LIGO-like IFO parameters, contrasted

With the ground-rules given below, the design of such an IFO is driven by one parameter only, which we can choose to be any one of: the ITM transmissivity or reflectivity T_ITM, the arm cavity finesse F, the arm cavity storage time tau_s, the arm cavity pole frequency f_pole, the arm cavity power gain G_arm, the PRC power gain G_prc, the DC shot noise strain sensitivity h_shot(0), etc. This is illustrated, for the 40 m, by the following busy plot:

Note that LIGO I operates with: T_ITM = 0.03, Finesse = 204, tau_s = 1734 usec, f_pole = 91 Hz, G_arm = 130, G_prc = 60, h_shot(0) = 7.4e-24.
(All shot-noise strain sensitivity numbers quoted here are uncertain as absolute numbers, but their ratios are meaningful).

Ground-rules:

Contrasting parameters

To make the 40 m most ligo-like, we would like to choose the LIGO number f_pole = 91 Hz, but as we can see in the figure above, this requires an ITM with transmission T_ITM ~ 1e-4, which is of the same order as the ITM loss; it will be difficult to attain such low transmission and high finesse with realistic optics. Also, G_prc is just about 1 in that case; which is decidedly non-ligo-like, and defeats are purpose (to demonstrate a reduction in G_prc using RSE).

So we back off, and consider a much higher f_pole. Let's consider f_pole = 2000 Hz, which leads to T_ITM = 6298 ppm, T_RM = 0.0869, finesse = 976, tau_s = 79 usec, G_arm = 611, G_prc = 13.8, h_shot(0) = 3.4e-22.

Now the logic is this: such a high G_prc, with a high-powered laser, will lead to significant PRC losses, and thermal lensing effects. We need to reduce it. This is where RSE comes in.

We can reduce G_prc by reducing f_pole to, say, 500 Hz. This leads to T_ITM = 1489 ppm, T_RM = 0.3165, finesse = 3915, tau_s = 318 usec, G_arm = 2314, G_prc = 3.7, h_shot(0) = 1.6e-22. Now G_prc has been reduced by a factor of 4, but the bandwidth of the IFO at high frequencies has shrunk due to the smaller f_pole.

Now we add the RSE signal mirror (SM) in the asymmetric port. The carrier is absent at the asymmetric port, so it doesn't see the SM. But the GW signal exits through the port, and it does see the SM. The compound mirror composed of the ITM/SM is in resonance for the carrier and the GW signal, producing a larger transmittance, and thus, a larger f_pole. We can choose a T_SM which reproduces f_pole = 2000 and h_shot(0) = 3.4e-22 while keeping finesse, tau_s, G_arm, G_prc at their f_pole = 500 values (T_SM = 0.619 will do it).

The value of h_shot(0) is, unfortunately, characteristic of the broad-band (f_pole = 2000 Hz) configuration; but we assume that, below f_pole, we are dominated by other noise sources, anyway, so we're not losing any sensitivity there.

That's not exactly true, for the 40m with the parameters we've chosen; it may be that we'll be shot-noise limited all the way down to 250 Hz. But that does not change the significance of the experiment. In fact, it will make it easier for us to demonstrate the expected change in the shot-noise limited response as one makes use of RSE.

SO, the goal of the experiment is as follows: First, establish the shot-noise limited response of a LIGO-like IFO (without RSE) with, say, f_pole = 2000 Hz. Then, reconfigure for f_pole = 500 Hz, but with RSE to bring it back to 2000 Hz, but with a factor 4 smaller G_prc. THAT'S THE GOAL.

Noise budget

The noise is dominated by:

To model the suspension noise, I assume test masses of 4" diameter (ok, 10 cm), 8.9 cm thickness; a suspension phi = 2e-7 * f (viscous damping), f_susp = 0.744 Hz.

Obviously, we can increase the test masses to reduce this noise. See the discussion of pros and cons, below.

See also discussions of the seismic noise, internal thermal noise, and radiation pressure, below.

We see that we are shot-noise limited above 300 Hz; we choose f_pole values of 500 Hz and 2000 Hz in the discussion above in order to stay clear of all other noise sources.

In the figure below, we show "optical readout noise" (photon shot noise and radiation pressure noise) for LIGO-like configurations with f_pole = 500, 1000, 1500, and 2000 Hz, along with the RSE curve designed to bring f_pole from 500 -> 2000 Hz, and the noise curve for all other sources.

We see that the RSE curve matches the f_pole = 2000 Hz curve above ~75 Hz, bue follows the f_pole = 500 Hz curve in the radiation pressure noise nominated region below that. This is because the radiation pressure noise is due to the carrier power in the arms, not the signal power out the dark port.

Radiation pressure noise will not be a dominant source of noise for the 40 m with 4" optics, or for LIGO with 10" optics.

PRC lengths and sideband (RF modulation) parameters

f_pole (Hz)          &   500  &  2000 \\
G_sb_APD (0.542 m)   & 0.782  & 0.895 \\
opt. asym (m)        & 0.878  & 0.455 \\
G_sb_APD (opt. asym) & 0.993  & 0.923 \\

Mirror radii of curvature, spot sizes

Optimize g = g1 * g2, g1 = (1 - R_ITM/L_arm), g1 = (1 - R_ETM/L_arm).

Spot size at the end mirrors, for FP cavity with L = 1 m, versus g1 and g2:

The structure near the g=1 hyperbolas are, I think, spurious.

White space means unstable FP resonator cavities. 45^o line is for symmetric cavities.

Circles correspond to present 40 m configuration (red, (1,0.38)), 40 m upgrade config (green, (.577,.577)), LIGO arms config (magenta, (0.46,0.72)), and LIGO/40m PRC config (black, (1,1)).

Modematching study: minimize mode mismatch due to imperfect radii of curvature of ITM, ETM mirrors:

We choose a stable (g = 1/3), symmetric arm cavity. The beam waist is in the middle of the 38.25 m arms. This gives:

waist =                 3.54 mm
R_ETM = 90.5 m, w_ETM = 3.98 mm    
R_ITM = 90.5 m, w_ITM = 3.98 mm    
R_BS =   infty, w_BS  = 4.16 mm
R_RM = 60.32 m, w_RM  = 4.18 mm

Note that LIGO operates with g = 1/3, but the cavity is a bit offset from the symmetric, with g1 = (1-4000/14500), g2 = (1-4000/7407), in order to keep the spot size at the ETM such that less than 1ppm of the light falls out of the 24cm aperture (w_ETM * 5.257 < aperture). This is NOT a problem at the 40 m with 4" optics!

Note that, like LIGO and the present 40m, the PRC is nearly unstable, with g approximately = 1. I don't know how to avoid this, or what its consequences are.

Mode Cleaner

We believe that the requirements of the 40 m upgrade with respect to initial laser pointing accuracy, jitter, higher order mode rejection, etc, can be met using the PSL pre-mode-cleaner (PMC) and a fixed-spacer 1 meter mode cleaner such as already exists at the 40 m (this is NOT a quantitative argument, as of yet!).

I hope we can use the existing 1 meter fused silica spacer currently in use at the 40m, as well as the existing spring mounts. I don't know anything about this, at present!

To provide the most suppression at high frequencies (eg, f_RF = 32.7 MHz), keep cavity pole f_pole as small as possible, ie, mirror transmissivity T as small as possible.

But not too small: to be roughly insensitive to uncertainties in losses, keep T >> Losses. Also, very small T means that even the desired TEM_00 mode is lost, with transmissivity << 1.

Want optimal coupling for TEM_00 mode; so, approximately, T_1 = T_2 + 2 * Losses.

Choose T_2 = 400 ppm, T_1 = 500 ppm;
This gives T_00 = 0.8, fpole = 12 kHz, suppression of 2e-7 at 32.7 MHz.

Now optimize the (symmetric) cavity g-factor to stay away from any HOM resonances:

We choose g_cav = 0.30.

Here is an optical design for a stable, symmetric cavity that provides good rejection of higher order modes.

Want to grade the transmission of the mirrors to be higher at larger radii, to increase suppression of HOMs which have <r> ~ waist * sqrt (n+m+1).

I do not know how to set, or address, the specs on pointing accuracy, jitter, etc.

I have to learn how to design telescopes for matching PSL <-> MC and MC <-> IFO.

Output mode cleaner?

Seismic noise

The five existing core-optics chambers (BS, SV, SE, EV, EE) have 4-stage, 3-leg/stage seismic stacks, fitted with viton springs.

We plan to replace the viton springs with LIGO metal springs and flurel seats.

We hope that this will reduce the amount of viton/flurel in the vacuum system (maybe not?). In any case, we expect that after the rebuild and bakeout, the seismic stacks, and thus the 40m vacuum, will be significantly cleaner. Our main concern is contamination of the mirror surfaces, NOT water vapor or other residual gas.

As a side benefit, the seismic isolation will be much better. Here is the expected contribution to the horizontal displacement, x_rms(f), from seismic motion. The contribution to the strain is (I believe) given by 2/L_arm * x_rms(f). (The factor 2 comes from the square root of 4 test masses).

The smooth curve is an envelope function that is (more-or-less) greater than the metal spring curve, everywhere, and is thus "conservative". It is: xseis = 1e-8./(1+(f/10).^12.5) (in meters).

These curves are modelled as follows:

I crudely estimate the masses of the stack: top optical table and plate, 275 kg; leg elements, 75 kg each.

The damped metal springs can support a maximum load of 100 lbs or 45 kg. We put in as many springs as we need to hold the weight, and no more. This translates to the following numbers of springs for each stage of the 3 legs, from top to bottom (so multiply by 3 to get the total per stage): 2,4,6,7. Total: 57 springs per stack.

The spring constant for the damped metal springs at 100 Hz is k = 379 lbs/in, or 67.7 kg/cm.

The resonant frequency for stage i, in Hz, is
F_i = sqrt(N_springs * N_legs * k * g / M) / (2pi)
where g = acceleration due to gravity. We get, for the stages from top to bottom: 6.1, 9.1, 10.8, 12.3 Hz.

Each stage has a simple pole transfer function,
tr_i = f_i^2./(f_i^2 - f.^2 + i*f_i^2/Q_i);
where we take Q = 300 (a total guess). The stack transfer function is the product: tr_1*tr_2*tr_3*tr_4.

Then we have the pendulum transfer function, a simple pole with f = 0.74 Hz and Q = 3.

Then we have the seismic spectrum itself. I don't know the spectrum at the 40m site (do you?). I use the "Hanford site noisy, w/ microseismic peak", and MUTLIPLY BY 10.

The product of these spectra give the curves shown above.

Imperfect Optics

The need to operate at arm cavity pole frequencies that are far below the arm FSR drives us to small ITM transmissivities, of the same order as the losses (50 ppm).

This suggests that small optical imperfections in the ITMs can lead to big changes in the IFO operation.

FFT (Bochner, LIGO-P980004) is designed to address this question with a full simulation of the (DC) E-fields in the IFO, with realistically-deformed mirror maps.

At the moment, I don't know how to make those mirror maps, so I've only run FFT with perfect optics.

FFT runs with lambda/1800, etc.

RMS deformation      &      0 &   Lambda/1800 & Lambda/1200 & Lambda/ 800 & Lambda/ 400 \\
RMS deformation (nm) &      0 &   0.59        & 0.89        & 1.33        & 2.66        \\
R_RM (\%)            &   68.4 &               &             &             &             \\
Opt. Asymm (cm)      &   45.5 &               &             &             &             \\
Mod Depth Gamma      &   0.36 &               &             &             &             \\
G_prc, Carr, TEM00   &    8.9 &               &             &             &             \\
G_arm, Carr, TEM00   &   2600 &               &             &             &             \\
G_APD, Carr, TEM00   &   5e-3 &               &             &             &             \\
G_APD, Carr, Total   &   6e-3 &               &             &             &             \\
1-C                  & 1.6e-3 &               &             &             &             \\
G_prc, SB,   TEM00   &    7.2 &               &             &             &             \\
G_APD, SB,   TEM00   &   0.95 &               &             &             &             \\
G_APD, SB,   Total   &   0.95 &               &             &             &             \\
R_ref,       Total   &  0.015 &               &             &             &             \\
f_{pole} (Hz)        &   2022 &               &             &             &             \\
h_SN(0)   (1e-22)    &    1.8 &               &             &             &             \\

Size of core optics