1. Introduction

Fabry-Perot interferometers (FPIs) are resonant cavities for light, which uses the resonance condition created by internal interference as a probe of the properties of the light. Traditional FPIs are made from two reflecting surfaces mounted parallel to each other. Light that enters the FPI forms a standing wave when the length between the reflecting surfaces is a half-integer multiple of the wavelength, when it is resonantly enhanced. Otherwise, the light destructively interferes. This property allows for precise measurement and filtering of wavelengths. The utility and precision of these cavities has seen them used in a range of applications, including as precise astro-photonic calibration sources [1], precision sensors [2][3], ultra-high finesse superconducting resonators [4] and for gravitational wave detection [5].

By temporally scanning the resonant condition of the cavity, FPIs are able to probe the mode spectrum of lasers. A laser with a single mode spectrum will show a periodic pattern of Lorentzian peaks separated by the cavity free spectral range, while multimode lasers show numerous, more closely spaced peaks with random separations. This makes FPIs ideal for monitoring the mode profile of lasers, meaning they are widely used in applications where this is important. For example, in cold atom experiments, the lasers used for doppler cooling [6], to create optical dipole traps [7] or for the generation of optical lattices [8,9] are all required to be single mode. If the laser is in multi mode operation this will lead to undesirable heating effects. This means that such experiments often use one or more FPIs as part of their laser locking and/or monitoring setup [10]. However, where the application is simply to monitor whether the laser output is single or multimode, the performance requirements are often much lower than the specifications of the FPI being used.

In traditional FPIs, the cavity is formed with free space between the end mirrors. These schemes are sensitive to external noise, such as vibrations or temperature fluctuations that can change the alignment of the mirrors [11], which couple into the signal and degrade the cavity performance. Free space cavities are also difficult to miniaturise [12]. Another approach is to construct the FPI inside an optical fibre. The mirrors of the FPI are embedded in the dielectric material, either on the end of the fibre or in a splice, in which case the length of the cavity is the length of the fibre. In-fibre based designs are much cheaper to manufacture, and are easier to align [13] than traditional FPIs. They are robust, simple to fabricate, and relatively insensitive to vibrations and pressure changes [11]. They are also easier to miniaturize and integrate into optical systems. Such FPIs are used in applications including elementary quantum networks [14], high temperature strain and flow sensing [15,16] and cavity quantum electrodynamics (CQED) [12].

An alternative device that operates on similar principles is a fibre ring resonator (FRR) [17,18], constructed by connecting two ports of a directional coupler or in-fibre beam-splitter. Two ports of the directional coupler function equivalently to the input and output mirrors of a FPI, with the fibre connecting the other two ports forming the cavity. These versatile devices are commonly employed as sensors, measuring temperature [19], optical wavelength [20], magnetic fields [21], strain [22] and as fiber-optic gyroscopes [23,24]. FRRs have also found uses in numerous other applications, including spectral filtering [25], laser stabilization [26], squeezed light generation [27] and as frequency combs [28].

In this paper, we present a method to incorporate the equivalent of a Fabry-Perot Interferometer as a scanning optical cavity by constructing a fibre ring resonator, and use it to monitor the longitudinal mode profile of a laser. This design is made from inexpensive and readily available materials. By changing the resonance condition of the cavity via varying the temperature of the fibre and then monitoring the cavity spectrum, we can determine whether a laser is single or multimode. We quantify the stability of the cavity by examining how the free spectral range (FSR) changes over time and determining the finesse of the cavity. The cavity is found to be well suited to applications with moderate performance requirement, such as monitoring whether a dipole trap or lattice laser in our metastable helium Bose-Einstein condensate (BEC) experiment [29] remains single mode over time.

2. Cavity theory

The free spectral range (FSR) of a cavity is a measure of the range of frequencies over which the cavity can operate. It is defined in terms of the refractive index ($n$), the length between mirrors ($L$), and the speed of light ($c$) as:

The finesse of a cavity is a measure of the round-trip loss independent of the length. The finesse of the cavity can be experimentally determined by considering the linewidth of the cavity, defined as the full-width half maximum ($\Delta \nu _{FWHM}$) of the Lorentzian peaks formed due to broadening around the resonant frequencies of the cavity. This relates to the finesse as:

As an alternative formulation, given a cavity with a total round trip loss of $1-\rho$, then if $\rho \approx 1$ the Lorentzian finesse is given by:

3. Method

The FRR is formed by connecting two ports of a 99:1 in-fibre beam splitter by splicing the fibre together (see Fig. 1), where the beam splitter takes the place of the mirrors in a standard FPI. This is similar to the in-fibre ring-down setups demonstrated in [30] [31]. The fibre is 47.5(5) cm in length and fabricated from fused-silica ($n = 1.444(1)$). The output signal is captured by an InGaAs photo diode (PD), since the FRR is designed to operate at 1550nm, and is monitored using an oscilloscope. From Eq. 1, the FSR of this cavity is $\Delta \nu _{FSR} = 437(5)$MHz.

 

Fig. 1. Experimental Setup: The cavity was made from a 47.5(5) cm length of single mode smf-28 optical fiber, and monitors a 1550nm laser. A 99:1 in-fibre beam splitter was used to split the beam path between the cavity and a photo diode, which detected the signal of the laser mode. The inset shows a cross section of the peltier sandwich construction used for tuning the fibre cavity. This consists of two 1.7 K/W heat sinks as the outer surface, which remove heat from two Peltier devices, connected in series for both heating and cooling of the cavity. Aluminium blocks were placed between the Peltier devices to form a channel, between which the bare cavity fiber and a thermistor probe are located. Each contact surface was coated with thermal paste to allow for optimal thermal conductivity.

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In a standard free space Fabry-Perot cavity, to scan the cavity resonant frequency (set by the length of the cavity) the length is typically changed using a piezo-actuator. For a ring-fibre set up like ours, this would correspond to changing the length of the fibre, which could be achieved using a linear piezoelectric fibre stretcher to directly change the cavity fibre length [32] or the spacing between two posts that the fibre is wrapped around [33] or by wrapping the cavity fibre around a piezoelectric ring device [34]. An alternative would be to introduce a strain on the fibre, but this is difficult to do in a fast and precise manner. For a ring resonator setup this can also be achieved by incorporating a phase modulator in the cavity [17].

Instead, we opt to change the temperature of the fibre in a controlled and reproducible manner. There are two competing mechanisms that change the resonance condition of the cavity ($L=\lambda q/2nL$, with $\lambda$ the laser wavelength and $q$ an integer) when the temperature is changed. Firstly, the length of the fibre $L$ (and hence the cavity) will change according to the thermal expansion of fused silica, which is $0.55\times 10^{-6}/K$ [35]. Secondly, the refractive index $n$ of the fibre will change at a rate of $\Delta n/\Delta T \approx -8\times 10^{-6}K^{-1}$ [36]. Comparing these effects, the change in $L$ is about an order of magnitude smaller than the change in $n$, and hence $\Delta n$ is the dominant factor for our case. Theoretically for our cavity parameters, to achieve a change of one FSR of the cavity by heating a 5cm length of the fibre (the length of the peltiers used) would require a change of $\approx$2$K$.

To systematically scan the cavity using the change of $n$ with temperature, we constructed a sandwich configuration of two Peltier devices (each with 17W of cooling power), through which $\approx$5cm of the fibre is contained, as shown in Fig. 1. Each Peltier is mounted to a 1.7 K/W heat sink on one side, which improves the temperature response. The configuration of the Peltier devices is set so that both will simultaneously heat or cool the fibre for the same polarity of drive current, which allows for simple heating and cooling of the fibre by changing the direction of the drive current. Two aluminium spacers form a channel for the cavity, with the fibre mounted between these sheets along with a thermistor to monitor the temperature of the system. All internal surfaces are coated with thermal paste to ensure optimal thermal contact.

The thermistor used to monitor the temperature is a YSI 400 Series temperature probe, which records a voltage that is non-linear with the temperature variation. A Wheatstone bridge circuit converts the output of the thermistor to be linear with temperature. The circuit is constructed such that 0 V corresponds to 293.15 K (room temperature), with a conversion rate of 4 K/V for the output. This allows us to monitor the temperature of the fibre in between the two Peltier devices, providing a linear voltage signal proportional to the temperature of the system.

An arbitrary waveform generator (AWG) is used to generate square waves as the drive signal for the Peltier devices. To increase the output of the AWG to the power required for the Peltier devices, the signal is amplified using a linear amplification circuit. An improvement on this would be to incorporate a feedback circuit between the temperature monitor probe and the drive signal, which would increase the stability of the fibre temperature. However, this was not found to be necessary for our application.

4. Results and analysis

A square wave pulse was used to drive the Peltier devices to create the temperature change that varied the length of the fibre cavity. The Peltier devices and attached thermal mass were empircally found to have a thermal response time of approximately 10 seconds. Hence the drive signal was chosen to have a frequency of 100mHz to account for this lag, as shown in Fig. 2. This drive signal created a temperature change in the cavity of $\sim$$\pm 3.5K$, as shown in Fig. 2(b). Due to the slow thermal response time of the device, the temperature response of the system appears to be slightly non-linear. However, over much of the scan range a linear approximation holds well. As we are working over small temperature differences, we expect the change in refractive index $\Delta n$ from the thermal expansion to be linear in temperature change $\Delta T$.

 

Fig. 2. Cavity drive and response. (a) The signal used to drive the Peltier devices for heating and cooling: 100mHz square waves with a peak-to-peak amplitude of 10V and a 0.5V offset. (b) Temperature response of the cavity to this drive signal, as measured by the temperature sensor (see Fig. 1). (c) Optical output of the cavity for the drive signal shown in (a), as measured by the output port photo-diode for a single-mode input 1550nm laser. Periodic peaks appear in the scan with small side peaks. $\Delta _{1,2}$ indicates the first peak pair used in Fig. 3.

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While this drive signal was applied, the output from the PD (monitoring the light output from the cavity) is shown in Fig. 2(c), for a single mode laser operating at a wavelength of 1550nm. The large transmission peaks in the cavity occur when the refractive index times the length of the cavity is a half-integer multiple of the wavelength of the laser. Note that this scan also shows small side peaks, which are due to different polarisation modes supported by the cavity [17], as is discussed in more detail below.

The difference in cavity resonance frequency between the large PD peaks corresponds to one FSR, which is constant. Due to the periodic drive signal causing $\Delta T$ (and hence also $\Delta L$) to have both a positive and negative slope, adjacent peaks within a scan, can arise from three cases. Either the cavity temperature will be the same for both peaks (i.e. $\Delta T=0$ - labeled $\Delta _{1,2}$ in Fig. 2(b)), or a $\Delta T$ that represents an actual change in the optical path length of the cavity (via either increase or decrease of $n$) equating to $\pm$ 1FSR, corresponding to both peaks lying on the same linear slope of the ramps in Fig. 2(b). We label these options $\Delta T = 0$ and $\Delta T =\pm 1FSR$ respectively and plot them for the data shown in Fig. 2(c) in Fig. 3. The data is located in three clear bands, corresponding to $\Delta T = 0, + 1$ and $-1 FSR$, as expected.

 

Fig. 3. Temperature difference between adjacent peaks (see Fig. 2). The horizontal axis is the adjacent peak pair, with the first point indicated in Fig. 2(c) by $\Delta _{1,2}$. The change in temperature is grouped into three categories: 0 temperature changes, meaning the length of the cavity is constant, and +-1 FSR when changing the length by 1FSR. The points in the $\pm 1$FSR regions are centred at $\pm 2.0(2)$K. Errorbars represent $1\sigma$ measurement uncertainties, which are dominated by temperature measurement and cavity peak location uncertainties.

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There are small delays in thermal response of the temperature probe and/or the cavity, which are observed in the cavity peak data not lining up with the temperature scan exactly as expected. To compensate for this, an optimisation parameter was introduced by shifting the data in order to ensure that the frequency difference between each pair of peaks had an FSR that was either 0 or consistent at $\pm 1$FSR. The optimal parameter indicated a shift between the temperature probe and cavity of 50(20)ms meaning that the PD signal has a delayed response compared to the temperature sensor. This observed shift could be due to the silica having a relatively low thermal conductivity and thus the effects of temperature changes on the cavity occurred at different times to the temperature probe, which is located close to the Peltiers, as well as non-uniform heating of the fibre.

Figure 4 shows the cavity PD signal over a small scan range to indicate the peaks and side peaks of the cavity for a single mode laser 4(a), and multimode laser 4(b). Over this range the change in temperature was approximately linear over time, which allows a conversion from time to frequency using the output of the temperature probe and the known cavity FSR as the frequency calibration. A Lorentzian curve has been fitted to the single mode scan in Fig. 4(a), from which we determine the FWHM of the measured cavity peak to be 0.484(2) MHz. Note that the linewidth of the laser is $\sim$100kHz and hence the cavity dominates the observed FWHM. This would suggest that the finesse of the cavity is 900(10), which from Eq. 3 would imply a round trip loss of $\sim 0.7 \%$. This is comparable to what is expected given the 99:1 beamsplitter used.

 

Fig. 4. Cavity photo diode signal for single and multimode lasers. These are essentially the same as in Fig. 2(c), just over a much smaller scan range. (a) A scan of a single mode laser over a Lorenztian peak (fitted with a red curve) is shown, with a small side peak also visible. In contrast, when the laser runs multimode the scan (b) shows multiple peaks, due to the different frequencies of the multiple longitudinal modes that the laser is operating in. Both scans were taken in the linear regime of the temperature graph, and thus the separation in time between the side peaks can be approximately converted to a separation in temperature and then frequency, via the known FSR between adjacent peaks. The side peaks visible in both spectra are due to the propagation of different polarisation modes in the cavity.

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In the case of a multimode laser (Fig. 4) we observe a less well defined distribution, which appears to be composed of multiple Lorentzians, each at different frequencies. This would indicate that the laser is operating a superposition of different frequency modes. These results demonstrate the ability of the cavity to determine the frequency distribution of incoming light (at least over the cavity FSR), which was our intended objective.

The side-peaks visible in both spectra in Fig. 4 are due to the cavity supporting the propagation of different polarisation modes simultaneously. They can be removed by polarisation filtering of the input [17]. However, noting that the spacing between the side-peaks is the same as the larger peaks, though with a much smaller amplitude, they do not impede the intended function of the cavity, as a simple and inexpensive mode monitor, so such filtering can be considered an optional extra.

To monitor the long term performance of the cavity, the drift of the FSR of the cavity over time is shown in Fig. 5. Each data point is an average of the FSR measured in a single scan, such as the one shown in Fig. 2. The uncertainties are determined from the standard deviation of the FSR over this period, and a data point was taken every five minutes. The data indicates that the FSR is stable, with $\Delta T_{+1FSR} = 2.1(1)$K, and $\Delta T_{-1FSR} = -2.0(1)$K, which agrees well with the $\approx 2 K$ we calculated for our cavity parameters. Note that the $\pm 1$ FSR values both agree within uncertainty. The short-term fluctuations of the FSR about the average lines are also within experimental uncertainty; these fluctuations may be due to ambient temperature changes. To reduce the uncertainty due to these fluctuations in future experiments, the ambient temperature could be mitigated by using a temperature feedback stabilisation circuit, and perhaps by changing the design so the peltiers heat up the entire fibre instead of only a small length.

 

Fig. 5. Drift in the cavity over an hour. The data at each point in time represents an average of the points in each band of Fig. 3, with uncertainties calculated from the standard deviation of the points. Errorbars represent measurement uncertainties, which are dominated by temperature and cavity peak location uncertainty.

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5. Conclusion

We have built and demonstrated a ring configuration, in-fibre, Fabry-Perot interferometer that can be scanned using the temperature change induced by Peltier devices. This device has been shown to be able to characterise the longitudinal mode structure of a laser. The cavity can be scanned over multiple FSRs and we have shown that the drift of the cavity over a long time is minimal (within uncertainty). We also determined the finesse of the cavity to be 123(2), corresponding to a round trip loss of 5%, and showed that the cavity could distinguish between single and multimode lasers.

These results make the proposed cavity a viable and very inexpensive scheme for applications such as mode monitoring, as is often required in cold atom experiments. Indeed, the designed use of this setup is on a BEC apparatus [29], where it operates as a mode monitor for a dipole trap laser that intermittently runs multimode. Relatively straightforward extensions could be made to allow the setup to be used for laser frequency stabilisation [26], or to operate as a relative temperature [19] or strain sensor [22], by using a laser to measure shifts in the cavity FSR.