Theoretical Description of the Nonlinear Optical Pulse Propagation
An electric field \(E(\vec {r},t)\) interacts with the charged properties of matter. As electrons possess considerably less mass than the ionic cores, their dynamics excited by the driving field evolve on shorter timescales than the induced lattice oscill
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Theoretical Description of the Nonlinear Optical Pulse Propagation
An electric field E( r , t) interacts with the charged properties of matter. As electrons possess considerably less mass than the ionic cores, their dynamics excited by the driving field evolve on shorter timescales than the induced lattice oscillation. The external field polarizes the material, inducing oscillating dipoles which themselves emit radiation. Therefore the propagation of the light inside the medium takes the form of a polariton, consisting of the electric field E( r , t) coupled to the polarization wave P( r , t). The propagation and interaction of E( r , t) and P( r , t) in the absence of free carriers and charges is described by the nonlinear wave equation [1] r , t) + − ∇ 2 E(
r , t) r , t) 1 ∂ 2 E( 1 ∂ 2 P( = − c2 ∂t 2 c2 0 ∂t 2
(2.1)
c is the speed of light in vacuum and 0 the vacuum permittivity. Equation (2.1) uses the MKS unit system. This convention will be employed throughout the entire thesis. If the amplitude of the external field becomes sufficiently large (≥ 1 V◦ in A dielectrics) the dipoles cannot follow the driving field linearly anymore. This results r , t) to the polarization response [2]. in a nonlinear contribution P N L ( r , t) + P N L ( r , t) P( r , t) = P L (
(2.2)
r , t) represents the linear material response and accounts for the linear disP L ( persion the different frequency components experience during propagation [1]. r , t) = 0 χ(1) ( r , t)E( r , t) P L (
(2.3)
r , t) is the lowest order susceptibility. In general χ(l) ( r , t) are tensors of the χ(1) ( order l + 1 and depend on the frequency of the applied field. For reasons of lucidity χ(l) will be treated as scalars in the following and the material response is assumed to be instantaneous.
© Springer International Publishing Switzerland 2016 A.M. Sommer, Ultrafast Strong Field Dynamics in Dielectrics, Springer Theses, DOI 10.1007/978-3-319-41207-8_2
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2 Theoretical Description of the Nonlinear Optical Pulse Propagation
If the amplitude of the nonlinear contribution to the polarization is sufficiently small, it can be considered as a small perturbation of the system and approximated in a Taylor expansion [1]. r , t) = 0 χ(2) E( r , t)2 + 0 χ(3) E( r , t)3 + · · · P N L (
(2.4)
The susceptibility and the first order material permittivity (1) are connected by = 1 + χ(1) . Additionally (1) is related to the linear refractive index n 20 = (1) . Under consideration of Eq. (2.1) and this dependency Eq. (2.2) can be written as [1] (1)
− ∇ 2 E( r , t) +
r , t) r , t) (1) ∂ 2 E( 1 ∂ 2 P N L ( = − c2 ∂t 2 c2 0 ∂t 2
(2.5)
Fourier transformation of Eq. (2.5) into the frequency domain results in a differω ( ential equations for every frequency component E r ). The individual equations are NL r ) which involves all frequencies as it coupled by the nonlinear source term Pω ( depends on the complete electric field. ω ( ∇2 E r) +
(1) ω 2 ω 2 N L r) = − 2 P ( r) E ω ( 2 c c 0 ω
with E( r , t)
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