Optical fibers. Weak guidance approximation in cylindrical waveguides

Transcription

1 Optical fibers Weak guidance approximation in cylindrical waveguides

2 Propagation equation with symmetry of revolution r With cylindrical coordinates : r ( r,θ) 1 ψ( r,θ) 1 ψ( r,θ) ψ + + r r + r θ [ ( ) ] k n r,θ -β ψ( r,θ) = core cladding q The refractive index only varies with r : n ( r,θ) = n( r) We search solutions under the form : ψ r,θ = R Radial r.φ θ ( ) ( ) ( ) Angular or azimuthal periodical

3 Solution of the angular dependance The propagation equation becomes : r d R R dr r dr + R dr +r 1 d [ kn ( r) -β ] = - Φ Φ dθ - 1 Φ d Φ = constant dθ Doesn t depend on q 1 d Φ - = Φ dθ F is p periodical with Doesn t depend on r + N r R d R dr r dr + R dr +r [ k n ( r) -β ] = Φ + dθ d Φ= 3

4 Angular dependance and modes degeneracy d Φ + dθ Φ= + N For each solutions : Φ( θ) = cos θ or Φ( θ) = sinθ For each solution : independant linear polarizations The modes have a degeneracy of 4 The fundamental mode ( = ) has a degeneracy of 4

5 Radial dependance r In the core d R dr dr +r dr < r <a + [( k n ( r) -β ) r - ] R = α = kn1 - β u=α a n n 1 core r cladding q layer step index fiber a In the cladding r > a κ =β - k n v =κ a u + v = V V =k a n1 - n = 5

6 Radial solutions in the core < r <a This equation has two solutions : Bessel functions of the first kind Bessel functions of the second kind Diverges when u r a No physical meaning 6

7 Radial solutions in the cladding r > a Modified Bessel functions of the first kind This equation has two solutions : Modified Bessel functions of the second kind R r = I l v r a R r = K l v r a No physical meaning 7

8 Form of the transverse fields The transverse field is continuous in r=a < r <a Oscillations in the core r > a Decay in the cladding 8

9 Relation of dispersion The normal derivative of the transverse field is continuous in r=a l = u J 1(u) J (u) = v K 1(v) K (v) l 1 u J l 1 u J l u = v K l 1 v K l v u + v = V Relation of dispersion of the LP modes in a step index fiber For each value of l, there are n possible values for u i.e. for the longitudinal propagation constant b.n is a function of V. (the higher V, the higher n) LP l m modes (Linearly Polarized) : l is linked to the index of the Bessel function periodicity of m tells that the mode refers to the m th solution in u Φ( θ) 9

10 Normalized notations Normalized propagation constant b(v) b(v) 1 Cut-off frequency defined by b(v)=, i.e v=, i.e u=v Effective refractive index Relation of dispersion J -1( V 1-b) l 1 V 1-b = -V J ( V 1-b) l = V 1 [ n + b( n n )] β neff = = 1 - k K b K -1 ( V 1-b) ( V 1-b) J1 1-b = -V J K b K n n eff n 1 1 ( V b) ( V b) ( V b) ( V b) Remarks : At low V propagation most in the cladding At high V propagation most in the core 1

11 u(v) Number and cut-off frequencies of the LP modes The number of modes increases with V. At the cut-off frequency, u=v. LP 1 LP 1 has no cut-off frequency and always exists. LP 1 is the single propagating mode as long as V<.45 11

12 Marcuse approximation for b(v) In the field of optical telecommunication, we frequently have 1.5< V <.4. LP 1 is the only propagating mode. In this case, with A=1.148 and B=.996 Example : n =1.45 D=.64 a=3 µm l =1.546 µm V~ => single mode fiber b~ n eff ~

13 Confinement factor The confinement factor G(V) is the fractional power in the core : For a step index fiber and for the LP l m mode, we can demonstrate : For a given mode : at low V, G. Most of the field is in the cladding. as V increases, G1. Most of the field is in the core. 13

14 Practical use of single mode fibers Single mode condition to avoid intermodal dispersion : V <.45 Maximize the fractionnal power in the core to minimize the effect of external perturbations : GV 6% Practically 1.5< V <.4 14

15 Structure of the LP l m modes R(r) 1 r a LP Core Cladding R(r) Core Cladding 1 r a Core : Amplitude oscillations in q : A p rotation at r fixed => l cancellations in r : m-1 cancellations inside the core (except in ) Cladding : pseudo exponential decay Uniform polarisation LP l 15

16 Aspect of the LP l m modes opticalengineering.spiedigitallibrary.org Intensity 16

17 l = fold degenerated Degeneracy of the LP l m modes LP 1 LP 3 r J u a yˆ r J u a l > 4 fold degenerated LP 11 xˆ r J u a yˆ r J u a xˆ J r u cos a 1 yˆ r J u cos a 1 xˆ J r u sin a 1 yˆ J r u sin a 1 xˆ 17

18 Optical Imaging through Multimode Fibres N. Dubreuil 18

19 Transmissions on multimode optical fibre e 1 (t) MUX Spatial Multimode Optical Fibre DMUX Spatial y 1 (t) e (t) y (t) e 5 (t) y 5 (t) N. Dubreuil Multiplexage de modes Fibre multi-modes/multi-cœurs y x - Multi-core Fibers (MCF) - Coupled-core multicore fibers (CC-MCF) - Few-Mode Fibers (FMF) 19

20 The Gaussian beam approximation of the LP 1 mode LP 1 mode : one maximum in the center, uniform angular distribution Gaussian approximation : The Gaussian mode closest to the LP 1 mode has a diameter : d=w obtained by optimizing the coupling efficiency of a Gaussian beam on the LP 1 mode : η = [ Ψ Ψ ds ] Ψ exact exact ds gaussian gaussian Ψ ds

21 Marcuse formula for the approximation of the LP 1 mode 1% precision for 1.<V<4 1

22 Applications of the Gaussian mode approximation The Gaussian mode approximation provides analytical expressions for connection losses between single mode fibers, caused by : mismatch of their physical parameters misalignment

23 Equivalence of single mode fibers with different parameters Fiber 1 n 1 = n = 1.45 a = 4.46μm λ = 1.3μm Fiber 1 is single mode as πa V = n1 -n λ =.3 The Gaussian closest to the LP 1 mode has a diameter of : d = ω=1.μm Fiber n 1 = n = 1.45 a = 4.7μm λ = 1.3μm Fiber is single mode as πa V = n1 -n λ =. The Gaussian closest to the LP 1 mode has a diameter of : d = ω=1.μm => No specific losses 3

24 Longitudinal connection gap D Fiber 1 gaussian mode F 1 () Free space propagation of gaussian beam F ( D 1 ) Projection of on fiber gaussian mode F () F ( D 1 ) F 1 (D)F * dxdy 1 1 D l 4p n w 4 Dl l (db)1 log 1 p nw Exemple : d w 1m D m l 1, 3 m n 1, 45 l A longitudinal gap is not a major source of losses ( db),6 4

25 Mismatch of physical parameters and lateral misalignment x F 1 x, y 1 w 1 p exp x y w 1 F y x, y 1 w p exp x x w Coupling efficiency : Insertion losses : F 1 F * dxdy n db log w 1w w 1 w x exp w 1 w 5

26 pertes ( db) Mismatch of physical parameters alone w 1w w 1 w x exp w 1 w If alignment is perfect x max w 1w w 1 w n db log w 1w w 1 w w1 w Insertion losses <,1 db if diameter mismatch < 14%

27 Lateral misalignement of matched fibers w 1w w 1 w x exp w 1 w matched fibers w w w 1 x exp w n db log 4,34 x w For w=1m Insertion losses <,1 db for lateral shift x <,76 m Fibers lateral alignement is critical 7

28 Angular misalignment alone x x' y z q y' z' In the absence of fibers mismatch a exp p n l w q l db 4,34 p n l w q l w 1 w w when w=1m Insertion losses <,1 db for angle misalignment q <,5 8

29 Example of limitations of the weak guidance approximation Nanofibres : diametre F< l SMF 8 F=15 µm The longitudinal components of the fields cannot be neglected Nanofibre F=5 nm Enhancement of the intensity Applications : realisation of supercontinuum, w, 3w Intense evanescent field Applications : sensors F=5 nm, l=53 nm : Intensity x 4 6% of the light in the evanescent field 9

30 Fabrication of optical fibres H 3 m T C fibre-optique.pagesperso-orange.fr Preform («macroscopic» fibre) : silica rod 1 m, F=1 cm v~km/mn Fibre length : hundreds of km 3