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 m. Assuming an input field
Ein = E0 exp (iω0t), (43 )
a sinusoidal phase modulation of the field can be described as
( ) E = E0 exp i(ω0t + m cos (Ωt)) . (44 )
This equation can be expanded using the identity [27]
∑∞ exp(izcosφ ) = ikJk(z)exp (ikφ ), (45 ) k=−∞
with Bessel functions of the first kind Jk(m ). We can write
∑∞ E = E exp (iω t) ik J (m ) exp(ikΩt ). (46 ) 0 0 k k=−∞
The field for k = 0, oscillating with the frequency of the input field ω0, represents the carrier. The sidebands can be divided into upper (k > 0) and lower (k < 0) sidebands. These sidebands are light fields that have been shifted in frequency by kΩ. The upper and lower sidebands with the same absolute value of k are called a pair of sidebands of order k. Equation (46View Equation) shows that the carrier is surrounded by an infinite number of sidebands. However, for small modulation indices (m < 1) the Bessel functions rapidly decrease with increasing k (the lowest orders of the Bessel functions are shown in Figure 16View Image). For small modulation indices we can use the approximation [2]
( m2)n ( m )k ∑∞ − 4 1 ( m )k ( k+2) Jk(m ) = 2- n!(k +-n)! = k! 2- + O m . (47 ) n=0
In which case, only a few sidebands have to be taken into account. For m ≪ 1 we can write
E = E0( exp (iω0t) ) (48 ) × J (m ) − iJ (m ) exp (− iΩt) + iJ (m ) exp(iΩt) , 0 −1 1
and with
J−k(m ) = (− 1 )kJk (m ), (49 )
we obtain
( m ( )) E = E0 exp (iω0t) 1 + i-- exp (− iΩt ) + exp (iΩt) , (50 ) 2
as the first-order approximation in m. 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 m is given:
( m2 m ( ) ) E = E0 exp (iω0t) 1 − --- + i-- exp (− iΩt) + exp(iΩt) , (51 ) 4 2
which shows that power is transferred from the carrier to the sideband fields.

Higher-order expansions in m can be performed simply by specifying the highest order of Bessel function, which is to be used in the sum in Equation (46View Equation), i.e.,

or∑der E = E0 exp (iω0t) ik Jk(m ) exp (ikΩt ). (52 ) k=−order
View Image

Figure 16: Some of the lowest-order Bessel functions Jk(x) of the first kind. For small x the expansion shows a simple xk dependency and higher-order functions can often be neglected.

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