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Abstract
We present a novel method allowing high-power single-frequency emission with sub-kHz linewidth from a compact multi-frequency diode laser locked to high-Q optical microresonator. Using high-Q MgF2microresonator and multi-frequency diode laser operating at 1535 nm with the output power of 100 mW and an emission spectrum consisting of approximately 50 lines with MHz linewidth, we observed a spectrum collapse to a single line or several lines with a sub-kHz linewidth and output power power of 50 mW. The Bogatov effect predicted more than 30 years ago was observed and studied in the spectrum of the locked laser. For analysis of the considered effect, original theoretical model taking into account self-injection locking effect, mode competition and Bogatov asymmetric mode interaction was developed and numerical modeling was performed. All numerical results are in a good agreement with our experimental data. Accurate analytical estimations for the parameters critical for the considered effect were obtained. The proposed method may be applied for different types of diode lasers operating in different spectral ranges.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Cost-effective narrow-linewidth lasers are required in many different industrial fields, such as spectroscopy [1], sensing [2], light detection and ranging [3], quantum optics [4], and atomic clocks [5]. The development of perspective integrated, chip-based solutions require compact laser systems. Moreover, for many important practical applications, including Kerr combs and dissipative Kerr solitons generation [6], especially using on-chip configuration, seeding single-frequency laser should have sufficient power to avoid additional amplifiers.
External-cavity diode lasers (ECDLs) provide single-frequency operation with a narrow linewidth. Also, their lasing frequency can be tuned, which is important for spectroscopy and frequency synthesis. However, ECDLs are not compact enough and typically emit moderate output power. Several approaches were proposed to overcome these obstacles [7] though are not commercialized.
Diode lasers are compact and cover practically the whole optical spectral range. However, a typical linewidth of distributed feedback (DFB) or distributed Bragg reflector (DBR) laser is about 1 MHz while sub-kilohertz level is required for many applications. Several methods of further linewidth reduction, e.g., using optical feedback with different external resonant optical cavities, may be applied [8–15]. However, single-frequency diode lasers, due to design features typically required for single frequency operation have insufficiently low optical output powers.
In this work, we exploit self-injection locking to high-Q whispering gallery mode (WGM) microresonators for narrowing the output spectrum of lasers [16–18]. This effect was demonstrated for various lasers, including quantum cascade [19], fiber-loop [20] and DFB lasers [21] achieving the instantaneous linewidth down to sub-Hz levels [22]. However, instead of initially single-frequency lasers we use compact and commercially available multi-frequency diode lasers, which can have significantly higher output powers. Various methods providing single-frequency emission from multi-frequency laser are well-developed [23], however most of them are characterized by the lack of the output power. We present a novel method allowing high-power single-frequency emission with sub-kHz linewidth from the compact multi-frequency diode laser locked to high-Q optical microresonator.
Due to the mode competition effect, the self-injection locking with single-frequency lasing is possible in this case, allowing much higher output power. In this work we provide additional theoretical models and numerical simulations to support recent experimental work [24,25] and obtain accurate analytical estimations for the parameters critical for the considered effect.
We show that in the regime of the self-injection locking of a diode laser to a high-Q WGM microresonator, wide (~10 THz) multi-frequency emission spectrum may collapse to a single line with a linewidth of at the kHz level or below. Due to the phenomenon of mode competition the total initial power redistributes in favor of the locked mode (or, in some cases, in favor of several locked modes) providing a single-frequency (few-frequency) regime with energy concentration reaching 96%. We demonstrate the transformation of a 100 mW multi-frequency regime into a 50 mW single-frequency regime. Some loss of efficiency is likely due to losses of the optical scheme used and can be further improved. It should be noted, that the Bogatov effect predicted in [26] was observed and studied with unprecedented accuracy in the spectrum of the locked laser. The proposed approach may be applied for different types of diode lasers operating at different wavelengths since WGM microresonators may possess ultra-high quality-factors from ultra-violet (UV) [27] to mid-infra-red (MIR) region [28].
If several laser modes are locked to different WGM modes simultaneous emission at different frequencies can appear. This effect opens a way, for example, to compact dual-frequency lasers that are required for spectroscopy [29,30], LIDARs [31–33], holographic interferometry [34], optical terahertz sources [35–37] and other applications.
We developed a theory of multi-frequency self-injection locking based on the model of multi-mode laser of Yamada [38] with the Bogatov asymmetric mode interaction [26] and Lang and Kobayashi model [39] introducing an optical feedback to the model of lasing. This combined model was shown to provide an adequate description of the dynamics and emission spectra of self-injection locked multi-frequency lasers. All numerical results obtained from the developed model are in a good agreement with our experimental data.
2. Experiment: self–injection locking
2.1. The experimental setup
The scheme of the experimental setup is shown in Fig. 1. In our experiments we used a multi-frequency (but spatially single-mode) diode laser operating at 1535 nm with a spectrum width of ~ 10 nm and maximum power of 200 mW (at a current of 350 mA applied to the diode). The distance between the Fabry-Perot lines of the diode’s emission spectrum was Δ fd= c /2Ln ~ 17.6 GHz (c is the speed of light, the chip length is L = 2.5 mm, n = 3.154 is the refractive index of the chip). The diode spectrum had ~ 50 incoherent lines. Mutual coherence of adjacent lines was about 1 MHz. The optical spectrum of the free-running diode laser with theoretical curve (red line) is shown in Fig. 2(a).
Fig. 1 Experimental setup. Diode laser at 1535 nm (multi-frequency); PD – photodetector; ESA – electrical spectrum analyzer; OSA – optical spectrum analyzer; OSC – oscilloscope; b: photo of the microresonator with the coupling prism.
Fig. 2 a: Experimental (blue line) and numerically calculated (red line) emission spectrum of the free-running multi-frequency diode laser [Eq. (6)–Eq. (7)]; b: Experimental (blue line) and numerically calculated (red line) emission spectrum of the self-injection locked multi-frequency diode laser [Eq. (6), Eq. (7), Eq. (14)]. The model parameters are shown in Table 1.
For self-injection locking we fabricated a crystalline MgF2disk microresonator of 5 mm in diameter (FSR Δ fWGR≈14 GHz) with the curvature radius of the rim of 500 μm. The intrinsic quality-factor of the microresonator was measured to be ~ 109. Qualitatively the same reproducible results were obtained for other central wavelengths (e.g. 1550 nm, 1650 nm) and with microresonators of other dimensions and materials (BaF2, CaF2).
The radiation from the diode laser was focused on the inner surface of the total internal reflecting prism for coupling with the microresonator. A microresonator was placed close to the focusing point, where the WGM modes could be efficiently excited. After the coupling prism the output radiation was sent to a single-mode fiber through the collimating lenses and was fed to photodetector and the oscilloscope (OSC), to the optical spectrum analyzer (OSA) and to the electrical spectrum analyzer (ESA).
No optical isolator was used between the laser and the microresonator so effective feedback was realized due to the resonant Rayleigh backscattering on the internal and surface inhomogeneities of the microresonator [40]. Due to this effect part of the incoming radiation, which was in resonance with the frequency of the WGM, was reflected back into the active region of the laser (black arrow). For single-frequency lasers, resonant feedback results in the self-injection locking effect leading to a radical narrowing of the linewidth of the laser [21]. The maximal value of the linewidth reduction depends of the quality-factor of the microresonator mode, used for the self-injection locking and the coefficient of reflection from the resonator [18].
In our experiments the self-injection locking regime with multi-frequency diode was also obtained by tuning the frequency of the laser (adjusting the diode’s current) close to one of the frequencies of the WGM modes. Varying the distance between the coupling prism and the microresonator and, consequently, adjusting the feedback strength, we observed different regimes of laser generation. It was found that at the particular distance range significant spectrum reshaping including switching from multi-frequency to single-frequency generation regime occurs. In this case the diode mode closest to the microresonator mode, was self-injection locked and the output spectrum collapsed to single line. Outside this range laser emission spectrum remained practically invariable.
Figure 2(b) demonstrates the effect of the self-injection locking of a multi-frequency diode laser on its output spectrum. One can see that an initially multi-frequency emission spectrum consisting of about 50 lines was transformed into a single-line spectrum with the characteristic asymmetric form. Such an asymmetric spectrum was first observed and explained by Bogatov [26]. The locked regime was stable in a standard laboratory conditions for several hours without any additional stabilization technique.
Such dramatic spectrum collapse is a result of the self-injection locking effect taking place when the frequency of one of the modes of the multi-frequency laser is close enough to a WGM resonance [18]. In this case, we observed that other modes are suppressed by 35 dB due to the complex dynamics of the laser gain. The most intense mode "burns out" the inverse population, thereby depriving the remaining modes of the energy source, leading to the energy competition of the modes [41]. In this way, an additional energy is pumped into the self–injection locked mode of the laser, and its intensity significantly increases [see Fig. 2]. The energy efficiency of such a spectrum conversion strongly depends on the reflection coefficient of the external resonator and allows the single mode energy concentration up to 100%. In our experiments we observed that significant part (~ 95%) laser power is radiated by the dominating mode. However, due to losses in the optical scheme (reflections on lenses’ and prisms’ surfaces) this power is reduced resulting in ~ 50% efficiency (compared to total power of free-running laser) of the single-frequency emission. Note, that the limitation on maximum working power depends on the stability of a diode laser at high driving current. Although the diode we used had maximum specified power 200 mW but stable regime was possible only up to 100 mW (typical value of the current was 200 mA). Above this level we observed thermal drifts of laser frequency, so that self-injection locking regime was not stable. Evidently, that the losses can be decreased with optimized coupling elements while the diode thermal stabilization can be improved. Nevertheless, the maximum power of the demonstrated single-frequency laser (up to 50 mW) was already larger than the power of a conventional single-frequency DFB diode laser.
Table 1. Parameters, used in the model. Several parameters were obtained by fitting the experimental spectrum [see Fig.4(b)] with the calculated one [Eq.6, 7 and 14] (labeled as "fit"). Parameters, labeled as "doc" were taken from the laser documentation. The parameters known in literature provided for comparison with fitted ones together with corresponding references.
2.2. Linewidth
The linewidth of the locked diode laser was measured by beating the self-injection locked laser with reference laser (NKT Koheras Adjustik, the specified linewidth is 0.1 kHz). Beatings of two lasers were detected by a high-speed detector, and the linewidth was measured with an electrical spectrum analyzer. Figure 3(a) shows the result of the linewidth measurements (red curve). To take into account the contribution of the white and flicker frequency noise to the linewidth, the Voigt function was used for the approximation (black line) of the beatnote profile [43]. Thus we decomposed the total linewidth into two components: Lorentzian ΔνLorentz = 0.3 kHz for the white frequency noise impact and Gaussian ΔνGauss = 1.7 kHz for the flicker frequency noise impact. It was found that the measured flicker frequency noise is determined by the reference laser.
Fig. 3 a: Beating signal (near 7.5 GHz) of the self-injection locked laser with the reference NKT Koheras Adjustik laser (red line) and Voigt approximation of this signal (black line). ESA resolution bandwidth is 20 kHz and video bandwidth is 20 kHz; sweep time is 2.4 ms, no averaging. The sweep time was optimally selected to minimize frequency drift and low-frequency noise losses [42]. b: The Hadamard and Allan deviations of the frequency difference between two lasers self-injection locked on different modes (beating frequency 2.8 GHz) in one microresonator and flicker frequency noise approximation ∝ τ0 (blue line) and ∝ τ1 approximation (red line), which corresponds to ∝ f−3 frequency noise.
Fig. 4 Optical spectrum of the several-frequency radiation of the self-injection locked laser (blue - the experiment, orange - the model): a: two-frequency regime; a: four-frequency regime; a: six-frequency regime. The model parameters are presented in Table 1; additional feedback is added to corresponding modes.
To analyze further the intrinsic frequency stability of the self-injection locked laser Allan and Hadamard variances of the relative frequency and [44] were measured by beating of two similar diode lasers locked to the different WGM modes (beating frequency 2.8 GHz) in the same microresonator [see Fig. 3(b)]. The lasers had L= 2.5 mm (FSR=17.68 GHz) and L=1.5 mm (FSR=31.7 GHz). The maximum output power for both diodes was 200 mW. The locking to the same resonator allows to suppress significantly the noise due to thermal fluctuations of this resonator, thus emphasizing the residual noise of the locked laser. We found that linear thermal frequency drift in the absence of thermal stabilization of the resonator dominates in the Allan variance for averaging times between 10−1 s and 1 s (dotted line in Fig. 3(b)). This does not change the overall frequency stability characteristics of the self-injection locked laser, but makes it difficult to analyze the frequency noise. Therefore the Hadamard variance was used for the purposes of time domain frequency stability characterization as its most important advantage is insensitivity to the linear frequency drift [44]. The Allan deviation is left for comparison with previous works [17, 22, 45]. It was found that at times less than 10−1 s, the Hadamard variance is ∝ τ0 [Fig. 3(b), blue line], which corresponds to the flicker noise of the frequency, while at times larger than 10−1 s the noise of higher frequency order ∝ τ2 dominates [Fig. 3(b), red line]. So the Hadamard variance has the following form:
The results of the frequency noise approximation of the Hadamard variance are presented by the red and blue lines in Fig. 3(b) corresponding to a2 = 6 × 10−23 s−2, a0 = 1.15 × 10−24. Taking into account white frequency noise from the Voigt approximation (ΔνLorentz = 0.3 kHz), the spectral density of the frequency noise for such type of the Hadamard variance can be reconstructed [46, 47]:
To find the contribution of the low-frequency noise to the linewidth, effective integral linewidth was introduced [47, 48]:
which gave us Δveff = 0.4 kHz. It was found that the contribution of the frequency noise to the integral linewidth is negligible.The expression for the reduction of the white frequency noise via self-injection locking could be derived from the Lang and Kobayashi model with the frequency-dependent optical feedback and is given by [18, 49]:
where Qd and Q are quality factors of the laser cavity and the external resonator, respectively; Γ ≈ 10−2 is the reflectivity coefficient of the external cavity and α is the linewidth factor and ηd is the ratio of the output mirror loss to the total loss. where Ro is the reflectivity of the output laser mirror, τd is the roundtrip time of the laser cavity, ν0is the laser frequency. The reflection coefficient given by the laser diode manufacturer Ro = 0.54 as well as α = 4. By substituting the experimental data into Eq. (4) it was obtained that the linewidth of the self-injection locked modes is of the order of 0.1 kHz, which is in correspondence with measured ΔνLorentz.2.3. Multiple locked modes
In addition to the collapse of the diode laser multi-frequency spectrum into one narrow line, simultaneous self–injection locking of a few laser modes to different microresonator modes was observed. This effect results in effective discrimination of these locked modes and transformation of initial spectrum with ~ 50 lines into a spectrum with just a few locked narrow lines. The locking occurs on modes, spaced by an integer number of the microresonator FSR interval ΔfWGR and laser FSR interval Δfd: Δfmult= M ΔfWGR= N Δfd. In this case, the mode competition near each diode frequency acts the same way as in the case of single-frequency self-injection locking: the spectrum in the vicinity of the resonant frequency is suppressed, and energy is redistributed in favor of the locked spectral line. Spectral width of the locked mode also decreases significantly. This situation is depicted in Fig. 4, which shows (a) two-frequency regime [panel (a)], four-frequency regime [panel (b)] and six-frequency regime [panel (c)]. Note that different mode families in the microresonator have slightly different FSRs [50, 51], providing different spacings between locked modes [Figs. 4(b)–4(c)].
3. Model
To analyze the experimental results, an original model describing the self-injection locking of a multi-frequency laser by a mode of a high-Q WGM microresonator was developed.
The standard multimode laser model can be represented as a system of differential rate equations [39, 52]:
where I is the diode current, e is the electron charge, N is the number of excited electrons, τsis the lifetime of the excited electron, Sl, Gl and are, correspondingly, the number of photons, the coefficient of the linear gain (stimulated emission of photons), and the stimulated emission coefficient in the laser mode l, Gth is the threshold gain, Fl is the spontaneous emission coefficient:is the spontaneous emission factor (normalized over the time of radiative recombination of excited electrons τr), Δλ is the spontaneous emission width, λl is the wavelength of the mode l, λpeak, is the central wavelength of the laser. The magnitude of the threshold gain is determined by the design features of a particular laser and in the simplest case for a laser cavity consisting of two mirrors with reflection coefficients R0 and Re, one can obtain the following expression:
where nD is the refractive index of the diode, c is the speed of light and αlossis material loss factor. The gain in each mode depends on a combination of such effects as stimulated emission of photons, spectral hole burning, spectral hole burning due to neighboring modes, and asymmetric mode interaction. For the gain factor Gl, the following expression can be written [38, 52]: where is the self-saturation coefficient (spectral hole burning), is the coefficient of symmetric cross-saturation (spectral hole burning due to neighboring modes), - the asymmetric mode interaction coefficient (Bogatov effect) [53].The coefficient is determined both by the number of excited electrons and by the dispersion of the linear gain:
where θ is the differential gain, Ng is the number of excited electrons at which the laser diode becomes optically transparent, D is the linear gain dispersion coefficient.The effect of asymmetric interaction of modes was first described by Bogatov in [26], where a model of stimulated scattering of laser light on the dynamic electron density inhomogeneities was introduced as the theoretical explanation of this effect. The model proposed by Bogatov describes the change in the permittivity δϵ, caused by the dynamic inhomogeneity of the electron density due to the stimulated emission of the excited electrons under the influence of mode interference. The expression obtained by Bogatov for the variation of the dielectric constant can be rewritten in terms of the gain of a laser active region. Thus we got that the coefficient of asymmetric gain (Bogatov coefficient) is described by the following expression:
where Ωl(k)= ωl − ωk are laser modes offsets, S = Σ Sl is the total number of photons, α is the linewidth enhancement factor.4. Laser model with optical feedback
A simplified laser model with the feedback generally includes three mirrors (the front and back facet mirrors of the laser diode and an external mirror). In this approach, a part of the laser beam, reflected from the external mirror, returns back into the laser, providing feedback. The dynamics of this system can be described via the equations of Lang and Kobayashi [39], where the feedback term for electric field amplitude El is introduced. Taking into account that the photon number , the expression for the δSfeedback contribution to the dynamics of the mode intensity in Eq. (7) can be obtained as follows:
where is total feedback rate, ϕl(t) is the phase of the mode, ψl = ωlτ + arg(Γ(ω), τ is the round trip time from laser to the reflector and back.To modify the system of the Lang-Kobayashi equations for the case when the high-Q optical microresonator acts as an external mirror, it is sufficient to replace the reflection coefficient of the mirror Γ(ωl) in the expression for the total feedback rate in Eq. (13) with an expression for the frequency-selective reflection coefficient of the WGM microresonator [40]. To simplify the analysis, we assumed that each laser mode interacts efficiently with only one mode of the microresonator. This assumption is evidently justified for the case when the FSR of the laser is larger than the resonator mode spacing.
By tuning the laser frequency one can achieve the regime when a certain mode of the laser ωl=p becomes close enough to some mode of the optical microresonator ωm. In this case, the feedback to this laser mode from the microresonator increases dramatically, and the locking of the laser mode to the high-Q mode of the optical microresonator occurs. In stationary regime we can assume that , so the feedback expression is simplified:
where δlp is the Kronecker symbol, meaning that the feedback is added only to the mode closest to the frequency of the WGM. We further assume that the feedback phase is tuned to integer multiple of 2π to maximize the feedback. The model parameters were estimated by fitting the observed self-injection locked spectra with numerically calculated curves [see Fig. 2(b)]. Then we checked that by removing the feedback we are getting the free-running laser spectrum [see Fig. 2(a)].5. Strong feedback and energy conversion
The emission spectrum envelope in the model of a free-running multi-frequency laser [Eq. (6)-Eq. (7)] is mainly defined by the linear gain dispersion [see Eq. (11)] and spontaneous emission [see Eq. (8)]. In the model of a laser with an optical feedback, the frequency-selective feedback coefficient introduced in Eq. (14) additionally to the dispersion of the linear gain also plays an important role. We revealed that if one laser mode p is self-injection locked to the resonator mode, the feedback coefficient of this mode can compensate the dispersion term of the linear gain. The total gain of this mode then exceeds the gain of the central mode (zero-dispersion mode) with wavelength λpeak. This enhances the power of the mode p to that comparable of the central mode. Further feedback enhancement can lead to the strong feedback – “complete” suppression of other modes
In this case, the mode p uses all the excited electrons produced by the pump current, which simplifies the electron/photon dynamics equation (6), allowing the summation to be omitted. Consequently, this process effectively transfers the energy of the laser modes into the locked mode. It is the strong feedback when a multi-frequency laser becomes effectively a single-frequency one.
To obtain the strong feedback condition, we derive N from from stationary form of Eq. (7) for the locked mode and substitute into in stationary form of Eq. (7) for other modes. In this way we find a relation between Sp and Sl. For the consistency of our initial assumptions and solution, it is necessary that the condition described by Eq. (15) should follow from this solution. In this way we get the following criterion for the strong feedback:
The physical meaning of this statement is that for the efficient spectrum conversion the strong feedback should be greater than the spontaneous emission rate.
A series of measurements of the self-injection locked multi-frequency laser emission spectrum at different feedback levels were carried out. Figure 5(a) shows several experimentally obtained states of the self-injection locking regime when the level of the optical feedback was controlled via the gap between the microresonator and the prism. Performing numerical modeling with different feedback levels we obtained theoretical curves shown in Fig. 5(a). A good correspondence between theory and experiment is observed.
Fig. 5 a: Experimentally obtained spectra of the self-injection locked laser at different feedback levels (colored solid lines) with numerically calculated envelopes (black lines) for Γ1= 1 × 10−2, Γ2= 1.2 × 10−2, Γ3= 1.5 × 10−2, respectively. The green spectrum is not approximated well; b: Numerically calculated dependence of the single mode energy concentration (η) on the feedback level (blue line) and experimentally obtained energy concentration points (squares). A circle corresponds to the green spectrum in upper figure (a) and a triangle corresponds to the free-running laser [Fig. 2(a)]. The model parameters are shown in Table 1.
In the strong coupling regime [the purple curve in Fig. 5(a)] a single narrow line and the maximum suppression of other spectrum lines are observed. When the gap increases, the suppression decreases. The green, red and blue lines in Fig. 5(a) show that at lower coupling efficiency, the intensity of the suppressed modes begins to grow. At a certain threshold level of the feedback, other lines start to appear in the optical spectrum [green curve in Fig. 5(a)], above which the locking is destroyed. The feedback is weak near the threshold level, in other words, the power of backward wave is not enough for the stable self-injection locking.
To estimate the efficiency of the spectrum collapse, a parameter describing concentration of the energy in the locked mode was introduced and calculated using the developed model. The curve in Fig. 5(b) shows the numerical estimation of η for different feedback levels Γ together with the experimental dots. Note, that after entering the strong feedback regime [according to Eq. (16)] the concentration quickly grows and reaches the value of about 96% that corresponds to transition to a single-frequency generation. The energy concentration near the threshold level can be considered as an estimated value only. All numerical results obtained from the developed model are in a good agreement with the experimental data. The model parameters are shown in Table 1.
The measured power feedback level |Γ(ωp)|2 (about ~ 10−4) was high enough for single frequency lasing. Similar measurements when the laser diode was stabilized by other WGM modes gave estimates of the optical feedback level of about 10−3 − 10−4. The same feedback level was demonstrated with a DFB laser locked to high–Q microresonator [56]. The obtained results suggest that a higher level single mode energy concentration and line narrowing can be achieved by developing a technique for increasing the feedback. It should be noted that arbitrary increase of the feedback will decrease the output power of the stabilized laser. However, according to Fig. 5 strong feedback can be small enough not to degrade the output power.
6. Conclusion
We proposed and demonstrated experimentally for the first time a technique that allows to create a powerful single-frequency narrow-linewidth compact coherent laser source. The technique is based on the self–injection locking of a multi-frequency diode laser with a comparatively broad spectrum consisting of a large number of lines to the mode of a high-Q microresonator resulting in a spectrum collapse to a single line with a sub-kHz linewidth. Besides that, total initial power redistributes in favor of the locked mode providing high-power single-frequency generation. The analytical model describing dynamics and characteristics of the considered laser was developed. Analytical expression for the threshold feedback level for the effective spectrum transformation was derived. Numerical results are in a good agreement with experimental data. Novel types of self-injection locked diode lasers operating simultaneously at several wavelengths with narrow linewidths are also demonstrated. The proposed method may be applied for different types of diode lasers operating in different spectral ranges.
Funding
Russian Science Foundation (17-12-01413).
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