  It has been suggested that electromagnetic cascade of very high energy
gamma-rays from gamma-ray bursts (GRBs) in the infrared/microwave background
can produce delayed MeV-GeV photons. This delay could be caused by the angular
spreading effect of the scattered microwave photons or deflection of the
secondly pairs due to intergalactic magnetic field. Very high energy TeV
photons of GRBs could be produced by a few mechanisms including the
proton-synchrotron radiation and electron inverse Compton emission from GRB
internal shocks as well as external shocks. We suggest that the information
provided by the delayed emission could give constraints on models for TeV
gamma-rays. A more accurate calculation of the delayed time caused by the
angular spreading effect is presented by considering recent observations of the
extragalactic infrared background and the theoretic high-redshift infrared
background. We also suggest that the dependence of the maximum time delay of
scattered photons on their energies, if determined by future GLAST detector,
could differentiate the two mechanisms causing the time delay.
