Self-injection locking of a laser diode to a high-Q WGM microresonator

N. M. Kondratiev; V. E. Lobanov; A. V. Cherenkov; A. S. Voloshin; N. G. Pavlov; S. Koptyaev; M. L. Gorodetsky

doi:10.1364/OE.25.028167

1. Introduction

Stable narrow-linewidth lasers are of great importance for many applications in science and technology, such as precision frequency metrology, high-resolution spectroscopy, optical sensors, atomic clocks as well as tests of fundamental physical theories. Compact and low-cost diode lasers cover nearly the entire optical spectrum. However, their natural linewidth and stability are not sufficient for such applications, while additional stabilization efforts strongly diminish their advantages.

Several methods to stabilize diode lasers are known. One possibility to achieve simultaneously high power and narrow linewidth is to transfer the narrow frequency spectrum of a well-stabilized but low-power master laser to a high-power broad-spectrum slave diode laser using optical injection [1]. However, such systems are rather complicated and very sensitive to ambient conditions. The linewidth reduction can also be achieved by locking a laser diode to an external high Q-factor reference cavity. Active locking, like Pound-Drever-Hall (PDH) technique [2–4], is the most used conventional method, requiring optical modulation and electronic feedback circuitry. The side-of-fringe stabilization [5] provides locking without optical modulation but requires stable laser intensity and the reference level.

Passive stabilization of semiconductor lasers uses resonant optical feedback from an external optical element [6–12] e.g. diffraction, Bragg or holographic gratings in Littrow or Littman configuration [13–17], high finesse cavities, such as Fabry-Perot (FP) resonators [18–21] and their combinations [22, 23].

High-finesse FP cavities successfully used for many laser stabilization applications are comparatively bulky, while high-finesse mirror coatings are specific to the chosen wavelength. These problems are absent for high-Q whispering gallery mode (WGM) microresonators [24] which look conveniently compatible with laser diodes. Current WGM microresonators are manufactured from different glass and crystalline materials such as quartz, alkaline earth fluorides (CaF₂, MgF₂, BaF₂, SrF₂), diamond, BBO, LiNbO₃ etc. and possess ultra-high Q-factor from UV [25] to MIR [26] region. Self-injection locking to a WGM microresonator uses resonant Rayleigh scattering [27, 28] on internal and surface inhomogeneities when a fraction of incoming radiation in resonance with the frequency of the selected WGM reflects back to the laser. This effect provides fast optical feedback and may result in a significant reduction of the laser linewidth. First demonstrated with fused silica spheres [29] this technique now reached commercial maturity by efforts of OEwaves company [30] with demonstrated linewidth narrowing and frequency stabilization of various lasers [31–33], including quantum cascade [34], fiber-loop [35] and DFB lasers [36] with the instantaneous linewidth down to sub-Hz level [37].

However, while the theory of self-injection locking with FP cavities is well developed (see for example [20]), only one early attempt for WGM microresonators is present [38] with a complicated system of equations and only analytical estimate for the coefficient of stabilization. We analyze theoretically the self-injection locking of laser diodes to a high-Q WGM microresonator and derive simple analytical expressions for the width of the locking band and the resulting laser linewidth. We also discuss the conditions of laser frequency tuning by variation of WGM microresonator resonance frequency, e.g., using microresonator temperature control, piezo or electro-optic tuning. All analytical estimations are in good agreement with experimental results.

2. Laser dynamics

We assume for simplicity a reduced model of the laser cavity consisting of two mirrors (FP cavity) which can be formed by cleavage, coating or distributed Bragg grating and consider the case of a single-frequency oscillation. This simple model allows to describe the self-injection locking mechanism adequately and may be further augmented if required. In this model the slowly-varying laser cavity field A(t)e^iϕ⁽^t⁾ with the amplitude A(t) and phase ϕ(t) reflected from the front laser mirror [see Fig. 1] can be found as a sum of the field B(t) that was transmitted back to the laser cavity due to backscattering in the WGM microresonator and retarded field reflected from both mirrors:

A (t) e^{i ϕ (t)} = i T_{o} B (t) + R_{e} R_{o} A (t - τ_{d}) e^{i ω τ_{d} + 2 (γ - α) L + i ϕ (t - τ_{d})},

where τ_d = 2nL/c is the round-trip time (n – refractive index, L – diode length, c – speed of light), R_o and R_e are amplitude reflection coefficients of the output and end laser mirrors,

T_{o} = \sqrt{1 - R_{o}^{2}}

is the output transmission – we used in Eq. (1) an agreement for the unitary symmetric scattering matrix S = ((R, iT)), (iT, R)) [39], ω is the instantaneous frequency of the generated light, γ and α are the material gain and loss coefficients.

Fig. 1 Schematic of the WGM microresonator-based self-injection locking.

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We now choose a laser cavity resonance frequency ω_dτ_d = 2πM (M is integer) so that it is the nearest to the generation frequency. Expanding $A (t - τ_{d}) ≃ A - τ_{d} \dot{A}$ and $ϕ (t - τ_{d}) ≃ ϕ - τ_{d} \dot{ϕ}$ in Taylor series of τ_d to the first order and assuming that the detuning is much smaller than the free spectral range (1/τ_d), we keep the terms up to the first order of smallness and rewrite Eq. (1) in the following form:

\dot{A} + i \dot{ϕ} + (\frac{κ_{d}}{2} - \frac{g (| A |^{2})}{2} (1 + i α_{g}) - i (ω - ω_{d})) A = i \frac{κ_{d o}}{T_{o}} B e^{- i ϕ} .

Here

κ_{d o} = \frac{T_{o}^{2}}{τ_{d} R_{e} R_{o}}

is the output beam coupling rate, κ_d is photon decay rate which is the sum of the decay rate due to the intrinsic losses κ_di and coupling with the environment κ_dc, g stands for the gain which depends nonlinearly due to saturation on internal intensity:

κ_{d i} = 2 α \frac{c}{n}, κ_{d c} = 2 \frac{1 - R_{e} R_{o}}{τ_{d} R_{e} R_{o}}, 2 γ \frac{c}{n} = (1 + i α_{g}) g (| A |^{2}) .

For high-reflection mirrors, when

R_{o, e} \approx 1 - T_{o, e}^{2} / 2

, one gets κ_dc ≈ κ_do + κ_de. Note, that for similar mirrors κ_dc = 2κ_do. The phase – amplitude coupling factor α_g, which is of the order of unity, was introduced to take into account the fact that in real diode lasers γ is complex because injected carriers provide not only the gain but also change the refractive index and hence the phase. For analysis of the self-injection locking effect, we consider only the simplest case of one laser mode and one WGM cavity mode.

2.1. Self-injection locking

First, we derive the expression for the optical field amplitude B(t) reflected from the WGM microresonator. Considering that the backscattering is weak (no observable microresonator’s mode splitting in doublets and taking only the leading terms, we get:

B (t) = \frac{i T_{o} Γ (ω)}{R_{o}} A (t - τ_{s}) e^{i ω τ_{s} + i ϕ (t - τ_{s})},

where τ_s is the round-trip time in free space from laser to reflector and back and Γ(ω) is frequency-dependent complex reflection coefficient of the WGM microresonator due to Rayleigh backscattering. Substituting (4) into (2) and separating real and imaginary parts we obtain two equations:

\dot{A} = (g (| A |^{2}) / 2 - κ_{d} / 2) A - \frac{κ_{d o}}{R_{o}} | Γ (ω) | A (t - τ_{s}) \cos ψ,

\dot{ϕ} = α_{g} g (| A |^{2}) / 2 + (ω - ω_{d}) - \frac{κ_{d o}}{R_{o}} | Γ (ω) | \frac{A (t - τ_{s})}{A (t)} \sin ψ,

where ψ = arg(Γ) + ωτ_s. Note that Eq. (6) reminds a famous Adler equation in the theory of injection locking [40], where ϕ (the difference between the pump and oscillator phases) stands in the sine function instead of ψ. However, in our case with passive resonator ψ does not depend on ϕ but only on its derivative in the next order of approximation as the initial phase of the laser is irrelevant. The system (5)–(6) has the form of Lang-Kobayashi equations [41, 42] without specifying a gain mechanism and with generalized frequency-dependent reflection. While the first difference is inessential for the analysis, the second difference is important.

Multiplying the stationary form of Eq. (5) by α_g and subtracting it from the stationary form of Eq. (6), we eliminate g(|A|²), obtaining the following equation:

ω - {\bar{ω}}_{d} = | Γ (ω) | {\bar{κ}}_{d o} \sin (ψ - \arctan α_{g}),

where

{\bar{ω}}_{d} = ω_{d} - α_{g} κ_{d} / 2

determines the free running laser frequency (hot resonance). For convenience we have also introduced an effective (hot) output beam coupling rate

{\bar{κ}}_{d o} = \frac{κ_{d o}}{R_{o}} \sqrt{1 + α_{g}^{2}}

.

For the reflection of the wave with frequency ω close to the WGM resonance ω_m from microresonator we use the steady state expression derived in [28]:

Γ (ω) = - \frac{i η κ_{m}^{2} β / 2}{{(κ_{m} / 2 - i (ω - ω_{m}))}^{2} + {(β κ_{m} / 2)}^{2}} .

Here κ_m = κ_mi + κ_mc is the microresonator’s mode decay rate with κ_mi determined by the intrinsic losses and κ_mc determined by the external coupling, η = κ_mc / κ_m characterizes coupling efficiency (η = 1/2 corresponds to the critical coupling, η ≈ 1 – to the strong overcoupling), and β is the dimensionless coupling rate between counterpropagating modes, normalized for convenience to κ_m/2. The parameter β is directly related to the mode splitting (if β > 1 then it equals to a distance measured in linewidths between resonant frequencies in doublets), first observed in silica microresonators [43]. However, in high-Q crystalline microresonators scattering and hence backscattering is much smaller (β ≪ 1) and mode splitting is nearly absent (usually not directly observable in experiments).

Then, from Eq. (7) we obtain the dependence of the generated frequency ω on the laser cavity frequency ${\bar{ω}}_{d}$ (stationary tuning curve):

{\bar{ω}}_{d} - ω_{m} = (ω - ω_{m}) + \frac{η κ_{m}^{2} β {\bar{κ}}_{d o}}{2} \frac{(ω - ω_{m}) κ_{m} \cos \bar{ψ} + ((1 + β^{2}) κ_{m}^{2} / 4 - {(ω - ω_{m})}^{2}) \sin \bar{ψ}}{{(κ_{m}^{2} / 4 (1 + β^{2}) - {(ω - ω_{m})}^{2})}^{2} + {(ω - ω_{m})}^{2} κ_{m}^{2}},

where

\bar{ψ} = ω τ_{s} - \arctan α_{g} + 3 / 2 π

is the phase delay. Note, that we have added 3π/2 to the phase delay (however, not losing the generality) to get the most advantageous regime corresponding

\bar{ψ} \approx 0

, shown further. This equation describes the considered effect of the self-injection locking to a high-Q microresonator. Eq. (9) can be significantly simplified by introducing relative detuning of the hot resonance in the laser cavity from the nearest resonance in the microresonator

ξ = 2 ({\bar{ω}}_{d} - ω_{m}) / κ_{m}

and the generated frequency detuning from the same resonance in the microresonator ζ = 2(ω − ω_m)/κ_m:

ξ = ζ + \frac{K}{2} \frac{2 ζ \cos \bar{ψ} + (1 + β^{2} - ζ^{2}) \sin \bar{ψ}}{{(1 + β^{2} - ζ^{2})}^{2} + 4 ζ^{2}} .

Here

K = 8 η β {\bar{κ}}_{d o} / κ_{m}

is the combined coupling coefficient. This parameter K is analogous to the injection parameter C used in a theory of self-mixing interferometers [41], where the self-injection is achieved with frequency-independent reflector forming an additional Fabry-Perot cavity.

Now we look at the influence of the delay time τ_s (e.g. laser-microresonator distance) on the self-injection locking, and the effect is twofold. It is important that in general $\bar{ψ}$ also depends on ω, and hence on ζ. Indeed, if we assume $\bar{ψ} (ω_{m}) = ψ_{0}$ , then we obtain:

\bar{ψ} = ψ_{0} + (ω - ω_{m}) τ_{s} = ψ_{0} + \frac{κ_{m} τ_{s}}{2} ζ .

First, the initial phase delay ψ₀ determining a locking regime may be adjusted by tuning τ_s or ω_m in small range |δ(ω_mτ_s)| < π. This moves the whole curve along the ζ = ξ line and may result either in additional detuning of the locked laser frequency from the WGM resonance [see Fig. 2(a)] or in frequency jump. Second, the coefficient κ_mτ_s before ζ depends on the total delay time τ_s (laser – microresonator distance). When this coefficient increases, unwanted fringes and regions of multistability on self-injection locking curve ζ(ξ) [see Fig. 2(b)] or equivalently new extrema on ξ(ζ) dependence appear. We’ve checked numerically that for ψ₀ = 0 the limiting condition may be well approximated as κ_mτ_s/2 < 4.7K^−0.36. In practice, however, this limit is not very relevant, as for microresonators with high Q-factor Q_m = ω_m/κ_m ~ 10⁹ it corresponds to distances of the order of tens of meters. Much more important, however, is the requirement on the stability of this distance δd:

\frac{δ ω}{δ d} = \frac{κ_{m}}{2} \frac{\partial ζ}{\partial ψ_{0}} \frac{2}{c} \frac{\partial ψ_{0}}{\partial τ_{s}} = {(\frac{\partial ξ}{\partial ψ_{0}} / \frac{\partial ξ}{\partial ζ})}_{ψ_{0} = 0} \frac{κ_{m} ω_{m}}{c} = \frac{K}{(K + 1)} \frac{ω_{m}^{2}}{2 Q_{m} c} .

As follows from Eq. (12) at Q_m = 10⁹, for high enough K and the required linewidth δω = 2π × 1kHz the stability of the distance should be kept at the level of δd = 2.5 nm.

Fig. 2 (a) Tuning curves (10) for different initial phases: I–IV correspond to phases ψ₀ = [0, π/3, 2π/3, π] with κτ_s ≪ 1. The envelope for the family of curves with different ψ₀ is shown with black dash-dotted line, the green solid line shows a tuning curve for a free running laser; (b) Tuning curves I-IV corresponding ψ₀ = 0 and long delay, so that κ_mτ_s/2 = [0, 1, 2, 3]; (c) Tuning curves for different K values: I–III correspond to ψ₀ = π, and K = [5, 3, 1], lines IV–VII correspond to ψ₀ = 0, K = 0, 2, 4, 6 ; (d) An optimal self-injection locking curve with ψ₀ = 0, κ_mτ_s ≪ 1, and K ≫4. Unstable branches are shown with dashed lines, and bistable transitions with blue arrows.

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The dependence of the stationary tuning curve (10) on parameters ψ₀, τ_s and K is demonstrated in Fig. 2. For illustrative Fig. 2 we assumed K = 30, β = 0.01. From Fig. 2 one can see that for large enough K the tuning curve for all values of the phase delay has a positive flat branch with comparable width and slope. For negative cos ψ₀ [Figs. 2(a), curves III and IV, and 2(c), curves I–III] the central branch is unstable and less efficient locking is only possible on upper or lower stable branches having a steeper slope. We are mostly interested in regimes, when ψ₀ ~ 0 and K ≫ 1 like the one in Fig. 2(d). Note, however, that other regimes are used in self-mixing interferometry [41].

It is well-known that in a three-solution region the middle branch is unstable. Let’s consider the curve in Fig. 2(d). If the free running laser frequency ${\bar{ω}}_{d}$ is initially far detuned from the WGM mode and then tuned by current or temperature from the left (low frequency) or right (high frequency) towards the WGM frequency, the generated frequency goes along the metastable branch of the tuning curve (solid line) closely following ${\bar{ω}}_{d}$ until a turning. At the turning point, it jumps (short blue solid arrows) to the inner stable branch with the ultra-low slope where it is now self-locked. Changes (fluctuations) of the laser cavity frequency ${\bar{ω}}_{d}$ on this branch result in negligible changes in the generated frequency ω until they are large enough to reach other turning points marked by long blue arrows. The width of the flat slope region is called the locking band [thick red solid line in Fig. 2(d)].

3. Analytical estimations

Now we assume that the phase delay is adjusted for the most preferential regime (cos ψ₀ = 1) and as κ_mτ_s, β ≪ 1 Then, Eq. (10) simplifies to

ξ = ζ + K \frac{ζ}{{(1 + ζ^{2})}^{2}} .

The function ξ(ζ) can have 0, 1, or 2 pairs of extrema (turning points), thus providing different regimes of the self-injection locking. These regimes are defined by the values of K and can be found solving the bi-cubic equation obtained from

\frac{d ξ}{d ζ} = 0

condition:

ζ^{6} + 3 ζ^{4} - 3 (K - 1) ζ^{2} + K + 1 = 0 .

Using the discriminant of the cubic Eq. (14) we find that K > 4 corresponds to strong coupling and locking with hysteresis [Fig. 2(c), curve VII]. For the locking regime two pairs of extrema exist for Eq. (13):

\begin{array}{l} ζ_{1, 2} = \pm \sqrt{2 \sqrt{K} \cos (\frac{π}{3} + \frac{1}{3} \arccos (\frac{2}{\sqrt{K}})) - 1} = \pm \sqrt{3} (\frac{1}{3} + \frac{32}{81 K}) + O (K^{- 2}), \\ ξ (ζ_{1, 2}) = \pm \sqrt{3} (\frac{3}{16} K + \frac{1}{3}) + O (K^{- 1}), \end{array}

\begin{array}{l} ζ_{3, 4} = \pm \sqrt{2 \sqrt{K} \sin (\frac{π}{6} + \frac{1}{3} \arccos (\frac{2}{\sqrt{K}})) - 1} = \pm (\sqrt[4]{3 K} - \frac{5}{6 \sqrt[4]{3 K}}) + O (K^{- 3 / 4}), \\ ξ (ζ_{3, 4}) = \pm (\frac{4 \sqrt[4]{3 K}}{3} - \frac{2}{3 \sqrt[4]{3 K}}) + O (K^{- 3 / 4}) . \end{array}

Thus

ζ_{1} - ζ_{2} \approx \pm 2 \sqrt{3} / 3

defines the stabilized generated frequency range, and the locking range can be found as

ξ_{1} - ξ_{2} \approx \frac{3 \sqrt{3}}{8} K

. This regime may provide significant stabilization since the maximal stabilization coefficient

{\frac{d ξ}{d ζ} |}_{ζ = 0} = K + 1

is rather high.

For K < 4 there are no real roots of Eq. (14) and the function (13) is monotonous. This corresponds to the weak coupling regime with some frequency pulling [Fig. 2(c), curves IV, V and VI]. In this case the stabilization coefficient has the same expression but as K < 4 its value is lower. The estimate for the locking band obtained above can also be used.

When $\cos {\bar{ψ}}_{0} = - 1$ , Eq. (10) simplifies to Eq. (13) with the opposite sign before K. Note that for K < 1 we still have positive slope at ζ = 0 but no stabilization. For K > 1 and $\cos {\bar{ψ}}_{0} = - 1$ the slope at ζ = 0 is negative and only first two extrema of Eq. (15) are possible (with K replaced by −K). However, as in this regime the middle branch is unstable [see Fig. 2(c), curves I and II], the locking range is effectively halved. Due to the same reason the maximal stabilization coefficient corresponding to the minimal slope of the upper or lower branch is approximately four times worse ${\frac{d ξ}{d ζ} |}_{ζ = 1} = K / 4 + 1$ (ζ = ±1 is the lowest bending point). The above estimates confirm that positive-slope regime provides the widest locking band and the best frequency stabilization.

Let’s rewrite the above results in a simplified form using physical quantities. We assume that we are in a positive slope regime which provides best frequency stabilization and K ≫ 4. From Eq. (8) we get Γ_m = |Γ(ω = ω_m)| ≈ 2ηβ – amplitude reflection coefficient from the WGM cavity at resonance. Therefore, for the locking range we get

\frac{Δ ω_{lock}}{ω} \approx \sqrt{1 + α_{g}^{2}} \frac{Γ_{m}}{Q_{d}} .

We assumed here that laser cavity losses are mostly determined by its output mirror, its cold quality factor is

Q_{d} = \frac{ω}{κ_{d o}}

, and R_o ≈ 1. Note that as β (and hence Γ_m) is proportional to the WGM resonance Q-factor and density of scatterers, the locking band is effectively proportional to the ratio of the WGM to laser cavity quality factors. As the line width enhancement equals to the inverse square of the stabilization coefficient [7, 20], for stabilized laser we get

\frac{δ ω}{δ ω_{free}} \approx \frac{Q_{d}^{2}}{Q_{m}^{2}} \frac{1}{16 Γ_{m}^{2} (1 + α_{g}^{2})},

where

Q_{m} = \frac{ω}{κ_{m}}

and δω_free is a linewidth of the free running laser. Note that here, following [20], we assume that the laser frequency noise is white.

Typical WGM and laser cavity achievable parameters and corresponding estimates are presented in Table 1. Note that for a laser with several MHz linewidth we can reach the locked linewidth less than a Hertz that is in a good agreement with the results demonstrated in [37].

Table 1. Typical laser and crystalline WGM parameters and locking properties estimates.

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4. Frequency pulling

Another important effect of the self-injection locking is a possibility of laser frequency tuning by varying the WGM frequency (the effect of frequency pulling). In [36] it was demonstrated that following the WGM frequency, the laser frequency can be pulled up to 4 GHz. As the axes in Fig. 2 are presented in terms of detunings from the WGM frequency, one may conclude that while the laser is in the locked state (on the thick red line), and the difference between the laser and the WGM eigenfrequency is less than a half of the locking band, the difference between the generated frequency and WGM frequency is close to zero. In other words, the generated frequency will follow the WGM frequency unless the laser leaves the self-locking state.

Since ψ₀ depends on ω_m this tuning may result in the change of the locking regime. To diminish this effect the time delay (laser-to-cavity distance) should be as small as possible. If we presume that the phase delay change δψ₀ < π/4 [see Fig. 2(a)] does not harm the locking regime, and we want to tune the ω_m over the whole locking band ∆ω_lock, we get:

\frac{\partial ψ_{0}}{\partial ω_{m}} δ ω_{m} = τ_{s} Δ ω_{lock} < \frac{π}{4} .

For the locking band over 4 GHz the maximum laser to cavity distance is less than

d_{\max} = \frac{π c}{8 Δ ω_{lock}} \approx 0.5 cm

.

Interestingly, the frequency pulling in the self-locking regime may be useful for the efficient generation of the dissipative Kerr soliton (DKS) combs [44]. The DKS may only exist if the pump is red detuned from a WGM resonance to compensate Kerr frequency pulling. Usually, this can be achieved by steady CW laser tuning from the blue to the red slope. In the locking regime the generated frequency is rather close to the WGM resonance and detuning ζ may be close but not equal to zero. Its value is defined by the laser-microresonator detuning ξ and initial phase delay ψ₀: ζ ≈ ξ/K – ψ₀/2 at strong coupling regime and |ψ₀| < π. In this way, by varying the laser diode frequency or laser-microresonator detuning ξ (inside the locking band) one can control the generated light detuning [with slower 1/K rate, see Fig. 2], and reach detuning value sufficient for soliton generation. When switching to the soliton state occurs, the circulating CW power in the resonator drops down, and the WGM resonance frequency may abruptly shift due to thermorefractive and thermal expansion effects. In this case, the pump frequency may leave the soliton existence domain and solitons may disappear if the frequency tuning and thermal relaxation rates are not specially matched. However, in the self-locking regime, the pump frequency will follow the thermally-induced shift of the resonance frequency, and solitons survive [45].

5. Numerical Modeling and experimental results

To confirm our analytical results we simulated the dynamics of the self-injection locking process numerically. Since for the dynamical problem we can not use the steady state solution (8), we solve the complete system of equations for both forward and backward waves A⁺ and A⁻ propagating in the microresonator [28]. Furthermore, we have to specify a gain model in Eq. (2) for the laser field amplitude A. We use a rate equations model (without spontaneous emission) analogous to that in [38] introducing the the carrier density N:

\frac{d N}{d t^{'}} = J_{N} - \frac{κ_{N}}{κ_{m}} N - N g_{N} | A |^{2},

\frac{d A}{d t^{'}} = ((1 + i α_{g}) N g_{N} - \frac{κ_{d}}{κ_{m}} + i (ξ_{0} - v_{ω} t^{'})) A - 2 \frac{κ_{d o}}{κ_{m}} \frac{T_{m}}{T_{o}} e^{i ω_{m} τ_{s} / 2} A^{-},

\frac{d A^{+}}{d t^{'}} = - A^{+} + i β A^{-} - 2 η \frac{T_{o}}{T_{m}} e^{i ω_{m} τ_{s} / 2} A,

\frac{d A^{-}}{d t^{'}} = - A^{-} + i β A^{+} .

Here

J_{N} = \frac{2 I}{e V κ_{m}}

is the normalized injection current, κ_N is the relaxation rate of inverse population,

T_{m} = \sqrt{η κ_{m} τ_{m}}

and τ_m is the WGM cavity round-trip time, and g_N is the normalized carrier gain coefficient so that the term Ng_N replaces

\frac{g (| A |^{2})}{κ_{m}}

in Eq. (2).

We use the term i(ξ₀ − v_ωt′) in Eq. (21) to model a laser frequency tuning with initial laser-microresonator detuning ξ₀ = 2(ω_m − ω_d₀)/κ_m, and normalized laser frequency sweep rate v_ω. As the optical frequency is fixed at ω_m in these equations for slowly varying amplitudes, the generated frequency difference may be obtained from temporal oscillation of these amplitudes as $ω = ω_{m} - \frac{d}{d t} \arg (A)$ . For the modeling we used laser parameters that do not correspond to a specific laser system, but are reasonable for various construction and geometry of laser diodes [see Table 2].

Table 2. Typical laser parameters were used for numerical simulation.

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The results of the numerical simulations were compared with analytical results. Numerical results presented in Fig. 3(a) are in a very good agreement with the theory. Analytical curves are shown as dashed lines and we can see that the numerically calculated frequency closely follows it until the turning points, where regime switching occurs, as it was described in the end of section 2. We found, however, that if the distance between metastable branches is small (K is very high), the transition to the inner stable branch may occur earlier than the turning point. Our numerical simulations also allows to plot the dependence of the intensity |A⁺|² (which is directly proportional to the generated light intensity) on the detuning [Fig. 3(b)] that is absent in our simple analytical theory not considering the gain model. However, this dependence is easily experimentally observable and is in direct connection to frequency locking boundaries. We see that the form of the intensity curve for ψ₀ = 0 has characteristic rectangular shape that roughly corresponds to one half of the locking band.

Fig. 3 Comparison of analytical theory and numerical modeling. a) Curves I–IV - dependencies of generated frequency detuning from the laser frequency detuning for different phases $ψ_{0} = [0, \frac{π}{2}, π, \frac{3 π}{2}]$ . Theoretical tuning curves (10) are shown with dashed for comparison. b) Simulated dependence of the intensity inside WGM microresonator over the laser frequency detuning for different phase delays for the same parameters.

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Our theoretical models are also in a good agreement with experimental observations which we demonstrate here only as an illustration. We used the same scheme as shown in Fig. 1, with free–space single mode DFB laser diode (linewidth ∆λ ≈ 10 MHz, P_max = 10 mW, λ = 1550 nm) coupled using glass (BK-7) prism with the MgF₂ WGM resonator with a diameter 5.4 mm corresponding to FSR ≈ 13 GHz. This microresonator was manufactured by the precision single-point diamond turning machine (accuracy < 0.1 µm). The ultra-high intrinsic Q-factor exceeding 10⁹ was achieved by polishing with diamond slurries. An oscillogram of the output intensity when the laser current was swept is shown in Fig. 4. As the detected output signal results from interference of the pump and output optical fields, the resonance peaks turn into resonant dips and the response curves are vertically inverted as compared to Fig. 3(b).

Fig. 4 Experimental laser generated intensity. a) The frequency of a DFB laser diode is swept by current through different WGM resonator modes. Arrows indicate modes that have different initial phase delays ψ₀ analogously to 3(b). b) Responses for the same mode with different adjusted phases ψ₀

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Besides the fundamental modes, many non–fundamental modes with different values of phase delay $\bar{ψ}$ were excited in the microresonator. One may notice good qualitative agreement of the curves shown in Fig. 3(b) – simulations, and in Figs. 4(a) and 4(b) – experiment. Note, that the output signal from the resonator due to interference with the input is inverted corresponding to the intracavity field.

6. Conclusion

We analyzed the process of the self-injection locking of a diode laser to a high-Q WGM microresonator and found simple analytical formulas for the width of the locking band and the resulting laser linewidth. The locking range is found to be proportional to the ratio of microresonator’s to laser’s quality factors which can be extremely high. Furthermore the laser linewidth enhancement was found to be square-inverse proportional to this ratio allowing for sub-Hertz line narrowing. The laser-to-microresonator distance stability restriction was estimated to be of the order of nanometers. The conditions of effective laser frequency tuning by variation of WGM microresonator resonance frequency (frequency pulling effect) was also found to be applying limitation on this distance. It follows from our theory that the laser and the microresonator should be as close as possible.

Funding

Russian Science Foundation (17-12-01413).

Acknowledgments

A.V. Cherenkov acknowledges support from the Foundation for the Advancement of Theoretical Physics “BASIS”.

References and links

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Parameters			Properties
Laser quality factor	Q_d	10⁴	Coupling coefficient	K	10⁴
WGM quality factor	Q_m	10⁹	Locking range	∆ω_lock	700 MHz
WGM reflection (ampl.)	Γ_m	10⁻²	Line narrowing	δω/δω_free	10⁻⁸
Line enhancement factor	α_g	2.5	Laser to cavity distance	d_max	3 cm

J_N	1400/π	α_g	2.5	ξ₀	−8	β	0.05
κ_N/κ_m	75	κ_d/κ_m	75	v_ω	π/2000	η	0.5
g_N	75	κ_do / κ_m	75	ω_mτ_s	3π/2 + 1.2	T_m/T₀	0.04

Parameters			Properties
Laser quality factor	Q_d	10⁴	Coupling coefficient	K	10⁴
WGM quality factor	Q_m	10⁹	Locking range	∆ω_lock	700 MHz
WGM reflection (ampl.)	Γ_m	10⁻²	Line narrowing	δω/δω_free	10⁻⁸
Line enhancement factor	α_g	2.5	Laser to cavity distance	d_max	3 cm

J_N	1400/π	α_g	2.5	ξ₀	−8	β	0.05
κ_N/κ_m	75	κ_d/κ_m	75	v_ω	π/2000	η	0.5
g_N	75	κ_do / κ_m	75	ω_mτ_s	3π/2 + 1.2	T_m/T₀	0.04

Self-injection locking of a laser diode to a high-Q WGM microresonator

Abstract

1. Introduction

2. Laser dynamics

2.1. Self-injection locking

3. Analytical estimations

4. Frequency pulling

5. Numerical Modeling and experimental results

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (4)

Tables (2)

Equations (23)

Optics Express