Reconstruction of multidimensional nonlinear polarization response of Pancharatnam-Berry metasurfaces

Pancharatnam-Berry (PB) metasurfaces have been considered as innovative optical devices for efficient manipulation of both the phase and the polarization of the electromagnetic field. Here, we present the simultaneous generation of circularly, elliptically, and linearly polarized states by the simplest PB metasurface with one-dimensional phase gradient unit cells. Furthermore, we propose that the reconstruction of a multidimensional nonlinear polarization response of a nanomaterial can be achieved in a single heterodyne measurement by active manipulation of the polarization states of incident light. Using multidimensional spectroscopy, we demonstrate the possibility to track both stationary and transient delocalized charge distributions via detecting plasmonic populations and coherences.

Figure 1

(a) Sketch of the generation of the third-order nonlinear response giving rise to five beams for a single four-wave-mixing process in a nonlinear PB metasurface. The arrowed circles represent the direction of the circular polarization state of each beam. (b) Schematic of the metasurface. The length and width of the antenna are $l_x$ and $l_y$, respectively. The distance between two antennas along the $x$ and $y$ axes is $a_1$ and $a_2$, respectively. Every nanopillar along the $x$ axis is rotated by $\frac{\pi }{5}$ relative to its neighbor. For the third-order nonlinear process, the unit cell of the periodic structure is depicted by the red dashed box and includes $M=5$ nanopillars. (c) Six-level system with ${\omega }_x=460$ $\mathrm{THz}$, ${\omega }_y=500$ $\mathrm{THz}$, ${\omega }_f=1075$ $\mathrm{THz}$, ${\omega }_{x^{\prime }}=989$ $\mathrm{THz}$, ${\omega }_{y^{\prime }}=1125$ $\mathrm{THz}$, ${\mu }_y=0.6{\mu }_x$, decay $\gamma =10$ $\mathrm{THz}$.

Figure 2

The nonlinear spectrum vs the signal (new mode) frequency. The resonance frequencies depicted as vertical dashed lines are ${\nu }_1={\omega }_x$, ${\nu }_2={\omega }_y$, ${\nu }_3={\omega }_{x^{\prime }}-{\omega }_x$, ${\nu }_4={\omega }_f-{\omega }_y$, ${\nu }_5={\omega }_f-{\omega }_x$, and ${\nu }_6={\omega }_{y^{\prime }}-{\omega }_y$ illustrated by the level scheme in Fig. 1. The frequency of the incident wave is ${\omega }_1={\omega }_2=600$ $\mathrm{THz}$. (a) One-dimensional spectrum of heterodyne signals from six transmission channels. The figure in the right side of (a) denotes the polarization configuration of three incident light in the four-wave-mixing process. (b) and (c) The real and imaginary parts of the nonlinear effective susceptibility tensor components reconstructed from the spectrum in (a).

Figure 3

Two-dimensional spectrum of the susceptibility components ${\chi }_{xxxx}$ (a), ${\chi }_{xyyx}$ (b), ${\chi }_{xxyy}$ (c), and ${\chi }_{yyyy}$ (d) displayed vs the frequency of the signal field and one of the incoming fields. The diagonal (${\omega }_s={\omega }_1={\nu }_j$) and off-diagonal (${\omega }_s={\nu }_i$, ${\omega }_1={\nu }_j$, $i\ne j$, $i,j=1,...,6$) cross peaks indicate the stationary and transient oscillating charge corresponding to plasmonic occupation (population) and coherences, respectively, between the resonant transitions indicated in Fig. 1. Simulation parameters are as follows: ${\omega }_2=600$ $\mathrm{THz}$, ${\nu }_7={\omega }_2+({\omega }_x-{\omega }_y)$, ${\nu }_8={\omega }_2$, and ${\nu }_9={\omega }_2+({\omega }_y-{\omega }_x)$. The configurations of the polarization state of the incident waves are the same as those in Fig. 2.

Figure 4

The nonlinear spectrum vs the signal (new mode) frequency for phase-matching case I. The resonance frequencies shown here are the same as those in Figs. 2 and 3. (a) One-dimensional spectrum of heterodyne signals from six transmission channels. (b) and (c) The real and imaginary parts of the nonlinear effective susceptibility tensor components reconstructed from the spectrum in (a). (d) and (e) Two-dimensional spectrum of the susceptibility components ${\chi }_{yxxy}$ and ${\chi }_{yyxx}$ displayed vs the frequency of the signal field and one of the incoming fields. (a)–(c) supplement Fig. 2 in the main text, and (d) and (e) supplement Fig. 3 in the main text.

Figure 5

The nonlinear spectrum vs frequency for phase-matching case II. The frequency of incident light is ${\omega }_1={\omega }_2=600$ $\mathrm{THz}$. (a) and (d) One-dimensional spectrum of six nonvanishing components for two different combinations of the polarization state of incident light. The polarization state of incident light is shown to the right of each row. (b), (c), (e), and (f) The real and imaginary parts of the nonlinear susceptibility tensor components reconstructed from the spectra in (a) and (d).

Figure 6

Two-dimensional spectrum of components ${\chi }_{xxxx}$ (a), ${\chi }_{xyyx}$ (b), ${\chi }_{xxyy}$ (c), ${\chi }_{yyyy}$ (d), ${\chi }_{yxxy}$ (e), and ${\chi }_{yyxx}$ (f) with ${\omega }_2=600$ $\mathrm{THz}$ for phase-matching case II.

Figure 7

The nonlinear spectrum vs frequency for phase-matching case III. The frequency of incident light is ${\omega }_1={\omega }_2=600$ $\mathrm{THz}$. (a) and (d) One-dimensional spectrum of six nonvanishing components for two different combinations of the polarization state of incident light. The polarization state of incident light is shown to the right of each row. (b), (c), (e), and (f) The real and imaginary parts of the nonlinear susceptibility tensor components reconstructed from the spectra in (a) and (d).

Figure 8

Two-dimensional spectrum of components ${\chi }_{xxxx}$ (a), ${\chi }_{xyyx}$ (b), ${\chi }_{xxyy}$ (c), ${\chi }_{yyyy}$ (d), ${\chi }_{yxxy}$ (e), and ${\chi }_{yyxx}$ (f) with ${\omega }_2=600$ $\mathrm{THz}$ for phase-matching case III.