3.5 Sidebands as phasors in a rotating frame

A common method of visualising the behaviour of sideband fields in interferometers is to use phase diagrams in which each field amplitude is represented by an arrow in the complex plane.
View Image

Figure 18: Amplitude and phase modulation in the ‘phasor’ picture. The upper plots a) illustrate how a phasor diagram can be used to describe phase modulation, while the lower plots b) do the same for amplitude modulation. In both cases the left hand plot shows the carrier in blue and the modulation sidebands in green as snapshots at certain time intervals. One can see clearly that the upper sideband (ω0 + Ω) rotates faster than the carrier, while the lower sideband rotates slower. The right plot in both cases shows how the total field vector at any given time can be constructed by adding the three field vectors of the carrier and sidebands. [Drawing courtesy of Simon Chelkowski]

We can think of the electric field amplitude E0 exp(iω0t) as a vector in the complex plane, rotating around the origin with angular velocity ω0. To illustrate or to help visualise the addition of several light fields it can be useful to look at this problem using a rotating reference frame, defined as follows. A complex number shall be defined as z = x + iy so that the real part is plotted along the x-axis, while the y-axis is used for the imaginary part. We want to construct a new coordinate system (′ x, ′ y) in which the field vector is at a constant position. This can be achieved by defining

x = x′ cosω0t − y′ sinω0t y = x′ sin ω t + y′ cosω t, (60 ) 0 0
x′ = x cos(− ω t) − y sin (− ω t) ′ 0 0 (61 ) y = x sin (− ω0t ) + y cos(− ω0t).
Figure 17View Image illustrates how the transition into the rotating frame makes the field vector to appear stationary. The angle of the field vector in a rotating frame depicts the phase offset of the field. Therefore these vectors are also called phasors and the illustrations using phasors are called phasor diagrams. Two more complex examples of how phasor diagrams can be employed is shown in Figure 18View Image [11].

Phasor diagrams can be especially useful to see how frequency coupling of light field amplitudes can change the type of modulation, for example, to turn phase modulation into amplitude modulation. An extensive introduction to this type of phasor diagram can be found in [39].

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