Signal Extraction
3.1 Optically Heterodyned Signals
3.1.3 Phasor Diagrams
Phasor diagrams are an extremely useful graphical aid to the intuition when thinking about optical heterodyne signal extraction. Discussion again will assume the phase modulated spectrum of Eq. (3.3). Referring back to Eq. (3.9), the real part of each term in the brackets can be seen to be the scalar product of two vectors whose real and imaginary parts make up the basis of the vector space.
3{{ab*}
=
3{{a}3{{b}+
~{a}~{b}=a. b
(3.22)The carrier phasor is made stationary by factoring the carrier frequency out. The RF sidebands are rotating either clockwise or counter-clockwise relative to the carrier, depending on whether they're lower or upper sidebands. Examining Eq. (3.9) again, the process of demodulation fixes the RF sidebands at some point in their rotation characterized by the demodulation phase {3. A change in demodulation phase gen- erates a static rotation of the sidebands in the opposite sense, as evidenced by the conjugation of the demodulation rotation on the lower sideband. Therefore, the pro- cess of signal extraction can be thought of as the scalar product of the carrier vector with the resultant RF sideband vector, at a point in their rotation which is controlled by the demodulation phase.
Eqs. (3.14) and (3.16) can be revisited in terms of phasor diagrams. Phase modulated light generates RF sidebands which are orthogonal to the carrier, as seen in Eq. (3.3) by the factor of i in their amplitudes. Thus, the RF sideband phasors start out both parallel to each other and orthogonal to the carrier. This is a more useful definition the "inphase" demodulation since it relates to the state of the RF sidebands relative to the carrier, rather than simply a numerical value for {3, and will be used as the functional definition for the rest of this thesis. Signal extraction is shown in Figure 3.1. Since the unperturbed resultant RF sideband and the carrier
Carrier signa{ :, \ Counter-rotating
\ sidebands
;,!:ri,;, modulated~- -- - -- - -- --- ---~ -- - --- --~-- _ I _:"tt=t '~eband
Signal= - •
Figure 3.1: Signal extraction in the inphase demodulation.
phasors are orthogonal to each other, no signal is measured. If any of the phasors is amplitude modulated, it can be seen that there still would be no signal. However, if the carrier is phase modulated, as shown in the figure, there is now a component of the carrier parallel to the resultant RF sideband phasor, and so the scalar product
is non-zero. This is the basic form of the Pound-Drever cavity locking technique, in which the carrier resonates in a cavity, and the RF sidebands are excluded. Length fluctuations of the cavity cause phase fluctuations of the carrier to first order, while the RF sidebands remain unperturbed. If the RF sidebands are also in the cavity, all frequencies will experience the same phase modulation, that is to say, the resultant RF sideband will be rotated by the same amount as the carrier, and no signal would be measured. A third situation of interest is if the cavity is dispersive and all frequencies are resonant. Then the phase shift for a cavity length perturbation is different for the carrier and the RF sidebands, and the phasors are rotated by different amounts.
A signal will be measured, either larger or smaller than if the sidebands are excluded from the cavity, depending on the direction of the phase shifts. This all goes back to the interpretation of Eq. (3.14), which stated that the inphase demodulation is sensitive to relative phase modulations between the carrier and RF sidebands.
In the quadrature demodulation, the RF sideband phasors are rotated by a quarter of a cycle, such that they're anti-parallel. The resultant sideband vector is null, so clearly any perturbation to the carrier doesn't generate a signal. In fact, about the only way a signal can be measured is when the amplitude of the RF sidebands is differentially modulated. The applicability for this demodulation phase is, for
Carrier-
-
-
Counter-rotating sidebands
Resultant sideband
./
- - - - - J_ ________________ _
Figure 3.2: Signal extraction in the quadrature demodulation.
example, in an asymmetric Michelson interferometer when the carrier is dark at the output. With the asymmetry, the two RF sidebands have some non-zero amplitude at the output. When the interferometer is pushed away from the dark fringe, one sideband will grow in amplitude, while the other will shrink, causing a differential amplitude modulation, consistent with the interpretation of Eq. (3.16). Another
possible use of the quadrature demodulation is for cavity locking, in which the carrier is resonant, but the RF sidebands are slightly off-resonant. Subsequent perturbations to the length of the cavity causes their amplitudes to be differentially modulated, and so generate signals in the quadrature demodulation.
The "signal" in these cases is due to an additional phasor which is either normal or parallel to the original phasor, depending on whether the perturbation is in phase or amplitude, respectively. This signal phasor will in general have some time depen- dence, which can be decomposed into its Fourier components. In the case of phase modulation, the original phasor is rotationally dithered, tracing out a swinging length phasor normal to the original. Amplitude modulation causes the original phasor to grow and shrink, generating a swinging length phasor parallel to the original phasor.
These swinging length phasors can be decomposed into two counter-rotating phasors, rotating at the frequency of the perturbation. This is shown in Figure 3.3. A clear
Phase fluctuations
¢ = 7T/2 ¢ = 7T ¢ = 37T/2
·.'--Fixed
Amplitude fluctuations
¢=7T/2 ¢=7T ¢ = 37T/2
'Fixed
Figure 3.3: Generation of the signal phasors.
example of this is the phase modulation of the input light which is decomposed into a carrier plus the two counter-rotating RF sidebands. This picture can be extended
further to generate the second order sidebands as well, although it will typically be assumed that the perturbations are small enough that the second order effects can be ignored.
In general, it's possible that the RF sidebands aren't equal in magnitude or phase.
The input light certainly has equal RF sidebands; however, the transmission functions which propagate them to the photodiode may not be the same. In this case, the RF sidebands are unbalanced. The resultant RF sideband no longer is a swinging length vector, but rather, it traces out an ellipse at the modulation frequency
n,
shown in Figure 3.4. Clearly, even without any perturbations, the value of the scalar product' . ,_
' '
/
'' /
' 'f
---~ ' I I IResultant sideband ellipse
--- ---
0 offset demodulation phase
Figure 3.4: Unbalanced RF sidebands. The RF sideband resultant traces out an ellipse as a function of demodulation phase. The only non-zero output from this configuration is at the demodulation represented on the right, where the resultant is orthogonal (as well as the phase 180° rotated from this).
is non-zero for all demodulation phases except two, when the resultant is normal to the carrier. The presence of a DC offset for arbitrary demodulation phase can in principle be nulled by a summed electronic offset after demodulation. This would have the benefit of a clear interpretation of any non-zero output being due to signal.
However, it's also clear that the value of the DC offset is entirely dependent on the state of the phasors remaining stable. For example, drifts in alignment, modulation depth, etc., would cause the required voltage offset to be periodically re-evaluated.
In order to avoid complications as much as possible, attention will be confined to the demodulation phase in which the resultant is normal to the carrier, and the optical offset is zero. This is seen in Figure 3.4 to be the functional definition of the inphase demodulation.
In this situation, signal extraction isn't nearly as clear. In the balanced sideband case, the inphase demodulation would only be sensitive to phase modulations. When the RF sidebands are unbalanced, however, an amplitude modulation of the two RF sidebands can also generate a signal phasor which has a component parallel to the carrier. On the other hand, the only type of measurable signal impressed upon the carrier is still due to phase modulations, since the resultant RF sideband is orthogonal to the carrier.