Observation of magneto-electric rectification at non-relativistic intensities

The subject of electromagnetism has often been called electrodynamics to emphasize the dominance of the electric field in dynamic light–matter interactions that take place under non-relativistic conditions. Here we show experimentally that the often neglected optical magnetic field can nevertheless play an important role in a class of optical nonlinearities driven by both the electric and magnetic components of light at modest (non-relativistic) intensities. We specifically report the observation of magneto-electric rectification, a previously unexplored nonlinearity at the molecular level which has important potential for energy conversion, ultrafast switching, nano-photonics, and nonlinear optics. Our experiments were carried out in nanocrystalline pentacene thin films possessing spatial inversion symmetry that prohibited second-order, all-electric nonlinearities but allowed magneto-electric rectification.


Theory of magneto-electric rectification
Both classical and quantum models have successfully described induced magneto-electric rectification in individual diatomic molecules [1]. At moderate intensities (non-relativistic), the nonlinear interaction of the electric and magnetic fields of light gives rise to a static electric dipole moment along the propagation direction. In this dual field interaction, the first photon induces electric polarization at the optical frequency and the second photon exerts magnetic torque on the excited state of the molecule, converting orbital angular momentum to rotational angular momentum that results in magneto-electric rectification.
Classically, the magneto-electric response can be described using a simple Lorentz oscillator model that includes molecular rotations (librations) of a diatomic molecule. A detailed description of the model has been published elsewhere [1,2]. Here, we briefly summarize the classical model before utilizing it to predict the rectification signal observed in this work. The Lorentz force and the corresponding equation of motion for a bound electron subjected to optical fields E, B are: where ( ) is the electron position at time t and A is the position of the point of equilibrium. Both coordinates are specified with respect to the center-of-mass. ω0 and γ are the resonant frequency and damping constants of the Hooke's Law oscillator. e and m are the charge and the effective mass of electron, respectively. ( ) is the total force caused by the electric and magnetic optical fields.
and its quasi-static component along the propagation axis can be computed.

Pulse-front tilt characterization
To find the zero of pump-probe delay and to measure the correlation between pump and probe pulses in cross-beam geometry, we placed a GaAs wafer at the sample position and monitored the sum frequency generation (SFG) signal along the bisector of the two beams while translating the delay stage, Fig. S3. The tilt angle was controlled by rotating the grating with respect to the incident laser beam. Since a cylindrical concave mirror was used, the angular direction of the laser beam was little affected by the adjustment of the grating. However, lateral displacement of the beam had to be corrected to pass through two alignment irises using a pair of mirrors. Figure S3. Schematic of the crossed beam pump-probe experiment. The tilted pulse-front was controlled by selecting the grating angle. To find zero delay the sum-frequency generation (SFG) signal was detected with a PMT positioned on the bisector of the incident pump and probe beams in reflection mode (GaAs is opaque at 800 and 400 nm). A 10 nm bandpass filter centered at  = 400 nm was used to isolate the SFG signal. Top-right box: An illustration of the paths followed by red-shifted and green-shifted components of the pulse after dispersion by the grating resulting in a pulse front tilt angle γ at the sample. Two lenses are needed to correct for the chirp. In the actual setup, a concave mirror (CM) and a cylindrical lens (CL) were used for this purpose.

Magneto-electric, field-induced second harmonic generation (ME-FISH)
The total second harmonic intensity generated by a probe pulse (propagating in the x direction) includes contributions from the surface, SSHG, and the symmetry breaking, ME-FISH, processes. The latter comes from four-wave mixing process: (3) ( ) ( ) (0). These contributions are in phase because both are formed from the incident probe wave. Hence the total intensity is given by: (2 ) = | SSHG (2 )+ ME−FISH (2 )| 2 / where is the electromagnetic impedance of the medium. where a, b, and c are constants that depend on geometry. S (2) and (3) are the surface and the third order susceptibilities, respectively. We can now replace (3) (0) by eff (2) , the effective magnetoelectric susceptibility for second harmonic generation, mediated by the pump-induced ME rectification field.
In our experiment, the pump beam propagates along the z-axis, which is also the rectification field direction. Note that inversion symmetry is only broken by the MER field for input field components polarized along the z-axis. The ME-FISH polarization therefore arises as a DC Kerrinduced signal polarized parallel to the symmetry-breaking rectification field according to Hence the total harmonic intensity is Note that the first term in this expression is not pump-induced, whereas the second is. The third term can be ignored since it is second order in the (small) rectification field (0).
The dependence of second harmonic intensity on the probe polarization angle  is different for the pump-independent SSHG and pump-induced ME-FISH signals. Noting that probe components along y and z are given by y ( ) = 0 cos and z ( ) = 0 sin , respectively, the two contributing nonlinear polarizations are found to vary with  according to

Exclusion of electric quadrupole interactions
In addition to the magneto-electric interactions, quadrupolar electric interactions can theoretically give rise to a second order nonlinear response. Consequently, it might be thought that such an interaction could explain rectification in the experiments reported in this work. However, this is not the case. Quadrupole interactions can lead to frequency-doubling but not to rectification, as we show below. Consider an x-polarized pump field x = 0x ( − ) + . .. A second order polarization that points along the pump propagation axis (z) and results from a quadrupole interaction [3] then has the form z (q) = 0 zxzx In an isotropic material the tensor susceptibility element zxzx (q) does not vanish. However, upon substitution of the pump field, the nonlinear polarization is found to be

Exclusion of charge liberation by electron-hole pair excitation
The pentacene thin film sample in our experiment had an absorption peak at 670 nm, with a tail that extended only as far as 750 nm. The absorption strength at 800 nm, which was the center wavelength of the excitation laser, was therefore negligible. Even so, if electron-hole pairs were excited as the result of absorption at 800 nm and resulted in a charge separation field, the field strength would exhibit a linear dependence on pump power. However, in the present work a quadratic dependence was observed of the pump-induced signal on excitation power.
A weak two-photon absorption could also produce charge carriers, which might cause a SHG signal with a quadratic dependence on pump power. However, the signal rise time would be as fast as the pulse duration in this case, contrary to the observed rise time of ~500 fs. The decay time of pump-induced SHG would then be either that of the singlet exciton lifetime (24 ns) or the singlet fission time scale (80 fs), See Refs. 28-29 of the main text. Since our signal decay time was ~600 fs, it is quite different from either of these possibilities. Furthermore, we would expect a polarization-independent signal from photo-induced charges, contrary to the experimental results.
Our polarization angle dependence was in excellent agreement with the prediction for a magnetoelectric rectification field, and distinct from the variation predicted for signals of all-electric origin.