3.7 Coupling matrices for beams with multiple frequency components

The coupling between electromagnetic fields at optical components introduced in Section 2 referred only to the amplitude and phase of a simplified monochromatic field, ignoring all the other parameters of the electric field of the beam given in Equation (1View Image). However, this mathematical concept can be extended to include other parameters provided that we can find a way to describe the total electric field as a sum of components, each of which is characterised by a discrete value of the related parameters. In the case of the frequency of the light field, this means we have to describe the field as a sum of monochromatic components. In the previous sections we have shown how this could be done in the special case of an initial monochromatic field that is subject to modulation: if the modulation index is small enough we can limit the amount of frequency components that we need to consider. In many cases it is actually sufficient to describe a modulation only by the interaction of the carrier at ω0 (the unmodulated field) and two sidebands with a frequency offset of ± ωm to the carrier. A beam given by the sum of three such components can be described by a complex vector:
( ) ( ) a(ω0) aω0 | a(ω − ω )| | a | ⃗a = |( 0 m |) = |( ω1|) (64 ) a(ω0 + ωm ) aω2
with ω0 = ω0, ω0 − ωm = ω1 and ω0 + ωm = ω2. In the case of a phase modulator that applies a modulation of small modulation index m to an incoming light field ⃗a1, we can describe the coupling of the frequency component as follows:
a = J (m )a + J (m )a + J (m )a 2,ω0 0 1,ω0 1 1,ω1 −1 1,ω2 a2,ω1 = J0(m )a1,ω1 + J−1(m )a1,ω0 (65 ) a2,ω2 = J0(m )a1,ω2 + J1(m )a1,ω0,
which can be written in matrix form:
( ) | J0(m ) J1(m ) J−1(m )| ⃗a2 = | J −1(m ) J0(m ) 0 | ⃗a1. (66 ) ( J1(m ) 0 J0(m ) )
And similarly, we can write the complete coupling matrix for the modulator component, for example, as
( ) ( ) ( ) a2,w0 J0(m ) J1(m ) J−1(m ) 0 0 0 a1,w0 || a2,w1 || || J −1(m ) J0(m ) 0 0 0 0 || || a1,w1|| | a2,w2 | | J1(m ) 0 J0(m ) 0 0 0 | | a1,w2| || a || || 0 0 0 J (m ) J (m ) J (m )|| || a || (67 ) || 4,w0 || || 0 1 −1 || || 3,w0|| | a4,w1 | | 0 0 0 J −1(m ) J0(m ) 0 | | a3,w1| ( a4,w2 ) ( 0 0 0 J1(m ) 0 J0(m ) ) ( a3,w2)

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