Theory of Optical Modes in Step Index Fibers

Transcription

1 Theoy of Optical Modes in Step Index Fibes 1/17

2 n = n n clad coe inside the coe inside the cladding E= Eˆ ˆ ˆ + Eφφ + Ez z H= Hˆ + H ˆ φ + Hzˆ φ z We find the modes by looking fo solutions of: Eφ E 2 E + ( nk0) E = E 2 2 ( 0) ˆ + nk E φ E φ 2 2 φ 2 + E 2 2 ( nk0 ) E ˆ φ + + φ E ( 2 ( ) ) 2 z 0 z + E + nk E zˆ = 0 The equations have a simple physical intepetation. φ (fom Pollock) 2/17

3 (fom Pollock) Since the equations fo E and E φ ae coupled, we fist solve fo E z. H z is a solution of the same Helmholtz equation and its solutions have the same fom. We find all othe field components fom E z and H z using Maxwell s equations. We look fo solutions of the fom: z ( ) ( φ ) ( ) E = R Φ Z z 3/17

4 In the coe we find: jβz jβz ( ) = + jνφ jνφ ( φ ) ce de ( ) = + Z z ae be Φ = + R ej fn ν ν ( n k ) 2 whee κ = β, and ν = 0,1, coe 0 We can simplify these noting that: Often we have only fowad going waves (b=0) The N ν (κ) solution goes to minus infinity at = 0 so it is unphysical (f=0) We need both the e jνφ and e -jνφ tems to descibe the φ dependence of the eigenmodes, but we can limit the discussion to the e jνφ solution with the undestanding 4/17

5 that a mode with e -jνφ dependence can be found fom the e jνφ mode by otating the fibe. Then we can wite: jνφ jβz E = AJ e e + c.. c z jνφ jβz H = BJ e e + c.. c In the cladding egion we find: z ν ν ( γ ) ( γ ) jνφ jβz E = CK e e + c.. c z jνφ jβz H = DK e e + c.. c z ν ν whee ( ) 2 γ = n k β 2 2 clad 0 Fom Pollock and Lipson 5/17

6 Fom Izuka Chaacteistic Equation fo an Optical Fibe We insist on continuity of the tangential field components E z, E φ, H z, and H φ and find: βν a + κ γ J ν ( κa) K ν ( γa) kn 0 coej ν ( κa) kn 0 cladk ν ( γa) = + + κjν ( κa) γkν ( γa) κjν ( κa) γkν ( γa) 6/17

7 This chaacteistic equation can be used with: ( γ ) = + = 0 V a a, whee V ka ncoe nclad to find values fo κ, γ, β, and n eff. Meidional Modes (ν=0): Fo modes that coespond to bouncing meidional ays, thee is no φ dependence. Modes ae of two types TE 0µ and TM 0µ with µ=1,2,. ( ) ( ) ( ) ( ) ( ) ( ) If we set this tem = 0, If we set this tem = 0, E = E = 0 and this is a TE mode H = H = 0 and this is a TM mode z z ( ) ( ) J ν κa K ν γa kn 0 coej ν κa kn 0 cladk ν γa + + κjν κa γkν γa κjν κa γkν γa = 0 Skew Modes (ν 0): These modes have adial stuctue. The modes have both E z 0 and H z 0 and thus ae called hybid modes. The hybid modes ae of two types labeled EH νµ and HE νµ, depending on the whethe E z o H z is dominant, espectively. 7/17

8 Field Distibutions in Optical Fibes Let s examine the mode pofiles in the plane z=0: E TE Modes: 0 1( κ0µ ) E J φ 1 0µ H J H φ 0 TM Modes: ( κ µ ) E J E H φ H J φ 1 0µ Thee is no azimuthal vaiation fo eithe type of mode. Example, TM 01 Mode: Figue All figues (unless noted) and the table in this lectue ae fom Elements of Photonics, Volume II. J 1 (κ 01 ) has a zeo at the oigin and one maximum in the coe. 8/17

9 EH νµ Modes: E J ν + νµ E J φ ν + νµ H J ν + 1 νµ H J φ ν + νµ HE νµ Modes: 1 E J ν 1 νµ E J φ ν νµ H J ν 1 νµ H J φ ν 1 νµ cosνφ sinνφ sinνφ cosνφ cosνφ sinνφ sinνφ cosνφ 9/17

10 Example - the HE 21 mode: 1( κ21 ) E J 1 21 E J φ H J 1 21 H J φ 1 21 cos2φ sin2φ sin2φ cos2φ E is puely adial fo φ = 0, π/2, π, and 3π/2. E is puely azimuthal fo φ = π/4, 3π/4, 5π/4, and 7π/4. H looks like E otated counte clockwise by π/4. J 1 (K 21 ) has a zeo at the oigin and one maximum in the coe. Field is puely adial hee Field is puely azimuthal hee Fields ae zeo hee Fields have a maximum hee φ Figue /17

11 Linealy Polaized (LP) Optical Fibe Modes It is customay in the theoy of optical fibes to make the weakly guiding appoximation n 1 = n 2 (the efactive index of the coe equal the efactive index of the cladding) because: 1. It simplifies the chaacteistic equation fo the modes. 2. It leads to the concept of linealy polaized modes. In the weakly guiding appoximation the lage steps in Figue become not jagged as modes become degeneate (i.e. they have the same popagation constant). The degeneate modes can be added togethe to fom new modes. 11/17

12 Can we constuct a set of linealy polaized modes? Yes. This is good because polaized light fom a lase would excite these supepositions of tue fibe modes. HE 11 is aleady linealy polaized. Figue in Elements of Photonics, Volume II. Othe LP modes can be constucted fom sums of the EH and HE modes that have the same popagation constant. 12/17

13 + = 13/17

14 Constuction and Labeling Rules: LP 0µ = HE 1µ LP 1µ = HE 2µ + TE 0µ o HE 2µ + TM 0µ LP mµ = HE m+1,µ + EH m-1,µ 14/17

15 Fibe Mode Degeneacy and Numbe of Modes Degeneacy of the Hybid Modes Fom Electomagnetic Theoy fo Micowaves and Optoelectonics, Keqian Zhang and Dejie Li The TE 0µ and TM 0µ modes ae not degeneate. The hybid EH νµ and HE νµ modes ae two-fold degeneate. Degeneacy of the LP Modes The LP 0µ modes ae the HE 1µ modes, so they ae two-fold degeneate. 15/17

16 The LP 1µ modes ae fomed by summing HE 2µ + TE 0µ o HE 2µ + TM 0µ, so they ae fou-fold degeneate. The LP mµ modes with m > 1 ae fomed by summing HE m+1,µ + EH m-1,µ, so they ae fou-fold degeneate. Two of the 4 LP 21 modes that can be fomed fom HE 31 and EH 11 modes. Fom Electomagnetic Theoy fo Micowaves and Optoelectonics, Keqian Zhang and Dejie Li 16/17

17 Fom Intoduction to Fibe Optics, Ghatak and Thyagaajan Numbe of Modes Fo lage V, the numbe of LP o hybid of modes is 4V 2 /π 2. 17/17