## Abstract

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

## 1 Introduction

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}=${\u220a}_{\mathrm{I}}^{2}$) and
optically denser layers (thickness *b*, refractive index
*n*
_{II}=${\u220a}_{\text{II}}^{2}$), and that
${n}_{\text{II}}^{2}$ is greater than
${n}_{\mathrm{I}}^{2}$. 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

$$+\mathrm{cos}\left({n}_{1}\mathrm{cos}{\theta}_{1}ka\right)\mathrm{cos}\left({n}_{2}\mathrm{cos}{\theta}_{2}kb\right)$$

$$+\mathrm{cos}\left({n}_{1}\mathrm{cos}{\theta}_{1}ka\right)\mathrm{cos}\left({n}_{2}\mathrm{cos}{\theta}_{2}kb\right),$$

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
*i*th layer, which corresponds to the region
*r*
_{i-1}<*r*<*r*_{i}
,
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
*E*_{z}
(*r*) and the magnetic field
*H*_{z}
(*r*) for the
*i*-th layer are assumed to be expressed by using the first kind
Bessel function *J*_{m}
and the second kind
*Y*_{m}
, as follows:

where ${q}_{i}^{2}$=*ω*
^{2}
*∊*_{i}*∊*
_{0}
*µ*
_{0}-*β*
^{2},
*m* is a non-negative integer number, and
*θ*_{m}
is an real number.
*∊*
_{0} and
*µ*
_{0} denote the permittivity and the
permeability of the vacuum.

The factor exp(j*ωt*-j*βz*),
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*=*r*_{i}
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 (*A*_{i}*,B*_{i}
) and the couple
(*C*_{i}*,D*_{i}
). The TM-polarized wave, in
which *E*_{z}
≠0, can be expressed by
*A*_{i}
and *B*_{i}
. The
TE-polarized wave, in which *H*_{z}
≠0, can be
expressed by *C*_{i}
and *D*_{i}
.
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

By using

we get the coefficients in the core
*A*
_{0},*B*
_{0},
*C*
_{0}, and *D*
_{0}, where
arbitrary values are put on *A*_{n}*,B*_{n}*,
C*_{n}
, and *D*_{n}
(*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 $\sqrt{{\in}_{\mathrm{II}}\u2044{\in}_{I}}=2.0$, 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*=1.2*k*
_{0} and
3.0*k*
_{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
(2*n*+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,
2*π/k*
_{0}=1.5*µm*
and *l*=7.5*µm*, 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 *r*_{f}
and the length
*l*_{f}
are defined as
*r*_{f}
=1*µm* and
*l*_{f}
=2.5*µm*. The outer
surface of the CPOW is assumed to be radation boundary, and that of the port is
perfect conductor. There are guided modes TE_{mn} and TM_{mn} 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 TE_{01} mode of
*k*=1.2*k*
_{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 TE_{21},
TM_{01} and TM_{21}, respectively. Guided modes exists in each
cases. The modes excited by TE_{21} and TM_{21} are corresponding to
the *m*≠0 cases, that is the hybrid modes. Transmittances
are smaller than that of TE_{01}, because the input and output ports are
optimized for the TE_{01} mode input.

## 5 Conclusion

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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