Guided modes in a dielectric waveguide structure with a coaxial periodic multi-layer are investigated by using a matrix formula with Bessel functions. We show that guided modes exist in the structure, and that the field is confined in the core which consists of the optically thinner medium. The dispersion curves are discontinuous, so that the modes can exist only in particular wavelength bands corresponding to the stop bands of the periodic structure of the clad. It is possible that the waveguide structure can be applied to filters or optical fibers to reduce nonlinear effects.
© 2000 Optical Society of America
Recently, photonic devices using periodic structures, such as photonic crystals, have received a great deal of attention due to their potential for use to perform new functions. Photonic crystal fibers (PCFs) of which clads are two-dimensional photonic crystal structures have been investigated, experimentally and theoretically [1, 2, 3, 4, 5, 6, 7, 8, 9]. We can categorize PCFs into two types. In the first, refractive index of the core is larger than that of the clad. A guided wave mode, which occurs as a result of the difference of the refractive indices and is similar to that of a conventional optical fiber, is called a conventional mode (C-Mode) in this paper. In the second type, the refractive index of the core is smaller than that of the clad, thus the propagating wave can not be confined in the core, as a result of the difference of the refractive indices. However, when the propagating wave is in the photonic band gap of the photonic crystal, a guided mode may exist, because the wave is reflected into the core due to Bragg reflection. We call this type a Bragg mode (B-Mode). In this type PCF, an air hole can be used as the core to reduce nonlinear effects.
In this paper, we investigate a cylindrical waveguide structure having a coaxial periodic structure as its clad. We call it a coaxial periodic optical waveguide (CPOW). Its periodic structure consists of optically denser and thinner media, and it is periodic with respect to the radius. The reflective index of the core is smaller than the average in the clad, therefore, there is not a C-Mode, but B-Modes may exist, and we can expect that the dispersion curves of these modes to be discontinuous. Previously, optical resonators and surface-emitting lasers using circular grating were studied [10, 11, 12, 13, 14, 15, 16]. Their structures are similar to the CPOW and they are also similarly affected by Bragg reflection. But, those structures have slab waveguide structures, and their modes are considered as modified slab waveguide modes by using perturbative methods such as the mode matching theory. Thus, the modes in those structures are basically C-modes, while in this paper the B-modes propagating along the coaxial axis will be discussed. When the EM field in a coaxial structure depends on azimuth angle, the EM field can not be resolved into TE and TM components, thus we should consider them to be hybrid modes as we do in conventional optical fibers.
In section 2, we discuss an optical waveguide with one-dimensional photonic bandgap structures to show the basic concept of the B-mode. In section 3, we give an formulation of an EM field in a CPOW by using Bessel functions, and give formal expressions for guided modes. In section 4, we show the numerical results obtained by using the expressions given in section 3, and the finite element method.
2 Waveguide structure with one-dimensional periodic structures
Consider a waveguide structure with one-dimensional periodic structures as shown in Fig. 1. Guided waves are reflected by the periodic structures, and propagate in the gap between the periodic structures [17, 18, 19, 20]. We assume that the periodic structure consists of optically thinner layers (thickness a, refractive index n I=) and optically denser layers (thickness b, refractive index n II=), and that is greater than . The wavenumbers of the Bloch wave for the TE- and TM-polarized waves in the periodic structure, denoted by k TE and k TM, can be expressed by
where k is the wavenumber in free space, and θ 1 and θ 2 are as defined in Fig. 2. When
k TE is a pure imaginary number, and corresponds to an evanescent wave. Similarly, when
k TM is a pure imaginary number. Eqs. (3) and (4) define the stopbands of the periodic structure, in which the wave propagating in the periodic structure is perfectly reflected. In addition, when Eq. (4) is satisfied, | cos k TE(a+b)| is always greater than | cos k TM(a+ b)|. Thus, when a TM-polarized wave is in the stopband, a TE-polarized one is also. The decay constant of a TE-polarized wave is greater than that of a TM-polarized wave. Assuming that the waves in an optical waveguide having a coaxial periodic structure have characteristics similar to a waveguide having one-dimensional periodic structures, Eqs. (3) and (4) can express an approximate necessary condition for guided modes to exist.
3 Electromagnetic field in a coaxial periodic structure
Consider a waveguide having a cylindrical structure like the one shown in Fig. 3. The core of the waveguide is an air hole, whose radius is denoted by r 0, at the center of the cross-section. The clad consists of a coaxial periodic structure, where the radius of the i-th boundary is defined by
As shown in Fig. 4, the relative permittivity for the ith layer, which corresponds to the region r i-1<r<ri , is defined by
We use the coodinate system in which the z-axis corresponds to the axis of the coaxial structure. The xy components are denoted by a polar form x=r cos ϕ and y=r sin ϕ. The z-direction components of the electric field Ez (r) and the magnetic field Hz (r) for the i-th layer are assumed to be expressed by using the first kind Bessel function Jm and the second kind Ym , as follows:
where =ω 2 ∊i∊ 0 µ 0-β 2, m is a non-negative integer number, and θm is an real number. ∊ 0 and µ 0 denote the permittivity and the permeability of the vacuum.
The factor exp(jωt-jβz), which expresses the wave propagation along the z-axis, is suppressed for brevity. The field in the core (0<r<r 0) is denoted by Eqs. (7) and (8) of i=0, and should be finite even at r=0, so that the coefficients of the second kind Bessel functions, B 0 and D 0, should be zero. The z- and ϕ-direction components are expressed in the following matrix form:
The boundary conditions for tangential components of the electric and magnetic fields at r=ri are expressed by
In the case of m=0, where the field is independent of ϕ, we can discuss the field in the waveguide by using the concept of TE- and TM-polarized waves, since there is no conection between the coefficient couple (Ai,Bi ) and the couple (Ci,Di ). The TM-polarized wave, in which Ez ≠0, can be expressed by Ai and Bi . The TE-polarized wave, in which Hz ≠0, can be expressed by Ci and Di . However, when m≠0, TE- or TM-polarized waves do not exist alone, thus the guided mode should be treated as a hybrid mode.
By using Eq. (13) iteratively with putting initial values for the coefficients A 0 and C 0 in the core, we can obtain, the EM field in the waveguide. However, because of numerical calculation error, it is difficult to obtain a solution which decreases exponentially with respect to the radius r. In general, there exists an exponentially increasing solution when an exponentially decreasing solution exists in a periodic structure. Numerical calculation error contains both decreasing and increasing solutions, so that the solution which increases along the forward direction of the calculation steps becomes dominant after several steps, and this solution becomes independent of initial values. Thus, we can obtain decreasing solutions, with respect to the radius r, by using Eq.(13) iteratively, toward the core from a layer whose radius is much greater than the wavelength of the propagating wave.
4 Numerical calculation
we get the coefficients in the core A 0,B 0, C 0, and D 0, where arbitrary values are put on An,Bn, Cn , and Dn (n=100). Solutions for the guided modes are obtained by searching the exponentially decreasing ones, with respect to r, for which coefficients B 0 and D 0 are nearly equal to zero, with varying parameters k=ω∊ 0 µ 0 and β/k. The solutions are normalized to be A 0=1 or C 0=1. The thickness and the refractive index of the layers in the periodic structure are assumed as follows: k 0 a=k 0 b=1.0 and , where k 0 is an arbitrary unit wavenumber of the structure. We can obtain dispersion curves by plotting the points where guided modes exist. Figs. 5 and 6, respectively, show the dispersion curves for k 0 c=1 and k 0 c=5. Here, we consider the case of m=0, so that the TE- and TM-polarized modes can be obtained separately. The solution which satisfies B 0=0 is TM-polarized, and for D 0=0 it is TE-polarized. Here the x- and y-axes show, respectively, k/k 0 and β/k.
Fig. 7 shows the points which correspond to guided modes for k 0 r=100. In this case, the radius of the core is much larger than the wavelength, so that the region where the points exist approximately corresponds to the stopband of the coaxial periodic structure. Actually, the points shown in Figs. 5 and 6 are in the stopband shown in Fig. 7. For comparison, the stopband of the waveguide with one-dimensional periodic structures, calculated by using Eqs. (1) and (2), is shown in Fig. 8. Except for the region of β/k~1, the stopbands shown in Figs. 7 and 8 are analogous. When β/k~1, the wavelength with respect to r is large, so that the difference between a plane and cylindrical waves becomes significant. In addition, we can see that the stopband of the TM-polarized wave is contained by that of the TE-polarized wave.
The fields for the cases of k=1.2k 0 and 3.0k 0 are shown in Figs. 9 and 10, where k 0 c=5.0 (see Fig. 6). To clarify the decay due to exponential decay, the y-axis is assigned to electric or magnetic field multiplied by the square root of r, so that we can compensate for the intrinsic decay of the Bessel functions. The hatched region indicates the optically denser (2n+1-th) layers. The derivatives of the fields with respect to r are discontinuous at the boundaries of the layers. The discontinuity becomes large when β/k is large, as we can see from Eq. (11). In the case of m≠0, we should deal with a couple of exponentially increasing solutions whose increasing factors are different. If the difference is not small, the solution for which factor is larger becomes dominant, so that it is difficult to obtain linear combinations of these two increasing solutions.
To discuss wave field distribution and propagation loss of guided modes in a CPOW, EM wave field was calculated by using the finite element method (FEM). We consider a CPOW model structure which has nine layers with parameters k 0 a=k 0 b=1.0, k 0 c=5.0, 2π/k 0=1.5µm and l=7.5µm, where l denotes the length of the model structure along the z-axis. In the FEM, EM wave can be excited by distinct guided modes of the ports which are defined in model structures. When the mode in the port has rotational symmetry order of m, the excited EM wave field in the CPOW also has the same symmetry. Thus, we can specify the structure of excited EM wave field by choosing the input wave mode. To carry out numerical calculations, the ports should be so simple that the input wave modes can be specified easily. We put circular waveguides at each ends of the CPOW as input and output ports, where the radius of the circular waveguide rf and the length lf are defined as rf =1µm and lf =2.5µm. The outer surface of the CPOW is assumed to be radation boundary, and that of the port is perfect conductor. There are guided modes TEmn and TMmn in the ports. Numerical results were obtained by using HP HFSS 5.4. Fig. 11 shows the EM wave intensity distribution in the model excited by the TE01 mode of k=1.2k 0 in the input port, where input wave power is assumed to be 1W. We can see that the EM wave power is confined in the core of the CPOW. The transmittance to the output port |T| and the reflectivity |R| equal 0.986 and 0.007, respectively. Thus, the loss due to radiation from the CPOWis approximately 0.007. Figs. 12, 13 and 14 show the distributions excited by TE21, TM01 and TM21, respectively. Guided modes exists in each cases. The modes excited by TE21 and TM21 are corresponding to the m≠0 cases, that is the hybrid modes. Transmittances are smaller than that of TE01, because the input and output ports are optimized for the TE01 mode input.
We investigated the EM wave in a coaxial periodic optical waveguide. Existence of TE- and TM-polarized guided modes and hybrid guided modes was demonstrated by using both Bessel functions expansion and the finite element method. The guided wave can propagate in the air hole with slight loss, and it can exist only in the stopband. By using the waveguide with the coaxial periodic structure, we can obtain fibers which have functions of filter, or which can reduce nonlinear effects.
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