Dynamical model of harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths

We study second and third harmonic generation in centrosymmetric semiconductors at visible and UV wavelengths in bulk and cavity environments. Second harmonic generation is due to a combination of spatial symmetry breaking, the magnetic portion of the Lorentz force, and quadrupolar contributions from inner core electrons. The material is assumed to have a nonzero, third-order nonlinearity that gives rise to most of the third harmonic signal. Using the parameters of bulk silicon we predict that cavity environments modify the dependence of second harmonic generation on incident angle, while improving third harmonic conversion efficiency by several orders of magnitude relative to bulk silicon. This occurs despite the fact that the harmonic signals may be tuned to a wavelength range where the dielectric function of the material is negative: A phase-locking mechanism binds the generated signals to the pump and inhibits their absorption. These results point the way to alternative uses and flexibility of materials such as silicon as nonlinear media in the visible and UV ranges.

Figure 1

Simple harmonic oscillator with generic charge $e$ on a spring of constant $k$, subject to external electric and magnetic forces, and to internal linear and nonlinear restoring forces. The origin is at ${\mathbf{r}}_{0;}$ r is the displacement from equilibrium. Higher poles are excited by distortions of the charge distribution due to nonuniform fields.

Figure 2

Typical geometry of a structure with two-dimensional symmetry, i.e., propagation occurs on the $y$-$z$ plane, and derivatives with respect to $x$ are suppressed. ${\mathbf{S}}_{\mathrm{TM}}$ and ${\mathbf{S}}_{\mathrm{TE}}$ are the Poynting vectors relative to TM and TE polarizations, and indicate direction of propagation. The rectangular box represents the sample.

Figure 3

Typical polarization profile as a function of position for different spatial integration steps, as indicated in the figure.

Figure 4

Reflected efficiency vs time for decreasing spatial and temporal step sizes. The calculations provide a fairly accurate answer even for the largest integration steps shown. In scaled units, $\delta \xi =z/{\lambda }_0=\delta \tau =ct/{\lambda }_0$.

Figure 5

Dielectric response of silicon at visible and near-IR wavelengths. The thin solid curves correspond to the data, as indicated. Empty, color-coded squares that retrace the complex data set correspond to a plot of Eq. (26). The data are reproduced reasonably well across the entire range.

Figure 6

SHG (a) and THG (b) as a function of incident angle for a bulk silicon substrate. The comparison in (a) with and without quadrupoles suggests that most of the SH signal may have quadrupolar origin (right axis). For THG (b), the signal originates mostly with the bulk ${\chi }^{\left(3\right),ed}$, and closely resembles the predictions in Ref. [30].

Figure 7

Linear transmittance from a silicon étalon 930 nm thick at normal incidence. At 780 nm transmittance is $\sim$80$\%$; wavelengths below 450 nm are absorbed, with no expectation to see harmonic signals pass through the étalon.

Figure 8

SHG without (a) and with (b) quadrupolar contributions, and (c) without the most dominant quadrupolar term. The shapes of the curves in (b) depend on the relative amplitudes of the matrix elements of the quadrupole tensor. Panel (c) shows that the curves can take on configurations intermediate between (a) and (b), with efficiencies similar to (a), so that the types of nonlinear sources discussed in the text contribute to similar degrees.

Figure 9

THG vs incident angle for the Fabry-Pérot étalon depicted in Fig. 7. When this curve is compared with Fig. 6, which depicts THG from bulk silicon, one can deduce that total efficiency is amplified by a factor of $\sim$30. Most of the enhancement comes from the transmitted signal, which is phase locked and resonates with the pump.

Figure 10

Linear transmittance and/or reflectance of a multilayer stack that contains a Si defect layer for right-to-left or left-to-right propagation. The resonance near 800 nm is $\sim$20 nm wide, and is resolved using subpicosecond, incident pulses [50].

Figure 11

(a) Peak transmitted and reflected conversion efficiencies as a function of incident pulse width. The pulses are tuned at resonance near 813 nm. THG saturates as longer pulses resolve the resonance. (b) Transmitted and reflected THG spectra when the incident pulse is $\sim$250 fs in duration. Even though the cavity $Q$ ≈ 40, the total conversion efficiency increases by more than two orders of magnitude compared to the Si étalon. The far left axis scale is shared by both (a) and (b).