Theoretical calculations of the carrier induced refractive index ...

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181111-2 Michael J. Connelly Appl. Phys. Lett. 93, 181111 2008 FIG. 2. Color online FCA induced refractive index change at 1550 nm for a strain of 0.15%. The parabolic approximation uses effective masses for unstrained InGaAsP. FIG. 4. Color online BF and FCA contributions to the refractive index change vs wavelength for no strain. The carrier density increases from 110 24 to 410 24 m −3 in increments of 110 24 m −3 in the direction of the arrows. There is no polarization dependence. TE,TM E,n = C g E b=LH,HH,SO0 0 k 2 sin M TE,TM,b k, 2 LE cv,b k,1−f c kf v,b k,d dk, 3 where “LH,” “HH,” and “SO” denote the light-hole, heavyhole, and split-off valence bands VBs respectively, C g =e 2 /2 0 cm 2 0 N r , N r is the refractive index with no injected carriers, M TE,TM,b k, are the matrix elements, and E cv,b k, are the conduction band CB to VB transition energies. k and are the magnitude and azimuth of the momentum vector. 7 Intraband relaxation and doping effects are taken into account using a sech line broadening function LE cv,b = r / sech⌊ r /E cv,b −⌋, where r the effective intraband relaxation time is taken to be 22.4 fs. 5 The absorption spectrum calculations take into account band-gap shrinkage. Calculated absorption spectra for a strain of 0.15% are shown in Fig. 1. Using Eq. 2 N BF,TE,TM can be FIG. 3. Color online BF and FCA contributions to the refractive index change at 1550 nm as a function of carrier density. The strain varies from 0% to 0.2% in increments of 0.05%. FIG. 5. Color online BF and FCA contributions to the refractive index change as a function of wavelength, with the carrier density as parameter, for a tensile strain of 0.15%. The carrier density increases from 110 24 to 410 24 m −3 in increments of 110 24 m −3 in the direction of the arrows. The solid and dotted lines of the BF contribution refer to TE and TM polarizations, respectively. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
181111-3 Michael J. Connelly Appl. Phys. Lett. 93, 181111 2008 determined as a function of wavelength, carrier density, and strain. The refractive index change due to FCA is usually given as 2 N FCA E,n =− e2 2 n + p 4 2N r 0 E m c m v, where parabolic bands are assumed. m c and m v are the effective masses of the CB and VB heavy-hole, respectively. p is the VB hole density. In general, the CB can deviate from being parabolic. Furthermore in tensile-strained material, the light-hole band is raised above the heavy-hole band and can contribute significantly to the FCA refractive index change. The VB effective masses are not constant but can vary significantly with momentum. Generalizing from Eq. 4, the CB contribution to N FCA is given by N FCA,c E,n =− e2 2 2N r 0 E 2 0 k 2 2 m c k f cE c kdk, where E c is the CB energy-momentum relationship. 5 m c k is the k-dependent effective mass obtained using a parabolic fit to E c k in a small region around k. k 2 / 2 m c k is the density of states and f c is the Fermi–Dirac distribution function. Similarly the VB contribution to N FCA is given by N FCA,v E,n =− e2 2 k 2N r 0 2 sin E 2 b=HH,LH0 dk0 2 2 m v k, f v E v,b k,d. 6 It is assumed that p=n, as is usually the case in SOAs. E v,b k, is the VB energy-momentum relationship and 5 m v k, is the k and dependent effective mass obtained using a parabolic fit to E v,b k, in a small region around k for a given . Equation 4 can be rewritten as N FCA E,n = N FCA,c E,n + N FCA,v E,n. 7 A typical carrier density dependence of N FCA showing the CB and VB contributions and comparison with the parabolic approximation is shown in Fig. 2. The BF and FCA contributions to the refractive index change at 1550 nm are shown in Fig. 3 as a function of carrier density with strain as parameter. The FCA component is effectively independent of the strain. However, the BF contribution does have significant polarization dependence. The BF and FCA contributions to the refractive index change as a function of wavelength, with the carrier density as parameter for strains of 0% and 0.15%, are shown in Figs. 4 and 5, respectively. The refractive index change has a significant wavelength dependency and when strain is present significant polarization dependency. 1 H. Soto, D. Erasme, and G. Guekos, IEEE Photon. Technol. Lett. 11, 970 1999. 2 L. Q. Guo and M. J. Connelly, J. Lightwave Technol. 23, 4037 2005. 3 L. Q. Guo and M. J. Connelly, J. Lightwave Technol. 25, 4102007. 4 C. Michie, A. E. Kelly, J. McGeough, I. Armstrong, and I. Andonovic, J. Lightwave Technol. 24, 3920 2006. 5 M. J. Connelly, IEEE J. Quantum Electron. 43, 472007. 6 P. K. Basu, Theory of Optical Processes in Semiconductors Oxford University Press, New York, 1997. 7 G. Jones and E. P. O’Reilly, IEEE J. Quantum Electron. 29, 1344 1993. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
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