ω
ph
±
=
ω
±
1
-
β
?
?
cos
?
?
θ
(
26
)
where ω
±=ω
0(1±nβ) and ω
0 is the central angular frequency of the driving laser. In Equation (26), ω
ph+ is due to electron interaction with the counter-propagating GP, whereas ω
ph? is due to interaction with the co-propagating GP. Note that the rightmost expression in Equation (1) reduces to ω
ph+(ω
ph?) when θ
i=π (θ
i=0).
The spectrum of the emitted radiation as a function of its frequency ω, azimuthal angle ? and polar angle θ, making the assumption of high confinement factors n>>1 to achieve a completely analytical result: