The simplest configuration of a microwave photonic oscillator involves two stable lasers demodulated on a photodiode [1,2]. One can use a single laser substrate-emitting dichromatic radiation [3,4] or can utilize two independent high-quality lasers [1,2,5]. Locking the lasers to two different modes of the same optical cavity improves their relative coherence and leads to generation of low-noise microwave signals [6–11].

A pure optical (self-injection) locking of the lasers to the cavity modes facilitated by the resonant reflection [12–15] or the same frequency scattering [16,17] of the laser light back to the laser chip is one of the most efficient techniques to achieve generation of high spectral purity light. A few percent of the narrowband backreflection locks the laser to the cavity mode. This configuration does not call for any electronics and does not involve any modulation of the laser light. The microwave signals generated by the self-injection-locked (SIL) lasers are characterized by high spectral purity and are spurious-free.

A ring cavity is a natural choice for SIL experiments [16,17]. The cavity of this type provides backscattering exclusively at the frequencies of the cavity modes. A Fabry–Perot cavity usually reflects light at all the frequencies but the eigenfrequencies. As the result, SIL using Fabry–Perot cavities calls for certain optimizations of the experiment. One can either utilize special cavity modes as well as special cavity geometry [6,12–15,18–20] or create low-finesse ring cavities involving the high-quality factor Fabry–Perot cavities for self-injection locking demonstration [21–23]. These methods become rather involved when one intends to lock more than one laser to the cavity.

In this Letter, we report on a demonstration of a simple and efficient technique of SIL of two lasers to a compact and monolithic cylindrical Fabry–Perot cavity [24]. We couple light to high-order modes of the cavity that diverge from the symmetry axis of the cylinder. By sending the light backwards along one of the lobes of the cavity far-field emission, we create a system that supports efficient resonant backreflection that locks the laser to the mode. By coupling two lasers at both sides of the cavity, we realize a highly coherent bi-chromatic optical source. The output emission of this optical source, sent to a fast photodiode, has produced a microwave signal with phase noise with levels comparable with the best achievements demonstrated in microwave photonic configurations of similar kind [5–11]. We observed spurious-free signals with a phase noise of $-110$ dBc/Hz at 10 kHz offset and 10–75 GHz carriers.

The proposed configuration has a few advantages if compared with other SIL schemes. The optical scheme can be aligned in a simple way and lends itself for the tight packaging. It is easier than the alignment and packaging of either a free-standing whispering gallery mode resonator or an integrated ring cavity. Our method works with any mirror configuration of the cavity and reaches its optimum performance with the essentially non-confocal cavities characterized with clean optical spectra. The monolithic Fabry–Perot cavity that we have utilized does not require evacuation needed to eliminate gas effects in hollow Fabry–Perot cavities. It also does not need clean optical environment required for the cavities based on the total internal reflection, because small contaminants, accumulated at the external cavity surfaces, do not degrade the cavity performance. The bulk Fabry–Perot cavity is mechanically and environmentally stable. It has a relatively large volume resulting in reduced fundamental thermodynamic fluctuations. On the other hand, the volume is small enough to ensure high frequency of the mechanical modes of the cylinder, and that reduces vibration sensitivity compared to hollow Fabry–Perot resonators requiring heavier spacers and frames.

The cavity is created using a recipe published previously [24]. It is made from a fused silica preform of 15 mm in diameter and 25.4 mm in length. One end surface of the cavity was polished flat, and the other was shaped to be convex with 1 m radius of curvature. High reflectivity ion-beam-sputtering coating centered at 1550 nm is deposited on both surfaces. As one can see, the cavity is non-confocal. Moreover, the placement of the mirrors is not symmetric because of the general manufacturing errors. We measured the optical axis of our cavity to be displaced 1.4 mm from the cylinder axis. In contrast with the bow-tie modes supported by the confocal cavities usually preferred for SIL, the high-order modes are robust and insensitive to small variations of the geometrical shape of the cavity like this. We were able to accommodate this eventual geometry and implement alignment necessary for our proposed design and achieve the results presented below.

To lock two lasers to the Fabry–Perot cavity, we utilize high-order modes along with a spatial filter (a pinhole). We noticed that the high-order modes of the cavity have lobes propagating at the angle with respect to the cavity mirror surface (see Fig. 1). We have matched the input laser beam with one of such lobes. The laser beam both pumps the cavity mode and is reflected from the mirror surface. We use spatial filtering to suppress the impact of the reflection on the laser (Fig. 2). Since the mode of the cavity supports the standing optical wave, the light confined in the mode propagates back to the laser via the same optical path. It happens only when the laser frequency coincides with the frequency of the mode. The non-resonant reflection of the light can be filtered out spatially. As a result, the laser locks to the mode by means of the SIL process. Since the locking requires only a few percent of light to be backreflected, we split the laser beam in two parts and use only one of the parts for the cavity interrogation, while the other part is utilized as the output for the generation of the microwave signal.

 

Fig. 1. Direct coupling of a semiconductor laser chip to a non-confocal FP cavity. A Fabry–Perot cavity, formed by concave and flat mirrors (1 and 2) bonded to the properly shaped and polished end surfaces of a cylindrical solid-state preform, is optically coupled to a Gaussian beam of a diode laser (3). The Gaussian beam of the laser is tilted at a predesigned angle (see Fig. 2). As a result of the tilt, the direct back-reflection from the cavity mirror is suppressed, and the direct coupling to a high-order Hermite–Gaussian (HG) mode is maximized. The image shows an example of coupling to the $HG_{1,4}$ mode. The optimized larger area Gaussian beam of the laser (b) is sent at the mirror imprint of the (c) $HG_{0,0}$ beam. As a result, the overlapping of the Gaussian beam of the laser with the (a) $HG_{1,4}$ mode is $10^4$ times higher than with (c) $HG_{0,0}$.

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Fig. 2. Coupling factor of the Gaussian beam of the laser to different modes at the laser beam waist of $1\times$ (a), $2\times$ (b), and $3\times$ (c) of the fundamental $HG_{0,0}$ mode of the resonator. The black lines show the backreflection efficiency of the laser’s Gaussian beam via direct reflection from the front mirror of the Fabry–Perot cavity. The blue lines and the red lines show the coupling efficiency of the tilted Gaussian beam of the laser to the transverse modes $HG_{1,2}$ and $HG_{1,4}$, respectively.

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The cavity has well-defined and stable spatial mode profiles (see Fig. 3). A numerical simulation can easily predict all the low-order modes. The very-high-order modes have deteriorating profiles because of the mode clipping by the cavity walls. We use higher-order modes, such as shown in Figs. 3(c) and 3(d). The fundamental and other modes are not excited because of their geometrical mismatch with the pumping beam, which we have discussed above, as well as due to the non-degeneracy of the cavity spectrum. The high-order modes have quality (Q-) factor exceeding a billion. The backscattering to the laser light from the selected cavity modes was measured to exceed 1%. The locking range of the laser (Fig. 4(a)) exceeds 2 GHz and is comparable with the cavity-free spectral range. The SIL is visualized with the reduction of the intensity of the light reflected from the cavity. When the laser is not locked to a cavity mode, its linewidth is too broad and light does not enter the cavity. However, when the laser is locked to the mode, the reflection drops as the result of resonance interaction with the mode. In case of critical coupling, the reflection drops nearly to zero. Because of the intentional incomplete overlap of the input beam and the cavity mode profile in our case, the reflection reduction is incomplete.

 

Fig. 3. Far-field mode profile of the Fabry–Perot cavity. Panels (a), (b), (c), and (d) show the modes HG₀₀, HG₀₁, HG₀₂, and HG₁₂, respectively, visualized with the InGaAs near-infrared camera. Figures (e), (f), (g), and (h) show the far-field of the same modes simulated numerically.

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Fig. 4. Measurement of the laser locking range and the cavity Q-factor. The laser frequency is quickly scanned through the SIL region. (a) Frequency width of the region can be estimated from the known frequency response of the laser to the current change. The lifetime of the light confined in the cavity can be characterized by the transient time of the laser unlocking from the resonator. (b) Time was measured to be $\tau =1.6\ \mathrm{\mu}$s. The corresponding quality factor of the cavity is $Q=\omega \tau = 2\times 10^9$, where $\omega$ is the angular frequency of the mode of the cavity. (c) Clean transmission spectrum of the cavity mode measured with a tunable fiber laser. (d) Clustered spectrum of an imperfect confocal cavity (not used in the experiment) is not optimal for SIL.

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To characterize the Q-factor of the SIL modes, we measured the mode ringdown time by scanning the frequency of the laser through the cavity mode of interest (Fig. 4(b)). When the laser unlocks from the cavity, its frequency jumps by a value in the gigahertz range with respect to the light confined in the cavity. As the result, we can filter out the light leaving the cavity and the light emitted by the unlocked laser and scattered off the cavity. We measured the intensity of the light leaving the cavity. Cavity ringdown measurement by the intensity does not require a high bandwidth photodiode and has better sensitivity if a higher than 50 Ohm load of the photodiode is utilized. The ringdown technique was validated using a standard spectroscopy measurement using a narrowband standalone laser (Fig. 4(c)).

To identify the best cavity configuration, we measured the spectrum of cavities with various mirror selections. In the case of the nearly confocal cavities, each mode experienced clustering (Fig. 4(d)). The essentially non-confocal cavities had a clean spectrum (Fig. 4(c)). The robust and selective SIL operation to the high-order modes is one of the major achievements of this study. To create the microwave photonic oscillator, we lock two lasers to the same cavity [6–11]. The SIL operation is achieved in the configuration depicted in Fig. 5. Both lasers are model L1550P5DFB from Thorlabs, in 5.6 mm TO-can packages, operating at 1549 nm wavelength and mounted into Newport LDM-4405 and Thorlabs LDM-56 mounting structures. Light from the lasers is collimated using C140TMD-C lenses and is matched with two cavity modes. The modes are selected to optimize the coupling and do not belong to the same mode family. The cavity is placed in the open air at our laboratory, and no thermal and mechanical stabilization of the cavity–laser assembly is involved. Simultaneous SIL regime for both lasers is maintained over hours. The experimental setup is illustrated in Fig. 6. We split the emission of the lasers in two parts. One part is utilized for the locking, and the other part is used for the generation of the microwaves. The cavity does not introduce any discrimination effect to the light that bypasses the cavity even in the case of not exact resonant locking of the lasers to the modes. The photonic oscillator generates signals characterized with low phase noise (Fig. 7). This noise level is at least 20 dB better than the noise of oscillators based on small whispering gallery mode cavities [5]. It is shown in what follows that the noise does not change with the increased laser beatnote frequency.

 

Fig. 5. Fabry–Perot cavity coupled to the lasers. A sturdy aluminum frame hosts the Thorlabs thermally controlled laser diode mounts, a Fabry–Perot cavity (a white cylinder in the middle), and the coupling lenses which match the laser field with the high-order modes of the cavity. The optical elements are permanently mounted with an Optocast UV-curable epoxy to matched glass plates.

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Fig. 6. Scheme of the experimental setup. Two mode-matched laser subassemblies (1, 2) are coupled to a Fabry–Perot cavity (3) with the direct backreflection aperture filter. The lasers are simultaneously self-injection-locked to the adjacent modes of the cavity (3). The fiber-coupled emission of the lasers is split three ways. The 3 dB coupler (4) feeds 50% of the original laser’s emission to a high-speed photodetector (8). The remaining 50% of the optical emission is equally distributed by the second 3 dB coupler (5) between the optical output (6) and the LI-curve monitor (7).

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Fig. 7. (a) Phase noise of a 10 GHz signal generated by two SIL lasers using two high-order modes of a monolithic Fabry–Perot cavity (blue line). To identify the improvement of the laser phase noise due to the SIL process, we unlocked one of the lasers and repeated the measurement of the phase noise of the beat signal (red line). The injection locking results in the noise improvement by more than 80 dB at 10 kHz offset. The thermodynamic limitation of the spectral purity that can be achieved using the particular cavity is shown as a solid black line. Certain improvement of the relative stability of the lasers with respect to the fundamental noise can be explained as the result of the incomplete overlap of the modes. The magenta curve illustrates the phase noise of the SIL laser measured using a reference laser electronically locked to an ultra-stable cavity. The hump results from the electronic lock. (b) Allan deviations of the $\omega _{RF}=$10 GHz signal (blue dots) and one of the SIL lasers are relatively high because the cavity is not thermally stabilized. The Allan deviation of the laser (magenta dots) is scaled with the frequency ratio $\omega _{laser}/\omega _{RF}$.

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The performance of the oscillator is limited due to the fundamental noise of the cavity resulting from the thermodynamics-mediated fluctuation of the refractive index of the host material as well as the cavity dimensions. We calculate the noise using the fluctuation–dissipation theorem [25]. The analytical solution for the noise takes into account the boundary conditions of the cylindrical resonator with finite size and standing wave. Using this approach, we find the fundamental phase noise limitation depicted in Fig. 7. The thermorefractive noise of the spacer [26] and the Brownian noise of the coating [27] dominate the noise.

One of the major advantages of a microwave photonic oscillator is its potential broad tunability. The phase noise of the generated signal depends mostly on the phase noise of the lasers, assuming that the fast photodiode produces microwaves of the same power independently on frequency. Changing the oscillation frequency is facilitated by tuning the frequency of either one or both of the lasers. In the case considered in this Letter, it is possible to tune the lasers among different modes of the cavity. This is not a continuous tuning, as the spectrum is discrete. To achieve continuous tuning, one has to lock each laser to an individual cavity [5]. We illustrate the tuning in Fig. 8. One of the lasers was kept locked. The other laser was re-locked to different cavity modes. The relative frequency of the lasers was changed without any degradation of the locking quality. For each frequency separation of the lasers, one should be able to generate low-noise microwaves if a proper photodiode is available. To conclude, we have demonstrated the experimental generation of low phase noise ($-110$ dBc/Hz at 10 kHz offset) microwave signals by frequency beating optical output from two semiconductor lasers simultaneously self-injection-locked to selected off-axis modes of a high-Q Fabry–Perot cavity. The cavity is constructed of a monolithic block of high-purity fused silica with deposited high-finesse mirrors and does not require environmental isolation for the production of low-noise microwaves. The oscillator is useful at high frequencies where regular electronic approaches have worse performance.

 

Fig. 8. (a) Phase noise at different RFs. (b) Optical spectra corresponding to some phase noise curves shown in (a). One of the lasers is permanently locked to a mode of the cavity, while the other one is tuned by the temperature of the laser chip from one mode of the cavity to another. When the mode is identified, the laser is locked to the mode via the SIL. The cavity temperature is not stabilized. A slight degradation of the phase noise at 105 GHz is attributed to lower responsivity of the photodiode and higher loss of the RF circuit at that frequency.

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