Step-Index Fiber
Introduction
The transmission speed of optical waveguides is superior to microwave waveguides becau optical devices have a much higher operating frequency than microwaves, enabling a far higher bandwidth.
Today the silica glass (SiO 2) fiber is forming the backbone of modern communication systems. Before 1970, optical fibers suffered from large transmission loss, making optical communication technology merely an academic issue. In 1970, rearchers showed, for the first time, that low-loss optical fibers really could be manufactured. Earlier loss of 2000 dB/km now went down to 20 dB/km. Today’s fibers have loss 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 ction, even the most simple shape is difficult and cumbersome to deal with analytically. A circular step-index waveguide is a basic shape where benchmark results are available (e 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 n 1 = 1.4457 and the cladding is doped, with a refractive index of n 2 = 1.4378. The values are valid for free-space wavelengths of 1.55 μm. The radius of the cladding is chon 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, thus 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 n eff is clo to the upper limit in this interval.
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n eff n 2
>n 2n eff n 1
<<
Model Definition
The mode analysis is made on a cross-ction 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 = −j β.
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As boundary condition along the outside of the cladding the electric field is t to zero. Becau the amplitude of the field decays rapidly as a function of the radius of the cladding this is a valid boundary condition.Results and Discussion
When studying the characteristics of 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
4.89
5. The electric and magnetic fields for this mode is shown in Figure 1 below.
E x y z t ,,,()E x y ,()e j ωt βz –()
=∇∇E ×()×k 02n 2
E –0
=n eff βk 0
-----=V 2πa λ0
----------n 12n 22–k 0a n 12n 22–==
Figure 1: The surface plot visualizes the z component of the electric field. This plot is for the effective mode index 1.4444.
Reference
1. A. Yariv, Optical Electronics in Modern Communications, 5th ed., Oxford University Press, 1997.
Model Library path: RF_Module/Tutorial_Models/step_index_fiber
Modeling Instructions化学性消化
From the File menu, choo New.
N E W
1In the New window, click Model Wizard.
M O D E L W I Z A R D
1In the Model Wizard window, click 2D.
英文读后感2In the Select physics tree, lect Radio Frequency>Electromagnetic Waves, Frequency Domain (emw).
3Click Add.
青藤的功效与作用4Click Study.
5In the Select study tree, lect Pret Studies>Mode Analysis.
6Click Done.
G E O M E T R Y1
1In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2In the Settings window for Geometry, locate the Units ction.
3From the Length unit list, choo µm.
Circle 1 (c1)
1On the Geometry toolbar, click Primitives and choo Circle.
2In the Settings window for Circle, locate the Size and Shape ction.
3In the Radius text field, type 40.
4Click the Build Selected button.
Circle 2 (c2)
中国登月计划1On the Geometry toolbar, click Primitives and choo Circle.
2In the Settings window for Circle, locate the Size and Shape ction.
3In the Radius text field, type 8.
4Click the Build Selected button.牡丹国际
M A T E R I A L S
Material 1 (mat1)
1In the Model Builder window, under Component 1 (comp1) right-click Materials and choo Blank Material.
2Right-click Material 1 (mat1) and choo Rename.
3In the Rename Material dialog box, type Doped Silica Glass in the New label text field.
4Click OK.
5Select Domain 2 only.
6In the Settings window for Material, click to expand the Material properties ction. 7Locate the Material Properties ction. In the Material properties tree, lect Electromagnetic Models>Refractive Index>Refractive index (n).
8Click Add to Material.
9Locate the Material Contents ction. In the table, enter the following ttings:
Property Name Value Unit Property group
Refractive index n 1.44571Refractive index
Material 2 (mat2)
1In the Model Builder window, right-click Materials and choo Blank Material.
2Right-click Material 2 (mat2) and choo Rename.
3In the Rename Material dialog box, type Silica Glass in the New label text field. 4Click OK.
5Select Domain 1 only.
6In the Settings window for Material, click to expand the Material properties ction. 7Locate the Material Properties ction. In the Material properties tree, lect Electromagnetic Models>Refractive Index>Refractive index (n).
8Click Add to Material.
9Locate the Material Contents ction. In the table, enter the following ttings:
Property Name Value Unit Property group
Refractive index n 1.43781Refractive index
E L E C T R O M A G N E T I C W A V E S,
F R E Q U E N C Y D O M A I N(E M W)助理值班员
Wave Equation, Electric 1
1In the Model Builder window, expand the Component 1 (comp1)>Electromagnetic Waves, Frequency Domain (emw) node, then click Wave Equation, Electric 1.
2In the Settings window for Wave Equation, Electric, locate the Electric Displacement Field ction.
3From the Electric displacement field model list, choo Refractive index.
M E S H1
1In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2In the Settings window for Mesh, locate the Mesh Settings ction.
