Analytical theory of second harmonic generation from a nanowire with noncentrosymmetric geometry

We analytically investigate the effect of a noncentrosymmetric geometry in the optical second harmonic (SH) generation from a nanowire made of a centrosymmetric material, in the interior of which quadratic optical processes are suppressed. We consider an infinite cylinder with a cross section that is slightly deformed away from a circle and with a radius much smaller than the wavelength. We calculate the induced linear and nonlinear fields perturbatively in terms of the deformation parameter, and we obtain the nonlinear dipolar and quadrupolar hyperpolarizabilities, whose spectra we evaluate for metallic and dielectric materials. We show that for very small deformations, the dipolar contribution to the response competes with the quadrupolar term, and may even be dominant. We explore the spectra of the hyperpolarizability and identify the contributions to its structure for metallic and dielectric nanowires. We also discuss the nature of the SH radiation at various frequencies, and we find that it may be dominated by the dipolar or the quadrupolar term, or that both may compete yielding nonsymmetric radiation patterns. Our calculation may be employed to assess, calibrate, and test numerical SH calculations.

Figure 1

Cross section of a deformed cylinder described by Eq. (1) for various values of the deformation parameter $d=0.0,...,0.2$.

Figure 2

Direction of the quadratic dipole moment induced in a deformed cylinder by an external field with different directions with respect to the horizontal. As the field rotates clockwise by an angle $\theta$, the dipole rotates counterclockwise by $2\theta$.

Figure 3

Normalized absolute value (solid) and phase (dashed) of the dipolar (upper panel) and quadrupolar (lower panel) nonlinear response functions ${\gamma }^d$ and ${\gamma }^Q$ for a cylinder with deformation parameter $d=0.01$ made of a Drude metal as a function of the normalized frequency $\omega /{\omega }_p$. The irrelevant abrupt $2\pi$ phase jumps are indicated with dotted vertical lines.

Figure 4

Normalized absolute value (solid) and phase (dashed) of the dipolar (upper panel) and quadrupolar (lower panel) nonlinear response functions ${\gamma }^d$ and ${\gamma }^Q$ for a cylinder made of a dispersive dielectric with a Lorentzian resonance characterized by a longitudinal ${\omega }_L$ and a transverse ${\omega }_T$ frequency with ${\omega }_L=\sqrt{2}{\omega }_T$, a lifetime $\tau =100/{\omega }_T$, and a deformation parameter $d=0.01$, as a function of the normalized frequency $\omega /{\omega }_T$. The irrelevant abrupt $2\pi$ phase jumps are indicated with dotted vertical lines.

Figure 5

Angular radiation pattern for a deformed metallic cylinder with deformation parameter $d=0.01$ described by a Drude response for frequencies $\omega$ approaching ${\omega }_{\text{sp}}$ or to its subharmonic: $\omega <{\omega }_{\text{sp}}/2$ (upper left), ${\omega }_{\text{sp}}/2<\omega$ (upper right), $\omega <{\omega }_{\text{sp}}$ (bottom left), and $\omega >{\omega }_{\text{sp}}$ (bottom right). As $\omega$ approaches a resonance, the total radiated power increases.

Figure 6

Angular radiation pattern for a deformed dielectric cylinder with deformation parameter $d=0.01$ described by a simple Lorentzian response with negligible dissipation for frequencies $\omega$ close to ${\omega }_{\text{spp}}$ or its subharmonic: $\omega <{\omega }_{\text{spp}}/2$ (upper left), ${\omega }_{\text{spp}}/2<\omega$ (upper right), $\omega <{\omega }_{\text{spp}}$ (bottom left), and $\omega >{\omega }_{\text{spp}}$ (bottom right). As $\omega$ approaches a resonance, the total radiated power increases.