3 Light with Multiple Frequency Components

So far we have considered the electromagnetic field to be monochromatic. This has allowed us to compute light-field amplitudes in a quasi-static optical setup. In this section, we introduce the frequency of the light as a new degree of freedom. In fact, we consider a field consisting of a finite and discrete number of frequency components. We write this as

∑ E(t,z) = aj exp(i(ωjt − kjz)), (41 ) j
with complex amplitude factors aj, ωj as the angular frequency of the light field and kj = ωj∕c. In many cases the analysis compares different fields at one specific location only, in which case we can set z = 0 and write
∑ E(t) = aj exp(iωjt). (42 ) j
In the following sections the concept of light modulation is introduced. As this inherently involves light fields with multiple frequency components, it makes use of this type of field description. Again we start with the two-mirror cavity to illustrate how the concept of modulation can be used to model the effect of mirror motion.
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Figure 15: Example traces for phase and amplitude modulation: the upper plot a) shows a phase-modulated sine wave and the lower plot b) depicts an amplitude-modulated sine wave. Phase modulation is characterised by the fact that it mostly affects the zero crossings of the sine wave. Amplitude modulation affects mostly the maximum amplitude of the wave. The equations show the modulation terms in red with m the modulation index and Ω the modulation frequency.
 3.1 Modulation of light fields
 3.2 Phase modulation
 3.3 Frequency modulation
 3.4 Amplitude modulation
 3.5 Sidebands as phasors in a rotating frame
 3.6 Phase modulation through a moving mirror
 3.7 Coupling matrices for beams with multiple frequency components
 3.8 Finesse examples
  3.8.1 Modulation index
  3.8.2 Mirror modulation

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