Step-Index Fiber Bend

The data transmission speed of optical waveguides is superior to microwave waveguides because optical devices have a much higher operating frequency than microwaves, enabling a far higher bandwidth.

Today the silica glass (SiO2) fiber is forming the backbone of modern communication systems. Before 1970, optical fibers suffered from large transmission losses, making optical communication technology merely an academic issue. In 1970, researchers showed, for the first time, that low-loss optical fibers really could be manufactured. Earlier losses of 2000 dB/km now went down to 20 dB/km. Today’s fibers have losses near the theoretical limit of 0.16 dB/km at 1.55 μm (infrared light).

One of the winning devices has been the single-mode fiber, having a step-index profile with a higher refractive index in the center core and a lower index in the outer cladding. Numerical software plays an important role in the design of single-mode waveguides and fibers. For a fiber cross section, even the simplest shape is difficult and cumbersome to deal with analytically. A circular step-index waveguide is a basic shape where benchmark results are available (see Ref. 1).

This example is a model of a single step-index waveguide made of silica glass. The inner core is made of pure silica glass with refractive index n1 = 1.4457 and the cladding is doped with a refractive index of n2 = 1.4378. These values are valid for free-space wavelengths of 1.55 μm. The radius of the cladding is chosen to be large enough so that the field of confined modes is zero at the exterior boundaries.

For a confined mode there is no energy flow in the radial direction, so the wave must be evanescent in the radial direction in the cladding. This is true only if

On the other hand, the wave cannot be radially evanescent in the core region. Thus

The waves are more confined when neff is close to the upper limit in this interval.

For a bent fiber, the mode is no longer completely guided by the refractive index structure. This can be qualitatively explained by considering that for a straight waveguide, the wavefronts (planes with a constant phase) are orthogonal to the fiber axis. For a circularly bent fiber, see Figure 1, the wavefronts rotate around the center point of the circle with a constant angular velocity. As a result, the wavelength and the propagation constant varies with the distance from the circle center point. At some distance from the center point, the wavelength is longer than the local material wavelength. Consequently the propagation constant is smaller than the local wave number defined by the vacuum wavelength and the refractive index of the cladding material. Beyond this radius, the wave cannot have a constant angular velocity and the wavefronts must bend, implying that the wave starts to radiate energy out from the fiber. For a more complete discussion about waves in bent waveguides, see Ref. 2.

Figure 1: Schematic of a bent waveguide with dashed phase fronts indicated.

Model Definition

The first mode analysis is made on a cross section in the xy-plane of the fiber. The wave propagates in the z direction and has the form

where ω is the angular frequency and β the propagation constant. An eigenvalue equation for the electric field E is derived from Helmholtz equation

which is solved for the eigenvalue λ = −jβ.

As boundary condition along the outside of the cladding, the electric field is set to zero. Because the amplitude of the field decays rapidly as a function of the radius of the cladding this is a valid boundary condition.

The second mode analysis is performed for a 2D axisymmetric geometry. In this case, the wave propagates in the azimuthal direction,

, and the electric field is expressed as

where r0 is an average radius for the mode in the bent fiber. The radius r0 is often slightly larger than the radius of curvature for the bent fiber. The eigenvalue solved for in this case is λ = −jβr0. As a consequence of this eigenvalue definition, the effective indices you provide as input to the eigenvalue solver and the effective indices that the solver returns are all scaled with the radius r0.

The geometry is defined as a rectangle surrounding the circular core domain. To absorb the radiating mode, there is a perfectly matched layer (PML) surrounding the rectangular cladding domain. The wavelength in the PML should correspond to the wave vector component normal to the PML–cladding boundary. Here we approximate this wave vector component with the radial wave vector component for the radiating wave in the cladding. The radial wave vector component can be obtained by first defining the azimuthal wave vector component as

where the radial coordinate ρ is measured from the center of the waveguide. As seen from the equation above, the azimuthal wave vector component equals the mode’s propagation constant within the waveguide core, where ρ ≈ 0, and it decreases when ρ increases.

We approximate the radial wave vector component, assuming that the squared sum of the radial and the azimuthal wave vector components should be equal to the wave number squared for the cladding material. So, for the radial wave vector component we get

and then the corresponding wavelength can be defined as

As an approximation, the effective index could be replaced by the refractive index of the core material. The coordinate ρ should here be the distance from the waveguide core to the PML boundary.

Results and Discussion

When studying the characteristics of straight optical waveguides, the effective mode index of a confined mode,

as a function of the frequency is an important characteristic. A common notion is the normalized frequency for a fiber. This is defined as

where a is the radius of the core of the fiber. For this simulation, the effective mode index for the fundamental mode, 1.4444 corresponds to a normalized frequency of 4.895. The norm of the transverse electric field components and the electric field polarization are shown in Figure 2 below.

Figure 2: The norm of the transverse electric field (surface plot) and the electric field polarization (arrows) for the fundamental mode.

As a comparison, the longitudinal components of the electric and magnetic fields for this mode is shown in Figure 3 below. Comparing the color bars for Figure 2 and Figure 3, it is clear that the mode has a predominantly transverse polarization.

Figure 3: The surface plot visualizes the z component of the electric field. This plot is for the effective mode index 1.4444.

Figure 4 shows the result for the bend fiber, indicating that the mode is leaky and radiates some power in the radial direction.

Figure 4: Surface plot of the z component of the electric field for the mode in the bent fiber. The contour plot shows the

component (in the direction of propagation) of the magnetic field and the arrow plot shows the electric field polarization.

References

1. A. Yariv, Optical Electronics in Modern Communications, 5th ed., Oxford University Press, 1997.

2. A.W. Snyder and J.D. Love, Optical Waveguide Theory, Chapman and Hall, 1983.

Wave_Optics_Module/Waveguides_and_Couplers/step_index_fiber_bend

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

First add some parameters that define the properties for the wave, the geometry, and the material.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lda0	1.55[um]	1.55E-6 m	Wavelength
f0	c_const/lda0	1.9341E14 1/s	Frequency
nClad	1.4378	1.4378	Refractive index, cladding
nCore	1.4457	1.4457	Refractive index, core
aCore	8[um]	8E-6 m	Core radius
aClad	40[um]	4E-5 m	Cladding radius
Rb	3[mm]	0.003 m	Bend radius
aSquare	100[um]	1E-4 m	Side length of square
tPML	20[um]	2E-5 m	PML thickness
dr	aSquare/2-tPML	3E-5 m	Distance from core center to PML boundary
ldaPML	lda0/sqrt(nClad^2-(nCore*Rb/(Rb+dr))^2)	1.1426E-5 m	Radial wavelength in PML

Silica Glass

Next, add two global material definitions. Those two materials will be used in both of the two model components that will be defined. The material properties will be added once these materials have been linked to the first model component.

In the window, under right-click and choose .

In the window for , type Silica Glass in the text field.

Doped Silica Glass

Right-click and choose .

In the window for , type Doped Silica Glass in the text field.

Straight Fiber

In the window, click .

In the window for , type Straight Fiber in the text field.

Geometry 1

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type aClad.

Click

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type aCore.

Click

Materials

Cladding

In the window, under right-click and choose .

In the window for , type Cladding in the text field.

Core

Right-click and choose .

In the window for , type Core in the text field.

Locate the section. From the list, choose .

Select Domain 2 only.

Global Definitions

Silica Glass (mat1)

Now, add the material properties for the globally defined materials.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	nClad	1	Refractive index

Doped Silica Glass (mat2)

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	nCore	1	Refractive index

Definitions

Add a variable for the norm of the transverse electric field. This will be used later in a plot.

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
normEt	(ewfd.Exewfd.Ex+ewfd.Eyewfd.Ey)/sqrt(ewfd.Exewfd.Ex+ewfd.Eyewfd.Ey)	V/m	Transverse electric field norm

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Study 1

Step 1: Mode Analysis

In the window, under click .

In the window for , locate the section.

Select the check box. In the associated text field, type nCore.

The modes of interest have an effective mode index somewhere between the refractive indices of the two materials. The fundamental mode has the highest index. Therefore, setting the mode index to search around to something just around the core index guarantees that the solver will find the fundamental mode.

In the text field, type f0.

In the window, click .

In the window for , type Study 1 (Straight Fiber) in the text field.

In the toolbar, click

Results

Electric Field (ewfd)

Click the

button in the toolbar.

Click the

button in the toolbar.

The default plot shows the distribution of the norm of the electric field for the highest of the 6 computed modes (the one with the lowest effective mode index).

Study 1 (Straight Fiber)/Solution 1 (sol1)

To study the fundamental mode, choose the highest mode index. Because the magnetic field is exactly 90 degrees out of phase with the electric field you can see both the magnetic and the electric field distributions by plotting the solution at a phase angle of 45 degrees.

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 45.

Electric Field (ewfd)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Surface 1

In the window, expand the node, then click .

In the window for , click in the upper-right corner of the section. From the menu, choose . This is the variable we added in a previous step. It is clear that the plot looks almost identical to the plot of the norm of the electric field, verifying that the electric field is predominantly polarized in the transverse direction (the xy-plane).

Electric Field (ewfd)

Indicate the polarization direction, using an arrow plot.

Arrow Surface 1