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Leonel Gonzalez*, General Dynamics Shekhar Guha, Air Force Research Lab

Propagation of High Intensity Light in Semiconductors. 16 July 2007 ICIAM07 – 6 th International Congress on Industrial and Applied Mathematics. Leonel Gonzalez*, General Dynamics Shekhar Guha, Air Force Research Lab Qin Sheng, Baylor University Srini Krishnamurthy, SRI International

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Leonel Gonzalez*, General Dynamics Shekhar Guha, Air Force Research Lab

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  1. Propagation of High Intensity Light in Semiconductors 16 July 2007 ICIAM07 – 6th International Congress on Industrial and Applied Mathematics Leonel Gonzalez*, General Dynamics Shekhar Guha, Air Force Research Lab Qin Sheng, Baylor University Srini Krishnamurthy, SRI International *email: leo.gonzalez@gdit.com 1

  2. Outline • Intro/background • Description of one and two photon absorption • Wave equation • Coupled PDE’s • Numerical modeling • Comparison with analytic solution • Conclusions

  3. Intro/background • Investigate how light propagates through bulk semiconductors: • Under conditions of low and high irradiance • Linear (1PA), Nonlinear (2PA) and Free Carrier (FCA) absorption present • Dependence on laser pulsewidth and FCA recombination lifetime • Applications include: • Optical diodes • Saturable/reverse saturable absorbers (laser mode lockers) • Solar cells

  4. 2PA: 2PA + FCA: 0.5 Egap < hn < Egap hn > Egap 0.5 Egap < hn < Egap Absorption Processes:1PA, 2PA & FCA 1PA:

  5. n2 within semiconductor: a 2PA b Instantaneous (Bound electronic) Dielectric breakdown Temperature Rise, DT Free carriers  N Recombination t dn/dT Melting Damage Absorption sa Refraction sr Refraction lattice & band gap expansion band filling /saturation, Moss-Burstein, plasma Incident Laser pulse / E-field (r, z, t) Pulsewidth dep. ( > psec) Refraction Absorption External: Exit Laser pulse / E-field* (r, z, t)

  6. Wave Equation Starting from Maxwell’s Equations, the nonlinear wave equation can be expressed as: Assuming an Electric field of the form where the Amplitude term, A(z, r, t) is slowly varying in space and time and is collimated through the sample, therefore:

  7. Field Amplitude & Irradiance The wave equation is now: Next, using the Poynting’s theorem relating irradiance and the Electric field: and

  8. Nonlinear Absorption The change in irradiance through the sample: Where I = beam irradiance a = linear absorption coefficient (1PA) [cm-1] sabs = Free carrier absorption cross-section [cm2] N = free carrier density [cm-3] b = nonlinear absorption coefficient [cm/MW]

  9. Coupled PDE’s • NLA: • NLR:

  10. Analytic Solution (no FCA) FCA not present: Solve for Iover z: 0 to crystal length Integrate I(l) over area to find total power, and over time to calculate total energy:

  11. Analytic Solution cont. • Ratio of output to input energies is transmission: • Where Q is: (dimensionless)

  12. Finite Difference Normalizing the NLA equation

  13. Finite Difference: I Expressed as an implicit Finite Difference approximation: which is a quadratic equation for with a solution of the form

  14. Finite Difference: N Expressing the Free Carrier rate equation as a finite difference:

  15. Procedure • For a given incident energy, calculate the peak irradiance, I0 • Loop over the radial dimension • For each radial point, use split-step method to propagate the beam through the sample updating I & N • Integrate the beam at the output of the sample over time to calculate the fluence as a function of radial position, F(r) • Integrate F(r) over the radial dimension to determine the output energy

  16. Boundary Conditions and Stencil lcrystal e-(t – tcenter)2 ..… At crystal end, integrate over time to find output F(ri).

  17. Comparison of FD and Analytic Solutions Good agreement between analytic and FD solutions for no FCA (dN/dt = 0) and FCA with t0 << t

  18. Stability Analysis • Stability analysis of the numerical method: • Investigating the relative error • With and without FCA present

  19. Stability Analysis cont. • No significant difference in k2 between FCA cases

  20. FCA Present: t0/t

  21. Comparison of NLA for CdTe Determination of material parameters: b, sabs, t Using two different pulse duration Nd:YAG lasers: ps << t & ns ~ t Material Characterization

  22. b, sabs, t : f(N, T) • For longer pulse durations the parameters: b, sabs, t are no longer constants but are dependent on both N and Temperature: f(N, T) • These are not general expressions but depend on the band structure of the material:

  23. Conclusions • Used an implicit Finite Difference scheme to model beam propagation through thin samples with FCA present during the pulse. • Continuing work to characterize interdependencies of parameters: b, sabs, t = f(N, T) • Beam propagation is also commonly done using Fast Fourier Transforms (FFTs), however the method used here easily handles cases of high nonlinearity and can be easily modified for the non-paraxial case. • Future work will include the effects of diffraction in the case of thick samples. Additional work by co-authors is investigating the non-paraxial cases.

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