Database of Dynamical Symmetries and Selection Rules in Nonlinear Optics

Improper-rotational symmetry of any order in the laser driving field (E(t)), while the medium is chiral and does not respect this symmetry. Here the symmetry can be a mirror plane (improper rotation of order 1), an inversion operator (improper-rotation of order 2), or any order improper rotation.

The PECD spectra respect the rotational operator, while the mirror part of the improper rotation inverts the PECD sign. For instance, for a simple mirror plane sigma, one would have the PECD function exhibit the selection rule: PECD(kx,ky,kz)=-sigma*PECD(kx,ky,kz), with k,x,y,z the photoelectron momenta. This would give rise for instance to forwards-backwards asymmetry in PECD.

More generally, for an improper rotation of order n one has PECD(kx,ky,kz)=-R_n*PECD(kx,ky,kz).

Note that the PES from each individual chiral molecule does not exhibit a selection rule, only the PECD.

Dynamical rotation symmetry (C_N,M)
Polarization rotation by 2πM/N + time translation by T/N leaves the field unchanged.

(±) circularly polarized Nq±M harmonics (integer q, all others forbidden)

Improper rotational microscopic dynamical symmetry, where E(t) is invariant under a coupled improper rotation of order 2n or 2n+1 (depending on even/odd integer), coupled to an accompanying time-translation, e.g. E(t)=R_{2n+1}*E(t-T/(2n+1)), with R a rotation operation in polarization space.
This symmetry can be obtained either by engineering multiple color fields, or using simple fields in materials with improper-rotational symmetries (e.g. aligned molecules or solids).

circular harmonics transverse to the rotational symmetry axis (i.e. in the mirror plane of the improper-rotation), of order (2nq-+m), with q any integer, and n,m the indices of the rotational operator R_{2n}^m for the even-order 2n case, while n(2q+1) harmonics all polarized linearly along the rotational axis. All other harmonics are forbidden.

For the odd order case with R_{2n+1}^m the selection rules are similar but with circular harmonics within the mirror plane of orders 2q(2n+1)+-m, and harmonics with linear polarization along the rotational axis of order 2n+1.

The electric field E(t) is invariant under a coupled operation of time-reversal, time translation by half an optical period (T/2), and a mirror operation (sigma) for its polarization components: E(t)=sigma*E(-(t+T/2)), where the time-reversal operator can be along any point in time in the laser field, t0.

The harmonics are generically elliptically-polarized, but the elliptical major/minor axis is normal to the mirror plane.

The electric field E(t) is invariant under a coupled operation of time-reversal, time-translation by half an optical period (T/2), and a two-fold rotation (R_2), such that E(t)=R_2*E(-(t-T/2)), where the time-reversal operation can be along any point in time t0 along the laser field.

Linearly-polarized only harmonics (suppression of ellipticity).

Static mirror plane, i.e. E(t)=sigma*E(t) where sigma is a mirror operation in the field's polarization space. For instance, for an xy mirror plane one would have E_z(t) = -E_z(t) (i.e. E_z=0).
Note any monochromatic laser field exhibits two such mirror planes within the dipole approximation, but more interesting geometries can also be engineered as long as the field polarization itself is contained within a plane.

The harmonics are allowed only within the mirror plane, and have forbidden polarizations transversely to it.

Such a selection rule might break in chiral media that does not respect mirror symmetries.

a combined effect of time-reversal and mirror plane (as appears in co-rotating w-2w laser fields).
can be written as T*sigma.

only transverse photocurrent to the mirror plane arises.

Rotational DS in the laser field polarization, such that E(t) is invariant under R_n (n'th order rotations) coupled to T/n time translations with T the optical cycle. Then overall E(t)=R_n*E(t-T/n).

This symmetry can be obtained also by driving with monochromatic circular fields in media with rotational intrinsic symmetries, e.g. aligned molecules or solids.

The total number of excited electrons (electrons that are not within the ground state orbitals/Bloch states) evolves dynamically in harmonics of the laser, but only in harmonics of order qn with q an integer (unlike in HHG where orders qn are forbidden).
That is, if one were to calculate the projected total conduction bands excitations in a solids during the laser-driven dynamics, that number would evolve in time in harmonics qn of the laser.