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We theoretically examine the bending loss of organic molecular-crystal nanofibers for which the light propagation is carried out by optically anisotropic exciton polaritons. Previous experimental studies showed that the leakage of light for bent thiacyanine nanofibers was negligibly small even for the radius of curvature of several microns. We formulate a finite-difference frequency-domain method stabilized by a conformal transformation to calculate the bending loss as a function of the radius of curvature and the propagation frequency. The present method is applied to the thiacyanine nanofiber and numerical results that support the previous experimental observation are obtained. The present study clearly shows that the polariton nanofiber gives a novel possibility for bent waveguides to fabricate optical microcircuits and interconnection that cannot be attained by the conventional waveguides based on the index guiding.
© 2013 Optical Society of America
1. Introduction
Bent waveguides are indispensable for optical circuits [1]. For straight waveguides, their electromagnetic eigenmodes propagate without optical leakage when their dispersion curves are located below the light line, which is given by the dispersion relation of the surrounding material. The eigenmodes are generally accompanied by evanescent components in the direction perpendicular to their propagation. The more distant from the light line their dispersion curves are, the more rapidly the evanescent components decay outside the waveguides. Such eigenmodes are generally robust against the bending loss, since they are less converted into propagating waves in the surrounding medium. In this respect, exciton polaritons, which are the mixture of the electromagnetic wave and the electronic polarization due to the excitation of the electron-hole pair, are advantageous, since their dispersion curves can be very far from the light line when their eigen frequency approaches the transverse exciton frequency ωT [2]. The speed of the polariton propagation or the group velocity vg, which is given by the slope of the dispersion curve, becomes small in this situation.
For the exciton polariton to be stable at room temperature, both the polariton gap, which is given by the difference between the longitudinal exciton energy h̄ωL and h̄ωT, and the exciton binding energy must be sufficiently larger than the thermal energy of approximately 30 meV. Frenkel excitons in organic molecular crystals generally have such large exciton binding energies, since both the electron and hole are located in the same molecule, which results in the large negative Coulomb energy. Organic molecular-crystal nanofibers, in particular, were recently discovered in which exciton polaritons were stable at room temperature [3]. The fluorescent emission generated by optical excitation was used to measure their propagation properties [4]. For example, clear Fabry-Perot peaks caused by the two ends of the nanofibers were observed in the emission spectra, from which the group index ng larger than 10 was found [3]. So, the group velocity of the exciton polariton was 10 times smaller than the speed of light in free space c.
Because the nanofibers were flexible due to the small inter-molecule force of organic crystals, bent waveguides were easily fabricated. It was observed that the optical leakage was surprisingly small even for sharply bent nanofibers with the radius of curvature of several microns [5]. Therefore, the application of organic molecular-crystal nanofibers to compact optical circuits is expected [6].
Unlike most of inorganic semiconductors, molecular-crystal nanofibers generally have large optical anisotropy due to their anisotropic unit structures. By exactly taking the anisotropic dielectric constant into consideration, we have successfully explained the basic properties of the anisotropic exciton polariton in thiacyanine nanofibers [7]. The large group index obtained by our calculation, in particular, agreed with the experimental observation quite well.
As a next step, in this paper, we theoretically evaluate the bending loss of the nanofiber as a function of the radius of curvature and the propagation frequency. Although the bending loss of waveguides composed of isotropic media was studied by many researchers [8–15], that of anisotropic media is much more difficult to evaluate and has scarcely been tried. We exactly treat the optical anisotropy and calculate the complex wave number, or the propagation constant, as a function of the frequency in the cylindrical coordinate system. From the imaginary part of the wave number, we obtain the bending loss.
This paper is organized as follows. In Sec. 2, we present the dielectric tensor appropriate for the anisotropic exciton polariton. We derive the Maxwell wave equation for the magnetic field in the cylindrical coordinate system and introduce a conformal transformation for efficient numerical calculation. In Sec. 3, after explaining optical properties of organic nanofibers, we present our numerical results for the bending loss. A brief summary is given in Sec. 4.
2. Theory
2.1. Dielectric tensor
We denote the three basic column vectors of the cylindrical coordinate system by er, eϕ and ez. According to reported experimental conditions, we assume that the nanofiber is located on a glass substrate as shown in Fig. 1(a). We found in Ref. [7] that the transition dipole moment of the constituent molecules is tilted by about 60 degrees from the fiber axis in the thiacyanine nanofibers, so we generally assume a tilt angle α and denote the orientation of the transition dipole moment by n = sinαer + cosαeϕ. The anisotropic dielectric tensor was derived in Ref. [7] and is given by the following equation in the cylindrical coordinate system.
where and Other components are equal to zero. In Eqs. (1)–(5), the superscript T denotes the transposed vector, ε∞ is the dielectric constant at high frequencies, ω is the angular frequency, and ωL (ωT) denotes the longitudinal (transverse) exciton frequency. While is a tensor, , where δij is Kronecker’s delta, is a scalar. On the other hand, inverse tensor satisfies .
Fig. 1 (a) Top and (b) side views of a bent organic molecular-crystal nanofiber on a glass substrate in the cylindrical coordinate system. The nanofiber has w = 500 nm and h = 150 nm. The tilt angle of the transition dipole moment is α = 60°. R is the radius of curvature. (c) Configuration of the nanofiber on the glass substrate after the conformal transformation.
2.2. Wave equation
The magnetic field H(r) satisfies the eigenvalue equation derived from the Maxwell wave equation:
where c is the speed of light in free space. As shown in Fig. 1, our geometry is independent of ϕ, so ε(r; ω) = ε(r, z; ω). To calculate the bending loss of the wave propagation, we assume a complex propagation constant β along the fiber axis and denote the magnetic field H(r) by where R is the radius of curvature, which is defined by the distance between the fiber axis and the origin (see Fig. 1). . Substituting Eq. (7) into Eq. (6), we obtain We have used ∇ ·H(r) = 0 → ∂(rHr)/∂r + r∂Hz/∂z + iβRHϕ = 0 in Eqs. (8) and (9). To remove the singularity at the origin, we introduce the following conformal transformation: which pushes the origin of coordinate r away to the minus infinity of coordinate u. Then Eqs. (8)–(10) are rewritten as where H̃r = (r/R)Hr and H̃ϕ = (r/R)Hϕ. In this description, the cylindrical coordinate system is transformed into the Cartesian one as shown in Fig. 1(c). Instead, e2u/R appears as a result of the conformal transformation. When R → ∞, e2u/R → 1, and then, Eqs. (12)–(14) are reduced to the case of straight waveguides.By discretizing these equations with respect to u and z, according to the Yee algorithm [16], we obtain their matrix equations.
where H̃r, Hz and H̃ϕ are column vectors composed of discretized H̃r, Hz and H̃ϕ, respectively, and Aij, Bij,Cij,Dij,Gij and Mij are coefficient matrices. In Eq. (15), for example, Arr corresponds to the discretization of the differential operator of the first, second and third terms on the left-hand side of Eq. (12). Likewise, Arz is for the fourth and fifth ones, Brz is for the sixth one, and Crϕ is for the seventh one. Other coefficient matrices in Eqs. (16) and (17) are also derived in the same manner. In the absence of the optical anisotropy (εrϕ = εϕr = 0), Eqs. (15) and (16) become This is an eigenvalue equation of β2. In the presence of the optical anisotropy (εrϕ = εϕr ≠ 0), however, H̃ϕ in Eqs. (15) and (16) has to be eliminated, using Eq. (17). Then, Eqs. (15) and (16) become where and (i, j = r, z). (Brr = 0) Due to the term proportional to iβ on the left-hand side, this is not an eigenvalue equation of β2. Therefore, Eq. (19) is transformed into This is the eigenvalue problem for the matrix on the left-hand side with an eigenvalue denoted by −iβ. Note that the factor iβ included in the notation of the eigen vectors on both sides does not cause a problem, since their mathematical definition is given in the third and fourth rows, so Eqs. (19) and (20) are exactly equivalent to each other. However, more computational time is required because the coefficient matrix is twice as large as that in Eq. (18). The coefficient matrix of Eq. (20) is non-Hermitian, so in general, eigenvalue β is a complex number: β = βr + iβi. The imaginary part βi represents the bending loss.In principle, Eqs. (8)–(10) and Eqs. (12)–(14) should give the same results [15]. But in practice, the numerical solutions of Eqs. (8)–(10) did not satisfy β(b) = −β(f) for anisotropic dielectric tensors as they should, where (f) and (b) denote the forward and backward waves, respectively. Since in the Yee algorithm the magnetic field is not set up in terms of εrϕ (= εϕr), it has to be substituted by the simple average of neighboring components. In our opinion, the simple average would cause serious numerical errors in the cylindrical coordinate system. For Eqs. (12)–(14), on the other hand, such errors do not occur. In the Cartesian coordinate system after the conformal transformation, the simple average would be still valid. In this paper, therefore, the conformal transformation is taken.
While there is an organic molecular-crystal nanofiber for R − w/2 < r < R + w/2 and 0 < z < h, a glass substrate is for z < 0. The cross section of the nanofiber is w×h. In the conformal transformation, there is an organic molecular-crystal nanofiber for u− < u < u+ and 0 < z < h, where u± = Rln[(R±w/2)/R], and the cross section is w′ × h, where w′ = u+ − u− = Rln[(R+ w/2)/(R − w/2)].
In a computational domain [Fig. 1(c)], perfectly matched layers (PML’s) are set up in the right, top and bottom regions in order to prevent reflection of leaked light at the domain boundaries. In the PML regions, the infinitesimal change of each coordinate is replaced by
where ξ = u, z, and σξ(ξ) is a conductivity [17]. σξ(ξ) is zero at the starting point ξs of the PML region, and gradually increases as ξ approaches the edge of the computational domain.When parabolic functions are used for σu(u), Eq. (22) can be calculated analytically.3. Numerical results and discussion
Figures 1(a) and 1(b) show the top and side views of the bent nanofiber on a glass substrate. The following values were assumed for the numerical calculation according to the experimental observation (see Ref. [7]): w = 500 nm, h = 150 nm, ε∞ = 2.34, α = 60°, h̄ωT = 2.85 eV, h̄ωL = 3.2 eV and εglass = 2.34. Figure 1(c) shows the configuration after the conformal transformation. In the transformed geometry, the overall computational domain is −2w′ < u − u0 < 3w′ and −10h < z < 10h, where u0 = (u+ + u−)/2(< 0) is the center of the nanofiber. (For R = 1μm, a smaller region of −1.5w′ < u − u0 < 3w′ was taken.) The nanofiber occupies −w′/2 < u−u0 < w′/2 and 0 < z < h. In the discretization of Eqs. (12)–(14), non-uniform meshes were used to reduce the computational memory and time [16]. Around the nanofiber, finer meshes were used. While the original discrete spacing for u and z was Δu = w′/10 and Δz = h/3, respectively, finer discretization of Δu/2 and Δz/2 was used for −w′ < u − u0 < w′ and −h < z < 2h, respectively. Outside the region surrounded by dashed lines, PML was set up for u0 + 2w′ < u < u0 + 3w′ (right region) and 6h < |z| < 10z (top and bottom regions). In Eq. (21), σu(us) = 0 at us = u0 + 2w′ and σz(zs) = 0 at |zs| = 6h. In these regions, light leaked from the nanofiber decays slowly without reflection.
Before presenting the numerical results of the bending loss, we briefly describe the optical properties of the thiacyanine nanofibers on the glass substrate clarified so far. By optical excitation, fluorescence emission of approximately 2.2–2.6 eV is generated and propagated through the nanofiber with small absorption, which results from the large separation between the absorption and emission bands [4]. A very small bending loss was observed for the radius of curvature of several microns as shown in Figs. 2(a) and 2(b), which cannot be understood by the total internal reflection caused by the small difference in the refractive indices of the nanofiber (1.5–1.7) and the surrounding medium. Although the radius of curvature around point A is approximately five microns, the intensity (solid line) is almost the same as that in a straight nanofiber (dashed line).
Fig. 2 (a) Microscope image of a bent thiacyanine nanofiber, (b) the signal intensity of the scanning optical microscope along the nanofiber, and (c) experimental result of the group refractive index estimated from Fabry-Perot peaks in fluorescence emission spectra. Experimental data were provided by Takazawa [3, 5].
Figure 2(c) shows the experimental result (dot) of the group refractive index ng estimated from the Fabry-Perot peaks in the fluorescence spectra [3] and the theoretical calculation (solid line) by the plane-wave expansion method in our previous study [7]. The group refractive index is more than 10 for photon energy larger than 2.55 eV. This unexpectedly large group index was brought about by the peculiar dispersion relation of the anisotropic exciton polariton in the nanofiber. As shown in the inset of Fig. 2(c), the strongest fluorescence was observed by the polarized excitation and detection (arrows) with a tilt angle of 60° from the fiber axis. This means that the transition dipole moment of the constituent molecules is oriented at α = 60°. By exactly considering the optical anisotropy, we succeeded in explaining the experimental result of the high group index [7] as shown in Fig. 2(c).
Now we proceed to the calculation of the bending loss. In Fig. 3, we show the dispersion curves of straight thiacyanine nanofibers on the glass substrate. The gray region is the light cone, in which light leaks into the glass substrate. Solid line and circle denote the numerical results of the plane-wave expansion (PWE) [7] and the present method, respectively. While in the PWE method the electromagnetic field is expanded with the basis of plane waves [7], in the present method the discretization of Eqs. (12)–(14) is used with a condition of e2u/R = 1. These two methods gave consistent results as shown in Fig. 3. Note that the propagation constant β is real under the light line for the straight waveguide. The group refractive index (solid line) in Fig. 2(c) was derived from ng = c/vg, where vg = ∂ω(β)/∂β is the group velocity and ω(β) is the eigen frequency of the lowest mode. Other modes do not contribute to the propagation of the fluorescence very much. For example, the second lowest mode has an anti-symmetric electric-field distribution that does not couple with the fluorescence emitted by the constituent molecules [7]. So, we will focus on the lowest mode in the following.
Fig. 3 Dispersion curves of the straight organic molecular-crystal nanofiber on the glass substrate.
Figures 4(a) and 4(b) show the real and imaginary parts of the wave number of the lowest mode, respectively, as a function of the radius of curvature. Only βi > 10−11μm−1 is shown in Fig. 4(b) because of the numerical accuracy. In the following, we focus on three representative cases: h̄ω = 2.3 eV (539.13 nm), 2.4 eV (516.67 nm), and 2.5 eV (496.0 nm). In Fig. 4(a), dashed lines are the wave number of the straight waveguide. βr gradually increases with decreasing R. On the other hand, βi decreases with increasing h̄ω, since the dispersion curve rapidly goes apart from the light line (see Fig. 3). Note that βi is very small even for R less than 10 μm. In general, the bending (power) loss per unit length, which is given by 2βi, satisfies 2βi = C1 exp(−C2R) for sufficiently large R, where C1 and C2 are constants. Dashed lines denote such an exponential fitting for which C1 and C2 were determined by the lowest and the second lowest points of βi. In other words, βi increases exponentially with decreasing R. This feature is the same as that derived analytically for bent planar waveguides [8]. For the planar waveguides, the dispersion relation of the guided modes and the bending loss can be derived analytically, which is accurate even for small R if the wave number is distant enough from the light line (see Appendix). With increasing h̄ω, the behavior of βi actually coincides with the dashed line. The right vertical axis in Fig. 4(b) is the bending loss in dB/μm, which is equal to 20log10e × βi = 8.6859βi. For R = 3 μm, the bending loss at h̄ω = 2.3 eV is as small as 10−4 dB/μm. Therefore, it is natural that the observed leakage of light was negligibly small in Fig. 2(a).
Fig. 4 (a) Real and (b) imaginary parts of the wave number of the lowest dispersion curve as a function of the radius of curvature.
Finally, we examine the leakage of light in the bent nanofiber with R = 1 μm. Figures 5(a) and 5(b) show |Hr(r, z)| and |Hz(r, z)| of the lowest mode, respectively, at h̄ω = 2.3 eV. |Hz(r, z)| is approximately five times as large as |Hr(r, z)|. The magnetic-field distribution shifts to the right-hand side and leaks into the glass substrate. Likewise, Figs. 5(c) and 5(d) are for h̄ω = 2.5 eV, for which the leakage of the magnetic field is negligible even for the very small radius of curvature. In the higher energy region, the bent thiacyanine nanofibers are robust against the leakage of light.
Fig. 5 Leakage of light in bent organic molecular-crystal nanofibers for R = 1 μm. While Figs. (a) and (b) show |Hr(r, z)| and |Hz(r, z)| of the lowest dispersion curve, respectively, at h̄ω = 2.3 eV, Figs. (c) and (d) are for h̄ω = 2.5 eV.
4. Conclusions
We theoretically examined the bending loss of optically anisotropic exciton polaritons in molecular-crystal nanofibers as a function of the radius of curvature R. In the cylindrical coordinate system, the wave number of the propagating mode is complex, and its imaginary part gives the bending loss. We removed the singularity of the wave equation at the origin by further introducing a conformal transformation, which allowed us perform an accurate numerical analysis by a finite-difference frequency-domain method. The present method could also be applied to straight fiber waveguides by setting R → ∞ and it gave the dispersion relation of a thiacyanine nanofiber that is consistent with our previous calculation by the plane-wave expansion method.
Although the bending loss of the nanofiber increases exponentially with decreasing R, it is still small for R less than 10 μm. On the other hand, the bending loss decreases with increasing ω, since the dispersion curve goes apart from the light line. This result supports the experimental observation by Takazawa that the leakage of light was negligibly small for bent thiacyanine nanofibers with the radius of curvature of several microns. So, the polariton nanofiber gives a novel possibility for bent waveguides to fabricate optical microcircuits and interconnection that cannot be attained by the conventional waveguides like Si waveguides based on the index guiding.
5. Appendix: Validity of the description 2βi = C1 exp(−C2R) for the bending loss
We refer to the discussion in Ref. [8]. It is assumed that a planar bent waveguide is configured like in Fig. 1(a). Inside and outside the waveguide, the phase velocity is given by Rdϕ/dt = ω/β and (R + x)dϕ/dt = ω/β0, respectively, where x = r − R is a distance from R, and β0 is the wave number in the background medium. Then, the critical radius xc = {(β/β0) − 1}R is obtained. Since electromagnetic waves for x > xc cannot follow the speed inside the waveguide, they leak into the background. Then, the bending loss is proportional to . In straight waveguides, electromagnetic waves exponentially decay for |x| > w/2, like H(x) ∝ exp{−q(|x| − w/2)}, where q is the decay rate. Only for xc > w/2 or R > w/[2{(β/β0) − 1}], 2βi ∝ exp[−2q{(β/β0) − 1}R] = C1 exp(−C2R) is derived. When the wave number β is distant enough from the light line β0 (β/β0 is large enough), this description is valid even for small R. A more accurate analytical description is given in Ref. [18].
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