Transcript Fiber Optics Communication

Fiber Optics Communication
Lecture 6
1
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
2
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
3
Overview
• Since n1-n2 << 1, the description of guided
and radiation modes is simplified from sixcomponent hybrid electromagnetic fields to
four field components.
• Modes in a planar dielectric slab waveguide
4
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
5
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
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mechanical phenomena tunnel effect
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
Modified Bessel first kind
Bessel Functions Second kind
Modified Bessel Second kind
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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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