### 3.2 Phase modulation

Phase modulation can create a large number of sidebands. The number of sidebands with noticeable
power depends on the modulation strength (or depth) given by the modulation index . Assuming an
input field
a sinusoidal phase modulation of the field can be described as
This equation can be expanded using the identity [27]
with Bessel functions of the first kind . We can write
The field for , oscillating with the frequency of the input field , represents the carrier. The
sidebands can be divided into upper () and lower () sidebands. These sidebands are light
fields that have been shifted in frequency by . The upper and lower sidebands with the same
absolute value of are called a pair of sidebands of order . Equation (46) shows that the
carrier is surrounded by an infinite number of sidebands. However, for small modulation indices
() the Bessel functions rapidly decrease with increasing (the lowest orders of the Bessel
functions are shown in Figure 16). For small modulation indices we can use the approximation [2]
In which case, only a few sidebands have to be taken into account. For we can write
and with
we obtain
as the first-order approximation in . In the above equation the carrier field remains unchanged by the
modulation, therefore this approximation is not the most intuitive. It is clearer if the approximation up to
the second order in is given:
which shows that power is transferred from the carrier to the sideband fields.
Higher-order expansions in can be performed simply by specifying the highest order of Bessel
function, which is to be used in the sum in Equation (46), i.e.,