**2. Principle of optical guidance**

Normally, optical fibers as cylindrical dielectric waveguides consist of core and cladding regions with at least two or more materials. Each region can be made of silica glass or other transparent materials such as plastic. Although practical optical fibers have several or a few layers of cylindrical cladding regions with cross-sectional ring shapes uniform along the light-travelling direction, these cladding regions can be functionally combined together and represented by the average cladding refractive index. The refractive index difference between the core and cladding can provide light confinement and hence guidance of light signal along the fiber. Upon travelling through the fiber, the optical signal is both attenuated and distorted. These effects impose a limit on how far the signal can travel in a fiber before it degrades beyond an acceptable level. The wavelength () of operation is an important consideration in fiber-optic communication systems. Practical fiber optic systems operate in a wavelength range of 0.8 μm to 1.6 μm, where glass attenuation is low. In particular, the silica glass fiber exhibits local minimums in the spectral attenuation curve at 1.3-μm and 1.55-μm regions, which are often referred to as the transmission windows. Many commercial fiberoptic communication systems are designed for operation at these two wavelengths.

Optical fibers, as any other electromagnetic waveguide, can support discrete modes of propagation [6]. Assuming the cladding is just air instead of the multiple cylindrical cladding layers with respective material parameters, a simplified step-index optical fiber can be considered as shown in Figure 1(a). For more specific values, the simplified step-index waveguide has a core radius (a) of 1.0 μm, core refractive index (n1) of 1.45, and cladding refractive index of 1.0. Then total internal reflection due to the difference between the core refractive index and the average cladding index can contribute to optical light guidance in the fiber. Analytical approach to calculate guided light propagation characteristics involves the method of separation of variables and the boundary conditions. For circularly cylindrical waveguides, the radial dependence of electromagnetic fields is governed by the Bessel differential equation, hence the solutions are described in terms of the Bessel and the modified Bessel functions. Figure 1(b) compares the effective refractive index ( ) of the fundamental mode obtained from the exact analytical solution and a numerical technique. The numerical approach will be described in the following section.

Showing the arrangement method of small-hole cladding on a transverse plane, modelling of the MHOF without and with an air core is addressed. For the electromagnetic analysis of the proposed microstructures with complicated geometries, the utilizations of a finite difference method (FDM) and a finite-difference time-domain (FDTD) technique are considered in the investigation of optical guidance properties, such as normalized propagation constant, mode field distribution, effective area, and chromatic dispersion. Employing the two numerical methods provides cross-verification and additive confidence in the accuracy of the results. Based on the FDTD and the FDM techniques, how the cladding structure of the HMOF affects optical guidance properties are analyzed in section 4. Then the investigation is extended to find out the effects of the size and shape of the core

Normally, optical fibers as cylindrical dielectric waveguides consist of core and cladding regions with at least two or more materials. Each region can be made of silica glass or other transparent materials such as plastic. Although practical optical fibers have several or a few layers of cylindrical cladding regions with cross-sectional ring shapes uniform along the light-travelling direction, these cladding regions can be functionally combined together and represented by the average cladding refractive index. The refractive index difference between the core and cladding can provide light confinement and hence guidance of light signal along the fiber. Upon travelling through the fiber, the optical signal is both attenuated and distorted. These effects impose a limit on how far the signal can travel in a fiber before it degrades beyond an acceptable level. The wavelength () of operation is an important consideration in fiber-optic communication systems. Practical fiber optic systems operate in a wavelength range of 0.8 μm to 1.6 μm, where glass attenuation is low. In particular, the silica glass fiber exhibits local minimums in the spectral attenuation curve at 1.3-μm and 1.55-μm regions, which are often referred to as the transmission windows. Many commercial fiber-

optic communication systems are designed for operation at these two wavelengths.

modified Bessel functions. Figure 1(b) compares the effective refractive index (

The numerical approach will be described in the following section.

fundamental mode obtained from the exact analytical solution and a numerical technique.

) of the

Optical fibers, as any other electromagnetic waveguide, can support discrete modes of propagation [6]. Assuming the cladding is just air instead of the multiple cylindrical cladding layers with respective material parameters, a simplified step-index optical fiber can be considered as shown in Figure 1(a). For more specific values, the simplified step-index waveguide has a core radius (a) of 1.0 μm, core refractive index (n1) of 1.45, and cladding refractive index of 1.0. Then total internal reflection due to the difference between the core refractive index and the average cladding index can contribute to optical light guidance in the fiber. Analytical approach to calculate guided light propagation characteristics involves the method of separation of variables and the boundary conditions. For circularly cylindrical waveguides, the radial dependence of electromagnetic fields is governed by the Bessel differential equation, hence the solutions are described in terms of the Bessel and the

in the holey fiber on optical propagation properties in section 5.

**2. Principle of optical guidance** 

**Figure 1.** (a) Simplified step-index optical fiber rod, (b) comparison of normalized propagation constants of the fundamental mode in the fiber of (a)

Based on the analytical and numerical results of the normalized propagation constants, it is reasonably found that the dielectric rod waveguide supports the first four modes at λ = 1.3 μm, which agrees well with the results from the Gloge's mode chart for this waveguide. And transverse electric field distributions, Ex and Ey, can be obtained by using the numerical method. Figure 2 illustrates the normalized field distributions of the first two modes for the optical rod waveguide at the operating wavelength of 1.3 μm. Here, it is noted that the side length of each square cell for the numerical analysis is 0.1 μm.

**Figure 2.** Normalized field distributions of the first two modes for the optical rod fiber

(a) (b)

Once the electric field distribution has been determined, the effective core area for the fundamental mode can be obtained using the following expression [7]:

$$A\_{eff} = \frac{\left(\int\_{-\infty}^{\infty} \int |\mathbf{E}(\mathbf{x}, \mathbf{y})|^2 \, d\mathbf{x} dy\right)^2}{\int\_{-\infty}^{\infty} \int \left|\mathbf{E}(\mathbf{x}, \mathbf{y})\right|^4 dx dy} \tag{1}$$

where **E**(,) *x y* is the electric field on a transverse plane. For the optical waveguide of Figure 1(a), the results of the effective area are 1.7925 m2, 2.0624 m2, and 2.2148 m2 at the wavelengths of 0.8 μm, 1.3 μm, and 1.55 μm, respectively.
