Arm Cavities

A reasonably good understanding of the arm cavities is important for generating ac- curate model predictions. The lack of adequate equipment did not allow a precise characterization of cavity losses and mode matching; however, some simpler measure- ments were used to place some acceptable constraints on these parameters.

4.3.1 Visibility

Measurements of the power reflected from a cavity in both the resonant and non- resonant state gives information about the losses and mode matching of the cavity.

The reflected powers are used to calculate the cavity visibility, defined as³ V

=

^{Vbright - Vdark}

vbright

(4.4)

Vbright is the measured DC voltage with the arms internally blocked, which assumes that the reflectance of the ITM is roughly equal to the reflectivity of the anti-resonant cavity, which is a fairly good approximation when ^Ritm is near 1. ^Vdark is measured when the cavity is locked, and aligned for optimum power build-up. Since the losses and mode matching of the arm cavities is applied to the carrier only, the voltages are corrected for the presence of RF sideband power. This is done by assuming the same amount of RF sideband power in both bright and dark measurements, and calculating this amount as a fraction of the bright measurement (13.8% power in the RF sidebands). This amount is subtracted off of the bright and dark measured voltages. The measured values then are modeled as follows.

Vbright ex ^Ritm

Vdark ex (1-M M)

+

M M x ^Rcavity

(4.5) (4.6)

M M is the mode matching fraction, and ^Rcavity is the reflectance of the cavity on resonance for the cavity mode, which is a function of the losses. A plot showing the mode matching as a function of the cavity loss is shown in Figure 4.6. This uses the measured visibility of 0.25 ± 0.005 for arm 2, and 0.22 ± 0.005 for arm 1.

4.3.2 Mode Matching

Typically, losses can be easily characterized by measuring the decay time of the cavity.

However, with cavities as short as the ones in the prototype, the decay time is on the order of 1 f.LS, and the required equipment to modulate the laser's intensity faster than this was not available. So, some tests were done to make an estimate of the

3Note that this is not the normal definition for visibility used in most optics texts. However, it does tend to give a good picture of what the losses of the cavity are, which, in the case of cavities with ideally unity reflectors for end mirrors, does define the ability to resolve a fringe.

c 0

0.95

n

~ 0.9

Ol c

=5 ^0.85

(ij

"8

E ^0.8

::::iE 0.75

1200 1300 1400 2000

Total cavitv losses loom)

Figure 4.6: Mode matching as a function of cavity loss, given the measured visibilities for arm 1 and arm 2.

modematching, and together with Figure 4.6, estimate the losses.

First, a simple approach was to unlock the cavity from the previous experiment, and sweep it through several fringes, and tabulate the height of the higher order modes (of course the RF modulation was turned off). For arm 2, approximately 84% of the total measured power was in the T EM₀₀ mode, with roughly 80% of the remaining 16% in the TEM₀₁ mode. If a more careful, painstaking additional alignment was done, arm 1 showed roughly 95% of the power in the 00 mode, while arm 2 was somewhat worse, at around 90% optimum. It's assumed that under normal circumstances, the modematching was more like 85%.

The next approach was to do a theoretical calculation, based on a measurement of the beam profile incident on the cavity and the mirror radii of curvature. First, the waist sizes and positions of the laser light were measured. It is assumed that the light is entirely in the 00 mode, such that the waist size and position completely determine the modal structure of the input light. This is probably a good approximation to the 90% level. Next, the radii of curvature of the mirrors was measured using a fiber optic illuminator, by measuring the distance at which the image of the light source is at the same distance as the light source itself. Surprisingly, the mirrors were significantly off the values ordered from REO. It's not known if this is a result of clamping the

II

Waist size (mm)

I

Waist position (em)

Horizontal waist 0.80 -27.4

Vertical waist 0.84 +4.3

Table 4.3: Measured beam parameters for the light input into the main interferometer.

Waist position is relative to the average power recycling cavity length position, from the power recycling mirror.

mirrors in their actuator mounts, or if the mirrors are simply not the radius that was specified. Table 4.4 summarizes the measured radii, as well as the waist of each arm cavity due to the ETM radius. The mode matching is calculated as an overlap

II

Mirror radius (m)

I

Waist size (mm)

I

ETM1 5.0

±

0.3 0.92

±

0.03 ETM2 5.8

±

0.3 0.99

±

0.03 PRM 4.4

±

0.2 0.85

±

0.03

Table 4.4: Measured arm ETM radii of curvature, and the resulting cavity mode waists, based on a length of 2.65 m and flat ITMs.

integral between the arm cavity mode from Table 4.4 and the mode of the input light from Table 4.3, where the waist position and size are all that is needed to characterize the fundamental mode.

using the Gaussian 00 mode structure

1 ^{- i (} 27rz -1)(z)+ ,.(x2+,,2)) - ( (x2+Y22))

Uoo(X, y, z)

=

- - e ^{T .xk(z)} e ^w(z) w(z)

w(z)

= wo-/1 ⁺

^z2/z5

R(z)

=

z(1

+

z5/ z²⁾ ry(z)

=

arctan(z/ zo)

zo

= 7rw5/

^>.

(4.7)

(4.8)

The overlap, or mode matching, between the laser light and arm 1, arm 2, and the

power recycling cavity is 98%, 96%, and 99.6%, respectively. Clearly, the input beam matches the individual cavities quite well.

It's also of interest to calculate the overlap between the sum of the arm cavity modes, and the power recycling cavity, which would give an estimate of the expected coupling of the input light to the power recycled Fabry-Perot interferometer that the carrier light will see.

The term ei⁸ arises due to the phase shift to the 00 mode that occurs when the waist is displaced longitudinally along the beam axis.[57] This phase shift is something the control system would null out, which also maximizes the integral. The maximum of this integral is about 97%.

Of course, the theoretical calculations are quite optimistic, and the measured value of approximately 85% modematching is assumed. The losses of the arms are taken to be 1280 ppm for arm 1, and 1520 ppm for arm 2, which includes the transmittance of the ETMs (300 ppm).