Wave Propagation in Optical Fibers1

8/2/2019 Wave Propagation in Optical Fibers1

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WAVEPROPAGATIONINOPTICAL

FIBERS

EC04 703: OPTICAL COMMUNICATION SYSTEMS

Nandakumar N P


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MAXWELLS EQUATIONS WAVEPROPAGATIONIN FIBERS

Determined by solving wave and Maxwells

equations in cylindrical coordinates

011

22

2

22

2

zzzz EqE

rrE

rrE

011 2

2

2

22

2

z

zzz

Hq

H

rr

H

rr

H


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KEY PARAMETERS

q2 is equal to 2-2 = k22. It is sometimescalled u2

is the z component of the wave propagation

constant k, which is also equal to 2/. Theequations can be solved only for certain valuesof , so only certain modes may exist. A modemay be guided if lies between n2k and n1k

V = ka(NA) where a is the radius of the fibercore. This normalized frequency determineshow many different guided modes a fiber cansupport


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SOLUTIONSTO WAVE EQUATIONS

The solutions are separable in r, , z and t. The and z functions are exponentials of the form ei.The z function oscillates in space, while the function must have the same value at (+2) that it

does at

The r function is a combination of Bessel functionsof the first and second kinds. The separate

solutions for the core and cladding regions mustmatch at the boundary


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RESULTING TYPESOF MODES

Either the electric field component (E) or the magneticfield component (H) can be completely aligned in thetransverse direction: TE and TM modes

The two fields can both have components in thetransverse direction: HE and EH modes

For weakly guiding fibers (small delta), the types ofmodes listed above become degenerate, and can becombined into linearly polarized LP modes

Each mode has a subscript of two numbers, where thefirst is the order of the Bessel function and the secondidentifies which of the various roots meets the boundarycondition. If the first subscript is 0, the mode ismeridional. Otherwise, it is skew


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MODE CHARACTERISTICS

Each mode has a specific

Propagation constant

Spatial field distribution

Polarization


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w-b Mode Diagram

Straight lines of dw/db correspond to the groupvelocity of the different modes

The group velocities of the guided modes all liebetween the phase velocities for plane waves in thecore or cladding c/n1 and c/n2


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MODESIN STEP INDEX FIBERS

Different values of b corresponds to differentsolutions of wave equation in circular waveguides

The solutions for b must be determined from theboundary conditions

Inside the core the factor q2 is given by

q2 = u2 = k12b2

And outside the corew2 = b2 - k2

2

0)()(21 waCKuaAJEEz z


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MODESIN STEP INDEX FIBERSCONTD


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MODESIN STEP INDEX FIBERSCONTD


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CROSS SECTIONAL VIEWOFTHE TRANSVERSEELECTRIC FIELD VECTORSFORTHE FOURLOWEST ORDER MODESIN SI FIBER


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LINEARLY POLARIZED MODES

Weakly guiding fiber approximation, index difference ismuch less than 1

The EM field pattern and the propagation constants ofthe mode pairs HEv+1,m and EHv-1,m are similar

Mode groupings {HE11}, {TE01, TM01, HE21}, {HE31,EH11}, {HE12}, {HE41, EH21} and {TE02, TM02, HE22}

These modes can be designated as LPjm modes ratherthan specifying TE, TM, HE and EH modes

Each LP0m mode is derived from an HE1m mode

Each LP1m mode is derived from TE0m, TM0m and HE2mmodes

Each LPvm mode (v >1) is from an HEv+1,m and an EHv-1,m modes


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THE CUTOFF

For each mode, there is some value of Vbelow which it will not be guided becausethe cladding part of the solution does not goto zero with increasing r

Below V=2.405, only one mode (HE11) canbe guided; fiber is single-mode.

Based on the definition of V, the number ofmodes is reduced by decreasing the coreradius and by decreasing


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FORMATIONOF LP MODES


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NUMBEROF MODES

Step Index Fiber

At low V, M4V2/2+2

At higher V, MV2/2

Graded Index Fiber

2

1

2aknM

Wave Propagation in Optical Fibers1

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