Difference in transfer functions of modulation of space and modulation of mirrors

Question:

l laser 1M 0 n0                        # Laser (P=1MW, 0 wavelength offset from 1064nm)           
s s0 1 n0 nBSb                         # Space from laser to beam splitter (1 m)

## Central beam splitter ##
bs BS 0.5 0.5 0 45 nBSb nBSy nBSx nBSd   
                                                                                                   
## X arm ##
s LX 4000 nBSx nMX1                    # Space: Length of X arm 
m MX 1 0 -45.003 nMX1 nMX2             # Mirror MX (R=1, T=0, -45deg tuning)

## Y arm ##
s LY 4000.01 nBSy nMY1                 # Space: Length of Y arm 
m MY 1 0 45.003 nMY1 nMY2              # Mirror MY (R=1, T=0, +45deg tuning)

## Output ##
s sout 1 nBSd nout                     

fsig sig1 LX  1 0 1
fsig sig1 LY 1 180 1
pd1 tf $fs nout 

xaxis sig1 f log 1 100k 1000
yaxis abs:deg

In reference to Task 1, for sheet 13, the above plot is in relation to modulating the space of the cavities, i.e., changing the length of both the cavities. The plot below is modulating the position of the mirror. What is the reason for the difference between the two plots? Theoretically, both the processes are identical.

(Also notice that the phase plots, might seem identical, but the range for the 2 plots are different…)

l laser 1M 0 n0                        # Laser (P=1MW, 0 wavelength offset from 1064nm)           
s s0 1 n0 nBSb                         # Space from laser to beam splitter (1 m)

## Central beam splitter ##
bs BS 0.5 0.5 0 45 nBSb nBSy nBSx nBSd   
                                                                                                   
## X arm ##
s LX 4000 nBSx nMX1                    # Space: Length of X arm 
m MX 1 0 -45.003 nMX1 nMX2             # Mirror MX (R=1, T=0, -45deg tuning)

## Y arm ##
s LY 4000.01 nBSy nMY1                 # Space: Length of Y arm 
m MY 1 0 45.003 nMY1 nMY2              # Mirror MY (R=1, T=0, +45deg tuning)

## Output ##
s sout 1 nBSd nout                     

fsig sig1 MX  1 0 1
fsig sig1 MY 1 180 1
pd1 tf $fs nout 

xaxis sig1 f log 1 100k 1000
yaxis abs:deg

Answer:

Length Modulation

First we’ll look at the case of modulated lengths. The maths behind this is described well in the Living Review, Chapter 5.5, which references a derivation in Charlotte Bond’s thesis, Appendix A.3. Here the extra phase accumulated due to a gravitational wave by light propagating through a space of length L is given as

\delta\varphi = \frac{\omega_0 h_0}{\omega_{gw}}\cos\left(\omega_{gw}t + \varphi_{gw} – \omega_{gw} \frac{L}{2c}\right)\sin\left(\omega_{gw}\frac{L}{2c}\right),

where ω is the gravitational wave frequency. On a round trip, light travels a distance of 2L, so from the sine term we see that there will be 0 accumulated phase when

\omega_{gw}\frac{2L}{2c} = n\pi\\ \therefore f_{gw} = n \frac{c}{2L}.

For 4 km arms, this first happens at a frequency of ≈ (3e8 m/s) / (8 km) ≈ 37.5 kHz, which is what we see from the first graph.

For a non-mathematical explanation of what’s happening, consider a sinusoidal gravitational wave, with period equal to twice the travel time of a photon along the arm (i.e. f = c / 2L, as describe above):

Here, L₀ is the unmodulated value of L (4 km). For some photon that departs from the central beamsplitter at t = 0, the integrated path length seen over a full round trip is simply L₀ – the reflection from the mirror at time t = L / c is unimportant. The photon therefore sees no net effect of modulation, and there is no differential phase between the x & y arms, leading to a null in the sensitivity of the Michelson.

Mirror Modulation

If we shake the end mirrors rather than modulate the arm length, instead of seeing the integrated path length over the whole round-trip, each photon now only samples the path length at a single point in time. The x & y arms are always 180° out of phase, regardless of signal frequency, so a pair of photons will that depart from the central beamsplitter at the same time will always experience some differential phase. This means a simple Michelson will have same response for any frequency.

The very slight decrease in sensitivity in the mirror modulation plot is due to the macroscopic difference in arm length between the x & y arms. At some very high frequency, the differential phase caused by this difference will be 180°, when

\Delta L = n\frac{\lambda_\mathrm{sb}}{2},\\\phantom{M}\\ \therefore f_\mathrm{sb} = n\frac{c}{2\Delta L}.\\

Here we have a 1 cm arm length difference, so there will be a null in sensitivity at

f_\mathrm{sb} \approx \frac{3\times10^8\,\mathrm{ms}^{-1}}{2\,\mathrm{cm}}\\\phantom{M}\\ \phantom{f_\mathrm{sb}.} \approx 1.5\times10^{10}\,\mathrm{Hz}.

If we change the xaxis of the file in question to reach this frequency, this is exactly what we see:

l laser 1M 0 n0                        # Laser (P=1MW, 0 wavelength offset from 1064nm)           
s s0 1 n0 nBSb                         # Space from laser to beam splitter (1 m)

## Central beam splitter ##
bs BS 0.5 0.5 0 45 nBSb nBSy nBSx nBSd   
                                                                                                   
## X arm ##
s LX 4000 nBSx nMX1                    # Space: Length of X arm 
m MX 1 0 -45.003 nMX1 nMX2             # Mirror MX (R=1, T=0, -45deg tuning)

## Y arm ##
s LY 4000.01 nBSy nMY1                 # Space: Length of Y arm 
m MY 1 0 45.003 nMY1 nMY2              # Mirror MY (R=1, T=0, +45deg tuning)

## Output ##
s sout 1 nBSd nout                     

fsig sig1 MX  1 0 1
fsig sig1 MY 1 180 1
pd1 tf $fs nout 

xaxis sig1 f log 1e8 6e10 1000
yaxis abs:deg

As for the phase difference between the two plots in the question, I think they’re actually the same. The space modulation case acquires 180° phase at 37.5 kHz, and the mirror modulation case acquires 360° phase at 75 kHz – it’s just the vertical ranges of the plots / wrapping of the phases that are different (I’m not sure if this is what you meant). However, I don’t know exactly why this occurs.

History

# Date User Information
201 2 years ago Philip Jones (current)
199 2 years ago Philip Jones
195 2 years ago Philip Jones
191 2 years ago Joy Sanghavi

One comment on “Difference in transfer functions of modulation of space and modulation of mirrors

  1. The phase evolution in both cases correspond to a time delay (phase = 2*pi*freq*timedelay). This is the amount of time it takes for us to see the effect of GWs or mirror motion at the output photo-detector. If we fit the phase we got above and calculate this time delay we get 13.34 microseconds which is the time required for light to travel along the 4 km arm. In the case of moving the mirror this makes sense, because the information about the moved mirror will be available at the photo-detector only after 13.34 microseconds (special relativity). In the case of GWs the effect happens throughout the arm and hence the delay of 13.34 microsecond is not obvious. But that’s what we get from finesse.

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