The interaction between a periodic and conventional dielectric waveguides is investigated theoretically for a two-dimensional model system. A modified coupled-mode theory is formulated for the considered system and found to agree well with rigorous numerical calculations. It is shown that in a certain wavelength range the contra-directional coupling between the two waveguides can be achieved with high efficiency. But the spectrum of the coupling efficiency is blue-shifted and thus the strongest coupling does not occur in the case when two individual waveguides have the same propagation constant. For such a contra-directional coupling system, the coupling efficiency grows with the coupling length and it tends to 100% (excluding insertion loss) when the coupling length is larger than a certain value, and the coupling window can be largely broaden by reducing the distance between the coupled waveguides.
© 2010 Optical Society of America
Backward waves in left-handed materials (LHM) are of interest because they are the foundation for a variety of novel phenomena . Backward waves are such electromagnetic (EM) waves whose Poynting vector and wave vector are in opposite directions. In the waveguiding systems, the counterpart of backward waves is backward mode with antiparallel energy and phase flows, which can be supported by a waveguide containing LHM [2, 3]. It has been suggested that coupled LHM waveguide and conventional waveguide can realize contra-directional coupling [4–8]. Such a contra-directional coupling is completely different from that achieved by grating-assisted couplers [9–12], because it has a broad bandwidth with high efficiency and is insensitive to coupling length. For the coupled LHM and conventional waveguides, the coupling efficiency can monotonously grow with coupling length and tends to 100% (exclude insertion loss) when the coupling length is larger than a certain value . This type of contra-directional coupling offers a new possibility in the design of optical components and circuits . So far, however, the contra-directional coupling based on LHMs has been well demonstrated only in the microwave regime [5–7] and its extension to the optical regime is severely restrained due to large dissipation and anisotropy in the LHMs .
It seems that the promising approach for realizing contra-directional coupling based on backward wave in optical regime is to exploit dielectric photonic crystals (PCs) [15, 16], in which backward-wave propagation is feasible at higher bands. The contra-directional coupling between a two-dimensional photonic crystal waveguide (PCWG) and a optical fiber has been demonstrated with high efficiency . A PCWG is formed by introducing a line defect into the lattice and it guides light based on the band-gap effect of the cladding PC, which must be wide enough in transverse direction to avoid the propagation loss. Compared to PCWGs, periodic dielectric waveguides (PDWGs), a special type of PCs, possess remarkable advantages such as compact transverse dimension, flexible coupling fashion, and broad bandwidth . Moreover, as the PDWG guides light based on the effective total internal reflection, it can naturally transforms into a uniform waveguide by gradually changing the dielectric filling fraction along the propagation direction, thus the link between a PDWG and a uniform dielectric waveguide is quite easy. So far, however, the PDWGs have not yet received much attention, especially on their property of backward wave propagation. In this paper, we will investigate theoretically the interaction between a PDWG and a CDWG and for this purpose a modified coupled-mode theory is formulated. We will analyze the coupling phenomena associated with the backward mode in the PDWG and show that the coupled PDWG and CDWG can makes a contra-directional coupling with high efficiency.
2. Modified coupled-mode theory
With the hope of eliciting essential physical properties, we choose to perform studies on two-dimensional (2-D) model systems, which are uniform in the y direction. Let us consider two dielectric guiding layers separated by a distance d in the x direction, as illustrated in the inset of Fig. 1(a). The upper layer is an array of rectangular dielectric columns of height a (a also represents the layer thickness), width b, and lattice constant p, while the lower one is a dielectric slab of thickness w. The columns with the relative permittivity εr 1 and slab with the relative permittivity εr 2 are surrounded by a third dielectric with the relative permittivity εr 3. As an illustrative example, we take the parameters of the waveguide system as follows: a = 0.26 μm, b = 0.28 μm, p = 0.34 μm, and w = 0.21 μm; εr 1 = 12.25 (Si), εr 2 = 6 (As2S3), and εr 3 = 2.1 (SiO2). Different values of d will be analyzed. In this waveguide system, waves travels along the z direction and they are assumed to be the E-polarization, i.e., the EM fields have the form of E = ŷ Ey and H = x̂ Hx + ẑ Hz. We investigate theoretically the contra-directional coupling between the two guiding layers, which correspond to a periodic dielectric waveguide (PDWG) and a conventional dielectric waveguide (CDWG), respectively. The dispersion relations for the individual PDWG (solid lines) and CDWG (dotted line) are shown in Fig. 1(a). The modes in the PDWG are solved by using Ho’s plane-wave expansion method (PWEM)  and supercell technique . In the numerical calculation, the period of the supercell in the x direction is chosen to be P = 6p+a. Ho’s PWEM has the property of fast convergence for the E-polarization  and we employ 201×31 plane waves. To examine the accuracy of the obtained results, for the propagation constant β = 0.412(2π/p) we calculate the accurate values of the normalized frequencies of (three) modes in PDWG with a large plane wave number (Npw = 601×91) and then find the corresponding results shown in Fig. 1(a) to be accurate within 0.1%. For the PDWG, there exist four modes, and the second one with negative group velocity is of our interest, which is a backward mode with antiparallel energy and phase flows. The dispersion band of this backward mode intersects the dispersion curve for the CDWG at β = 0.412(2π/p), i.e., at a free-space wavelength of λ = 1.55 μm. In what follows, we restrict ourselves to the wavelength range of the second band for the PDWG.
The coupled-mode theory (CMT) is commonly used to study coupled waveguide systems as it is intuitive and insightful . The conventional CMT has also been successfully applied to the studies of grating-assisted couplers [9–11], in which the fields are expressed as a linear superposition of basis vectors, which represent the fundamental modes of the unperturbed waveguides in isolation. As the field profile of a propagating mode is insensitive to the index modulation in the waveguide core, such a field expression for the grating-assisted coupler is still valid, and the grating effect primarily leads to the conversion between the fundamental modes propagating in the opposite directions in the same waveguide. However, the conventional CMT seems not to be applicable to the waveguide system considered here. In the PDWG of the system, the field (transverse) profile of the Bloch mode greatly changes along z within the unit cell [see Fig. 3(c)], and it is very difficult to find a uniform waveguide as a reference waveguide whose fundamental mode can effectively represent the modal field profile of the PDWG in the whole unit cell. On the other hand, as the PDWG has core layer sections whose permittivity is the same as that of the cladding, we cannot physically introduce the concept of local mode for the PDWG to represent transversely localized fields as for conventional nonuniform waveguides . To circumvent these problems, we properly modify the conventional CMT for the present system. In this modified CMT, we treat the whole fields in the unit cell of the Bloch mode of the individual PDWG as a basis vector, and correspondingly introduce discrete modal coefficients for the cells of the PDWG. In this way, the modified CMT is formulated as a set of linear algebraic equations, which are not the same results obtained from the discretization of the coupled mode differential equations for the conventional CMT, and it is worth stressing that the coupling coefficients in the two cases are quite different.
For the coupled PDWG and CDWG, we think that the CDWG only has a perturbation to the PDWG and its modal field profile is almost preserved in the whole unit cell. The interaction between the two waveguides mainly leads to the slow variation of their modal amplitudes along the propagation direction. The structure of the coupled waveguide system is also periodic along z direction, and we divide it into many cells centered at zn = np, where n are integers. The guided fields in each cell of the entire structure with the (relative) permittivity profile εs(x, z) may be approximately expressed as a linear superposition of the two modes of the individual PDWG of the profile ε 1(x, z) and the individual CDWG of the profile ε 2(x), i.e., E(x, z) = A (n) 1 E 1(x, z)+A 2(z)E 2(x, z) and H(x, z) = A (n) 1 H 1(x, z)+A 2(z)H 2(x, z), where zn - p/2 ≤ z ≤ zn + p/2. The fields of the backward mode in the PDWG have the form of E 1 = ŷuy(x, z)e iβ1z and H 1 = v(x, z)e iβ1z, where uy and v = (vx,0,vz) are periodic in the z direction, and β 1 is the Bloch wavevector limited to the first Brillouin zone, i.e., ∣β 1∣ ≤π/p. The mode in the CDWG is a fundamental mode, whose fields are expressed as E 2 = ŷey(x)e iβ2z and H 2 = h(x)e iβ2z, where h = (hx,0,hz) and β 2 is the propagation constant. The modal amplitudes A (n) 1 and A 2(z) vary slowly with n and z, respectively, which accounts for the coupling. From the reciprocity theorem , we have
where F c = E×H * m+E * m × H, with m = 1 or 2 denoting the PDWG or the CDWG. In the right side of Eq. (1), the integration with respect to z is related to two neighbouring cells centered at z = zn and z n+1. As the field E is actually continuous at z = z n+1/2, we approximately take E(x, z) = (1/2)(A (n) 1 + A (n+1) 1)E 1(x, z)+(1/2)[A 2(zn)+A 2(z n+1)]E 2(x, z) for the region of zn ≤ z≤z n+1 when we evaluate this integral. Also, as A (n) 1 and A 2(z) vary very slowly, this expression for the field is an excellent approximation. Substituting all field expressions into Eq. (1) and neglecting small terms in it, we thus obtain discrete coupled-mode equations
where A (n) 1 denotes A 1(zn), Δβ = (β 2-β 1)/2, and the coupling coefficients are given by
where x c1 and x c2 denote the x coordinates of the PDWG and CDWG axes, respectively. N 1 and N 2 represent the modal powers of the PDWG and CDWG, respectively. It should be noted that N 1 and β 1 have the opposite signs since the mode in the PDWG is a backward mode.
As seen from Eq. (4) and Eq. (5), the integration in them is conducted with respect to both the transverse and longitudinal variables (x and z), therefore the coupling coefficients for the present CMT are conceptually different from those for the conventional CMT , which corresponds to an integration only with respect to the transverse variable. The effective coupling often requires a phase matching between two coupled waveguides, i.e., β1 ≈ β2. In this situation, the quantities N 1 and N 2 have opposite signs. It is interesting if the coupling coefficients K 12 and K 21 then have opposite signs, as in the case of the coupled LHM waveguide and CDWG . To clarify this, the coupling coefficients for the case of d = 0.75p are plotted as a function of wavelength in Fig. 1(b). As expected, K 12 and K 21 have opposite signs and their magnitudes are almost equal in the neighborhood of λ = 1.55 μm, at which β 1 = β 2.
3. Supermodes in the entire structure
To validate the discrete coupled-mode equations, i.e., Eq. (2) and Eq. (3), we first use them to solve for the eigen modes of the entire structure (referred to as supermodes) and make a comparison with the accurate results obtained from the PWEM. Let A (n) 1 = Ā(n) 1 e iΔβzn and A (n) 2 = Ā(n) 2 e −iΔβzn, then Eq. (2) and Eq. (3) are rewritten as
The coefficients in Eq. (6) and Eq. (7) are all constants independent on n. In terms of Ā1 and Ā2, the electric field in the entire structure is expressed as Ey(x, zn) = [Ā(n) 1 uy(x, zn)+Ā(n)2 ey(x)]e iβ̅zn, where β̅ = (β 1 +β 2)/2, thus for a supermode with propagation constant β we have Ā(n) m = Cm eiδβzn = Cmqn (m = 1, 2), where q = e iδβp and δ β = β - . Substituting into Eq. (6) and Eq. (7) and eliminating the coefficients Cm, we obtain
where . Evidently, there exist two supermodes in the entire structure, and β ±= +δ β ±= - i ln(q ±)/p.
From Eq. (8) we can easily analyze the guiding properties of the entire structure. In the case of our interest, where Δβ ≈ 0, Eq. (8) reduces to q ±= (1±K)/(1∓K), thus q + > 1 and q - = 1/q + < 1, indicating that δ β ± are equal and opposite imaginary numbers. Therefore, in this case two supermodes are a pair of evanescent modes that are decaying in the opposite directions [but with the same Re(β)]. Obviously, in a wavelength interval where ∣ sin(Δβ p)∣ < 2K/(1+K 2), the supermodes are always evanescent and the maximum of the decay rate ∣δ β∣ occurs at a wavelength for which Δβ = 0. In the case when ∣sin(Δβ p)∣ > 2K/(1+K 2), q ± become complex and we find ∣q ±∣ = 1, thus δ β ± are real numbers and correspondingly the supermodes are two propagating modes with different propagation constants. These guiding properties of the entire structure are well illustrated in Fig. 2, where solid and dotted lines respectively represent the real and imaginary parts of β calculated with Eq. (8).
To obtain the accurate results of the supermodes, we again adopt the PWEM and the supercell technique. But here, the PWEM is formulated as an eigenvalue problem in the form of ℋX = β X, where X is a vector composed of the discrete Fourier coefficients of both Ey and Hx. In the formulation, Maxwells equations are converted into algebraic equations in the discrete Fourier space, and the product of ε Ey is Fourier factorized by Laurents rule, so the formulated PWEM has a merit of fast convergence . With such a PWEM, we can even solve for evanescent supermodes with complex β. In the numerical calculation, the period of the supercell in the x direction is taken to be P = 6p+(a+ d + w) and we employ 251×31 plane waves. The obtained results are also plotted as (solid or open) circles in Fig. 2. To examine the convergence of the PWEM and the accuracy of the obtained results, we calculate the propagation constant (β) of the supermodes as a function of plane wave number (Npw) for the case of d = 0.75p and λ = 1.55 μm. The values of the real and imaginary parts of β converge quickly as Npw grows and they are almost constant when Npw is larger than 301×41, and the values at Npw =251×31 are found to be accurate within 1%. Evidently, the agreement of the results from the modified CMT and the PWEM are remarkable, especially in the case with d = p, for which the coupling between the two waveguides is weaker.
However, different from what predicted by the modified CMT, the maximal decay rate of the evanescent supermodes actually occurs at a wavelength a little bit smaller than λ = 1.55 μm, i.e., in the case of β 2 > β 1, as shown in Figs. 2(d)–2(f). The actual spectrum of the evanescent supermodes is blue-shifted and the blue-shift becomes more evident with decreasing d. The modified CMT is an approximate theory and the primary approximation made in it is to represent the fields in the entire structure by a linear superposition of the modal fields of the individual waveguides. Evidently, this approximation neglects the influence of the spatial variation of the modal fields (uy,v) on the phase matching between the coupled PDWG and CDWG. On the other hand, the blue-shift implies that the field pattern in the unit cell of the PDWG is somewhat modified due to the interaction with the CDWG. To examine this, the electric field amplitudes of the supermode with β + (at λ = 1.55 μm) calculated from the modified CMT and the PWEM are presented in Fig. 3, where the field pattern of the backward mode of the individual PDWG is also plotted for comparison. If the field in Fig. 3(b) (obtained from the PWEM) is still viewed as a linear sum of the modal fields of the CDWG and PDWG, then the modal field of the PDWG is evidently modified in the presence of the CDWG, as seen from the comparison of Fig. 3(b) and Fig. 3(a) and Fig. 3(c). Here, we should point out that the blue-shift in the spectrum of the evanescent supermodes is substantially less than the spectral width, so the modified CMT is still useful for analyzing the contra-directional coupling between the PDWG and CDWG, especially when it is associated with the evanescent supermodes.
4. Coupling characteristics
We now consider the coupling between the PDWG and CDWG. Suppose that an initial power with wavelength λ is injected into the left end of the CDWG at z = -p/2 and the PDWG is terminated at z = (N + 1/2)p. In this case there are (N + 1) cells of the PDWG located in the coupling region, and the coupling length is L = (N +1)p. Evidently, if L → ∞ and λ lies in the wavelength range of the evanescent supermodes, only a single supermode decaying in the positive z direction is excited in the coupling region of the waveguide system. As the total power for an evanescent supermode is zero, the energy flow in the CDWG is completely coupled into the PDWG and then outputs backward at z = -p/2. In this situation, the fields in both the PDWG and CDWG decay along z and evidently, the effective coupling length is Leff = 1/Im(β −). The shorter Leff is, the stronger the coupling between the PWD and CDWG is. Clearly, the strongest coupling actually does not occur at λ = 1.55 μm where β 1 = β 2, but is blue-shifted to a smaller wavelength (as seen from Fig. 2), where β 2 is a little larger than β 1. This means that for the CDWG to effectively couple with the PDWG, its phase matching is not only related to the Block wavevector β 1 but also to the spatial variation of the modal field of the PDWG within the unit cell. The blue-shift behavior makes the present contra-directional coupling quite different from that between the CDWG and the LHM waveguide.
In a general case with finite L, both supermodes are excited in the coupling region and the coupling process can be treated as the interference of two supermodes [12, 21]. This is mathematically reflected in Eq. (6) and Eq. (7), which have the general solution Ā(n) m = C − m e iδβ-zn +C + m e iδβ+zn , where C m ± (m = 1, 2) are constants. From the boundary conditions of A (N) 1 = 0 and A (0) 2 = A 0, we find the coupled-mode solution for the coupled waveguide system
where τ ±= (i/2)K 12(e iΔβp/2 + q ± e −iΔβp/2)/(q ± e iΔβp/2 - e −iΔβp/2). Thus, the contra-directional coupling efficiency, which is defined as η = ∣N 1∣∣A (0) N 1||A (0) 1|2/[N 2|A (0) 2|2], is found to be
Figure 4(a) shows the coupling efficiency as a function of L calculated with Eq. (11) for the case of d = 0.75p. When the supermodes are evanescent modes at λ = 1.55 μm, η grows with L and it tends to 100% when L ≥ 30 μm. When the supermodes are propagating modes at λ = 1.52 μm (or 1.58 μm), η varies periodically with L and its maximal value is much less than 100%. Figure 4(b) shows the dependence of the coupling efficiency on L for different d values, and the wavelength is fixed at λ = 1.55 μm. For smaller d, η grows quicker with L and it tends to 100% at a smaller L, e.g., in the case of d = 0.5p, η tends to 100% when L ≥ 20 μm, which is considerably smaller than that for the case of d = 0.75p. It is desired that the coupling efficiency can be accurately computed using the exact supermodes obtained from the PWEM. To do so, we spatially divide the fields of each supermode in such a way that the (absolute) values of the energy flows on both sides of the division interface reach their maximum. But this division method is feasible only for two evanescent supermodes as they almost have the same division interface. In this way, the coupling efficiency as a function of L for λ = 1.55 μm is calculated numerically and also plotted as circles in Fig. 4(b), and it agrees well with that obtained from Eq. (11) for each value of d. To demonstrate the coupling behaviors described above, we simulate a coupling system with d = 0.75p and L = 85p using the commercial finite element software COMSOL. In the simulation, the scattering boundary condition is used at the boundaries of the computation domain. We choose the left end of the CDWG as an input port and apply a source there by setting the amplitude of Ey to be unity. Note that the right end of the CDWG is tapered to avoid the end reflection. Figure 5 shows the amplitudes of the electric fields for the wavelengths λ =1.52, 1.55, and 1.58 μm. For the cases of λ =1.52 and 1.58 μm, a large fraction of power in the CDWG travels through the coupling region and finally outputs at the right end. In contrast, for λ = 1.55 μm, almost no power outputs at the right end of the CDWG. All these agree well with our above analysis.
The spectral information of the coupling efficiency is of particular interest, and this is displayed in Fig. 6, where solid lines correspond to the results obtained from Eq. (11) and dotted lines with circles to the accurate values from the numerical calculations. The coupling length is fixed at L = 44.2 μm and our numerical calculations show that at λ = 1.55 μm, the (accurate) coupling efficiency is η = 99% for d = p, η = 99.95% for d = 0.75p, and η = 100% for d = 0.5p. If we define a coupling window for which η ≥ 90%, then it has a width Δλ = 16.7 nm for d = p, Δλ = 26.4 nm for d = 0.75p, and Δλ = 38 nm for d = 0.5p. The coupling window is broadening when d is decreased. The center of the coupling window calculated from the modified CMT is always at λ = 1.55 μm and it is rather accurate for the case of d = p. But when d is reduced, the actual center is obviously blue-shifted and so is the coupling window, as seen in Fig. 6. The central wavelength of the window is actually λ c = 1.548 μm for d = 0.75p and λ c = 1.546 μm for d = 0.5p. Though the coupling window is blue-shifted, the wavelength of λ = 1.55 μm, at which β 1 = β 2, always lies within the window.
A modified CMT has been formulated and with which the interaction between the PDWG and
This work was supported by the Ministry of Education (Singapore) under Grant No. R263000485112.
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