Mode Theory for Optical Fiber

1. Mode Theory for Optical Fiber A. Gaiwak

2. Cylindrical Fiber Modes • As with dielectric waveguide TE and TM modes are common • The cylindrical fiber is bound in two directions hence two integers l and m are used to specify the modes • For cylindrical waveguides we refer to TElm, TMlm, HElm and EHlm modes • The EH and HE modes are due to nonzero Ez and Hz components

3. Please note that TE and TM modes are due to Meridional rays and EH and HE modes are due to Skew rays • This complicated analysis even for Step index fiber can be simplified for communication OF which support weakly guiding approximation, where Relative RI is very small as compared to unity.

4. The value of Relative RI is less than 0.03 and this results in small grazing angle θ • For these guiding structures the mode theory gives dominant transverse field components, i.e. the approximate solutions for the full set of HE, EH, TE and TM modes may be given by two linearly polarized components called LP modes.

5. Please note that LP modes are not the exact fiber modes but they are resultant modes due to approximation. They are also called degenerate modes • Like basic modes LP modes also use subscripts l and m. • In general there are 2l field maxima around circumference of core and m field maxima along radius vector.

6. LP01 HE11 • LP11 HE21,TE01, TM01 • LP21 HE31, EH11 • LP02 HE12 • LP31 HE41, EH21 • LP12 HE22, TE02, TM02 • LPlm HE2m, TE0m, TM0m • LPlm(l>1) HEl+1,m, EHl-1,m

8. Previous slide shows electric field intensity profile for the lowest three LP modes and their constituent exact modes

9. Using the previous equation in Ψ and the weakly guiding approximation we can write the scalar wave equation for the cylindrical homogeneous core as • d2Ψ/dr2 +(1/r) (dΨ/dr) +1/r2 d2Ψ/dΦ2 + (n12 k2-β2)Ψ = 0

10. Ψ is the field E or H, k is propagation constant in vacuum, The propagation constant of the guided mode in vacuum β lies in the range • n2k < β < n1k • Solution of the wave equation for cylindrical fiber are is given in the form

11. Ψ = E(r) [ (cos (lΦ)/sin (lΦ)) exp (ωt - βz)] • Above Ψ represents the dominant electric field component • The periodic dependence on Φ following cos and sin functions gives a mode of radial order l. The fiber supports a finite number of guided modes of the form given by above equation.

12. If we introduce the solution given by this equation in wave equation we get a differential equation: • d2E/d2r + (1/r) dE/dr + [(n1k2 – β2) – (l2/r2)]E = 0 • For step index fiber the above equation is a Bessel differential equation, for core region the solutions are Bessel function denoted by Jl. A graph of this with r is shown in the next slide

14. Please note by observation that the field is finite at r = 0 and it is represented by zero order Bessel function J0 • As r increases and goes towards infinity the field vanishes and in the cladding the solution is given by another function called Modified Bessel functions denoted by Kl. These functions varies exponentially with r

16. For this case the electric field may be given by • E(r) = GJl (UR) for R < 1 (core) • And • E(r) = G J1(U) [Kl(WR)/Kl(W)] • G is amplitude coefficient • R = r/a is the normalized radial coordinate and a is the radius of the fiber core

17. U and W are the eigenvalues in the core and cladding respectively defined as • U = a(n12k2 – β2)1/2 • W = a(β2 – n22k2)1/2 • U is also referred as the radial phase parameter or the radial propagation constant. W is also called as cladding decay parameter

18. A very useful relation using U and W can be given by • V = (U2 + W2)1/2 • V = ka(n12 – n22)1/2 V is called the normalized frequency or simply the V number a very useful parameter in fiber design. It is a dimensionless parameter

19. V = (2π/λ) a(NA) • V = (2π/λ) an1(2RRID)1/2 • Please see that this expression relates core radius, relative reflective index difference and wavelength.

20. It is also possible to define the normalized propagation constant b for a fiber as • b = 1 – (U2/V2) • b = [(β/k)2 – n22] / (n12 – n22) • = [(β/k)2 – n22] / 2n12RRID • The limit of β is between n1k and n2k but b lies between 0 - 1

21. Boundary Condition require continuity of transverse and tangential components of Electric field at r = a Using bessell’s function relations outlined the Eigenvalue equation for LP mode may be written as U [Jl±1(U)]/[Jl(U)] = ±W [Kl ±1(W)]/[Kl(W)] Solution of above equations gives U and β as a function of V

22. When you have U and β as function of V, we can say that propagation characteristics of modes and their dependence on frequency and fiber geometry can be obtained

23. When β = n2k, then mode phase velocity is equal to the velocity of light in the cladding and mode is no longer properly guided • In this case mode is said to be cutoff and W = 0 • Unguided or radiating modes have frequencies below cutoff where β < n2k and W is imaginary. These modes are also called leaky modes

24. As β moves toward n2k less and less power is propagated in cladding and when as β = n1k all power is confined to cladding • Range of β signifies the vale for guided modes in the fiber

26. Previous slide shows the lower order modes for a cylindrical homogeneous core waveguide both notations are shown • Here Bessel’s functions J0 and J1are plotted w.r.t. Normalized frequency. Where they cross the zero gives cutoff points for the modes • Please note that Vc is different for different modes

27. First zero crossing of J1 occurs at V = 0 and this corresponds to cutoff for LP01 mode • The first zero crossing for J0 is when V = 2.405 giving Vc for LP11 mode • Second zero crossing for J1 occurs at V = 3.83 giving Vc for LP02 of 3.83

28. Hence we can have fibers manufactured with a particular V no. for limiting certain modes • It may please be further noted that cutoff value of V i.e. Vc occurs when β = n2k correspond to b = 0

30. Please refer to the previous slide • The electric field distribution of different modes gives distribution of light intensity within the fiber core. These patterns may give indication of predominant mode propagating in the fiber • These distinctive patterns are difficult to observe in multi mode propagation

31. MODE COUPLING • Previous discussions were for the perfact dielectric fibers • In actual fibers we have: • Fiber axis deviation • Variations in core diameter • Irregular core-cladding interface • Refractive index variations are found

32. These give rise to coupling energy from one mode to another mode • As depicted in the next slide the ray is not able to maintain the same angle with the axis in both the cases • In electromagnetic mode theory this corresponds to change in mode

34. Individual modes do not propagate throughout the length of fiber and there is energy transfer amongst different modes. This is called mode coupling or mode mixing. This has pronounced effect on fiber transmission characteristics

36. For step index fiber n(r) = n1 for r < a (core) n2 for r ≥ a (cladding) For MM fiber core diameter is 50μm or more As shown in the next sketch the Single Mode fiber is with 2 – 10 μm Please see how single mode is represented with only one axial ray

37. MM fibers are used where BWXLength requirement is low. It has advantages in the form of • Spatially incoherent sources can be used • Large NA facilitates efficient coupling • Low tolerance requirement for fiber connectors

38. Number of modes supported by the fiber are determined by physical parameters Total no. of guided modes may be given by Ms≈ V2/2 For a MM SI fiber core dia 80μm and RRID = 1.5%, at 0.85 μm if core RI is 1.48 estimate V no. and no. of guided modes

39. V = (2π/λ) an1(2RRID)1/2 V = 75.8 and Ms≈ V2/2 Ms = 2873

40. Graded Index Fibers • n(r) = n1(1 - 2RRID(r/a)α)1/2 r < a and n(r) = n1(1 - 2RRID)1/2 r ≥ a Here α is the profile parameter which gives characteristic RI profile for the GI fiber core

42. The best results in terms of intermodal dispersion are obtained with parabolic profile fiber with α = 2 • Unless specified otherwise all GI fibers are considered to be parabolic profile fiber • They exhibit far less Intermodal dispersion as SI fibers

45. For GI fibers following relations hold true Mg = (α / α +2) (n1ka)2 RRID and V = n1ka(2RRID)1/2 Mg ≈ (α / α +2) (V2/2) when α = 2, Mg ≈V2/4

46. Single Mode Fibers When 0 ≤ V <2.405 only one mode LP01 or HE11 propagates and rest all are suppressed If RRID = 1.5%, at 0.85 μm if core RI is 1.48 what should be the core diameter for single mode operation. If RRID is reduced by a factor of 10 what will be core dia.

47. V = (2π/λ) an1(2RRID)1/2 hence a = (V λ) / (2πn1(2RRID)1/2) a = 1.3 dia = 2.6 μm if we reduce the RRID by a factor of 10 then a = 4 and dia is 8 μm

48. Both the factors reduction of a or reduction of RRID pose problems in Single Mode fibers • If we choose Graded RI profiling for SM operation then Vc = 2.405 (1 + 2 / α) • For GI parabolic RI profile has RI at core axis 1.5 with RRID 1%. Estimate max. possible core diameter which allows SM operation at 1.3 μm wavelength

49. V = 2.4 (1 + 2 / α)1/2 = 2.4√2 and max core radius may be obtained from the eq. as 3.3 μm • This factor of √2 increase on V number in GI is significant as it allows to increase the core diameter by similar factor keeping the other parameters constant • If GI fiber with triangular profile is used then V

50. number for SM operation can further be increased this increase is by a factor of √3 over the comparable SI fiber For low RRID SM fibers for low V values the EM field associated with the dominant mode extends appreciably in the cladding i.e. for V no. values less than 1.4 around half the power propagates in cladding