Simulation of Phonon-Polariton Generation and Propagation in Ferroelectric LiNbO 3 Crystals
- PDF / 287,693 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 43 Downloads / 143 Views
C11.60.1
Simulation of Phonon-Polariton Generation and Propagation in Ferroelectric LiNbO3 Crystals David W. Ward, Eric Statz, Nikolay Stoyanov, and Keith A. Nelson The Massachusetts Institute of Technology Cambridge, MA 02139, USA ABSTRACT We simulate propagation of phonon-polaritons (admixtures of polar lattice vibrations and electromagnetic waves) in ferroelectric LiNbO3 with a model that consists of a spatially periodic array of harmonic oscillators coupled to THz electromagnetic waves through an electric dipole moment. We show that when this model is combined with the auxiliary differential equation method of finite difference time domain (FDTD) simulations, the salient features of phononpolaritons may be illustrated. Further, we introduce second order nonlinear coupling to an optical field to demonstrate phonon-polariton generation by impulsive stimulated Raman scattering (ISRS). The phonon-polariton dispersion relation in bulk ferroelectric LiNbO3 is determined from simulation. INTRODUCTION We consider electromagnetic wave propagation (1-dimensional and scalar for simplicity) in crystalline LiNbO3, in which there is an optic phonon resonance of the form, ω2 ( ε − ε ′ ) ′ + 2TO 0 2 ∞ , (1) ε(ω) = ε∞ ωTO − ω − i ωΓ
′ is the where ω is the electromagnetic wave frequency, ε 0 is the low frequency permittivity, ε∞ high frequency permittivity, ωTO is the transverse optic phonon frequency, and Γ is a phenomenological damping rate [1]. Introducing the permittivity in equation (1) into the expression for the macroscopic linear polarization P (ω) =∈0 ( εr (ω) − 1 ) E (ω) , we identify an auxiliary parameter Q as follows: ′ ) ∈ ω 2 ( ε ′ − ε∞ ′ − 1 ) E (ω) + 0 2 TO 20 ′ − 1 ) E (ω) + ωTO ∈0 ( ε0′ − ε∞ ′ )Q(ω ). (2) P (ω) =∈0 ( ε∞ E (ω) =∈0 ( ε∞ ωTO − ω − i ωΓ
Q may be evaluated independently of equation (2) to determine the total macroscopic polarization by Fourier transformation and solution in the time domain: d 2Q dQ 2 (t ) + Γ (t ) + ωTO (3) Q (t ) = ωTO ∈0 ( ε 0′ − ε ∞′ ) E (t ). 2 dt dt The coupling constant, appearing as a coefficient to Q, is determined by explicitly specifying a model for the mechanical component of the polarization [2]. We consider a transverse optic phonon mode in ferroelectric LiNbO3 scaled by the oscillator density N, and the reduced mass of the unit cell µ : Q = N µ ( x + − x − ), (4) where x + − x − is the ionic displacement of the normal mode. The system can then be thought of as a lattice of radiating harmonic oscillators coupled to each other only through the electromagnetic radiation they generate when oscillating. Introducing equation (3) into Maxwell’s equations through the electric displacement field, the complete system of equations
Downloaded from https://www.cambridge.org/core. Iowa State University Library, on 08 Jan 2019 at 03:40:01, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1557/PROC-784-C11.60
C11.60.2
Figure 1: Phonon-polaritons generated through ISRS by a 50 fs optical pulse with spot size a) 50
Data Loading...
