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Step-Index Fiber Bend
Introduction
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.
Application Library path: Wave_Optics_Module/Waveguides_and_Couplers/step_index_fiber_bend
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Mode Analysis.
6
Geometry 1
First add some parameters that define the properties for the wave, the geometry, and the material.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
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.
1
In the Model Builder window, under Global Definitions right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Silica Glass in the Label text field.
Doped Silica Glass
1
Right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Doped Silica Glass in the Label text field.
Straight Fiber
1
In the Model Builder window, click Component 1 (comp1).
2
In the Settings window for Component, type Straight Fiber in the Label text field.
Geometry 1
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type aClad.
4
Click  Build Selected.
Circle 2 (c2)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type aCore.
4
Click  Build Selected.
Materials
Cladding
1
In the Model Builder window, under Straight Fiber (comp1) right-click Materials and choose More Materials>Material Link.
2
In the Settings window for Material Link, type Cladding in the Label text field.
Core
1
Right-click Materials and choose More Materials>Material Link.
2
In the Settings window for Material Link, type Core in the Label text field.
3
Locate the Link Settings section. From the Material list, choose Doped Silica Glass (mat2).
4
Global Definitions
Silica Glass (mat1)
Now, add the material properties for the globally defined materials.
1
In the Model Builder window, under Global Definitions>Materials click Silica Glass (mat1).
2
In the Settings window for Material, locate the Material Contents section.
3
Doped Silica Glass (mat2)
1
In the Model Builder window, click Doped Silica Glass (mat2).
2
In the Settings window for Material, locate the Material Contents section.
3
Definitions
Add a variable for the norm of the transverse electric field. This will be used later in a plot.
Variables 1
1
In the Model Builder window, under Straight Fiber (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose All domains.
5
Locate the Variables section. In the table, enter the following settings:
Mesh 1
1
In the Model Builder window, under Straight Fiber (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Finer.
Study 1
Step 1: Mode Analysis
1
In the Model Builder window, under Study 1 click Step 1: Mode Analysis.
2
In the Settings window for Mode Analysis, locate the Study Settings section.
3
Select the Search for modes around 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.
4
In the Mode analysis frequency text field, type f0.
5
In the Model Builder window, click Study 1.
6
In the Settings window for Study, type Study 1 (Straight Fiber) in the Label text field.
7
In the Home toolbar, click  Compute.
Results
Electric Field (ewfd)
1
Click the  Zoom Extents button in the Graphics toolbar.
2
Click the  Zoom In button in the Graphics 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.
1
In the Model Builder window, expand the Results>Datasets node, then click Study 1 (Straight Fiber)/Solution 1 (sol1).
2
In the Settings window for Solution, locate the Solution section.
3
In the Solution at angle (phase) text field, type 45.
Electric Field (ewfd)
1
In the Model Builder window, under Results click Electric Field (ewfd).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Effective mode index list, choose 1.4444.
Surface 1
1
In the Model Builder window, expand the Electric Field (ewfd) node, then click Surface 1.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Straight Fiber (comp1)>Definitions>Variables>normEt - Transverse electric field norm - V/m. 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
1
In the Model Builder window, right-click Electric Field (ewfd) and choose Arrow Surface.
2
In the Settings window for Arrow Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Straight Fiber (comp1)>Electromagnetic Waves, Frequency Domain>Electric>ewfd.Ex,ewfd.Ey - Electric field.
3
In the Electric Field (ewfd) toolbar, click  Plot. As the geometry has rotation symmetry, the polarization direction is arbitrary.
Surface 1
Now visualize the longitudinal components of the electric and magnetic field.
1
In the Model Builder window, click Surface 1.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Straight Fiber (comp1)>Electromagnetic Waves, Frequency Domain>Electric>Electric field - V/m>ewfd.Ez - Electric field, z-component.
3
In the Electric Field (ewfd) toolbar, click  Plot.
Electric Field (ewfd)
Add a contour plot of the H-field.
Contour 1
1
In the Model Builder window, right-click Electric Field (ewfd) and choose Contour.
2
In the Settings window for Contour, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Straight Fiber (comp1)>Electromagnetic Waves, Frequency Domain>Magnetic>Magnetic field - A/m>ewfd.Hz - Magnetic field, z-component.
3
In the Electric Field (ewfd) toolbar, click  Plot. The distribution of the transversal E and H field components confirms that this is the HE11 mode. Compare the resulting plot with that in Figure 3.
Solution 1 (Straight Fiber)
Rename the dataset and the plot group to refer to the Straight Fiber model component.
1
In the Model Builder window, under Results>Datasets click Study 1 (Straight Fiber)/Solution 1 (sol1).
2
In the Settings window for Solution, type Solution 1 (Straight Fiber) in the Label text field.
Straight Fiber
1
In the Model Builder window, under Results click Electric Field (ewfd).
2
In the Settings window for 2D Plot Group, type Straight Fiber in the Label text field.
Root
Now add a 2D axisymmetric model component to model the bent fiber.
Add Component
In the Model Builder window, right-click the root node and choose Add Component>2D Axisymmetric.
Bent Fiber
In the Settings window for Component, type Bent Fiber in the Label text field.
Geometry 2
Add a circle representing the fiber core.
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type aCore.
4
Locate the Position section. In the r text field, type Rb.
Square 1 (sq1)
Add a square cladding region, representing the domain the mode essentially is propagating in.
1
In the Geometry toolbar, click  Square.
2
In the Settings window for Square, locate the Size section.
3
In the Side length text field, type aSquare.
4
Locate the Position section. From the Base list, choose Center.
5
In the r text field, type Rb.
Rectangle 1 (r1)
Finally, add three rectangle domains, where you will define perfectly matched layers (PMLs).
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type aSquare.
4
In the Height text field, type tPML.
5
Locate the Position section. In the r text field, type Rb-aSquare/2.
6
In the z text field, type aSquare/2-tPML.
7
Click the  Zoom Extents button in the Graphics toolbar.
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type tPML.
4
In the Height text field, type aSquare.
5
Locate the Position section. In the r text field, type Rb+dr.
6
In the z text field, type -aSquare/2.
Rectangle 3 (r3)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type aSquare.
4
In the Height text field, type tPML.
5
Locate the Position section. In the r text field, type Rb-aSquare/2.
6
In the z text field, type -aSquare/2.
7
Click  Build All Objects.
Bent Fiber (comp2)
Now, add the Electromagnetic Waves, Frequency Domain interface.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd).
4
Click Add to Bent Fiber in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Electromagnetic Waves, Frequency Domain 2 (ewfd2)
Add a Perfect Magnetic Conductor (PMC) exterior boundary condition.
Perfect Magnetic Conductor 1
1
Right-click Bent Fiber (comp2)>Electromagnetic Waves, Frequency Domain 2 (ewfd2) and choose Perfect Magnetic Conductor.
2
Materials
Now link to the previously defined materials.
Cladding
1
In the Model Builder window, under Bent Fiber (comp2) right-click Materials and choose More Materials>Material Link.
2
In the Settings window for Material Link, type Cladding in the Label text field.
Core
1
Right-click Materials and choose More Materials>Material Link.
2
In the Settings window for Material Link, type Core in the Label text field.
3
4
Locate the Link Settings section. From the Material list, choose Doped Silica Glass (mat2).
Definitions (comp2)
Next add the PML domains.
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
3
In the Settings window for Perfectly Matched Layer, locate the Geometry section.
4
From the Type list, choose Cylindrical.
5
Locate the Scaling section. From the Coordinate stretching type list, choose Rational.
6
From the Typical wavelength from list, choose User defined.
7
In the Typical wavelength text field, type ldaPML. This wavelength setting approximates the transverse (in the radial direction) wavelength for the mode.
Definitions (comp2)
Define a variable for the mode’s averaged radial position, by first defining the integration operator and then the actual variable.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
3
In the Settings window for Integration, locate the Advanced section.
4
Clear the Compute integral in revolved geometry check box.
Variables 2
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Mesh 2
1
In the Model Builder window, under Bent Fiber (comp2) click Mesh 2.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
In the table, clear the Use check box for Electromagnetic Waves, Frequency Domain 2 (ewfd2).
4
Locate the Sequence Type section. From the list, choose User-controlled mesh.
Size
Add the same mesh element size for the core and the cladding domains, as for the straight fiber model component.
1
In the Model Builder window, under Bent Fiber (comp2)>Mesh 2 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Finer.
Free Triangular 1
1
In the Model Builder window, click Free Triangular 1.
2
In the Settings window for Free Triangular, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Mapped 1
Add an extremely fine mapped mesh for the PML that resolves also the shortest transverse (radial) wavelengths.
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Size 1
1
Right-click Mapped 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extremely fine.
4
Click  Build All.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Mode Analysis.
4
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Electromagnetic Waves, Frequency Domain (ewfd).
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Mode Analysis
1
In the Settings window for Mode Analysis, locate the Study Settings section.
2
Select the Desired number of modes check box. In the associated text field, type 2.
3
Select the Search for modes around check box. In the associated text field, type nCore*Rb.
4
In the Mode analysis frequency text field, type f0.
5
In the Model Builder window, click Study 2.
6
In the Settings window for Study, type Study 2 (Bent Fiber) in the Label text field.
7
In the Home toolbar, click  Compute.
Results
Solution 3 (Bent Fiber)
Rename the dataset to refer to the Bent Fiber model component.
1
In the Model Builder window, under Results>Datasets click Study 2 (Bent Fiber)/Solution 2 (3) (sol2).
2
In the Settings window for Solution, type Solution 3 (Bent Fiber) in the Label text field.
Electric Field (ewfd2)
1
In the Model Builder window, expand the Results>Electric Field (ewfd2) node, then click Electric Field (ewfd2).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Effective mode index list, choose 0.0043364.
The default plot shows the norm of the electric field. The selected solution has the field mainly polarized in the z direction.
Surface 1
1
In the Model Builder window, click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type ewfd2.Ez.
4
Locate the Coloring and Style section. Click  Change Color Table.
5
In the Color Table dialog box, select Wave>Wave in the tree.
6
7
In the Settings window for Surface, locate the Coloring and Style section.
8
Clear the Color legend check box.
Contour 1
1
In the Model Builder window, right-click Electric Field (ewfd2) and choose Contour.
2
In the Settings window for Contour, locate the Expression section.
3
In the Expression text field, type ewfd2.Hphi.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Black.
6
Clear the Color legend check box.
Arrow Surface 1
1
Right-click Electric Field (ewfd2) and choose Arrow Surface.
2
In the Settings window for Arrow Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Bent Fiber (comp2)>Electromagnetic Waves, Frequency Domain 2>Electric>ewfd2.Er,ewfd2.Ez - Electric field.
3
Locate the Coloring and Style section. From the Arrow length list, choose Logarithmic.
4
From the Color list, choose Black.
Solution 3 (Bent Fiber) (sol2)
In the Model Builder window, under Results>Datasets click Solution 3 (Bent Fiber) (sol2).
Selection
1
Right-click Solution 3 (Bent Fiber) (sol2) and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Solution 3 (Bent Fiber) (sol2)
1
In the Model Builder window, click Solution 3 (Bent Fiber) (sol2).
2
In the Settings window for Solution, locate the Solution section.
3
In the Solution at angle (phase) text field, type 45.
Bent Fiber
1
In the Model Builder window, under Results click Electric Field (ewfd2).
2
In the Settings window for 2D Plot Group, type Bent Fiber in the Label text field.
3
Click the  Zoom Extents button in the Graphics toolbar. Compare your result with Figure 4.
Animation 1
1
In the Results toolbar, click  Animation and choose File.
2
In the Settings window for Animation, locate the Target section.
3
From the Target list, choose Player.
4
Locate the Scene section. From the Subject list, choose Bent Fiber.
5
Locate the Animation Editing section. From the Sequence type list, choose Dynamic data extension.
6
Check the Repeat check box in the Playing section of the Player settings.
7
Start the Player by clicking the Play button in the Graphics window.
8
Stop the Player by clicking the Stop button in the Graphics window.
r0
Evaluate the power-averaged mode radius. You should find that the average mode radius is 4 μm larger than the prescribed fiber radius of curvature (3 mm).
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, type r0 in the Label text field.
3
Locate the Data section. From the Dataset list, choose Solution 3 (Bent Fiber) (sol2).
4
From the Effective mode index selection list, choose From list.
5
In the Effective mode index list, select 0.0043364.
6
Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Bent Fiber (comp2)>Definitions>Variables>r0 - m.
7
Click  Evaluate.
n_eff_geometry
1
In the Results toolbar, click  Global Evaluation.
Evaluate the effective index, based on the prescribed radius of curvature of the fiber.
2
In the Settings window for Global Evaluation, type n_eff_geometry in the Label text field.
3
Locate the Data section. From the Dataset list, choose Solution 3 (Bent Fiber) (sol2).
4
From the Effective mode index selection list, choose From list.
5
In the Effective mode index list, select 0.0043364.
6
Locate the Expressions section. In the table, enter the following settings:
7
Click  Evaluate.
n_eff_power
1
In the Results toolbar, click  Global Evaluation.
Evaluate the effective index, based on the power-averaged mode radius.
2
In the Settings window for Global Evaluation, type n_eff_power in the Label text field.
3
Locate the Data section. From the Dataset list, choose Solution 3 (Bent Fiber) (sol2).
4
From the Effective mode index selection list, choose From list.
5
In the Effective mode index list, select 0.0043364.
6
Locate the Expressions section. In the table, enter the following settings:
7
Click  Evaluate.