Bragg waveguides. Leaky modes 1D Bragg reflector waveguides

Transcription

1 Outline Bragg waveguides These are covered etensively in Nanophotonics. Let s look closely at the nature of the modes that are common to all such structures. Leaky modes 1D Bragg reflector waveguides 142

2 Leaky modes Slab W slab Assume a solution that is continuous at = a, b, Re(k )>0 so only outgoing, even solution: E() = E(-) Identical to slab ecept include both eponentials Set derivates = at = a, b: Same function as core, same inde = a Eliminate φ to get transcendental equation = b Jonathan Hu and Curtis R. Menyuk, "Understanding leaky modes: slab waveguide revisited," Adv. Opt. Photon. 1, (2009) 143

3 Eample solution e n 2 =1.39 n 1 = 1.45 a = 0.39 λ V = 1 b = 5 a k α N = co ( N k0 ) = k0( j ) ( N k ) k = k ( j ) k 0 cl 0 j Field decays in z All quantities now have (small) imaginary part. 144

4 Shape of leaky mode E 2 looks like a mode. Re[E] indicates power is flowing outward a >b The magnitude of the field beyond >b is also growing eponentially. 145

5 Energy conservation The amplitude is decreasing in z. The material is not lossy, so where is the power going? Calculate power being lost between two planes z = z 1 to z 2 over two widths = ±A or = ±B A B Re@ED êl z Re@ED [ ( ) ] A 2Im β z z e E( ) d êl A A B z 1 P entering z 1, eiting z 2 between = ±A [ ( ) ] 2Im β z z e E( ) d B B P entering z 1, eiting z 2 between = ±B Second quantity is larger than first. Thus power flowing out through sides = ±B must be > than power flowing out through sides = ±A. The only way this is possible is if ( B) E( A) E > 146

6 Another eplanation for eponential transverse growth Power in mode is decreasing eponentially in z. The energy propagates at angle θ=arccos(β/k z ) At a plane z, this energy arrives at = tan(θ) z Intensity at z must increase eponentially with n co n cl n co k β k r co E(, z) 2 n cl n co [ E( )] Re,0 E( 0, z) 2 Foundations of Photonic Crystal Fibres, Zolla et al, Imperial College Press, section z 147

7 Real and Fourier-space Free-space and slab n=1.45 Real space Fourier-space (radiation modes) n cl =1.39 n co =

8 Real and Fourier-space Leaky mode Radiation modes have form below. k is real. Have both ingoing and outgoing waves. E(k,) = E*(k,-) Solve for form of radiation modes, epand same Gaussian input in this basis set: Fourier-space Real space (radiation modes) k r Spectrum is very nearly a Lorentzian = single pole (+cc) a ( k ) k ( k ) jk a( k ) jkia r i r + ( kr + jki ) k + ( kr jki ) The generalized (see ref) Fourier transform of this function is ep [( jk k ) ] r Thus k r is the oscillation wave # in and k i is the eponential gain factor. i 149

9 Loss and evolution of leaky modes 150

10 The troubles with leaky modes Not square integrable, so can not be treated as basis functions of a Hilbert space Not orthogonal No guaranteed ordering of effective inde as in the case of bound and radiation modes. In fact, N of a leaky mode typically lands in the middle of the radiation spectrum. Foundations of Photonic Crystal Fibres, Zolla et al, Imperial College Press, section

11 Bragg reflection waveguides Bragg reflector waveguides In the light of leaky modes By the Floquet theorem E ( ) = EK c cos EK jk ( ) e ( k) ( ) e jk < a < a > a E ( ) jk ( ) where is the Bloch mode, is periodic. K e E K Satisfying the EM boundary conditions at the interfaces yields a 22 matri equation whose determinant yields the transcendental characteristic equation for the modes. When K is comple, the field decays eponentially away from the core. For a finite thickness Bragg stack surrounded by a medium with inde equal to that of the core, the wave in the eternal medium must have transverse wave vector k and thus these modes behave eactly as the leaky modes of the W guides. 152