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We formulated an efficient numerical method for the dispersion relation of polaritonic channel waveguides and applied it to ZnO (wurzite) and ZnSe (zinc-blende) waveguides. The dispersion relation obtained by our calculations is distinct from that of bulk crystals. We found that important contributions to light propagation were made by two modes in the frequency range below the transverse exciton frequency, which was confirmed by comparing the group index obtained by our calculation with Fabry-Perot interference experiments. The numerical error of our method was estimated to be less than 1 % by comparing it with an analytical solution for a model structure. Our calculations predict an extremely small bending loss, which was estimated from the spatial decay rate of evanescent waves outside of the waveguide.
© 2017 Optical Society of America
1. Introduction
Excitons are the bound state of electron and hole via the Coulomb force, while exciton polaritons are the coupled state of exciton and photon. Exciton polaritons are characterized by their peculiar dispersion relation. For example, their group velocity slows down near the transverse exciton frequency (ωT) because of the strong coupling between the exciton and the photon [1]. Excitons are stable at room temperature when their binding energy is larger than the room-temperature thermal energy (kBTR ∼ 26 meV). Because some inorganic and organic semiconductors satisfy this condition, their exciton polaritons have been studied extensively from both fundamental and practical view points [2–8]. In particular, one-dimensional (1D) polaritonic waveguides are expected to be a promising candidate for future compact optoelectronic devices.
In this paper, we theoretically demonstrate the propagation properties of the 1D polaritonic channel waveguides by numerically solving the Maxwell wave equation. In our numerical analysis, the electromagnetic field is expanded in Fourier series by assuming a periodic array of the channel waveguides, which is often called the supercell method. When the distance between the adjacent waveguides is sufficiently larger than the lateral size of the channel waveguides, the electromagnetic field, which is strongly confined inside of the waveguide, can be accurately calculated. For given frequencies, the propagation constant is obtained by solving the eigenvalue problem of a relatively small matrix, as we shall see later.
The polaritonic channel waveguide has an infinite number of dispersion curves. However, in the frequency range of our interest, that is, for ω < ωT, important contributions are often given by only the first and second lowest dispersion curves, whose frequencies are slightly shifted from each other because of the asymmetric geometry of the chanel waveguide. We compare the group refractive index obtained from our numerical solution with that from Fabry-Perot peaks in experiments. As an approximate solution, we also use an analytical method for dielectric cylindrical waveguides placed in a uniform background medium. We applied these methods to polaritonic channel waveguides with anisotropic dielectric functions, which we obtained from experimental data by curve fitting using the least squares method.
This paper is organized as follows. In Sec. 2, we introduce frequency-dependent dielectric tensors without off-diagonal terms and explain about numerical and analytical methods. In Sec. 3, we present our numerical results for two kinds of polaritonic channel waveguides and discuss their accuracy. A brief summary is given in Sec. 4.
2. Theory
2.1. Frequency-dependent dielectric tensor
We consider a dielectric tensor without off-diagonal terms because of the high symmetry of our assumed crystal structures, so, ε(ω) = diag[εxx (ω), εyy (ω), εzz (ω)]. For wurzite and zinc-blende crystals, εxx = εyy ≠ εzz and εxx = εyy = εzz, respectively. Although more complicated dielectric functions were assumed in previous papers [9–12], we use the following simple formula for each of the diagonal terms, since it reproduces experimental data well as far as our calculation below ωT is concerned:
where ε∞ is the dielectric constant at high frequencies and ωL is the longitudinal exciton frequency (ωL > ωT). ε∞, ωT, and ωL were obtained from experimental data by curve fitting using the least squares method (see Appendix A). Since we focus on the frequency range where absorption loss is small, the imaginary part of the dielectric constant is ignored. Such a frequency region is lower than the electronic band gap of the constituent semiconductors. In that frequency range, Eq. (1) coincides quite well with experimental data. In what follows, ε(ω) > 0 is considered, since we only deal with ω smaller than ωT.2.2. Numerical solution
We consider a cylindrical waveguide oriented in the z direction placed on a dielectric substrate to model previous experiments [6, 8]. We take the x and y axes in the parallel and normal directions to the substrate surface, respectively. The electric field E(r) satisfies the Maxwell wave equation:
where c is the speed of light in free space. Because the waveguide is uniform in the z direction, E(r) has the following form: where β is the propagation constant. Ez (x, y) in Eq. (2) can be eliminated by using which is derived from ∇ · [ε(r; ω)E(r)] = 0. The waveguide is located in the center of the supercell defined by |x| ≤ Lx/2 and |y| ≤ Ly/2, and εii (x, y; ω) and Ei (x, y) are expanded in the Fourier series with respect to the reciprocal lattice vectors G of the supercell. where G = (Gx, Gy) and Gi = (2π/Li)Ni (i = x, y, Ni: integer). εii;G(ω) for a cylindrical waveguide on a substrate is derived in Appendix B. By substituting Eqs. (5) and (6) into Eq. (2), we obtain the eigenvalue matrix equation: where Mij (ω) is the coefficient matrix and Ẽi is the eigenvector composed of Ei;G. When we use NG reciprocal lattice vectors in Eqs. (5) and (6), the size of the eigenvalue matrix is 2NG. The typical size is several thousands. The detailed derivation is given in Appendix C and the accuracy of this method will be discussed in Sec. 3.3. β is obtained for given ω. Although there is an alternative numerical method to obtain ω for given β, a larger coefficient matrix has to be solved [13]. Another eigenvalue matrix equation can also be obtained by discretizing Eq. (2) according to the Yee algorithm [14]. However, more computational time is required because the size of the eigenvalue matrix may be tens of thousands for this case.2.3. Approximate analytical solution
For simplicity, we consider a uniform background outside the cylindrical waveguide. Since this problem can be solved analytically [15], we use this method as an approximate analytical solution. Then, optical modes are characterized by the z component of the electric and magnetic fields (Ez and Hz, respectively). When Ez (Hz) is zero, such a mode is called the TE (TM) mode. On the other hand, when both Ez and Hz are nonzero, the optical mode is called the HE (EH) mode if Hz (Ez) is dominant. We use the HE11 mode (the lowest dispersion curve) as an approximate analytical solution. A more detailed discussion is given in Appendix D. In what follows, we apply the approximate analytical method to two kinds of uniform backgrounds: air and SiO2.
3. Numerical results and discussion
3.1. ZnO cylindrical nanowire
The exciton binding energy of ZnO is 60 meV, which is sufficiently larger than kBTR ∼ 26 meV, so its exciton is stable at room temperature. In [6], Plumhof et al. reported that ZnO single-crystalline nanowires were epitaxially grown on a sapphire substrate with their wurzite c axis along the growth direction. The ZnO nanowires were able to be transferred onto a glass (SiO2) substate, so we consider a cylindrical ZnO nanowire lying on a glass substrate. In the xy plane, the cylindrical ZnO nanowire and the glass substrate occupy the regions of x2 + y2 ≤ R2 and y ≤ −R, respectively, where R is the radius of the cylindrical wire. We assume that the background is air. The refractive index of SiO2 (air) is n = 1.53 (n = 1). The dielectric constant of bulk ZnO has already been reported [9, 10, 16]. We consulted the experimental data of [10], which most accurately reproduced the behavior of exciton polaritons in the 1D ZnO nanowire.
Figure 1 shows the experimental data of the (real) dielectric constants of bulk ZnO for E ⊥ c (black circles) and E ‖ c (white circles) at room temperature. The electronic band gap of ZnO is approximately 3.4 eV, and the imaginary part of the dielectric constant is less than 0.04 for ħω < 3.25 eV. The solid and dashed lines denote the fitting for E ⊥ c and E ‖ c, respectively, using the least squares method. In Eq. (1), ε∞ = 3.9636, ωT = 3.3645 eV and ωL = 3.4304 eV for E ⊥ c, while ε∞ = 3.9406, ωT = 3.4197 eV and ωL = 3.5013 eV for E ‖ c. We take εxx (ω) = εyy (ω) = ε⊥(ω) and εzz (ω) = ε‖(ω).
Fig. 1 The (real) dielectric constants of bulk wurzite ZnO for E ⊥ c and E ‖ c at room temperature [10]. The electronic band gap of ZnO is approximately 3.4 eV, and the imaginary part of the dielectric constant is less than 0.04 for ω < 3.25 eV.
We calculate the dispersion curves of the ZnO nanowire for R = 100 nm. In the supercell of Sec. 2.2, Lx = Ly = 0.6 μm and NG = 2401(= 492). Figure 2 shows the dispersion curves of the ZnO nanowire on a glass substrate. The gray region denotes the light cone where light leaks into the glass substrate. The solid lines denote the numerical solutions in Sec. 2.2. The lowest and second lowest modes have dominant Ey and Ex components, respectively. They are slightly shifted from each other because of the asymmetric structure. These modes are degenerate in the absence of the glass substrate. The green line denotes the bulk dispersion curve [ ]. The two dashed lines denote the approximate analytical solutions in Sec. 2.3 (see also Appendix D) for the uniform background of air (upper curve) and SiO2 (lower curve), respectively. When the uniform background has the same dielectric constant as the cylindrical ZnO waveguide, the dispersion curve coincides with the green line (bulk material). The dispersion curve shifts upward with decreasing background dielectric constant. It is natural that the first and second lowest solid lines are located between the upper and lower dashed lines. This is because the glass substrate in the air is an intermediate structure between uniform backgrounds of air and SiO2.
Fig. 2 Dispersion curves of the ZnO nanowire for R = 100 nm on a glass substrate. The solid lines denote the numerical solutions. The upper and lower dashed lines denote the approximate analytical solutions for the uniform backgrounds of air and SiO2, respectively. The green line denotes the bulk dispersion curve.
The first and second lowest modes (solid lines) are distant from the light cone because of the large exciton oscillator strength. By using the propagation constant, we can obtain the spatial decay rate of evanescent waves in the normal direction of the cylinder surface, which in turn is a character of bending loss. The spatial decay rate κ is given by
where β and β0 are the propagation constant of the nanowire and the substrate (SiO2), respectively. Generally speaking, large κ results in small bending loss. In our previous paper, we reported an extremely small bending loss of thiacyanine polaritonic nanofibers placed on a glass substrate [17] derived by calculating the complex propagation constant for curved fibers. We found a bending power loss as small as 10−4 dB/μm even for the radius of curvature of 3 μm at ħω = 2.3 eV. For this case, κ was 12.4 μm−1. We may use this value for a rough estimation of the bending loss. In Fig. 2, κ of the lowest mode is 14.7 μm−1 at ħω = 3.05 eV (β = 27.8 μm−1 and β0 = 23.6 μm−1). Since this value is larger than our previous one, we can expect a still smaller bending loss for ZnO nanowires at 3.05 eV < ω < 3.25 eV.Clear Fabry-Perot interference peaks were observed in the transmission spectra of ZnO nanowires measured with fluorescent light generated by optical excitation [6]. From those Fabry-Perot peaks, their dispersion curves were estimated. In Fig. 2, the squares, circles and triangles denote the dispersion curves derived from the Fabry-Perot peaks for specimens with different lengths L = 2.1, 4.9 and 7.5 μm, respectively [6]. These three experiments gave similar and consistent results, which are also very close to our numerical solutions. (Note that there is an ambiguity in the propagation constant obtained from the Fabry-Perot peaks by a multiple of Δβ = π/L. The experimental data shown in Fig. 3 were adjusted in the horizontal direction to realize the best fit to the numerical and analytical results.) According to our calculations, two kinds of Fabry-Perot peaks should have been observed due to the presence of two dispersion curves in the relevant frequency range. However, the difference in the frequency interval between the two kinds of Fabry-Perot peaks is not very large. At ħω = 3.1 eV, for example, it is approximately 0.01 eV, so their independent detection may be difficult. We propose a comparison between transmission spectra with x and y polarizations, which characterize the main polarization component of the two propagated modes.
Fig. 3 Group refractive index of the ZnO nanowire for R = 100 nm on a glass substrate. The solid and dashed lines were derived from vg of the numerical and analytical solutions, respectively. The upper and lower dashed lines stand for the uniform backgrounds of air and SiO2, respectively. The green line stands for bulk ZnO.
Figure 3 shows the group refractive index ng of the ZnO nanowire on a glass substrate for R = 100 nm. ng is defined by the ratio c/vg, where vg = dω/dβ is the group velocity. The solid and dashed lines were derived from vg of the numerical and analytical solutions given in Fig. 2, respectively. The upper and lower dashed lines were obtained for background materials of air and SiO2, respectively. The green line denotes ng for bulk ZnO. On the other hand, the squares, circles and triangles were derived from Fabry-Perot peaks observed in experiments. Our numerical results represented by the solid line is close to the experimental data. The upper dashed line is an overestimation, while the lower dashed line is close to the solid line. ng is about 15 near ħω = 3.25 eV. This frequency is 4.4 % distant from the electronic band gap of ZnO, 3.4 eV. This means that the speed of light in the ZnO nanowire is 15 times as slow as that in free space. In addition, the black solid (nanowire) and green (bulk) curves are unexpectedly nearly the same in spite of the very small radius of the former. The relative error of the green curve to the black solid curve is only 6.3 % at ħω = 3.1 eV. When the dispersion curve is distant enough from the light cone (Fig. 2), its slope is relatively close to that of the bulk material. However, it should not be forgotten that there is a big difference between their dispersion curves.
Figure 4 shows the electric-field distribution of the first, second and third lowest modes (solid lines in Fig. 2) for ħω = 3.21 eV. Figures 4(a)–4(c) show the x, y and z components of the electric field of the lowest mode. The same color scale is used for the electric field amplitude in these figures. Similarly, Figs. 4(d)–4(f) and Figs. 4(g)–4(i) are the second and third modes, respectively. While the x and y components are real, the z component is purely imaginary, i.e., its phase is shifted by π as Eq. (4) shows. In the first and second modes, the y and x components are dominant, respectively. The origin of the z component is the evanescent wave near the surface of the waveguide. For example, the evanescent wave of the y polarization mainly appears near the top and bottom surfaces (Fig. 4(b)). According to Eq. (4), ∂(εyy Ey)/∂y near these boundaries yields the z component (Fig. 4(c)). In Fig. 4(d), similarly, ∂(εxx Ex)/∂x near the left and right boundaries yields the z component (Fig. 4(f)). The third mode is hardly excited, because its x and y components are mostly antisymmetric, so there is a mismatching between their mode profile and uniform incident waves.
Fig. 4 Electric-field distribution of the first, second and third lowest bands for ħω = 3.21 eV in Fig. 2. Figures (a)–(c) are x, y and z components of the electric field of the first lowest band, respectively. Similarly, Figs. (d)–(f) and (g)–(i) are the second and third bands, respectively.
3.2. ZnSe cylindrical nanowire
The exciton binding energy of ZnSe is 20 meV, which is comparable to kBTR ∼ 26 meV. In [8], van Vugt et al. reported that ZnSe nanowires were epitaxially grown on a Si substrate and were transferred onto a glass substrate. However, there was no detailed information about their crystal structures and growth direction. We looked for papers about ZnSe nanowires in similar growth conditions. In [18], zinc-blende single-crystalline ZnSe nanowires were grown on a Si substrate by metalo-organic chemical vapor deposition. So, we assumed the zinc-blende crystal structure for ZnSe nanowires. Experimental data of the dielectric constant of bulk zinc-blende ZnSe were reported in [11,12]. We examined whether these data would successfully explain the experimental results of the ZnSe nanowire on the glass substrate.
Figure 5 shows the experimental data (black circles) of the (real) dielectric constant of bulk ZnSe at room temperature [11]. The electronic band gap of ZnSe is approximately 2.7 eV, and the imaginary part of its dielectric constant can safely be ignored for ħω < 2.6 eV [12]. The solid line denotes the curve fitting using the least squares method. In Eq. (1), ε∞ = 4.9446, ħωT = 3.2244 eV and ħωL = 3.5395 eV.
Figure 6 shows the dispersion relation of the ZnSe nanowire for R = 100 nm on a glass substrate. The computational condition for the supercell was the same as Fig. 2. The solid and dashed lines denote the numerical and analytical solutions, respectively, while the black circles denote the experimental data for L = 2.08 μm [8]. The green line denotes the bulk dispersion curve. The numerical solution is very close to the experimental data. The upper and lower dashed lines denote the approximate analytical solutions for the uniform background of air and SiO2, respectively. The behavior of these dashed lines is almost the same as in Fig. 2. For ħω = 2.2 eV, the lowest solid line has κ = 13.0 μm−1 (β = 21.4 μm−1 and β0 = 17.0 μm−1). So, the bending loss of the ZnSe nanowire is expected to be extremely small at 2.2 eV < ħω < 2.6 eV.
Fig. 6 Dispersion curves of the ZnSe nanowire for R = 100 nm on a glass substrate. The solid lines denote the numerical solutions. The upper and lower dashed lines denote the approximate analytical solutions for the uniform backgrounds of air and SiO2, respectively. The green line denotes the bulk dispersion curve.
Figure 7 shows the group refractive index of the ZnSe nanowire. The numerical solution for the lowest solid curve shows the same tendency as the experimental data. The upper and lower dashed curves stand for the uniform background of air and SiO2, respectively. The green line stands for the bulk material. The relative error of the green curve to the solid curve is 15.5 % at ħω = 2.2 eV. When the dispersion curve is close to the light cone (Fig. 6), its slope is somewhat different from that of the bulk material.
Fig. 7 Group refractive index of the ZnSe nanowire for R = 100 nm on a glass substrate. The solid and dashed lines are derived from vg of the numerical and analytical solutions, respectively. The upper and lower dashed lines are for the uniform backgrounds of air and SiO2, respectively. The green line is for bulk.
3.3. Accuracy of the numerical solution
We examine the accuracy of the numerical solution of Sec. 2.2. Figures 8(a) and 8(b) show the dispersion curves for the isotropic ZnO nanowire for R = 100 nm in uniform backgrounds of air and SiO2, respectively. Unlike in Fig. 2, an isotropic dielectric constant [εxx (ω) = εyy (ω) = εzz (ω) = ε⊥(ω)] was hypothetically taken for the comparison with the analytical method. The computational condition for the supercell was the same as in Fig. 2. The dark gray regions indicate the light cones of the uniform backgrounds. The solid lines and black circles denote the analytical and numerical solutions, respectively. The lowest solid lines in Figs. 8(a) and 8(b) are the same as the two dashed lines in Fig. 2. The first, second, third and fourth lowest dispersion curves are the HE11, TE01, TM01 and HE21 modes, respectively. The numerical solutions can successfully reproduce the analytical ones. In Fig. 8(a), for example, the relative errors of these four modes are as small as 0.069 %, 0.060 %, 0.924 %, 0.472 %, respectively at ħω = 3.21 eV.
Fig. 8 Dispersion curves for the isotropic ZnO nanowire for R = 100 nm in a uniform background of (a) air and (b) SiO2. The solid lines and black circles denote the analytical and numerical solutions, respectively.
We should note the influence of the boundary condition at the interface between different dielectric materials on the accuracy of the numerical solution. The parallel and perpendicular components of the electric field are continuous and discontinuous, respectively. In general, discontinuous field distributions require a larger number of Fourier components to reproduce the original distribution with the same accuracy. For the TE01 (azimuthal polarization) and TM01 (radial polarization) modes, the electric field is always parallel and perpendicular to the boundary of the cylindrical waveguide, respectively. So, the numerical accuracy of the TM01 mode is less than that of the TE01 mode. (For the HE11 and HE21 modes, the electric field is partially perpendicular to the boundary.) In addition, higher-order modes also require a larger number of Fourier components to reproduce their complicated electric-field distributions. When the dispersion curve is close to the light cone, evanescent waves decay slowly outside the waveguide. It is difficult to accurately reproduce such slowly decaying evanescent waves because we need a larger supercell. These are the main reasons for errors in the numerical solutions.
4. Conclusion
We numerically calculated the dispersion relation of two kinds of polaritonic channel waveguides made of ZnO and ZnSe placed on a SiO2 substrate. In our calculation, the electric field was expanded in the Fourier series using the supercell method. As an approximate analytical solution, we also considered cylindrical waveguides in a uniform background.
Because the lowest and the second lowest modes of the channel waveguides were slightly shifted from each other due to their asymmetric structure, two kinds of Fabry-Perot peaks were expected in the transmission spectra. To observe them clearly, we proposed to measure each of them by using a different polarization of the electric field.
The dispersion curves of these channel waveguide modes were sandwiched by two approximate analytical solutions assuming air and SiO2 for the uniform background. So, the approximate analytical solutions can be used as a simple estimation of the two lowest dispersion curves of the polaritonic channel waveguides.
The group refractive index estimated from the interval of Fabry-Perot peaks often coincides well with the dispersion of the bulk exciton polariton. However, the waveguide dispersion is actually quite different from the bulk dispersion. So, we need accurate numerical studies to determine the former. The accuracy of our numerical method was verified by comparing it with analytical solutions for cylindrical waveguides.
Outside ZnO and ZnSe nanowires, the spatial decay rate of evanescent waves was larger than our previous value for thiacyanine polaritonic nanofibers. So, very small bending loss is expected in these nanowires.
Appendix A. Least squares method for Eq. (1)
We transformed Eq. (1) into
Suppose that we have N sets of data: (ωi, εi) (1 ≤ i ≤ N). We search for A = ε∞, and that minimize the following residual: ∂f/∂A = 0, ∂f/∂B = 0 and ∂f/∂C = 0 yield the following coupled linear equations: By solving these equations, we obtained ε∞ = A, and .Appendix B. Fourier coefficient of Eq. (5)
We denote the dielectric constants of the cylindrical waveguide, background and substrate by εii;1(ω), εii;2 and εii;3, respectively. Then, the Fourier coefficient of εii (x, y; ω) is given by
where , δ is Kronecker’s delta, J1 is the first-order Bessel function of the first kind.Appendix C. Derivation of the numerical solution
By substituting Eq. (3) into Eq. (2), we obtain
By substituting Eqs. (5) and (6) into Eqs. (13) and (14), we obtain where δG,G′ is Kronecker’s delta. is the (G, G″) element of the inverse matrix of [MG,G″] = εzz;G−G″(ω).Appendix D. Approximate analytical solution
We briefly explain about the analytical solution for dielectric cylindrical waveguides in the uniform background. We denote the dielectric constants of the cylindrical waveguide and background by ε1 and ε2, respectively (ε1 > ε2). Then, the boundary condition of the cylindrical waveguide at r = R (radius) gives [15]
where , , Jn is the n-th order Bessel function of the first kind, and Kn is the n-th order modified Bessel function of the second kind. For fixed ω, we search for β satisfying Eq. (21). For n = 0, the TE and TM modes satisfy and respectively. We have used J′0(x) = −J1(x) and K′0(x) = −K1(x). TE0m (TM0m) has the m-th order β. For n ≥ 1, on the other hand, Eq. (21) has to be solved, and HEnm (EHnm) has the m-th order β. The lowest dispersion curve is the HE11 mode.References and links
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