5-4: SI Fiber Modes5-4: SI Fiber Modes Consider the cylindrical Consider the cylindrical

coordinatescoordinates Assume propagation Assume propagation

along z,along z,

Wave equation resultsWave equation results

Using separation of variablesUsing separation of variables is integeris integer

5-4: SI Fiber Modes5-4: SI Fiber Modes Wave equation resultsWave equation results

Solutions are Bessel Solutions are Bessel functionsfunctions

Using boundary conditions, modal equations resultsUsing boundary conditions, modal equations results

5-4: SI Fiber Modes5-4: SI Fiber Modes There will be m roots for each There will be m roots for each value designated value designated mm

A mode is cutoff when it is no longer bound to the coreA mode is cutoff when it is no longer bound to the core

Corresponding modes are: TECorresponding modes are: TEm m TMTMm m EHEHm m HEHEmm

Fiber modes are hybrid except those for which n=0, Fiber modes are hybrid except those for which n=0, i.e. TEi.e. TEm m (E(Ez z =0),=0), TMTMm m (H(Hz z =0)=0)

V, normalized frequency, is a parameter connected to V, normalized frequency, is a parameter connected to the cutoffthe cutoff

ModesModes Mode chartMode chart HEHE1111 has no cutoff unless has no cutoff unless

a=0a=0

Linearly polarized modesLinearly polarized modes

When When <<1, we can <<1, we can introduce weakly guiding introduce weakly guiding fiber approximationfiber approximation

Under such Under such approximation, similar approximation, similar modes can be groupedmodes can be grouped

{HE{HE1111},{TE},{TE0101, TM, TM0101, HE, HE2121},{HE},{HE3131, EH, EH1111} etc.} etc.

ModesModes Using:Using:

Conclude with Conclude with jmjm and LP and LPjmjm

Mode chartMode chart

Naming ModesNaming Modes TE (TM):TE (TM):

E (M) perpendicular to Z, small component of M (E) in ZE (M) perpendicular to Z, small component of M (E) in Z

Ray is meridionalRay is meridional

TEM:TEM: E & M are perpendicular to ZE & M are perpendicular to Z Only mode of a single mode fiberOnly mode of a single mode fiber

Helical (Skew) Modes (HE and EH)Helical (Skew) Modes (HE and EH) Travel in circular pathsTravel in circular paths Components of both E and M in Z directionComponents of both E and M in Z direction

Linearly Polarized Modes (LP)Linearly Polarized Modes (LP) Summarizes all aboveSummarizes all above

Mode numberingMode numbering TE, TM, and TEM: numbers correspond to # of nulls in TE, TM, and TEM: numbers correspond to # of nulls in

their energy patterntheir energy pattern

LPLPjmjm: m is number of maxima a long a radius of a fiber, and j : m is number of maxima a long a radius of a fiber, and j is half the number of maxima around the circumferenceis half the number of maxima around the circumference

Modal Intensity distributionsModal Intensity distributions

LP01

LP03

LP11

LP12

LP21

LP22

LP41

Radial Intensity DistributionRadial Intensity Distribution

Effective index of refractionEffective index of refraction

Number of modesNumber of modes For EM radiation of wavelength For EM radiation of wavelength , the number of modes per unit solid angle , the number of modes per unit solid angle

is:is:

Area is the one the fiber enters or leaves, Area is the one the fiber enters or leaves,

Total number of modes: Total number of modes:

Solid angle: Solid angle:

Angle: Angle:

ApproximationApproximation

Solid angle: Solid angle:

Number of modesNumber of modes

But V is:But V is:

Finally:Finally:

Valid for large V (> 10)Valid for large V (> 10)

Single Mode PropagationSingle Mode Propagation Occurs when waveguide supports single Occurs when waveguide supports single

mode onlymode only Refer to modal curves, V<2.405, or Refer to modal curves, V<2.405, or

a/a/<2.405/2<2.405/2(NA)(NA) Actually two degenerate modes existActually two degenerate modes exist Due to imperfect circular fiber, they travel at Due to imperfect circular fiber, they travel at

different velocities exhibiting fiber Birefringencedifferent velocities exhibiting fiber Birefringence Small effect in conventional fibers (~10Small effect in conventional fibers (~10-8-8))

Single modeSingle mode Index profiles and Index profiles and

modal fieldsmodal fields Gaussian fitGaussian fit

Mode fieldMode field Mode field: measure of Mode field: measure of

extent of region that extent of region that carries powercarries power

w/a=0.65+1.69Vw/a=0.65+1.69V-3/2 -3/2

+2.879V+2.879V-6-6, for , for 1.2<V<2.41.2<V<2.4

SMF: MFD ranges 10.5 SMF: MFD ranges 10.5 – 11@ 1550 nm– 11@ 1550 nm

This Gaussian This Gaussian approximation helps in approximation helps in calculating important calculating important parameters of SMFparameters of SMF

Modes in GRINModes in GRIN nn22≤n≤neffeff≤n≤n11

We will consider parabolic profileWe will consider parabolic profile

Number of modes, N=VNumber of modes, N=V22/4/4 Transverse field patternsTransverse field patterns

Single mode conditionSingle mode condition

5.6: Pulse Distortion5.6: Pulse Distortion Pulse distortion:Pulse distortion:

Power limitedPower limited BW limitedBW limited

SI fibersSI fibers Modal distortionModal distortion DispersionDispersion

• MaterialMaterial• WaveguideWaveguide

SI fibers: Modal distortionSI fibers: Modal distortion Was found to be: Was found to be: ((/L)=n/L)=n11/c/c Typical for glass fibers~67 ns/kmTypical for glass fibers~67 ns/km Practical: 10-50 ns/km?Practical: 10-50 ns/km?

• Mode mixingMode mixing• Preferential attenuationPreferential attenuation• Propagation lengthPropagation length

SI fibers: Modal distortion: mode mixingSI fibers: Modal distortion: mode mixing

Exchange of power between modesExchange of power between modes

How it reduces distortion?How it reduces distortion? It increases attenuationIt increases attenuation

Pulse DistortionPulse Distortion

SI fibers: Modal distortion: propagation lengthSI fibers: Modal distortion: propagation length

SI fibers: Modal distortion: preferential attenuationSI fibers: Modal distortion: preferential attenuation

Higher order modes suffer greater attenuationHigher order modes suffer greater attenuation How it reduces distortion?How it reduces distortion? It increases total attenuationIt increases total attenuation

Small length not enough to excite high order modesSmall length not enough to excite high order modes

SI fibers: Dispersion: WaveguideSI fibers: Dispersion: Waveguide

: source linewidth: source linewidth

DispersionDispersion Waveguide dispersionWaveguide dispersion Material dispersionMaterial dispersion Total Dispersion:Total Dispersion:

((/L)/L)disdis=-(M+M=-(M+Mgg) ) Waveguide dispersion can Waveguide dispersion can

be neglected except for be neglected except for ~1.2-1.6 um~1.2-1.6 um

Total pulse spread, Total pulse spread,

Modal distortion is dominant in MMSI fiberModal distortion is dominant in MMSI fiber

Narrowing the source linewidth is ineffective, LED is usedNarrowing the source linewidth is ineffective, LED is used

Single Mode FiberSingle Mode Fiber No modal distortionNo modal distortion Material and Material and

waveguide dispersionwaveguide dispersion For short wavelength, For short wavelength,

material is dominantmaterial is dominant Fig 5-26 (MD only)Fig 5-26 (MD only) For l~1.3 um, For l~1.3 um,

waveguide dispersion waveguide dispersion should be consideredshould be considered

Single Mode FiberSingle Mode Fiber Fig 5-27: total dispersionFig 5-27: total dispersion -ve MD cancels +ve WD-ve MD cancels +ve WD Long high-data-rate systems Long high-data-rate systems

can be constructed @ these can be constructed @ these wavelengthswavelengths

Dispersion shifted fiberDispersion shifted fiber Dispersion flattened fiberDispersion flattened fiber Index profilesIndex profiles Polarization mode dispersion: 2 Polarization mode dispersion: 2

orthogonal polarizations of HEorthogonal polarizations of HE1111

Single Mode FiberSingle Mode Fiber In conventional SMF, dispersion exist at In conventional SMF, dispersion exist at

1550 nm: Requires dispersion compensation1550 nm: Requires dispersion compensation Dispersion compensating fiber: has opposite Dispersion compensating fiber: has opposite

dispersion at higher order modesdispersion at higher order modes Cutoff wavelength:Cutoff wavelength:

For nFor n11=.., n=.., n22=.., a/=.., a/<3.17 for SM condition. @<3.17 for SM condition. @=0.8 =0.8

um > a=2.54 um. If um > a=2.54 um. If is changed to 1.3 um, same is changed to 1.3 um, same fiber still SMfiber still SM

@@=1.3 um > a=4.12 um, which is not SM at 0.8=1.3 um > a=4.12 um, which is not SM at 0.8 @ which SM equation is equality is cutoff wavelength @ which SM equation is equality is cutoff wavelength cc

ccwill excite MM propagation will excite MM propagation cc=2.61 a NA =2.61 a NA

GRIN fiberGRIN fiber Smaller modal distortion than SISmaller modal distortion than SI

((/L)=n/L)=n11/2c/2c Comparing with SI, reduction of 2/Comparing with SI, reduction of 2/ For nFor n11=1.48, n=1.48, n22=1.46, =1.46, =0.0135 >> 2/ =0.0135 >> 2/ =148 =148 SI typical modal is 67 ns/km, GRIN is 0.45 ns/kmSI typical modal is 67 ns/km, GRIN is 0.45 ns/km

MD is dominant at 0.8-0.9 um >> LD is usedMD is dominant at 0.8-0.9 um >> LD is used At higher wavelengths, MD is small >> LED can be usedAt higher wavelengths, MD is small >> LED can be used

Total Pulse DistortionTotal Pulse Distortion άά L, is expected L, is expected άά L L1/21/2, is found, is found Equilibrium length, LEquilibrium length, Lee

Modal pulse distortion:Modal pulse distortion: =L=L/L) for L≤ L/L) for L≤ Lee

=(L L=(L Lee))1/21/2/L) for L≥ L/L) for L≥ Lee

LLee ά ά 1/mode mixing 1/mode mixing Little mode mixing >>LLittle mode mixing >>Lee is large >> good fiber is large >> good fiber No mode mixing >>LNo mode mixing >>Lee is ∞>> linear dependance is ∞>> linear dependance Lots of mode mixing >>LLots of mode mixing >>Lee is small >> poor fiber is small >> poor fiber

M&WD is independent of mode mixing >> M&WD is independent of mode mixing >> άά L L Care should be taken when computing Care should be taken when computing tottot