The final form of this equation makes it clear that (to first order) we now have carrier light of amplitudeE0J0(Γ) at the original carrier frequencyωas well as two sidebands at frequencies (ω+Ω) and (ω−Ω), each of amplitudeE0J1(Γ).
The light that we will be present at the reflection port of the cavity will be given by
Erefl =Eincrcav (B.5)
where rcav is the amplitude reflectivity of the cavity, including both the prompt reflection and leakage of the circulating power in the cavity. rcavis a function of the frequencyωof the light, and is given by
rcav(ω)=−ri+ t2iree−i2Lcω
1−riree−i2Lωc . (B.6) As defined in Chapter3,riis the amplitude reflectivity of the input mirror of the cavity,re is the amplitude reflectivity of the end mirror, andtiis the amplitude transmission of the input mirror. L is the length of the cavity andcis the speed of light. EquationB.5expands to
Erefl =E0
J0(Γ)eiωtrcav(ω)−iJ1(Γ)ei(ω+Ω)trcav(ω+ Ω)+iJ1(Γ)ei(ω−Ω)trcav(ω−Ω)
. (B.7)
Photodiodes are not capable of measuring the electric field directly. Rather, they detect the power of the light. For example, at the reflected port,
Prefl =E∗reflErefl. (B.8)
whereE∗is the complex conjugate ofE. Explicitly, this expands to Prefl=E20
J0(Γ)e−iωtr∗cav(ω)+iJ1(Γ)e−i(ω+Ω)tr∗cav(ω+ Ω)−iJ1(Γ)e−i(ω−Ω)tr∗cav(ω−Ω)
×
J0(Γ)eiωtrcav(ω)−iJ1(Γ)ei(ω+Ω)trcav(ω+ Ω)+iJ1(Γ)ei(ω−Ω)trcav(ω−Ω)
. (B.9)
Multiplying this out, and dropping the 2Ωterms (because, as described below, we will demodulate atΩand then low-pass the resulting signal) will leave us with
Prefl=E20[J20(Γ)r∗cav(ω)rcav(ω)
+J21(Γ)(|rcav(ω+ Ω)|2+|rcav(ω−Ω)|2)
−iJ0(Γ)J1(Γ)eiΩtr∗cav(ω)rcav(ω+ Ω)+iJ0(Γ)J1(Γ)e−iΩtr∗cav(ω)rcav(ω−Ω)
−iJ0(Γ)J1(Γ)eiΩtrcav(ω)r∗cav(ω−Ω)+iJ0(Γ)J1(Γ)e−iΩtrcav(ω)r∗cav(ω+ Ω)]
(B.10)
Rearranging, this is
Prefl=E20[J20(Γ)r∗cav(ω)rcav(ω)
+J21(Γ)(|rcav(ω+ Ω)|2+|rcav(ω−Ω)|2)
+iJ0(Γ)J1(Γ)e−iΩt(r∗cav(ω)rcav(ω−Ω)+r∗cav(ω+ Ω)rcav(ω))
−iJ0(Γ)J1(Γ)eiΩt(r∗cav(ω)rcav(ω+ Ω)+r∗cav(ω−Ω)rcav(ω))]
(B.11)
We would like to simplify this equation into something proportional to either cos(Ωt) or sin(Ωt).
For this, we need only examine the cross terms on the last 2 lines of EquationB.11. For ease of notation, we will let
A≡r∗cav(ω)rcav(ω−Ω) and B≡r∗cav(ω)rcav(ω+ Ω). (B.12) The cross terms of EquationB.11are then
c.t.=iJ0(Γ)J1(Γ)e−iΩt(A+B∗)−iJ0(Γ)J1(Γ)eiΩt(B+A∗) (B.13)
which, moving around and expanding gives c.t.=J0(Γ)J1(Γ)
−1
iAe−iΩt−1
iB∗e−iΩt+1
iBeiΩt+1 iA∗eiΩt
. (B.14)
Note that, sinceAandBare complex numbers,
A≡ |A|eiα, A∗≡ |A|e−iα, B≡ |B|eiβ, B∗≡ |B|e−iβ. (B.15) Utilizing this,
c.t.=J0(Γ)J1(Γ)
|A|1 i
ei(Ωt−α)−e−i(Ωt−α) +|B|1
i
ei(Ωt+β)−e−i(Ωt+β)
. (B.16)
Recall that
sin(x)= ex−e−x
2i (B.17)
to simplify the cross terms to
c.t.=J0(Γ)J1(Γ)2|A|sin(Ωt−α)+2|B|cos(Ωt+β). (B.18)
Using the fact that
sin(u±v)=sin(u)cos(v)±cos(u)sin(v) (B.19)
we can expand the cross terms to c.t.=J0(Γ)J1(Γ)
2|A|(sin(Ωt)cos(α)−cos(Ωt)sin(α))+2|B| sin(Ωt)cos(β)+cos(Ωt)sin(β)
(B.20)
which rearranges to
c.t.=J0(Γ)J1(Γ)2sin(Ωt) |A|cos(α)+|B|cos(β)+2cos(Ωt) |B|sin(β)− |A|sin(α). (B.21)
Recalling that, for an arbitrary complex numberz=|z|eiζ,
Re(z)=|z|cos(ζ) and Im(z)=|z|sin(ζ), (B.22)
we can simplify one final time to find
c.t.=2J0(Γ)J1(Γ) [sin(Ωt)Re(A+B)+cos(Ωt)Im(−A+B)]. (B.23)
Putting this back together with EquationB.11, we have
Prefl=E20
J02(Γ)r∗cav(ω)rcav(ω)
+J12(Γ)(|rcav(ω+ Ω)|2+|rcav(ω−Ω)|2) +2J0(Γ)J1(Γ)
sin(Ωt)Re(r∗cav(ω)rcav(ω−Ω)+r∗cav(ω)rcav(ω+ Ω)) +cos(Ωt)Im(r∗cav(ω)rcav(ω+ Ω)−r∗cav(ω)rcav(ω−Ω))
(B.24)
Recall that for EquationB.4we have restricted ourselves to|n|<1. In general, and in particular for the 3f analysis discussed in Section4.4, we will need to keep several more terms in the series. In that case, EquationB.24must be expanded to include these terms.
EquationB.24is the power incident on the photodiode. Some amount of photons will, via the photoelectric effect create a “photocurrent”, which is just the current of electrons ejected by the photons. The fraction of photons that will eject electrons is described by the quantum efficiency of the photodiode, and the responsivity of the photodiode which has units of Amps/Watt. The transimpedance of the electronics surrounding the diode will convert the current to a voltage.
The transimpedance need not be the same for all frequencies (and in general LIGO uses resonant photodiodes to emphasize the 1Ωcomponents, and de-emphasize other order harmonics), but for the rest of this analysis we will ignore this frequency dependence. So, the output of the photodiode and accompanying electronics will be
Vrefl∝Prefl. (B.25)
We are next interested in demodulating this voltage to a lower frequency, so that it is a more
tractable error signal. We use the same frequency as the original RF modulation for to create a
“local oscillator” reference signal. To maintain generality, we must multiply by either a sine or a cosine, to extract all of the information fromVrefl. To create both the sine and cosine we will split the local oscillator signal, and phase shift one component by 90◦. We lowpass the resulting signal to eliminate the high frequency terms, to get (assuming the constants are absorbed in the proportionality factor of EquationB.25)
VIrefl'J0(Γ)J1(Γ) Re(r∗cav(ω)rcav(ω+ Ω)+r∗cav(ω)rcav(ω−Ω)) (B.26) and
VreflQ 'J0(Γ)J1(Γ) Im(r∗cav(ω)rcav(ω+ Ω)−r∗cav(ω)rcav(ω−Ω)). (B.27) The version ofVreflthat contains the real parts ofrcavis referred to as “in-phase”, or the “I-phase”, which has been denotedVreflI . The other component which contains the imaginary parts ofrcavis referred to as the “quadrature phase”, or the “Q-phase”, which has been denotedVQrefl. Note that often, the component resulting from Vrefl being multiplied by a cosine will result in the I-phase component of the error signal, and the component resulting fromVrefl being multiplied by a sine will result in the Q-phase. This difference in notation is a result of the choice to use sine as the modulation in EquationB.2rather than cosine. If the phase of the local oscillator is set properly, the Q-phase signal will vanish, and all of the information about the length of the cavity relative to the laser frequency will be contained in the I-phase signal.