Fiber Optics Communication

Lecture 6

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Mode Theory for Circular Waveguides

• To understand optical power propagation in fiber it is necessary to solve Maxwell’s equation subject to cylindrical boundary conditions

• Outlines of such analysis will be studied here

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Overview• When solving Maxwell’s equations for hollow

metallic waveguide, only transverse electric (TE) and transverse magnetic (TM) modes are found

• In optical fibers, the core cladding boundary conditions lead to a coupling between electric and magnetic field components. This results in hybrid modes

• Hybrid modes HE means (E is larger) or HM means H is larger

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Overview

• Since n1-n2 << 1, the description of guided and radiation modes is simplified from six-component hybrid electromagnetic fields to four field components.

• Modes in a planar dielectric slab waveguide

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Overview

• The order of a mode is equal to number of field zeros across the guide

• Field vary harmonically in guiding region and decay exponentially outside this region

• For lower order modes, fields are concentrated towards the center of the slab

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Overview• Modes

– Guided• Modes travelling inside fiber along its axis. They are finite

solutions of Maxwell equation ( 6 hybrid E and H field)

– Radiated• Modes that are not trapped in core. These result from optical power

that is outside the fiber acceptance being refracted out of the core. Some radiation gets trapped in cladding, causing cladding modes to appear

• Coupling between cladding and core (radiation not confined)• Cladding modes are suppressed by lossy coating

– Leaky• Partially confined to core region and attenuates by

radiating their power. This radiation results from quantum mechanical phenomena tunnel effect

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V number• V number

– Cut off condition that determines how many modes a fiber can support

– Except for lowest mode HE11, each mode exists only for values of V that exceed a limiting value

– Modes are cut off when . This occurs when » (for 8 microm diameter fiber)

– Number of modes M in multimode fiber when V is large

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Modal Concepts

• For step index fiber, the fractional power flow in the core and cladding for a given mode

• M is proportional to V, power flow in cladding decreases as V increases.

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Maxwell’s Equations ……..1 (Faraday’s Law)

………2 (Maxwell’s Faraday equation)

………3(Gauss Law)

………4(Gauss Law for magnetism)

and . The parameter Є is permittivity and μ is permeability.

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Maxwell’s Equations

• Using vector identity ……(6)• Using (3), …….(7)• Taking the curl of 2,

………(8)

• (7) and (8) are standard wave equations

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Maxwell’s Equation

• Using cylindrical coordinates .…..(9) ……(10)• Substituting (9) and (10) in Maxwell’s curl

equation ….(11) ….(12)

….(13)

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Maxwell’s Equation• Also ----------(14) ----------(15) ----------(16)

• By eliminating variables, above can be written such that when Ex and Hz are known, the remaining transverse components can be determined

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Maxwell’s Equation …………..(17) ……………(18)

…...........(19)

.………… (20)Substituting (19) and (20) into (16) results in

….…(21) …….(22)

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Maxwell’s Equation• (21) and (22) each contain either Ez or Hz.

– Coupling between Ez and Hz is required by boundary conditions

– If boundary conditions do not lead to coupling between field components, mode solution will such that either Ez=0 or Hz=0.

– When Ez=0, modes are called transverse electric or TE modes

– When Hz=0, modes are called transverse magnetic or TM modes

– Hybrid modes exist if both Ez and Hz are nonzero designated as HE or EH

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Wave Equations for Step Index Fibers

• Using separation of variables ………..(23)

• The time and z-dependent are given by ………..(24)

• Circular symmetry, each field component must not change when Ø is increased by 2п. Thus

…………(25)

• Thus, (23) becomes

….(26)

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Wave Equations for Step Index Fibers• Solving (26). For inside region, the solution must

remain finite as r->0, whereas on outside the solution must decay to zero as r->∞

• Solutions are – For r< a, Bessel function of first kind of order v (Jv)

– For r> a, modified Bessel functions of second kind(Kv)

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Bessel Functions First Kind Bessel Functions Second kind

Modified Bessel first kind Modified Bessel Second kind 17

Propagation Constant β

• From definition of modified Bessel function

• Since Kv(wr) must to zero as r->∞, w>0. This implies that

• A second condition can be deduced from behavior of Jv(ur). Inside core u is real for F1 to be real, thus,

• Permissible range of β for bound solutions is

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Meaning of u and w

• Both u and w describes guided wave variation in radial direction– u is know as guided wave radial direction phase

constant (Jn resembles sine function)– w is known as guided wave radial direction decay

constant (recall Kn resemble exponential function)

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