Optical Communications4

8/11/2019 Optical Communications4

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DR. M. SherifOptical Fiber Communications

Electromagnetic Wave Theory (Mode Theory)Mode is a term used in the study of standing waves. A standing waveis produced when two waves with the same frequency and amplitudemoving in the opposite direction interfere. A standing wave has nonet energy transfer in any direction. The mode of the standing waveis given by the number of loops within the standing wave.Meridional ray propagates as plane waves along the z-axis. Considerthat the Electric/Magnetic Component of the wave will be in the x-

direction. Now, we have two identical waves that are bouncing fromthe two upper and down interface. This yields to a Standing Wave or(Mode). If we have Electric component in the x-direction we call itTE-Mode (Transverse Electrical Field). If we have Magneticcomponent in the X-direction we call it TM-Mode.


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TE 0


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A mode of propagation is only observed when the angle between the propagation vector and the interface (boundary of the cladding and thecore) has particular values.

Depending on the mode of propagation an electric field distribution isformed. The electric field decays towards the boundaries. For all modesof propagation the self-consistency condition has to be satisfied whichmeans that the wave in the waveguide reproducing itself ( i.e. the samewave pattern !!! ).

TE m: m denotes the number of occurrence of zero electric fieldintensity along x-axisTEMi,j: is a circular mode, exists in a single mode fiber, which has

both E and M components perpendicular to z-axis.


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TEMij


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Propagation, attenuation and phase constants

The propagation constant is separated into two components that havevery different effects on signals:

A wave propagating in space or down awave guide suffers from bothattenuation and phase shift both are

proportional to the length travelled andgiven by the equation:

Where,P o is the output power P i .. is the Input power l .. is the distance travelled


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Then,

The Attenuation of the Signal is given by =

i.e.Attenuation in [dB] = 8.686 * Attenuation in [Neper]

Or:


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And,the Phase Shift in Radian, [Rd] =

is called the AttenuationConstant.

And, is called the Phase ShiftConstant

Both and are dependenton the mediumcharacteristics and the signalfrequency.


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Electromagnetic Wave Theory:

In free space, the wave number k is defined as:

The number of complete wave cycles per unit length [m]

k may be expressed also in radians as:

Where, c is the speed of EM propagation in free space, approximatelyequals to 2.99792 x 10 8 meters per second.


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Now, Considering a monochromatic plane wave light that propagates

along the z-axis. Then, the wave number of this light is given by:

Where, is the wave phase constant. And, n 1 is refractive index of the fiber core.

In media other than free space, c must be replaced by the phasevelocity and, in fiber glass k is changed to .

Considering fiber glass with refractive index n, then:

Thus:

And,


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In a chromatic plane wave , we have more than one wave. We have awave bundle. Every wave of the bundle has a phase velocity. Theresultant is a group velocity.

If the phase velocity is given by:

If the group velocity is given by:

In fiber glass the refractive index n is dependent on the wavelength


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Also,

=>

Thus,

Thus,


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For generality consider n, N in place of n 1 and N 1 N is called the Group Index of the fiber guide

Define the material dispersion coefficient M() :

then:


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Example:Consider N is constant for certain wavelength domain (0.6 1.6) m andequals to N = 1.3.At = 0.85 m, it was found that n = 1.4. What is the value for n?

at = 1.33 m. Also, find M(= 0.85 m). Solution:

=> =>


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At = 0.85 m => n = 1.4 then, 1.4 = 1.3 + 0.85 k

k = (0.1/0.85) = 0.1176 m -1

Then, n= 1.3+0.1176

n (1.33) = 1.3 + 0.1176 * (1.33) = 1.456Also,

And, Differentiate both sides of the equation:


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Example:Find an expression for n() in the given regions: a) = ( 1.3 1.55 ) m

b) = ( 0.85 1.3 ) m

c) = ( 0.4 0.6 ) m


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Optical Communications4