Back to basics : Maxwell equations & propagation equations

Transcription

1 The step index planar waveguide Back to basics : Maxwell equations & propagation equations

2 Maxwell equations Propagation medium : Notations : linear Real fields : isotropic Real inductions : non conducting non magnetic Inhomogeneous : n(x,y,z) : transverse distribution : complex amplitude -7 μ0 = 4 π 10 ε0 = ε μ = 1 0 0c -1

3 Impact of the index inhomogeneity and Vectorial identity : In other words : 3

4 The propagation equations Similarly : 4

5 Symmetrical step index plane waveguide x Inhomogeneous medium along x d z n(x) y Infinite along y and z z is the direction of propagation n 1 n n Step refractive index : d - d x Almost everywhere n = 0 + continuity conditions at the interfaces 5

6 Continuity conditions When written as generalized fonctions (distributions), the Maxwell equations translate into continuity equations (cf 1A). n 1 1 in the plane of the interface At the interface between media 1 and continuity of 6

7 Guided modes : definition Guided Modes are distribution of fields which transverse structure remains invariant by propagation : real propagation constant Components of the fields : 7

8 Components of the fields From Maxwell equations, we can deduce 6 coupled equations between the components of the fields (transverse structure): Demonstration (to do) : 6 coupled equations 8

9 Components of the fields Solutions : 9

10 Plane waves in free space Consequences : Transverse Electro Magnetic fields (TEM) : Dispersion relation 10

11 Guided modes in a planar waveguide x d y z Infinite along y and z z is the direction of propagation Rewrite the 6 equations => we find independant systems of equations 11

12 TE and TM modes of the planar waveguide, and independant systems, and Transverse Electric modes TE Transverse Magnetic modes TM 1

13 Study of the TE modes Symmetrical planar waveguide : n(x) n 1 From the first system of equations, we find : n n d d x - with Inside the waveguide : Outside the waveguide : Continuity of the tangential components and Continuity of and 13

14 Radiative and guided modes n 1 n(x) n n d - d x Modes Oscillating Oscillating Radiating Oscillating Exponential Guided Exponential Exponential 0 solution Guidance condition NB : Guided and radiating modes form a complete basis on which we can decompose any propagative field 14

15 Study of the TE modes Inside the waveguide : Out of the waveguide : Theorem : In a symmetrical waveguide, the field is either symmetrical or anti symmetrical 15

16 Inside the waveguide : Study of the symmetrical TE modes Out of the waveguide : Normalized notations : Reduced propagation constant Reduced extinction coefficient Reduced frequency 16

17 Inside the waveguide : Study of the symmetrical TE modes Out of the waveguide : Normalized notations : Reduced propagation constant Reduced extinction coefficient Reduced frequency From the continuity of and we can demonstrate : Relation of dispersion 17

18 Study of the antisymmetrical TE modes Inside the waveguide : Out of the waveguide : Using the normalized notations we find : TO DO Relation of dispersion 18

19 Graphic solution for the existence of TE modes f(u) mode 0 mode 1 mode mode 3 Symmetrical modes (even #): Antisymmetrical modes (odd #) : u For a given symmetrical planar waveguide, how to determine the number of symmetrical and antisymmetrical modes and their longitudinal propagation constant? 0 / / 19

20 Graphic solution for the existence of TE modes f(u) mode 0 mode 1 mode mode 3 Symmetrical modes (even #): Antisymmetrical modes (odd #) : u For each value of V : determine the number of existing symmetrical and antisymmetrical modes Ex : V=5 => 1 symm. mode and 1 antisymm. mode determine the corresponding discrete values of u deduce the corresponding propagation constant 0 / / 0

21 Graphic solution for the existence of TE modes f(u) mode 0 mode 1 mode mode 3 Remarks : 1) the mode 0 always exists : this is the fundamental mode called TE 0 (symmetrical) ) Definition of the cut off frequency 3) Domain of variations of u for a TE m mode u 4) For a given mode TE m : at the cut off frequency, the mode is deconfined 0 / / 1

22 Solutions for TE modes 8 u(v) u=v/ m=5 m=4 6 m=3 4 m= m=1 m= V 0

23 Shape of the modes 8 u(v) 6 u=v/ m=5 m=4 m=3 4 m= m=1 m= V 0 m=0 m=1 m= m=3 Remark : The # of zero in the field profile =the # of the mode 3

24 Shape of the modes First symmetrical mode Mode 0 (u in [0, /]) u=0.5( /) u=0.1( /) d u=0.9( /) When u /, mode 0 becomes more confined in the core 4

25 Shape of the modes First antisymmetrical modes u=0.1( /) u=0.5( /) u=0.9( /) Mode 1 (u in [ /, ]) Mode 3 (u in [3, ]) d d When u, mode 1 becomes more confined in the core When u, mode 3 becomes more confined in the core 5

26 Normalized propagation constant and effective refractive index (TE modes) Effective refractive index : Normalized propagation constant: Modes n eff <n b<0 Oscillating Oscillating Radiating n n eff n 1 0 b 1 Oscillating Exponential Guided n 1 <n eff b>1 Exponential Exponential 0 solution 0 b 1 n n eff n 1 Guidance condition 6

27 Effective refractive index (TE modes) ndice effectif I n 1 =1.48 n = V 15 0 For b : same curves, normalized between 0 and 1 7

28 The guided TM modes TO DO d H x y H k n x y x d H y H x y 0 κ = β - k 0 n Continuity of H y x and Symmetrical TM Modes 1 n H y x x Antisymmetrical TM Modes n 1 V utanu n u 1/ ucotu n 1 V n 1/ u 8

29 Birefringence of the step index planar waveguide A single mode planar waveguide actually supports : 1 TE mode (full lines) 1 TM mode (doted lines) 1 b Ex : n 1 =1.5 n =1 d=0.555 µm 0 =1.3 µm 0.5 V 3 b TE b TM TE: n eff TM: n eff V Strong birefringence : n 0.09 =>Phase shift of after a propagation distance L b of 7.5 µm L b = 9

30 Transported power (TE) Flux of power = time average of the Poynting vector No flux of power in the directions perpendicular to propagation TO DO Transported power (TE, symm. and antisymm.) 30

31 Longitudinal components of the fields (TE) with Guided mode : n n eff n 1 with n eff = /k 0 Ex : n 1 = n = n 1 n This is the so called weak guidance approximation =>The longitudinal components of the fields may be neglected 31