Semiconductor Optoelectronics (Farhan Rana, Cornell University) Chapter 13 Distributed Feedback (DFB) Structures and Semiconductor DFB Lasers 13.1 Distributed Feedback (DFB) Gratings in Waveguides 13.1.1 Introduction: Periodic structures, like the DBR mirrors in VCSELs, can be also realized in a waveguide, as shown below in the case of a InGaAsP/InP waveguide. InP a InGaAsP waveguide InGaAsP/InP grating region InP z=0 z=L In the waveguide shown above, periodic grooves have been etched in the top surface of the InGaAsP waveguide before the growth of the top InP layer. Such periodic grating structures are examples of one dimensional photonic bandgap materials. The relative dielectric constant is a function of the z- coordinate and can be written as, (x,y,z) avg(x,y) (x,y,z) The average dielectric constant avg(x,y) corresponds to the waveguide structure shown below in which the grating region has been replaced by a layer with a z-averaged dielectric constant. InP InGaAsP InGaAsP/InP grating region replaced by a layer with a z-averaged dielectric constant InP z=0 z=L Semiconductor Optoelectronics (Farhan Rana, Cornell University) The z-average of the part (x,y,z) is therefore zero. If the period of the grating is a , then one may expend (x,y,z) in terms of a Fourier series, ipGz (x, y,z) f x,y dpe p0 where the reciprocal lattice vector (also called the grating vector) G equals 2 a . If (x,y,z) is * real then, dp dp . f x,y equals one in the grating region and equals zero everywhere else. In the above Fourier series for (x,y,z) , usually the fundamental harmonic dominates and therefore we will assume that, iGz iGz (x,y,z) f x,y d1e d1e A wave travelling in the waveguide with a wavevector can Bragg scatter from the periodic grating provided the conditions for Bragg scattering are satisfied, G final final The only way these conditions can be satisfied in one dimension is when final , i.e. the wave is reflected in the opposite direction, G G 2 a 2neff So a forward traveling wave will be Bragg reflected if its wavevetor is close to G 2 a . If we call this special wavevector o then o G 2 a . We can write (x,y,z) as, i2o z i2oz (x, y,z) f x, y d1e d1e 13.1.2 Wave propagation in a DFB Grating Waveguide – Coupled Mode Technique: One can analyze wave propagation in a DFB grating waveguide in two steps discussed below. Step 1: First consider the waveguide corresponding to avg(x,y) shown in the Figure above and solve for the eigemodes and the propagation vectors (eigenvalues) for all frequencies of interest. The eignemodes, Ex,y eiz and Hx,y eiz satisfy Maxwell’s equations, iz iz Ex,y e ioH x,y e iz iz Hx,y e ioavg x,y E x,y e or, t izˆ Ex,y ioHx,y t izˆ Hx,y ioavg x,y E x,y The above equations can be solved to give the mode effective index neff . Given a grating structure, we can now find the frequency o that will Bragg scatter from the relation, o n o c eff o o a If the wavevector is very different from o then the grating structure will likely not affect the solution much (there will be not much scattering). The interesting case is when o . This case is discussed below. Semiconductor Optoelectronics (Farhan Rana, Cornell University) Step 2: We treat the part (x,y,z) as a perturbation. The perturbation will have the strongest affect when o . For o , we write the solution as, iz * iz Ex,y,z A z E x,y e A z E x,ye iz * iz Hx,y,z A z H x,y e A z H x,y e Here, the functions A z and A z are assumed to be slowly varying in space. The form of the solution allows for coupling between the forward and backward going waves because of Bragg scattering from the grating. The technique described below is called coupled mode theory. Plugging the assumed form of the solution in Maxwell’s equations gives, iz * iz iz * iz A z E x,y e A z E x,y e io A z Hx,y e A zH x,y e iz * iz i2oz i2oz A z H x,y e A z H x,y e io avg x,y f x,y d1e d1e iz * iz A z E x,y e A z E x,y e Using the Maxwell’s equation satisfied by the eigenmode we get, dA z dA z zˆ Ex,y eiz zˆ E* x,y eiz 0 dz dz dA z iz * dA z iz i2oz i2oz zˆ Hx,y e zˆ H x,y e iof x,y d1e d1e dz dz iz * iz A z E x,y e A z E x,y e We multiply the first equation above by H* x,y and multiply the second equation above by E * x,y and then subtract the two equations, and keep only the terms that are approximately phase matched to get on left and right hand sides to get, * * dA z i2 z f x,y E x,y . E x,y dxdy i d e o A z o 1 * * dz Ex,y H x,y E x,y H x,y . zˆ dxdy If instead of subtracting, we add the two equations then we obtain, dA z i 2 z f x,y Ex,y . Ex,y dxdy i d e o A z o 1 * * dz Ex,y H x,y E x,y H x,y . zˆ dxdy * If (and only if) (x,y,z) is real and d1 d1 , then the above two equations can be written as, d A z 0 i ei2 o z A z i2 z dz A z i * e o 0 A z where the coupling constant is, M * * nng E x,y . E x,y dxdy M * d Grating region nng E x,y . E x,y dxdy o 1 M M * * 2nng nng E x,y . E x,y dxdy Re Ex,y H x,y . zˆ dxdy d 1 M G 2nng vg M Here, nng is the product of the index and the (material) group index of the grating region, vg is the group velocity of the mode, and the overlap integral G is the usual mode confinement factor for the grating region provided the mode electric field is real (for example, the mode electric field will be real Semiconductor Optoelectronics (Farhan Rana, Cornell University) if avg(x,y) is real and the z-component of the field is negligible). The coupling constant couples the forward and the backward propagating waves. To solve the above set of equations, we assume, i( o )z B (z) A (z)e i( o )z B (z) A (z)e B (z) and B (z) satisfy, d B (z) i( o ) i B (z) dz B (z) i * i( o )B (z) i i B (z) i * i B (z) We have a 2x2 linear system of equations. The eigenvalues of the matrix on the right hand sided are s where, 2 2 s 2 iq s 2 iq The corresponding eigenvectors are, s s is is The most general form of the solution is, B (z) iqz iqz iq s C1 e C2 e B (z) is is iq s The constants C1 are C2 are determined by the boundary conditions. Note that in terms of B (z) and B (z) the electric field can be written as, ioz * ioz E(x,y,z) E(x,y)B (z)e E (x,y)B(z)e From the expression above, the effective propagation vector of, say the forward going wave, at frequency is not anymore but is k where, 2 2 2 2 k o q o o o The difference between the modal dispersions and k is depicted in the Figure below. () () Bandgap g (k) (k) Note that a frequency gap (or a bandgap) opens in the dispersion relation of magnitude given by, g 2vg For values of frequency that fall in this bandgap, no real value of the propagation vector k satisfies the dispersion relation given above. Semiconductor Optoelectronics (Farhan Rana, Cornell University) 13.1.3 DFB Waveguide Mirror (or a Distributed Bragg Reflector (DBR)): Consider a DFB structure as shown in the Figure below. We need to calculate the reflectivity of the mirror for a wave coming in inside the waveguide from the left side. The reflection and transmission coefficients are, B 0 B (L) i L r t e o B 0 B (0) The boundary conditions are, B L 0 and B 0 0 . B (0) B (L) + + B (0) B (L) - - z=0 z=L We need to find the constants C1 are C2 in, B (z) iqz iqz iq s C1 e C2 e B (z) is is iq s in order to satisfy the above boundary conditions. The final result is, sinh[s(z L)] is cosh[s(z L)] B (z) B (0) sinh(sL) is cosh(sL) * sinh[s(z L)] B (z) B (0) sinh(sL) is cosh(sL) The reflection coefficient is, B (0) * sinh(sL) r B (0) sinh(sL) is cosh(sL) The transmission coefficient is, B (L) i L is i L t e o e o B (0) sinh(sL) is cosh(sL) The magnitude of the reflection coefficient is maximum when the wavevector of the incident wave is equal to o and 0 , * r i tanh L 0 2 Rmax tanh L The Figure below plots the reflectivity of a DBR mirror as a function of the wavelength (or wavevector) for different values of the coupling constant.