## Abstract

We utilize a high quality calcium fluoride whispering-gallery-mode resonator to passively stabilize a simple erbium doped fiber ring laser with an emission frequency of 196THz (wavelength 1530nm) to an instantaneous linewidth below 650Hz. This corresponds to a relative stability of 3.3 × 10^{−12} over 16μs. In order to characterize the linewidth we use two identical self-built lasers and a commercial laser to determine the individual lasing linewidth via the three-cornered-hat method. We further show that the lasers are finely tunable throughout the erbium gain region.

© 2014 Optical Society of America

## 1. Introduction

Compact and tunable stable light sources are in high demand in metrology [1] and biochemical sensing [2, 3] to mention just two predominant fields. Optical resonators are at the heart of narrow frequency light sources. A special type of resonators are whispering gallery mode (WGM) resonators, i. e. dielectric cavities that confine light due to total internal reflection at their dielectric interface [4]. Their quality factor *Q* is mainly limited by surface scattering and by material absorption. For highly transparent materials such as calcium fluoride (CaF_{2}) quality factors up to 10^{11} [5, 6] have been demonstrated. Operability exists throughout the whole transparency window of the host material, in case of CaF_{2} from 150nm to 10μm. With their resulting very narrow linewidth, these resonators serve as excellent optical frequency filters [7] and are suitable to enhance the lasing modes of a conventional primary lasing module [8, 9].

Before reaching the fundamental thermal noise floor limit [7, 10–12], the fractional stability of a monolithic reference cavity is mainly determined by its deformation due to mechanical vibration or thermal effects, whose influence scales with the cavity’s dimensions [13]. Recently, there has been progress in reducing the effect of thermal noise by utilizing the birefringence of magnesium fluoride [14, 15] or by increasing the resonator size [16, 17]. Therefore, and due to their compact sizes, WGM resonators are eminently suitable as a frequency reference [18, 19].

Here we report the setup and characterization of a free running widely tunabe fiber ring laser providing lasing linewidths below 1kHz. This is achieved by resonantly filtering the broad emission spectrum of an erbium doped fiber via the narrow-linewidth modes of a WGM resonator [20, 21]. Only these narrow modes can pass the resonator and achieve gain in the following round trip. This establishes a narrow linewidth lasing behavior which can be tuned throughout the gain region. In order to quantify the lasing linewidth two identical systems were built which allow for the first time to determine the individual linewidth of the setups.

## 2. Experimental setup

We set up a conventional fiber ring laser using an erbium doped fiber with a broad band emission spectrum in the telecommunication C-band wavelength regime (∼1530nm). It is pumped by a 980nm laser diode with 200mW output power through a wavelength-division-multiplexer. The benefit of such a ring laser system is that it is not affected by spatial hole burning. Light is coupled out via a 99/1 fiber coupler; clockwise circulation is prevented by an optical Faraday isolator. By inserting the WGM resonator into the fiber loop the final whispering gallery laser (WGL) is set up, as sketched in Fig. 1(a). We fabricate the WGM resonators on a home-built diamond lathe. Mono-crystalline CaF_{2} is cut into disks with an optimized surface curvature via diamond turning. The disks are 4mm in diameter. An optimal surface quality is achieved by means of diamond polishing with grain sizes down to 50nm. Loaded cold cavity *Q* factors of our millimeter sized resonators, measured in the frequency and time domain, read a few 10^{8}. Evanescent coupling through the polarization dependent WGMs can be achieved via a pair of piezo controlled coupling prisms (SF11 glass), depicted in Fig. 1(b). Prism coupling allows for a more rigid construction, which is less sensitive to mechanical vibrations of the environment. In comparison, previous approaches utilized tapered and angle polished fibers for the resonator coupling [22]. For the coupling into and out of the fiber loop a pair of gradient index (GRIN) lenses is used, where the numerical aperture and the focal point are adjustable. The coupling efficiency from the fiber loop transmitted through the WGM resonator was approximately 20%, measured by including the loss of the fiber coupling and GRIN lenses. For optimal coupling to distinct whispering gallery modes a fiber polarization controller is necessary. Laser output power is in the range of tens of microwatts. This solely passively stabilized lasing system provides a straightforward setup design, featuring easy assembly and tight packaging.

As our main task is the linewidth measurement, the tunability of the lasing frequency is paramount. With a broadband (1nm FWHM) optical bandpass filter a coarse tuning of the lasing mode’s wavelength over the whole emission spectrum of the erbium doped fiber in steps of the WGM resonator’s free spectral range (FSR) (∼20GHz) is possible due to mode competition. Further fine tuning can be achieved via temperature control of the resonator. Carefully combining both methods yields the possibility to continuously tune over more than a FSR.

In our experiment we observed a refinement from a multimodal lasing behavior spanning over 50GHz to a sub-kHz lasing line by including the WGM filter. Furthermore, in comparison with the cold cavity linewidth of the resonator (sub-MHz) the active lasing reduces the linewidth and improves the *Q* by three orders of magnitude. In order to verify these results, we modeled our ring laser setup analytically following and extending the approach by Wang [23]. We modeled the filtering mechanism introduced by the WGM resonator and implemented this model in an iterative numerical simulation, taking into account gain saturation. The experimentally observed characteristics could be reproduced by appropriate choice of parameters, the most crucial being the fiber cavity’s and the WGM resonator’s *Q* factors and the saturated intra-cavity power. The laser’s emission spectrum narrows with increasing intracavity power, which agrees well with a fully analytic approach [24]. Our simulation does not cover further aspects regarding the WGM resonator’s instability due to an increased lasing power, and is therefore not able to determine the optimal intracavity power.

## 3. Measurement procedure

As the frequency stability of the individual lasers cannot be measured directly, two identical systems were built. The beat note generated by mixing the emission spectra of the two lasers allows us to reconstruct the frequency stability of the combined system. For a later analysis we also recorded the beat note of both self-built systems with a commercial external cavity stabilized diode laser (Toptica DL pro design). We recorded 50ms (sampling rate is 200 megasamples/second, with 8 bit vertical resolution) of the beat note traces of the two approximately 10MHz detuned WGLs. First we analyze the phase noise of the system. By modeling the time-domain beat signal as *V*(*t*) = (*V*_{0} + *ε*(*t*))cos(2*πf*_{0}*t* + *φ*(*t*)), where *f*_{0} is the carrier frequency, *φ*(*t*) the time-dependent phase noise and *ε*(*t*) the time-dependent amplitude noise, we can perform a demodulation to extract both the phase noise and amplitude noise. For this purpose we define the two quadratures *I* = *V*(*t*) · cos(2*πf*_{0}*t*) and *Q* = *V*(*t*) · sin(2*πf*_{0}*t*). The resulting terms are filtered from all the high frequency components of the quadratures and the time-domain phase noise can be written as
$\phi (t)=\text{arctan}\frac{I}{Q}$ [25]. In Fig. 2(a) we can see the frequency-domain phase noise extracted from the measured beat note traces. It clearly follows the usual *f*^{−2} Lorentzian behavior.

For the computation of the beat note’s frequency, contiguous, non overlapping basic time intervals of 1μs length are chosen from the trace in order to obtain a frequency stream. This basic time interval is chosen such that it covers at least ten periods of the beat note signal. We perform a nonlinear least-square fit of a sine function over these separate time domain traces. The only fitting parameters in the used model are frequency, amplitude, offset and phase, which are all assumed to be constant over the basic 1μs fit interval. The resulting time resolved oscillation frequency values of all three measured beat note traces are depicted in Fig. 2(b). We prefer this method in comparison to a conventional frequency counter because of the higher flexibility and robustness in case of a low signal to noise ratio. Furthermore, we avoid ambiguity in the interpretation of the resulting values and we are free to compute different types of variances [26]. In order to determine the frequency stability of our lasers with respect to the averaging time the Allan deviation [27, 28]

is used, where*σ*is the Allan deviation of the lasing frequency,

*f*denotes the value of the lasing frequency referring to the

_{i}*i*-th basic time interval and

*M*is the total amount of basic time intervals. Pre-averaging of the frequency stream over multiple basic time intervals allows us to compute Allan deviations for different averaging time scales.

## 4. Results

The resulting Allan deviation values are shown in Fig. 3(a). They directly correspond to the lasing linewidth. The relative stabilities comparing the laser’s linewidth to its emission frequency of 196THz are also presented, they correspond to the inverse lasing *Q* factor. For the deviation values of the beat note trace evaluation a minimum of 1056Hz is reached for an averaging time of 18μs. Since the individual WGM laser setups are slightly different we are reporting an upper limit for the more stable WGL. This can be estimated by
$1056/\sqrt{2}\text{Hz}=750\text{Hz}$. Thereby we assume no negative correlation in the lasing behavior. The graph reveals a *τ*^{−1/2}-slope in the short timescale regime (less than 18μs), before the optimal averaging time is attained. This slope can be associated with a white frequency noise behavior [27]. After the optimal averaging time a drift to larger Allan deviation values is predominant. The timescale suggests that temperature fluctuations alone cannot be responsible. Also the curve’s slope does not fit to the therewith related random frequency walk (*τ*^{1/2}). A directed frequency shift due to heating of the modal volume seems more plausible here. The measurement noise curve in Fig. 3(a) is a measure for the quality of the signal and of the measurement procedure as a whole. It combines the frequency errors computed from the least square residuals of the sine fit procedure and thus reflects the signal to noise ratio of the acquired beat note, time jitter and quantization errors of the oscilloscope. Herein we also take into account errors due to the method of frequency calculation as well as the short term (less than 1μs) instability of the lasers, which is ignored by the fit model.

In order to extract the individual frequency stabilities we add a third lasing system, namely a commercial Toptica DLpro design laser and perform a three-cornered hat measurement [29,30]. The individual linewidth is determined by recording the beat notes from the three possible combinations of laser pairs simultaneously and solving for the single laser variances

The more stable WGM resonator laser reaches a relative stability of (1.67 ± 1.60) × 10^{−12} for an averaging time of 16μs. Equivalently, this corresponds to a lasing linewidth of (328 ± 314) Hz at the laser’s emission frequency of 196THz. In a conservative estimate we report a relative stability of 3.3 × 10^{−12} and a linewidth of 650Hz, respectively. These values agree well with the results of the prior estimation using the combined frequency stability directly. Compared to the phase noise considerations presented in Fig. 2(a) we find a noise stability which is one order of magnitude worse.

## 5. Conclusions

We demonstrated a sub-kHz linewidth lasing behavior in a solely passively stabilized erbium doped fiber ring laser. The stabilization arises through filtering via high-*Q* modes of a crystalline calcium fluoride whispering gallery mode resonator. This system allows for particulary compact device sizes and also offers possibilities for further improvements by means of additional active stabilization techniques. The final lasing linewidth can be influenced either by the cold cavity *Q* factor of the filtering resonator or by an increased circulating intracavity power.

Our evaluation method for the lasing stability is based on the analysis of the digitized time domain beat note traces and avoids the standard frequency counter approach. The reason for the different performances of the phase noise considerations and the Allan deviation lies in the fact that the latter takes into account the instantaneous frequency components which are determined from the beat trace, whereas the former assumes a perfect theoretical carrier frequency. Thus the Allan deviation gives a more complete noise measure.

## Acknowledgments

The authors would like to thank Dmitry V. Strekalov and Josef U. Fürst for stimulating discussions and Gerd Leuchs for his support. We acknowledge support by the Deutsche Forschungsgemeinschaft and the Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding program Open Access Publishing.

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