A Technique for In-situ Measurement of Free Spectral Range and Transverse Mode Spacing of Optical Cavities
Length and g-factor are fundamental parameters that characterize optical cavities. We developed a technique to measure these parameters in-situ by determining the frequency spacing between the resonances of fundamental and spatial modes of an optical cavity. Two laser beams are injected into the cavity, and their relative frequency is scanned by a phase-lock loop, while the cavity is locked to either laser. The measurement of the amplitude of their beat note in transmission reveals the resonances of the longitudinal and the transverse modes of the cavity and their spacing. This method proves particularly useful to characterize complex optical systems, including very long and/or coupled optical cavities, as in gravitational wave interferometers. This technique and the results of its application to the coupled cavities of a 40-meter-long gravitational wave interferometer prototype are here presented.
LIGO Laboratory, MS 100-36, California Institute of Technology, Pasadena, California, 91125, USA
Università degli Studi di Siena, Dipartimento di Fisica, Via Roma 56, 53100 Siena, Italy
Centre for Gravitational Physics, Department of Quantum Science, The Australian National University, Canberra, Australian Capital Territory 0200, Australia
Corresponding author: email@example.com
120.3180, 120.2230, 120.3940, 350.1270
The absolute length and mirror curvatures are defining parameters of an optical cavity. Together these quantities uniquely determine the Free Spectral Range (FSR) and the Transverse Mode Spacing (TMS): respectively the frequency spacing between different resonances of the cavity’s longitudinal modes and the frequency spacing between the resonances of transverse and longitudinal modes.
FSR and TMS are usually accurately specified and it is thus very important to measure them with great precision. Over time, monitoring cavity length and mirror curvatures can track changes affecting the cavity geometry. For instance, variations of the cavity length may reveal drifts of the mirrors’ positions caused by ground displacement ; changes of the mirrors’ curvatures can be evidence of deformations due to thermal effects . Precise measurements of FSR and TMS can also allow the modeling of the cavity reflected and transmitted power by predicting the mode distribution around the cavity working point.
Measuring these parameters proves particularly challenging in long and complex optical cavities, such as those in gravitational wave interferometers [3, 4, 5, 6]. Precisely characterizing the optical cavities in these detectors is crucial to achieve their best sensitivity. Often three to five or more coupled optical cavities are controlled simultaneously through the sensing of RF phase modulation sidebands added to the main laser. Resonance or anti-resonance conditions of the sidebands in each part of the interferometer must be ensured at all times for optimal decoupling among the degrees of freedom . When the RF modulation frequencies and the cavity lengths do not match, controlling the interferometer becomes more difficult. Also, a mismatch of more than a few centimeters in the arm cavities, or 1 mm in the recycling cavities may reduce the sensitivity of the detector. Noise from the laser and the RF modulator may then enter the interferometer’s auxiliary degrees of freedom and leak into the gravitational wave channel by cross-couplings [8, 9, 10].
Ye  showed that, in principle, sub-wavelength length measurement precision could be obtained by using a femtosecond laser. However this scheme is difficult to implement in situations where a dedicated ultra-short pulsed laser system may not bet readily available. Several different approaches have been tried in the past. In what was probably the simplest, Rakhmanov et al.  measured the length of a cavity with a precision of 4 mm by an optical vernier obtained by swinging the end mirror. In a later experiment, Rakhmanov et al.  measured the length of a 4 km cavity with 80 m precision by measuring the cavity’s frequency response by frequency modulating the laser. In a similar way, Uehara and Ueda  measured the radius of curvature of the end mirror of a plano-concave cavity. Additionally, Araya et al. , following DeVoe and Brewer , estimated the length of a 300 meter cavity with a relative precision of by simultaneously locking to the cavity the laser’s carrier and a phase-modulation sideband.
These techniques are difficult to scale and adapt to systems of very different lengths, or included in complex, coupled-cavity configurations. For example, frequency modulation techniques cannot be used in short gravitational wave interferometers since any modulation of the laser at frequencies near the cavity FSR would be suppressed by the input mode cleaner cavity. Tuning the sidebands’ frequency as in Araya’s technique, would not be possible in systems including additional input cavities.
To circumvent these limitations, we developed an alternative interferometric technique to measure the FSR and TMS of optical cavities with a larger range of lengths or in complex optical setups. We then tested this technique on the optical cavities forming the LIGO 40 m gravitational wave detector prototype at the California Institute of Technology. In the following, we describe this technique and present the results obtained in our experiments.
1 Principles of the Technique
The FSR of an optical cavity determines the frequency spacing between resonances of any given pair of cavity longitudinal modes. It is defined as :
where is the cavity length and the speed of light. This definition can then be used to infer the cavity length from the direct measurement of the FSR. In a similar way, the mirrors’ curvatures are inferred from the measurement of the TMS and the estimated cavity length.
Each Hermite-Gaussian mode is characterized by a different Guoy phase determining its specific resonant frequency in the cavity . Because of this phase, the set of resonances of a generic TEM mode is shifted from the resonances of the fundamental TEM mode by an integer multiple of the so-called Transverse or Spatial Mode Spacing. For a linear cavity this is defined as
where and are the g-parameters of the mirrors, with , representing their respective radii of curvature, and the absolute length of the cavity. The product of the g-parameters is often referred to as the cavity g-factor:
It follows from (2) that if the cavity mirrors are astigmatic, the resonances of complimentary modes, TEM and TEM are split. If we assume, for simplicity, that axes of the astigmatism for the two mirrors are aligned, different g-factors and can be associated with each of the two transverse spatial directions and , respectively111For more general astigmatic cavity cases, see .. In case of astigmatic cavities, the definition of transverse mode spacing is then generalized as
where , , and , represent the radius of curvature of the i-th mirror of the cavity in the and direction, respectively.
With this definition, the direct measurement of the TMS can then be used to estimate the cavity g-factor.
1.1 Measurement technique
The technique presented in this article determines the FSR and the TMS of a cavity from the resonances appearing in transmission as the laser frequency is scanned. Two lasers are used for the measurement: the first, serving as a master laser, is set to resonate in the cavity in its TEM mode; the second, phase-locked to the first, is held at an arbitrary offset frequency set by a local oscillator (LO) in the loop . The phase-locked loop (PLL) ensures that the relative frequency of the two lasers remains constant.
At first, the optical cavity under test is locked to the fundamental mode of the master laser by using the Pound-Drever-Hall technique . The slave beam, after being appropriately mode matched, is injected into the cavity together with the main beam. At the cavity transmission, the two beams interfere producing a beat note at their differential frequency as set by the PLL’s LO frequency. Finally, the cavity resonance profile is measured by tracking the beat note’s amplitude as a function of the LO frequency.
1.2 Cavity absolute length measurement
As the frequency of the PLL’s LO is swept, a resonance peak appears in transmission every time the relative frequency of the slave laser reaches a multiple of the cavity FSR. The FSR is then estimated by a least squares fit of this set of measured resonant frequencies by the linear function , where is the resonance’s order.
1.3 Cavity g-factor measurement
Following the FSR measurement, the TMS is measured by coupling the laser’s fundamental mode into both the TEM cavity mode and the TEM or TEM spatial modes. This is obtained by introducing a small misalignment between the input beam and the cavity axis [19, Sec. II]. In particular, the coupling into the cavity’s TEM mode is obtained by a tilt of the cavity axis in yaw; a coupling into the cavity’s TEM mode is obtained by a tilt in pitch.
The measurement begins with the relative frequency of the two lasers set to a resonant frequency equal to an integer multiple of the cavity FSR: . Then, the frequency of the auxiliary laser is scanned, while the cavity is held locked to the main laser. As their relative frequency equals the cavity’s TMS, the auxiliary laser resonates in the cavity’s TEM (or TEM) mode and is transmitted by the cavity. The interference between the main beam’s TEM mode and the auxiliary beam’s TEM (or TEM) mode produces a beat note on a photodetector at a frequency . The TMS is then measured by the frequency of the corresponding peak.
Both the TEM and TEM modes have antisymmetric phases around the beam axis. As a result, their interference with the TEM mode has zero net power when integrated over the photodetector’s transverse plane. To detect the beat note it is necessary to break the symmetry of the beam spot just before the photodetector. This was done by partially clipping the beam with a razor blade in front of the photodetector but could also be done with a broadband quadrant photodetector.
2 Characterization of the arm cavities of a gravitational wave interferometer
This technique was tested on the arm cavities of the Caltech 40 m prototype, a testbed facility for the Laser Interferometer Gravitational-wave Observatory (LIGO) . For the experiment, the interferometer was set in a dual-recycled Fabry-Perot Michelson configuration. In this setup, the two (40 m long) Fabry-Perot cavities, are connected in a Michelson configuration. Similar to the Advanced LIGO interferometers , the beam splitter’s symmetric and anti-symmetric outputs are coupled to the so-called recycling cavities: the Power Recycling Cavity (PRC) at the symmetric port and the Signal Recycling Cavity (SRC) at the anti-symmetric port. All the cavity mirrors are suspended and hang on wires as simple pendulums, for seismic noise isolation.
Figure 1 shows the optical layout of this experiment. The main beam illuminating the interferometer was provided by a Non-Planar-Ring-Oscillator laser (NPRO), amplified by a Master Oscillator Power Amplifier (MOPA) up to about 2 W at the time of the experiment . Dedicated frequency and power pre-stabilization subsystems were enabled on the laser. After these stages, phase modulation sidebands were added to the beam by electro-optic modulators (EOM) for cavity locking. Finally, prior to entering the interferometer, the laser beam passed through a 13-meter-long triangular mode cleaning cavity. The PM sideband frequencies are chosen to coincide with one of the FSRs of the mode cleaning cavity.
The auxiliary beam was injected into the anti-symmetric port of the interferometer, through the signal recycling cavity’s mirror (SRM). This solution allowed the beam to bypass the input mode cleaner cavity.
The auxiliary beam was provided by a 700 mW NPRO, located outside the interferometer’s vacuum envelope, on the same optics table hosting the dark port’s optical setup. A simple flipper mirror was then used to enable the auxiliary beam to enter the interferometer at the time of the measurement.
A phase-lock optical system was set up on the main laser table by interfering pick-off beams from the two lasers on an RF photodiode (Thorlabs PDA255 or New Focus 1611). The PLL ensured that the auxiliary laser’s frequency followed that of the main laser with an arbitrary tuneable offset, by suppressing the relative phase fluctuations between the two lasers.
3 Measurement results
The FSR and TMS were measured separately. In both measurements, the two laser beams were resonated only in the cavity to be examined by misaligning the other parts of the interferometer. The cavity was then locked to the main laser by the Pound-Drever-Hall (PDH) technique .
A computer controlled the PLL’s local oscillator via a GPIB I/O interface. A script swept the LO frequency and monitored a spectrum analyzer connected to the PD in transmission. The power of the PD signal at the LO frequency was recorded at each step of the frequency scan.
3.1 Arm length measurement
The arm cavity to be measured was first locked to the main laser by controlling the end mirror with electromagnetic actuators. Then the auxiliary laser was injected into the interferometer. The PLL frequency was scanned by 20 MHz, a range corresponding to 5 cavity FSRs, by sweeping the auxiliary laser’s frequency first below and then above the main laser’s frequency. The data obtained recording the power of the PD signal versus the PLL frequency was plotted as in Fig. 2. Resonance peaks were observed at multiples of the cavity FSR. The frequency of each peak was determined by a nonlinear least-squares fit of the PD signal amplitude with the function
where , and are additional fit parameters.
By fitting 10 resonances of the X arm and 5 resonances of the Y arm, was estimated with an error of and , respectively. The residuals from the fitting show that these errors are statistical and not systematic.
The cavity FSR was estimated by a linear least squares fit of these resonant frequencies (right plot of Fig. 2). Table 1 shows the result of the fitting, as well as the derived values of the cavity length. The FSRs were estimated with statistical standard errors of 0.6 Hz and 3 Hz in the X and Y arm, respectively. Accordingly, the cavity lengths were determined with a precision of and , over an absolute length of 38.5 m. However, the residuals in the linear fitting show deviations which are likely due to causes other than statistical fluctuations. In fact, deviations were observed at the level of Hz, equivalent to a displacement of mm (see plotted residuals in Fig. 2). These fluctuations may be due to drift of the cavity alignment or actual length changes during the course of the measurement due to the thermal expansion of the concrete slab.
3.2 Arm cavity g-factor
The arm cavity g-factor was measured by introducing a small misalignment in the input beam’s axis, first in pitch, then in yaw in order to detect the resonances of the TEM mode and the TEM mode, respectively. The measurement started by locking the cavity to the main laser’s fundamental mode. The amplitude of the PD signal was recorded and plotted against the PLL frequency.
The frequency scan revealed additional resonance peaks due to higher order modes. The first peak was produced by the TEM mode resonance; the second corresponded to the spatial mode resonance.
The TMS was determined by measuring the frequencies of the TEM and TEM modes and then comparing them with the frequency of the TEM mode.
The TEM (or TEM) peak was identified by matching it with its expected location as calculated from the mirror’s nominal radius of curvature. The fitting of the TEM and TEM resonances allowed us to estimate with errors of 15 and 46 Hz. The fitting residuals indicate that statistical fluctuations are responsible for these deviations.
It should be noted that the g-factor measurements presented here and the values obtained by direct measurements of the mirrors’ radii of curvature do not agree. Phase map measurements of the mirrors obtained by a Fizeau interferometer estimated the radii of curvature of the end and input mirrors of the X and Y arms to be 57.57, 57.68, 7280, and 7210 (all in meters), respectively. These numbers, combined with the measured cavity lengths give us g-factors of larger than the measured values. In general, the individual radii of curvature of mirrors in a two-mirror cavity cannot be directly derived from measured g-factors. However, we can still learn something about the mirror curvatures if we take into account that typically phase map measurements of flat mirrors are more accurate than those of highly curved mirrors [23, 24]. For instance, phase maps measurements of the nominally-flat mirrors in use in our lab have estimated radii of curvature ranging between and . Finite radii like these affect the g-factors by less than , compared to an ideally flat mirror. For this reason, most likely the observed discrepancies are due to the curvature of the end mirrors. Astigmatism in the end mirrors could explain the TEM/TEM mode splitting and justify the difference in the g-factors by . In particular, by assuming perfectly-flat input mirrors, our measurements could be explained by an astigmatism of 1 m over end mirrors with radii of curvature of 5657 m.
The two astigmatic modes were observed in the vertical and horizontal main axes in both arms cavities, rather than along arbitrary axes in each of them. This might occur if the astigmatism was related to the wedge angle on the cavity mirrors. Both the input and the output mirrors have wedge angles of 1 deg and 2.5 deg, respectively, and they were both installed with the angles on the horizontal axis. This non-degeneracy of the cavity’s TEM/TEM resonances was also confirmed by an experiment in which the cavity was locked to either of these modes. By misaligning the cavity in pitch or yaw, it was possible to lock the cavity in either the TEM mode or the TEM but never on both at the same time. This was evidence that the modes were indeed separated by much more than the cavity half line-width of 1.6 kHz.
3.3 Power recycling cavity
Determining the length of the other cavities, such as the recycling cavity is also very important in order to achieve a stable control of the interferometer. For this reason, we tested this technique on the Power Recycling Cavity (PRC). This cavity is formed by the Power Recycling Mirror (PRM) at one end and at the other end by the Michelson “compound mirror”, an effective mirror defined by the beam splitter and the input mirrors (ITM) of the arm cavities. The asymmetry in this short Michelson causes it to have a frequency-dependent reflectivity .
For this measurement, the auxiliary beam was introduced through a misaligned SRM. The transmitted beam was detected at the reflection port of the interferometer using the output of the Faraday isolator. The Michelson and the power recycling cavity were locked to the main laser beam, by actuating on the beam splitter and the PRM. Because the PRC length is much shorter than the arms’, the laser frequency had to be scanned by several hundreds of megahertz in order to explore one cavity FSR. This was easily accomplished thanks to the large dynamic range of the PLL, in contrast to previous length measurement techniques, which had been limited to only very long cavities with smaller FSRs.
The measured profile of the transmitted power showed a complex pattern, as expected from the frequency-dependent response of the Michelson. The data were fit with an analytical model for the cavity transmissivity (details in Appendix A), obtaining an estimate of the cavity absolute length and of the Michelson differential length (asymmetry) with a precision of 2 mm and 3 mm, respectively (see Table 1).
3.4 Measurement Precision
The precision of the FSR and TMS measurements listed in Table 1 was determined by the statistical errors in the parameters of the non-linear least squares fit of the data. The difference between the two arms was due to a larger number of data points measured for the X arm.
In principle, since the cavity is locked to the main laser, the best precision of the FSR and TMS measurements achievable by this technique is set by the laser frequency and intensity noise and by the noise in the PDH loop. In reality, fluctuations of the cavity stored power due to oscillations of the cavity alignment may affect the beat note’s amplitude and thus increase the errors in the fit parameters. In our case, since the angular degrees of freedom of the cavities were uncontrolled, the mirrors’ angular motion was likely the main cause of the measured fluctuations in the beat note signal.
In the future the technique could be greatly improved by measuring the phase of the beat note rather than its amplitude. The cavity resonance could then be measured with more accuracy and precision by identifying the frequency at which the beat note’s phase flips by 180 degrees. This would render the measurement intrinsically immune to unwanted fluctuations of the beam power. For instance the phase could be detected by measuring the transfer function between the PLL’s local oscillator signal and the beat note.
We demonstrated a new interferometric technique to measure the free spectral range and the transverse mode spacing of optical cavities of various lengths. It allowed measurements of the length and the g-factor of a Fabry-Perot cavity with a precision of 1 ppm and 10 ppm, respectively.
Compared to previous methods, this technique can be applied to optical cavities of lengths ranging from a few meters to several kilometers. With our experiment we proved that the technique is compatible with a complex optical system, comprising coupled cavities. Not requiring structural modifications to the optical system on which it is applied, it provides a convenient option in circumstances in which an in-situ and non-invasive measurement tool for cavity length and g-factor is needed.
We thank Yoichi Aso, Robert Ward and Hiro Yamamoto for illuminating discussions on this technique. We also thank the rest of the 40 m team for building and maintaining the interferometer during the course of this work.
The LIGO Observatories were constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation under cooperative agreement PHY-0757058. This paper has been assigned a LIGO Document Number of P1200048.
Appendix A PRC Transmissivity
When the primary laser is resonant, the transmissivity of the Power Recycling Cavity can be written as a function of the PLL’s local oscillator frequency in the following way:
where is the PRM transmittance; the ITM reflectance; the asymmetry (i.e the Michelson’s differential length); and the effective length of the power recycling cavity defined as where is the distance between the beam splitter (BS) and the PRM, and and are the lengths of the Michelson arms. The amplitude of the beat note in transmission is then proportional to .
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