  Modern optical microscopy methods have been advanced to provide super
resolution at high imaging speed, but not chirality discriminative. We recently
proposed chiral structured-illumination microscopy (SIM) method to image chiral
fluorescent domains at sub-wavelength resolution. Chiral SIM is suitable for
imaging chiral domains at sub-wavelength resolution but loses the high spatial
frequency of the achiral ones. In order to obtain the full picture of all
fluorescent domains at sub-wavelength resolution, we propose an advanced
version of chiral SIM, termed double SIM, which enables simultaneous imaging of
achiral and chiral domains at sub-wavelength resolution. In double SIM, the
illumination field must be spatially structured both in the intensity and
optical chirality so that moir\'e effects can be concurrently generated on the
achiral and chiral fluorescent domains of a sample. This allows down-modulating
the high spatial frequency of both domains at the same time and thus provides
sub-wavelength details after image reconstruction. To generate the illumination
field with concurrently structured intensity and optical chirality, we propose
interfering two coherent circularly polarized light with the same handedness at
the sample plane. We introduce the working principle of double SIM and
theoretically demonstrate the feasibility of this method using different kinds
of synthetic samples.
