  We present new theoretical models for quantum optical mode-selective filters
and mirrors using continuous-variable teleportation. We call these devices
telefilters and telemirrors respectively. Both devices act as the identity
channel on a mode of interest from an input multi-mode field while filtering or
reflecting all the orthogonal modes. We utilise these models to analyse a
causality problem in relativistic quantum optics, specifically the apparently
acausal transmission and propagation of temporally delocalised wavepackets
through mode-selective mirrors. Firstly, we show how telemirrors - and thus
mode-selective operations generally - enact a fundamental time-delay on such
wavepackets, which is necessary in order to prevent violations of causality. In
an attempt to circumvent this delay, we next consider teleporting the
independent temporal components of the input field separately and continuously,
that is, performing operations on-the-run. In this scenario, the telemirror
transmits the mode of interest as well as orthogonal modes which carry with
them uncorrelated noise. In this scenario, the device may be considered
mode-discriminating but not mode-selective.
