## Abstract

It has become a significant challenge to accurately characterise the properties of recently developed very high finesse optical resonators (*F*>10^{6}). A similar challenge is encountered when trying to measure the properties of cavities in which either the probing laser or the cavity length is intrinsically unstable. We demonstrate in this article the means by which the finesse, mode-matching, free spectral range, mirror transmissions and dispersion may be measured easily and accurately even when the laser or cavity has a relatively poor intrinsic frequency stability.

©2009 Optical Society of America

In recent times researchers have been able to construct optical resonators with finesses of up to 10 million [1], with commercial suppliers able to provide conventional Fabry-Perot cavities with finesses approaching 1 million [2, 3]. This poses two substantial challenges to a researcher who wishes to directlymeasure the bandwidth (*Q*~10^{11},*BW*~1 kHz) of these potentially very narrow resonances in the frequency domain: first, the result of a frequency scanned measurement is the convolution of the laser linewidth with the cavity bandwidth, which results in a measured bandwidth that appears to be broader than the underlying true bandwidth. Second, the effect of frequency instability in the probing laser causes fluctuations in the measured transmission and reflection coefficient of the cavity. Examples of both these effects are shown in Fig. 1 where we have scanned a resonance in a moderately high finesse cavity with a high quality stabilized commercial Ti:sapphire laser (Coherent MBR-110). In cavities with suspended mirrors, such as those used by the gravitational wave detection community [4], or with high frequency drift rates or thermal or vibrational sensitivity [1] one encounters a similar challenge because fluctuations in the cavity resonant frequency during themeasurement lead to distortions of the observed lineshape.

The usual method to overcome this challenge is to use a time domain method to measure the energy decay of the transmitted field by quickly switching either the frequency or amplitude of the incident laser as it comes into resonance with a mode of the cavity [4, 5, 6]. This technique can yield accurate measurements of the total optical losses but cannot separately determine losses associated with transmission of the mirrors as against scattering or absorption losses. Neither can this type of approach deliver information on any other characteristic of the cavity, or on the quality of the coupling between the laser beam and the cavity mode. An alternative time domain technique using a swept frequency source and monitoring the reflected field can yield additional information on the input coupling of the cavity because of the interference between the directly reflected laser field and the leaking energy from the cavity mode [7, 1]. However, all these time domain approaches only yield the energy absorption, which is only one term in the determination of the width of the resonance [5]. For example, scattering of energy out of the initial spatial mode-shape may lead to broadening of the mode bandwidth but if the scattered energy is still guided in the cavity it may not lead to a reduction in circulating energy. In the past a very impressive complete characterisation of one high finesse cavity has been performed [8] although in that particular application the cavity length was very short (~40*µ*m) so that the challenges associated with narrow bandwidthswas essentially circumvented. More recently advanced techniques using frequency comb technology have enabled broadband characterization of the dispersion of cavities [9] or both wavelength dependent loss and phase shifts of cavities [10] although to date these techniques have not been deployed to generate complete cavity information.

In contrast, we propose a simple technique to directly measure all the defining characteristics of an optical cavity: this approach can be performed even when the intrinsic frequency instability of the cavity or of the probing laser exceeds the width of the resonant mode itself. The fundamental approach is to probe the cavity with an incident laser field which has been phase-modulated at a frequency of the order of the free spectral range of the cavity. We then simply measure the average transmitted and reflected power as a function of this modulation frequency while the carrier is frequency locked to a resonance of the cavity. Our approach shares some of the features of a couple of earlier techniques: Bondu and Debieu [11] theoretically suggested an optical transfer function approach based around perturbing a Pound-Drever-Hall frequency locking system with some additional phase modulation and then measuring the response of the locking system. This was implemented experimentally by the Virgo collaboration and shown to generate stable values for many of the defining characteristics of the cavity [4]. The challenge with this technique is that the PDH locking system must have a bandwidth larger than the fundamental mode spacing of the cavity, which is achievable for the long cavities used in the gravitational wave detection community, but will not be the case in reference cavities used for frequency stabilization experiments. In a separate experiment, Slagmolen et al [12] made use of a network analyzer to perform synchronous detection of reflected amplitude modulation at frequencies near the inter-mode spacing of the cavity. In contrast to both of these previous techniques we avoid the use of synchronous detection, which simplifies the measurement substantially. Furthermore, the technique delivers additional information over those techniques which allows an immediate complete characterisation of the cavity without the need for auxiliary measurements. For example, the measurement yields separate estimates of the transmission, the losses associated with scattering and absorption as well as the mode-matching of the input light field.

In order to properly define the parameters to be measured we derive the characteristics of the cavity from first principles following initially the approach of Siegman [13]. In the case of a monochromatic incident laser field, an expression for the steady-state field circulating in the cavity can be found by requiring the circulating, *E*̃_{circ} to remain unchanged after one full round trip:

where *g*̃_{rt} is the full round trip complex transfer function that the light experiences in the cavity. Without loss of generality we will describe the cavity in terms of only three loss parameters: the power transmission at the front mirror, *T*
_{1}, the power transmission at the back mirror, *T*
_{2}, and a generic loss term, *δ*
_{0}, which includes the effect of all other non-useful losses that the circulating field encounters (including that associated with scattering and absorption on the mirrors). The sum of all the losses (transmission through both mirrors and other losses) will be termed *δ _{T}*. In this first description we assume that the spatial mode-matching of the inputmode and the cavity mode is perfect in order to maintain clarity (we will relax this constraint in the next section). In the limit of small losses we can thus write

*g*̃

_{rt}=(1-

*T*

_{1}/2-

*T*

_{2}/2-

*δ*

_{0}/2)exp[2

*πjL*/

_{opt}*λ*

_{0}]=(1-

*δ*/2)exp[2

_{T}*π jL*/

_{opt}*λ*

_{0}] where

*L*is the full optical round trip length and

_{opt}*»*

_{0}is the free space wavelength.

Solving Eq. 1 in terms of these losses we can write the fields on resonance (i.e.*L _{opt}*/(2

*λ*

_{0}) is an integer) as:

which allows us to write the four on-resonance measurable parameters of a cavity: its Free Spectral Range (FSR) (frequency spacing of fundamental modes), Finesse *𝓕* (the ratio of the FSR to the bandwidth of amode), Contrast,C (ratio of the absorbed power to the incident power on resonance), and transmitted power on resonance, *P*
_{trans}=|*E*̃_{trans}|^{2}:

where we define Λ^{2} as the fractional power lost to non-useful losses (scattering and absorption). Given a measurement of the four experimental parameters on the left hand side of Eqs. 4–7, we can solve to derive unique values for *L _{opt}, T*

_{1},

*T*

_{2}and

*δ*: the four defining characteristics of the individual optical elements and the cavity. In circumstances where the components exhibit some Group Delay Dispersion (GDD) then one will find that

_{T}*L*will be frequency dependent. As we will show below, careful measurement of this effect (frequency dependent FSR) one can obtain a measure of the cavity dispersion as well [14, 15].

_{opt}We begin by considering an incident phase modulated input laser field of the form:

$$+{J}_{1}\left(\varphi \right)\left({e}^{j\left({\omega}_{c}+{\omega}_{m}\right)t}-{e}^{j\left({\omega}_{c}-{\omega}_{m}\right)t}\right)$$

$$+{J}_{2}\left(\varphi \right)\left({e}^{j\left({\omega}_{c}+2{\omega}_{m}\right)t}-{e}^{j\left({\omega}_{c}-2{\omega}_{m}\right)t}\right)$$

where *ω _{c}* is the (angular) carrier frequency,

*ω*is (angular) phase modulation frequency and the phase modulation index is

_{m}*ϕ*. In Eq. 9 we have expanded Eq. 8 up to the second order sidebands. In the experiment below we have used phase modulations of the order of 1 radian. In order to properly fit the high resolution measurements it is necessary to include the effect of these second order sidebands.

In our technique we tune *ω _{m}*/(2

*π*) around the

*FSR*, while the carrier has been frequency locked to a mode using the conventional technique (described in more detail below). Two photodetectors monitor the transmitted and reflected fields from the cavity. In our proposal we detect just the stationary (dc) terms of these expressions (in contrast to the synchronous detection approaches reported in the earlier experiments [11, 12, 4]). In the limit where the losses are small and the field is perfectly mode-matched, we can write an expression for the average transmitted and reflected power as follows:

where *S* expresses the lineshape of the resonance:

and where *ϕ _{s}*=2

*π*[

*f*-

_{m}*FSR*]/

*FSR*is the angular frequency detuning normalized in terms of the FSR.We see that the form of the lineshape function is that of a constant plus two Lorentzian functions: the Lorentzian associated with the first sideband has a bandwidth equal to that of the mode resonance, while the second Lorentzian exhibits a bandwidth which is half that of the mode resonance.

An example of Eq. 10 and 11, together with their limits, is plotted on Fig. 2. When the modulation frequency equals the free spectral range then the shape function, *S*, has a value of one, and the transmittance and reflectance is equivalent to the simple monochromatic measurement described earlier (Eqs. 2 and 6 respectively) where we interpret the full power of the phase modulated signal as the incident power. However, when the modulation sidebands are well detuned from the resonances, the shape function approaches a value of *J*
_{0}(*ϕ*)^{2} and we essentially transmit just the carrier of the signal, while on the other hand, for reflection we obtain:

$$=1-{J}_{0}^{2}\left(\varphi \right)\left({T}^{2}+{\Lambda}^{2}\right)$$

One sees superficially that the shape of the transmission and reflection coefficient displayed in Fig. 2 is similar to that which one would obtain from measuring a frequency stable resonance with an ideal monochromatic tunable laser; however, the benefit with this new method is that argument of the transfer function is not the optical frequency detuning, but the difference between the modulation frequency and the free spectral range of the cavity. Here we see the twin advantages of the technique: the modulation frequency can be determined with radio-frequency accuracy by using a high quality modulation source, which is not the case for the optical frequency of a free running laser, while simultaneously the absolute frequency instability of the free spectral range of a mechanically unstable cavity will be substantially lower than the frequency stability of a single mode. In particular, the absolute frequency noise in the FSR is lower than the frequency fluctuations of a mode by the ratio of the mode-frequency to the FSR. Since this later number is of the order of 105 for a typical cavity we can see that this type of measurement has the potential for much higher accuracy than a simple measurement of a single mode.

In the absence of mode-matching effects, a measurement of the type shown in Fig. 2 yields all of the parameters necessary to characterise the cavity. A fit to the fractional transmitted power yields measures of the (i) total losses of the cavity, (ii) the phase modulation index, and (iii) the product of the transmissions of the two mirrors. The individual mirror transmissions and the dissipative losses can be separately determined by making use of the reflection measurement on just a single port of the cavity. In addition, the reflection measurement yields independent measures of the total loss and phase modulation index. These twin measurements also deliver accurate estimates of the free spectral range of the cavity by finding the modulation frequency which maximizes the transmission and minimizes the reflection. By repeating this process as a function of wavelength we can build up a picture of the dispersion of the cavity.

In general, when attempting to measure the characteristics of the cavity the complex mode shape of the input field may not match that of the eigenmode of the cavity [16, 17]. In addition, there may be frequency components of the input beam which are not resonant with the cavity. To take account of these effects we include an additional efficiency degradation in the measured fields we presented earlier in Eqs. 11 and 10:

$$=\left(1-S\epsilon \right)+\epsilon {\Gamma}^{2}S$$

and

where *ε* is a measure of the fraction of the incident laser power which matches the spatial shape of the cavity mode and is in the frequency component which is being tuned over the cavity. As we will show below, our measurement technique allow determination of the numerical value of this factor and hence estimate the quality of the mode-matching without the need to measure the amount of power seen in other higher order modes [4, 16, 17]. In order to obtain separate estimates of the mirror transmission, dissipative losses and the mode-matching fraction from a single-portmeasurementwe have assumed that themirrors have identical characteristics (which is reasonable given they came from the same mirror coating run). It is only necessary to make this additional assumption if one wishes to separately estimate the mode-matching factor. Alternatively, one could make an additionalmeasurementwith an incident field on the second port of the resonator.

By way of demonstration of the technique we have measured the properties of a vibrationimmune, high-finesse, low dispersion and ultra-low expansion (ULE) cavity using a Titanium: Sapphire tunable laser. The ULE cavity was supported from four special points by rubber spheres which are intended to minimize the acoustic sensitivity. The entire cavity is placed in vacuum and is temperature controlled at the level of 0.3mK over time scales of a few hours. The intrinsic frequency instability of the probing laser was of the order of 3–500 kHz over 1s dominated by a few strong modulation peaks at low frequency. By scanning this laser over one of the resonances of the cavity we can obtain an estimate of the finesse (Fig. 1) as 23,000 although it is clear that the quality of the data is limited by the frequency instability of the Ti:S laser.

To implement the technique described here we passed the Ti:S laser through a high frequency electro-optic modulator to generate ~1.5GHz sidebands with a phase modulation index of approximately 1. Additional modulation sidebands at 1MHz were applied to lock the carrier of the cavity-stabilized Ti:S laser to the cavity using the standard Pound-Drever-Hall technique [18] with fast corrections (control bandwidth of ~80 kHz) sent to an acousto-optic modulator that shifted the laser output frequency while slow corrections were used to tune the laser’s internal reference cavity with a bandwidth of ~1 kHz. The transmission and reflection data obtained for a cavity mode at 779.96772 nm as a function of modulation frequency is shown in Fig.3.

It is immediately apparent that the quality of the data is very substantially superior to that obtained from a direct scan of the optical frequency (Fig. 1). It should be noted that the quality of the lock to centre of resonance can be poor as we measure the frequency instability of the FSR and not that of a single mode (an advantage of 10^{5} in this case.) The fits to the transmission and reflection data yield a finesse of 40,100±50 and 38,400±70 and a phasemodulation index of 0.97±.0003 and 0.98±.0004 respectively where all the quoted errors are statistical. The product of mirror transmission and mode-matching losses is 2.59±0.01×10^{-9} which when combined with the reflection measurement yields a mode-matching efficiency of 43%, and a mirror transmission of 78±1 ppm. The mirror transmission was measured by the manufacturer as of the order of 80 ppm which shows the good agreement between our cavity measurement and the mirror values. From these measurements of the total loss (*δ _{T}*) and the transmission one can separately estimate a dissipative loss component (

*δ*

_{0}, from scattering and absorption) of 4±4 ppm where the majority of the error in this case arises from estimated uncertainties in power calibrations of the two measuring photodiodes as well as uncertainty in the transmission and loss measurements themselves.

The FSR was measured as 1518262250±100Hz, which yields a total optical length of 197457625±20nm. By measuring the FSR together with the total loss (*δ _{T}*) of a number of modes over a range of wavelengths we can generate the plot shown as Fig. 4. The change in FSR evident on this plot results from dispersion in the mirror coatings [10, 9, 14, 15] which changes the effective length of the cavity. The solid curve shown on the upper portion of Fig.4 is derived from a theoretical modelling of the excellent low dispersion properties of the coating which has a maximum value of less than 2fs

^{2}across this wavelength range. The solid curve on the lower panel of the plot compares the measured transmission of the mirrors provided by the manufacturer [2] to the measured loss of the cavity at these same wavelengths using our technique. Once again good agreement is seen between the manufacturer [2] mirror data and our measurements of the cavity.

As mentioned earlier, the dissipative loss at ~780 nm is small in comparison with the transmission losses. In these circumstances, further examination of the equations presented earlier indicates that the measurement technique can be simplified even further, which allows the complete elimination of any prior calibration process of the photodiodes used in the measurement. In a cavity with negligible dissipative losses (i.e. Λ=*δ*
_{0}~0) the sum of the reflected and transmitted power can be seen to equal to unity (i.e. compare Eqs. 14 and 15) independent of any imperfection in mode-matching. Using this information, one can scale the voltage observed on the transmitted photodiode so that when summed to the voltage measured on the reflected signal photodiode no resonant dip is observed when the modulation frequency is tuned to the free-spectral range. Furthermore, the sum of these two voltage curves then sets the unity reflection value in terms of observed voltage on the reflection photodiodewhich allows normalization of this reflected signal. One then proceeds with the analysis as shown above to generate values for the cavity parameters but where the complexity and potential error of any calibration processes have been circumvented. We performed this for the 24 measurements on Fig. 4 to obtain mode-matching and mirror transmission factors without any need for calibration and these were in agreement with the values generated in Fig. 3 for the ~780 nm mode.

In conclusion, we have described a technique that can differentiate between transmission through the mirrors, dissipative losses and mode-matching inefficiency; yielding more cavity parameters than is possible using a simple energy ring-down measurement. We believe this technique will find increasing utility as cavity bandwidths become ever smaller than the frequency stability of the best laser sources. In circumstances where the cavity itself is unstable (such as when the mirrors are suspended or when the cavity has high thermal or vibration sensitivity) this technique still allows accurate measurement of cavity parameters as the carrier will circumvent the associated frequency modulations while the free spectral range has a high suppression factor for this length modulation. We have demonstrated the ability to measure a mode bandwidth with a precision of the order of 5% using a laser that has an frequency instability that is more than a factor of 2 larger (worse) than the bandwidth itself.

## Acknowledgments

We thank all members of the Frequency Standards and Metrology Group at UWA for their help and support. In addition, we thank Advanced Thin Films for providing us with the mirror transmission data and the modelled dispersion characteristics. This work was funded by the Australian Research Council. We identify a commercial product in this article merely for completeness and do not imply any endorsement of the product over any other device.

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