$. Thus, the infinite periodical array possesses guided modes within the frequency domain, which obeys the kinematic criterion

[TABLE]

where the single waveguide allow only the radiating modes. Note that, since $n\omega(\beta,k)<\sqrt{\beta^{2}+k^{2}},$ a guided mode may not exist for small $k$ at all.

Then, let us explain qualitatively the cubic dependence for the $Q$-factor found in the previous section. First, let us consider the infinite array of the waveguides. Let $A_{j}(t)$ be the effective time-dependent partial amplitude for the $j$-th waveguide, which characterizes the waveguide as a whole. The time evolution of $A_{j}(t)$ may be approximately described by the equation which similar to a Schrödinger one

[TABLE]

Here $V$ is the effective coupling between the nearest waveguides. Let us find the solution for this equation in the form

[TABLE]

Substituting (27) into (26) one obtains the dispersion law:

[TABLE]

Now let us turn to the finite array. As found above, the infinite array possesses the infinite $Q,$ while the finite array possesses a large but a finite $Q.$ (see Fig. 3). For this reason, it is natural to assume that this is connected with the availability of the *edge *waveguides in the array which are responsible for the radiation of the photon. Based on this fact, one can write for the finite array the equation similar to Eq. (26):

[TABLE]

The parameter $\gamma\ll V$ is responsible for the free photon emission. Using (27) one obtains from (29)

[TABLE]

Note that $\omega=\omega^{\prime}+i\gamma$ may be complex. For the particular $j=N,$ this equation takes the form

[TABLE]

So,

[TABLE]

The dependence for $a_{j}$ in Fig. 4 approaches zero at the edges of the array and resembles a cosine one. For this reason, let us seek the solution to Eq. (30) in the form

[TABLE]

$k$ being close to $\pi-\pi/N$. Substituting (28) and (33) into (32), one gets

[TABLE]

Let

[TABLE]

where $x$ is complex and $|x|\ll\pi/N$. For the sake of simplicity, let us assume $N$ to be even. Then, substituting (35) into (34), one gets

[TABLE]

Taking into account a smallness of the arguments in the trigonometric functions in (36) one obtains:

[TABLE]

Then, since $\gamma\ll V$, one has

[TABLE]

Substituting (38) and (35) into (28), one gets:

[TABLE]

Then, the quality factor

[TABLE]

reveal the sought-for cubic dependence.

V Conclusion

In this paper we investigated the guided modes in the array of coupled waveguides below the cutoff frequency of a single waveguide, i.e. in the frequency domain where the single waveguide possesses only the radiating modes. It is shown that the infinite periodic array possesses a band of the guided modes with the infinite $Q$-factor. In the case of the finite array, the modes below the cutoff frequency are weakly radiating ones. Their quality factor increases with the number of waveguides as $Q(N)\sim N^{3}.$These results are obtained numerically using the multiple scattering formalism. A clear physical interpretation of the numerical results is given.

Acknowledgements

The study is partially supported by the Russian Fund for Basic Research (Grant 16-02-00660).