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Abstract
Optical loss plays a significant role in optical experiments involving optical cavities such as recycling cavities and filter cavities in laser interferometer gravitational-wave detectors. For those cavities, modal frequency degeneracy, where the fundamental and a higher order mode resonate inside the cavity simultaneously, is a potential mechanism which may bring extra optical loss to the cavity thus degrade detection sensitivity. In this paper, we report observation of modal frequency degeneracy in a large-scale suspended Fabry-Pérot cavity. The cavity g-factor is tuned by a CO2 laser heating one test mass, and the cavity finesse is obtained from a ring-down measurement of the transmitted light. We demonstrate that the modal frequency degeneracy can cause a reduction of the cavity finesse by up to ∼30%, corresponding to a ∼2-fold increase in total optical loss. To minimize optical loss in gravitational-wave detectors, the effect of modal frequency degeneracy needs to be taken into account in the design and operation of the detector.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Gravitational-waves generated from binary black hole coalescences [1–3] and binary neutron-star systems [4] have been detected by ground-based gravitational-wave interferometers, such as Advanced LIGO [5] and Advanced Virgo [6]. In observation runs 1 to 3 the low frequency (<100 Hz) detection sensitivity is limited by a combination of seismic noise, suspension thermal noise, mirror coating thermal noise, quantum noise and other technical noises, while at high frequencies (>100 Hz) it is mainly limited by quantum noise [7,8]. The Advanced LIGO and Advanced Virgo are going to upgrade with new coating materials to reduce the coating thermal noise [9] and injection of frequency-dependent squeezed vacuum via filter cavities to reduce the quantum noise in broadband frequencies [10,11]. The quantum measurement schemes such as speed meter [12] and white-light-cavity [13–15] are proposed to further improve the quantum noise limited sensitivity. In addition, advanced techniques are proposed in the third generation detectors [16,17] to improve the detector sensitivity and expand the reach of gravitational wave astronomy to the edge of the universe.
Many of these advanced techniques require low optical loss. Particularly, high optical loss in the arm cavities and recycling cavities results in reduced optical power gain and increased quantum noise, while in filter cavities for squeezed vacuum it results in a reduced level of frequency dependent squeezing, also increasing the quantum noise [18]. Ideally, the laser beam inside the cavities would only circulate in the fundamental optical mode. However, as illustrated in previous works [19,20], the optical loss may increase, mainly caused by the scattering of light off the imperfect mirror surfaces. A proportion of the light in the fundamental optical mode may be scattered into a higher order transverse mode, which can resonate inside the cavity if the cavity length satisfies their resonance conditions. This effect is called modal frequency degeneracy, where the fundamental mode and some higher order transverse mode circulate inside the cavity simultaneously. Modal frequency degeneracy has been observed from small-scale optical resonators [21,22]. This effect was predicted to have the potential to degrade LIGO sensitivity and as a result the power recycling cavity was designed to avoid degeneracy between higher order optical modes from the coupled cavities comprising two arm cavities, signal and power recycling cavities [23,24]. Recently, excess loss from scattering of the fundamental mode off into a resonant higher order transverse mode has been observed in Advanced LIGO due to imperfections in the mirror surface [25]. However, to best of our knowledge, extra optical loss induced by complete modal frequency degeneracy between the fundamental and a higher order mode has not been observed in large-scale suspended optical cavities, such as those used in Advanced LIGO and Advanced Virgo.
In this paper, we report the first observation of optical loss caused by modal frequency degeneracy in a large-scale suspended optical cavity. In Section 2 we describe the experimental method used to characterize the cavity and measure the finesse as a function of mode spacing. In Section 3 we present the results of cavity tuning and finesse measurement, and the observation of modal frequency degeneracy. Section 4 is a discussion of the limitations of this experiment, while Section 5 summarizes our findings. The theory used in the paper is presented in Appendix.
2. Methods
2.1 Experimental setup
The schematic of the experimental setup is shown in Fig. 1(a). The experiment was implemented in a 74-m-long suspended Fabry-Pérot cavity, consisting of two fused silica mirrors: the input test mass (ITM) and the end test mass (ETM). The nominal radii of curvature (RoC) of ITM and ETM are 37.4 m and 37.3 m, respectively. The nominal power transmissivity of ITM and ETM are 200 ppm and 20 ppm, respectively, at the laser wavelength of 1064 nm. Each of the two test masses weighs 0.88 kg and has a cylindrical form (100 mm in diameter and 50-mm-thick), with modifications of a flat surface and two holes on each side of the mirror to allow attachment to a low-loss modular suspension system [26]. Each test mass was suspended, through four niobium wires, from a control mass, which was feedback-controlled by a digital signal processor controller [27]. The control mass was further suspended from a 4-stage vibration isolation system minimizing environmental vibrations [28]. Each test mass and its suspension system were installed in a 3-m-high stainless-steel vacuum tank, minimizing vibrational noise from air disturbance. An infra-red (IR) camera was installed next to each vacuum tank to monitor the circulating mode shape on the test masses. The two vacuum tanks are connected by a 74-m stainless-steel vacuum pipe. The vacuum level for the tanks and the pipe was kept at $\sim$10$^{-6}$ mbar.
Fig. 1. (a) Schematic diagram of the experimental setup. $\frac {\lambda }{4}$, quarter-wave plate. $\frac {\lambda }{2}$, half-wave plate. ITM, input test mass. ETM, end test mass. L1, lens 1, f = 100 mm. EOM, electro-optical modulator. BS, beam splitter. PDH locking setup is framed in green dashes. Optical cavity is shaded in gray color. (b) Schematic diagram of modal frequency degeneracy in frequency domain.
The laser source in our experiment is a 500 mW non-planar ring oscillator (NPRO) laser (InnoLight, Model Mephisto 500NE, 1064 nm Nd:YAG laser). The fundamental mode of the laser beam was locked to the cavity using Pound-Drever-Hall (PDH) locking technique [29]. An auxiliary optical beam reflected from each mirror was incident on a quadrant photodetector producing an error signal in control servo that maintains the cavity alignment. The transmission beam from the cavity was focused by a f = 100 mm lens onto a quadrant photodetector (QPD). The total optical intensity and the horizontal and vertical differential intensity components of the transmitted laser beam were recorded by the QPD, and the signals were then processed by a spectrum analyzer (SA) to measure the cavity ring-down and mode spacing respectively.
To allow observing modal frequency degeneracy, the cavity mode spacing was thermally tuned by injecting a CO$_2$ laser (Synrad, 48-series) onto the ITM [30]. Figure 1(b) is a schematic diagram of frequency components of the optical cavity. The long vertical lines represent the frequencies of ${\rm TEM}_{0,\,0}$ modes while the short vertical lines represent higher order transverse modes. The mode spacing $\Delta f$ is thermally tuned until a higher order transverse mode is degenerate to a fundamental ${\rm TEM}_{0,\,0}$ mode from the next longitudinal order. The dashed lines represent higher order transverse modes after the CO$_2$ thermal tuning. The free spectral range, $f_{\rm FSR}$, is the frequency difference between adjacent longitudinal modes. In the experiment, the CO$_2$ beam mode was tuned to be close to the same size as the circulating beam in the cavity such that it mimicked the thermal effect caused by the high circulating laser power in the large-scale gravitational-wave detectors. The maximum output power of the CO$_2$ laser was 10 W. As the cavity is near-concentric, the mode spacing is very sensitive to the change of RoC of the test mass. In the experiment, we used up to 8% of the maximum power of the CO$_2$ laser, corresponding to 0.8 W laser power, before the cavity became unstable due to excessive thermal-tuning.
2.2 Characterization of the cavity length
In the experiment, an important parameter to be determined is the cavity length. We used the following two methods to accurately measure the cavity length.
The first method is to measure the response function of the PDH error signal at the reflection port of ITM. When the fundamental mode ${\rm TEM}_{0,\,0}$ is resonant inside the cavity, the resonant condition is satisfied at every multiple of the free spectral range, $f_{\rm FSR}\equiv \frac {c}{2L}$, where $c$ is the speed of light and $L$ is the cavity length. The response function of the PDH error signal to a frequency modulation of the laser source can be expressed as [31]:
where $r_{a},~r_{b}$ are the amplitude reflectivities of the ITM and ETM, respectively. $T=\frac {L}{c}$ is the transit time of the light inside the cavity, $s\equiv i\omega$ is the Laplace transform parameter, and $\omega$ is the angular frequency. Figure 2(a) shows a response function of the PDH error signal at the first free spectral range, which is measured to be 2.028965 MHz $\pm$ 2 Hz for the Lorentzian-fitted peak. The cavity length is calculated to be 73.87817 $\pm$ 0.00007 m.Fig. 2. (a) Response function of the PDH error signal measured at the first $f_{\rm FSR}$. (b) Spectrum of the transmitted light measured at the first $f_{\rm FSR}$. The blue dots in (a) and (b) are the measurement data, and the red curves are fitted Lorentzian functions. The signal-to-noise ratio is better in (a) than (b).
The second method to measure the cavity length is via the spectrum of the transmitted light. When the cavity is locked to the fundamental mode, laser noise at the free spectral range frequency is resonantly enhanced, resulting in a beating signal on the QPD. Figure 2(b) shows a peak at the first spectral range, which is fitted to be $2.02897$ MHz $\pm$ 10 Hz. The cavity length is then calculated to be 73.8780 $\pm$ 0.0004 m, consistent with the result derived from the first method.
2.3 Measurement of the mode spacing
The modal frequency degeneracy between the fundamental mode and a higher order mode happens when the cavity is tuned to satisfy the following condition:
where $N_d$ denotes the order number of the degenerate higher order mode, $m$ is an integer, $\Delta f$ is the cavity mode spacing, which is defined as the frequency difference between the fundamental and first order transverse modes in the same longitudinal order: where $f_0$ and $f_1$ are the frequencies of the fundamental and the first order mode, respectively. $g_{1,\,2}\equiv 1-\frac {L}{R_{1,\,2}}$, where $R_{1,\,2}$ are the radii of curvature of ITM and ETM. The product $g_1\cdot g_2$ is called the cavity g-factor.To tune the mode spacing, a CO$_2$ laser was used to heat the ITM, changing its RoC by thermal expansion. When Eq. (2) is satisfied, the fundamental mode and a higher-order transverse mode resonate inside the cavity simultaneously. The cavity g-factor can be monitored by measuring the mode spacing via resonantly enhanced laser noise on the QPD at the transmission port of ETM. The beating signal between the fundamental and first order modes was received by the QPD as the differential signals at the mode spacing frequency. Figure 3 presents two peaks at $\sim$100 kHz, without the CO$_2$ laser heating the ITM. The two peaks, separated by $\delta f_1=$3.882 $\pm$ 0.017 kHz, are likely due to the slightly asymmetric RoC in the orthogonal directions of the two first order transverse modes, ${\rm TEM}_{0,1}$ and ${\rm TEM}_{1,0}$.
Fig. 3. Power spectral density of the differential transmission signal measured by the QPD. Two peaks are measured with a frequency difference $\delta f_1$=3.882 $\pm$ 0.017 kHz. Blue dots are measurement data, and the fitted double-peak Lorentzian curve is in red.
The mean value of the two peaks in Fig. 3 is measured as $99.3$ kHz. The nominal RoCs of the ITM and ETM are $37.4$ m and $37.3$ m respectively. Using Eq. (3) and the cavity length measured in Section 2.2, a nominal mode spacing of 130 kHz is calculated for the cavity without CO$_2$ laser heating. As the cavity is near-concentric, the difference between the measured and the theoretical values can be due to a small change of 0.15 m from the nominal RoC value of both the ITM and ETM, which is within their nominal uncertainty range. In our experiment, the measurement of the mode spacing can accurately monitor the change of the RoC of the ITM, when it is tuned by the CO$_2$ laser.
2.4 Measurement of the cavity finesse
The cavity finesse, which is a quantity indicating the total optical loss, is defined as:
where $\tau$ is the time constant of the exponential ring-down, which can be measured from the cavity transmission received by the QPD when the laser is switched off. Figure 4 shows an example of cavity ring-down measurement in a non-degenerate condition. Initially the laser was locked to the cavity and the power in transmission was stable. The Nd:YAG laser was then switched off at t = 0, where the cavity transmission power started to decay exponentially. The measurement data is shown in blue and the exponential fit in red. From the measured data in Fig. 4, $\tau$ is estimated to be 1.14 $\pm$ 0.02 ms. In Fig. 4, it is clear that the laser switch-off time is significantly smaller than the ring-down time constant, which is negligible in fitting the ring-down data to an exponential curve. According to Eq. (4), the cavity finesse is calculated to be 14,500 $\pm$ 300. As explained in Appendix A.1, the total optical loss can be estimated from the finesse measurement. Here, the calculated optical loss is 213 $\pm$ 8 ppm.Fig. 4. Ring-down measurement of the transmitted light in ${\rm TEM}_{0,0}$ mode. The reference time ‘0’ is the point when the laser was switched off. The blue dots are the time series data, the red curve is the fitted exponential ring-down. The time constant $\tau$ is calculated from the fitted curve.
In modal frequency degeneracy, small mirror imperfections couple a proportion of the fundamental mode to a higher order mode which is resonant simultaneously inside the cavity. The optical loss of the degenerate higher order mode is higher than that of the fundamental mode mainly due to the finite mirror size and diffraction loss, resulting in a lower cavity finesse than non-degenerate conditions. A theoretical model of the effect of modal frequency degeneracy on cavity finesse is described in Appendix A.2.
3. Results
3.1 CO$_{\it 2}$ laser thermal tuning
To observe modal frequency degeneracy, the cavity g-factor was tuned by the CO$_2$ laser heating. Figure 5 shows the measurement of the mode spacing as a function of the CO$_2$ laser power heating the ITM. The CO$_2$ laser power was first increased from 0 W to 0.2 W, which is the minimum operating output power. Then it was increased in increments of 0.05 W, which is the minimum output power increment, until 0.75 W. There were 5 measurements of the mode spacing acquired at each CO$_2$ laser power level. Each data point in Fig. 6, marked by the red ‘+’, is the average of the 5 measurements at the same CO$_2$ laser power level, and the error bar of each point is its standard deviation. This figure demonstrates that the cavity mode spacing increases continuously with increased CO$_2$ laser power heating the ITM.
Fig. 5. Measurement of the mode spacing $\Delta f$ as a function of the CO$_2$ laser output power. Each data point is an average of 5 measurements at the same laser power level. Error bars represents the standard deviation of the 5 measurements.
Fig. 6. Photographs of the ITM displaying modal frequency degeneracy between the fundamental mode and (a) a ${\rm TEM}_{12,5}$ mode, at 0.25 W CO$_2$ laser power, and (b) a ${\rm TEM}_{5,6}$ mode, at 0.75 W CO$_2$ laser power.
3.2 Observation of modal frequency degeneracy
As described in Section 2.1, the beam spots on the ITM and ETM were monitored by IR-cameras when the cavity g-factor was tuned. Optical mode frequency degeneracy was observed at some CO$_2$ laser power levels, when the cavity g-factor satisfied the resonant conditions for both the fundamental and higher order modes simultaneously. Figure 6 shows photographs of modal frequency degeneracy acquired by the IR-camera at the ITM. In each of Figs. 6(a) and 6(b), the resonant fundamental mode appears as the bright central spot, while a higher order mode degenerate with the fundamental mode can also be identified. In Fig. 6(a), the higher order optical mode is determined to be ${\rm TEM}_{12,\,5}$ by counting the small bright spots in the two orthogonal directions. This was measured at a CO$_2$ laser power of 0.25 W. In Fig. 6(b), the higher order optical mode is determined to be ${\rm TEM}_{5,\,6}$, measured at a CO$_2$ laser power of 0.75 W. Note that the camera was not simultaneously employed with the cavity finesse measurement due to difficulties in synchronizing the camera acquisition with the photodetector measurement.
3.3 Cavity finesse measurement during CO$_{\it 2}$ thermal tuning
To examine the optical loss of the degenerate cavity, the cavity finesse was measured using the ring-down method when the cavity g-factor was tuned using CO$_2$ laser heating from 0 to 0.8 W. Figure 7 is the measurement of the cavity finesse as a function of mode spacing. In the experiment, the CO$_2$ laser power was increased from 0 W to 0.2 W, then increased in increments of 0.05 W until 0.8 W, when the optical cavity became unstable. The heating power was maintained at each level for $\sim$3 minutes, which is the time scale of the response of mirror RoC to a change in CO$_2$ laser heating. Measurements of the mode spacing were synchronized with ring-down measurements such that each ring-down measurement had an associated mode spacing value. An average of 10 ring-down measurements were made at each CO$_2$ laser power. The finesse, calculated from each ring-down measurement is plotted as a function of the measured mode spacing in Fig. 7.
Fig. 7. Finesse measurement at different mode spacings during the CO$_2$ laser thermal tuning. Each red cross corresponds to a mode spacing and ring-down measurement at the same time.
In Fig. 8, the maximum and minimum finesse are 16,700 and 10,950, respectively. There are 4 drops in finesse, marked by green circles, most likely corresponding to modal frequency degeneracy. Table 1 lists the finesse value, the measured mode spacing $\Delta f_{\rm meas}$, the nearest theoretical mode spacing $\Delta f_{\rm theo}$ calculated from Eq. (2), and the expected order number of the degenerate higher order mode, for each of the drops. The theoretical mode spacing $\Delta f_{\rm theo}$ was calculated assuming $m=1$ in Eq. (2), when the order of the degenerate higher order mode is lowest. Note that in Fig. 8 there is another drop in finesse at 184.3 kHz which is likely from the same modal frequency degeneracy as the drop at 185.4 kHz, due to fluctuations of mode spacing modulation that may not be time resolved by the spectrum analyzer which requires some integration time. It was noted these fluctuations in mode spacing became larger at times with increased residual motion in the angular degrees of freedom [32]. In Table 1, we use the measurement of the lower drop to represent the modal frequency degeneracy with the 11th order mode. As shown in Table 1, the measured values of mode spacing match with the theoretical values to within 3%. In addition to the previous point (non-time resolved mode spacing modulation), another possible reason of this error is that as the beam size of the CO$_2$ laser is similar to that of the fundamental optical mode on the mirror, when the mirror is heated, the higher order mode could see a slightly different change of the RoC due to its larger beam size on the mirror. This small difference in RoC could introduce systematic errors in the estimation of the theoretical mode spacing. In addition, the residual difference between $\Delta f_{\rm meas}$ and $\Delta f_{\rm theo}$ could be due to the mirror figure errors.
Fig. 8. Calculation of finesse as a function of $r_{nm}$ at four different values of coupling efficiency $\beta$.
Table 1. List of the 4 Drops in Finesse, Measured Mode Spacing, Theoretical Mode Spacing, and the Order of the Degenerate Higher Order Mode
According to Eqs. (8) and (14) in the Appendix, the total optical loss can be calculated from the finesse measurement. Specifically, excluding the four finesse drops, the mean finesse value measured in Fig. 7 is 15,300, corresponding to a total optical loss of 190 ppm for non-degenerate conditions. The minimum finesse value, 10,950, was measured in a degenerate condition, corresponding to a total optical loss of 353 ppm, $\sim$2-fold higher than that in non-degenerate conditions. These measurements indicate that degenerate high order optical modes can have a significant impact on optical loss.
4. Discussion
In this study, optical loss induced by modal frequency degeneracy was observed in a near-concentric 74-m suspended Fabry-Pérot cavity. A CO$_2$ laser was used to thermally deform the test mass surface to tune the cavity g-factor across a number of degeneracies. During thermal tuning, the cavity finesse reduced significantly around some expected mode spacing, while it stayed fairly constant otherwise, as presented in Fig. 7. This measurement confirmed that the optical loss from the fundamental mode was stable at different cavity g-factors, and the coupling to higher order modes contributed extra optical loss at degeneracy.
In Fig. 7, the measured mode spacing ranges from 100 kHz to 190 kHz. Theoretically, there should be 9 values of mode spacing where frequency degeneracy can happen in this range. However, only four drops of the finesse were found, indicating that not all frequency degeneracies were observed. The reason that we did not observe these five missing higher order modes could be caused by the rough steps in tuning cavity g-factor , due to the minimal 0.05 W increments of the CO$_2$ laser power, and that the mode spacing measurement may not be time resolved by the spectrum analyzer. Another possible reason is that there were no resonant higher order modes as the scattering from the fundamental mode to those higher order modes was very small due to mismatches between the mirror surface profile and the spatial distribution of those higher order mode. Finesse measurements were acquired during the thermal transition to improve g-factor resolution, however resolution was still limited as the thermal time constant of two minutes does not allow many measurements to be made in each transition. In addition, each of the four drops in finesse in Fig. 7 was measured at a mode spacing close to the theoretical value where modal frequency degeneracy should happen. However, it is not guaranteed that the finesse was measured when the degenerate higher order mode was fully resonant. An uncertainty with respect to this measurement could potentially turn the derived loss value into lower limits for modal frequency degeneracy. Future experiments should use a CO$_2$ laser source with finer power increments or add a neutral density filter at the output of the existing CO$_2$ laser, to allow a fine tune of the mode spacing. For instance, if we use a 10 dB neutral density filter, to change the mode spacing across the same range as in Fig. 7, we will be able to increase the CO$_2$ laser power at a minimum increment of 0.005 W, enabling us to observe more degeneracies.
Another issue with the current CO$_2$ laser tuning method is that the camera was not synchronized with the cavity finesse measurement. It was difficult to synchronize the photograph acquisition with each mode spacing and finesse measurement, due to the rapidly changing cavity g-factor. As a result, the transverse mode number (n, m) of the degenerate higher order mode monitored from the photograph varies frequently. In addition, it is a technical challenge to electronically synchronize camera acquisition and photodetector measurement. With a finer tuning of the CO$_2$ laser power, the cavity can be maintained in stable conditions, to allow the simultaneous acquisition of photograph, mode spacing and finesse. This method should allow us to identify which higher order mode is degenerate with the fundamental mode when a decrease of the cavity finesse is measured.
In addition to the ring-down measurement described in this study, an alternative way of monitoring the optical loss is to measure the relative power buildup of the cavity. During modal frequency degeneracy, the relative power buildup will significantly drop as a proportion of the input optical power is coupled to the higher order mode, resulting in an increase of the shot noise in gravitational wave detection. In future work, the relative power buildup and the cavity ring-down can be measured concurrently, allowing decoupling of the higher order mode loss and the coupling coefficient between the degenerate modes. In gravitational-wave detection, the interferometer response to gravitational waves is shaped by the optical cavities. In the simplest case, it is parameterized by the cavity gain (shot noise) and the cavity linewidth (frequency response). To accurately model the interferometer to predict the detection sensitivity of gravitational waves, we need both of these parameters, requiring both measurements of the cavity power buildup and the cavity ring-down.
5. Conclusion
In this paper, we present the first observation of the modal frequency degeneracy in a large-scale suspended Fabry-Pérot cavity. We have demonstrated that the cavity g-factor can be tuned with CO$_2$ laser heating one of the test masses, and the cavity mode spacing can be accurately measured. We have verified that optical loss can double due to modal frequency degeneracy. This discovery indicates that the modal frequency degeneracy could be a significant source of extra optical loss for the large-scale suspended optical cavities, such as the arm cavities and filter cavities in gravitational-wave detectors. Our method provides a useful way of examining the optical loss of those cavities and may help improve the detection sensitivity in gravitational-wave detectors.
Appendix
A.1 Finesse at non-degeneracy case
When the cavity is tuned far away from degeneracy, only the fundamental mode is resonant in the cavity. The equilibrium state equation of the fundamental mode is given by:
where $T_1$ is the power transmissivity of the ITM, $E_{\rm in}$ and $E_{00}$ are the electric field amplitudes of the input light and the fundamental mode inside the cavity, $r_{00}$ is the amplitude reflectivity of the fundamental mode in one round trip inside the cavity, which is given by: where $T_2$ is the power transmissivity of the ETM, and $\delta$ denotes the total optical loss of the fundamental mode at non-degenerate conditions. In the cavity described in the paper the main loss contributions are optical scattering and absorption in the test mass high reflectivity coatings. Clipping loss is negligible as the spot size of the fundamental mode is much smaller than the test mass diameter. We also know that the relationship between $r_{00}$ and the half-life period $\tau$ in the ring-down measurement is:Combining Eqs. (4), (6) and (7), the finesse of the cavity can be expressed as:
In Section 3.3, the average finesse in non-degenerate conditions was calculated to be 15,300. To calculate the optical loss at non-degenerate conditions, we use $\mathcal {F}$ = 15,300 and the nominal transmittivity values of $T_1$ = 200 ppm and $T_2$ = 20 ppm in Eq. (8). The optical loss is then calculated to be 190 ppm for non-degenerate cavity.
A.2 Finesse at degeneracy case
When the RoC of ITM is tuned with the CO$_2$ laser, light scattered from the fundamental mode to the higher order mode can be resonantly enhanced, increasing loss coupling. Assuming the fundamental mode is degenerate with a ${\rm TEM}_{n,\,m}$ mode, similar to what is expressed in [33], the equilibrium state equations of the light fields resonating inside the cavity are:
where $E_{00}$ and $E_{nm}$ are the electric field amplitudes of the fundamental and degenerate higher order mode, respectively, $r_{00}$ and $r_{nm}$ are the amplitude reflectivities of these two modes, respectively, in one round trip of the cavity, $\beta$ is the field coupling coefficient between the degenerate fundamental and higher order modes. The amplitudes of the fundamental and degenerate modes can be derived by solving Eqs. (9) and (10) and expressed as:According to Eqs. (11) and (12), the effective cavity reflectivity in the degenerate conditions is:
Like Eq. (8), the finesse in the degenerate conditions can be expressed as:
Substituting Eq. (13) into (14), the finesse in the degenerate condition is shown as a function of high order mode round trip reflectivity $r_{nm}$ and coupling coefficient $\beta$ between the fundamental and high order optical mode. Figure 8 presents a plot of $\mathcal {F}^{\prime }$ as a function of $r_{nm}$ at four different values of $\beta$. From this figure we note that the finesse in degenerate conditions drops with higher coupling coefficient between the two degenerate modes. The curves with different coupling coefficients converge to the same finesse value 15,300 when $r_{nm}=r_{00}$.
Furthermore, we can calculate the total optical loss at degeneracy. Like Eq. (8), we can express the finesse at degeneracy as:
where $\delta ^{\prime }$ denotes the total optical loss at degeneracy. According to Eq. (15), the minimum finesse in our measurement is 10,950, corresponding to a total optical loss $\delta ^{\prime }$=353 ppm at this degenerate condition. This optical loss is about 200% of the average loss at non-degenerate conditions. The excess optical loss at degeneracy is mainly contributed by the clipping loss due to finite mirror size and the scattering and absorption loss of the mirror coatings for the higher order mode. For instance, if the degenerate high order optical mode is resonant off the center of the mirrors, the clipping loss will increase. In addition, the coating is not well characterized in the outer $\sim$10 mm of the mirror and extra loss in this region can explain the observed excess loss of the high order mode. In future work, we can analyze the potential sources of the optical loss. More specifically, we can implement simultaneous monitoring of the mode shape from the camera which allows us to calculate the clipping loss based on the beam location on the mirror and the transverse orders of the higher order mode. This method can also indicate if the mirror coating is well characterized.Note that according to Eqs. (11) and (12) the relative power buildup of the cavity will significantly drop during modal frequency degeneracy. This indicates that modal frequency degeneracy can largely increase the shot noise in gravitational wave detection. As previously mentioned in Discussion, combining the power buildup and cavity ring-down measurements would provide detailed analysis of the cavity loss, enabling accurate modelling of the interferometer to predict the detection sensitivity of gravitational-wave detectors.
Funding
Australian Research Council Centre of Excellence for Gravitational Wave Discovery.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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