1. 5-4: SI Fiber Modes  Consider the cylindrical coordinates  Assume propagation along z,  Wave equation results  Using separation of variables  is integer

2. 5-4: SI Fiber Modes  Wave equation results  Solutions are Bessel functions  Using boundary conditions, modal equations results

3. 5-4: SI Fiber Modes  There will be m roots for each value designated  m  A mode is cutoff when it is no longer bound to the core  Corresponding modes are: TE m TM m EH m HE m  Fiber modes are hybrid except those for which n=0, i.e. TE  m (E z =0), TM  m (H z =0)  V, normalized frequency, is a parameter connected to the cutoff

4. Modes  Mode chart  HE 11 has no cutoff unless a=0  Linearly polarized modes  When  <<1, we can introduce weakly guiding fiber approximation  Under such approximation, similar  modes can be grouped  {HE 11 },{TE 01, TM 01, HE 21 },{HE 31, EH 11 } etc.

5. Modes  Using:  Conclude with  jm and LP jm  Mode chart

6. Naming Modes  TE (TM): E (M) perpendicular to Z, small component of M (E) in Z E (M) perpendicular to Z, small component of M (E) in Z Ray is meridional Ray is meridional  TEM: E & M are perpendicular to Z E & M are perpendicular to Z Only mode of a single mode fiber Only mode of a single mode fiber  Helical (Skew) Modes (HE and EH) Travel in circular paths Travel in circular paths Components of both E and M in Z direction Components of both E and M in Z direction  Linearly Polarized Modes (LP) Summarizes all above Summarizes all above

7. Mode numbering  TE, TM, and TEM: numbers correspond to # of nulls in their energy pattern  LP jm : m is number of maxima a long a radius of a fiber, and j is half the number of maxima around the circumference

8. Modal Intensity distributions LP 01 LP 03 LP 11 LP 12 LP 21 LP 22 LP 41

9. Radial Intensity Distribution

10. Effective index of refraction

11. Number of modes  For EM radiation of wavelength, the number of modes per unit solid angle is:  Area is the one the fiber enters or leaves,  Total number of modes:  Solid angle:  Angle:  Approximation  Solid angle:  Number of modes  But V is:  Finally:  Valid for large V (> 10)

12. Single Mode Propagation  Occurs when waveguide supports single mode only  Refer to modal curves, V<2.405, or a/ <2.405/2  (NA)  Actually two degenerate modes exist  Due to imperfect circular fiber, they travel at different velocities exhibiting fiber Birefringence  Small effect in conventional fibers (~10 -8 )

13. Single mode  Index profiles and modal fields  Gaussian fit

14. Mode field  Mode field: measure of extent of region that carries power  w/a=0.65+1.69V -3/2 +2.879V -6, for 1.2<V<2.4  SMF: MFD ranges 10.5 – 11@ 1550 nm  This Gaussian approximation helps in calculating important parameters of SMF

15. Modes in GRIN  n 2 ≤n eff ≤n 1  We will consider parabolic profile  Number of modes, N=V 2 /4  Transverse field patterns  Single mode condition

16. 5.6: Pulse Distortion  Pulse distortion: Power limited Power limited BW limited BW limited  SI fibers Modal distortion Modal distortion Dispersion Dispersion MaterialMaterial WaveguideWaveguide  SI fibers: Modal distortion Was found to be:  (  /L)=n 1  /c Was found to be:  (  /L)=n 1  /c Typical for glass fibers~67 ns/km Typical 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 mixing Exchange of power between modes Exchange of power between modes How it reduces distortion? How it reduces distortion? It increases attenuation It increases attenuation

17. Pulse Distortion  SI fibers: Modal distortion: propagation length  SI fibers: Modal distortion: preferential attenuation Higher order modes suffer greater attenuation Higher order modes suffer greater attenuation How it reduces distortion? How it reduces distortion? It increases total attenuation It increases total attenuation Small length not enough to excite high order modes Small length not enough to excite high order modes  SI fibers: Dispersion: Waveguide  : source linewidth  : source linewidth

18. Dispersion  Waveguide dispersion  Material dispersion  Total Dispersion:  (  /L) dis =-(M+M g )   (  /L) dis =-(M+M g )   Waveguide dispersion can be neglected except for ~1.2-1.6 um  Total pulse spread,   Modal distortion is dominant in MMSI fiber  Narrowing the source linewidth is ineffective, LED is used

19. Single Mode Fiber  No modal distortion  Material and waveguide dispersion  For short wavelength, material is dominant  Fig 5-26 (MD only)  For l~1.3 um, waveguide dispersion should be considered

20. Single Mode Fiber  Fig 5-27: total dispersion  -ve MD cancels +ve WD  Long high-data-rate systems can be constructed @ these wavelengths  Dispersion shifted fiber  Dispersion flattened fiber  Index profiles  Polarization mode dispersion: 2 orthogonal polarizations of HE 11

21. Single Mode Fiber  In conventional SMF, dispersion exist at 1550 nm: Requires dispersion compensation Dispersion compensating fiber: has opposite dispersion at higher order modes Dispersion compensating fiber: has opposite dispersion at higher order modes  Cutoff wavelength: For n 1 =.., n 2 =.., a/ a=2.54 um. If  is changed to 1.3 um, same fiber still SM For n 1 =.., n 2 =.., a/ a=2.54 um. If  is changed to 1.3 um, same fiber 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 c  @ which SM equation is equality is cutoff wavelength c  c  will excite MM propagation  c  will excite MM propagation c  =2.61 a NA c  =2.61 a NA

22. GRIN fiber  Smaller modal distortion than SI  (  /L)=n 1   /2c  (  /L)=n 1   /2c Comparing with SI, reduction of 2/  Comparing with SI, reduction of 2/  For n 1 =1.48, n 2 =1.46,  =0.0135 >> 2/  =148 For n 1 =1.48, n 2 =1.46,  =0.0135 >> 2/  =148 SI typical modal is 67 ns/km, GRIN is 0.45 ns/km SI typical modal is 67 ns/km, GRIN is 0.45 ns/km  MD is dominant at 0.8-0.9 um >> LD is used  At higher wavelengths, MD is small >> LED can be used

23. Total Pulse Distortion   ά L, is expected   ά L 1/2, is found  Equilibrium length, L e  Modal pulse distortion:   =L  /L) for L≤ L e   =(L L e ) 1/2  /L) for L≥ L e  L e ά 1/mode mixing Little mode mixing >>L e is large >> good fiber Little mode mixing >>L e is large >> good fiber No mode mixing >>L e is ∞>> linear dependance No mode mixing >>L e is ∞>> linear dependance Lots of mode mixing >>L e is small >> poor fiber Lots of mode mixing >>L e is small >> poor fiber  M&WD is independent of mode mixing >>  ά L  Care should be taken when computing  tot