1. INTRODUCTION

In recent years, optical microfibers (OMs) have attracted attention for a range of nonlinear applications such as supercontinuum generation [1,2], pulse shaping [3], second harmonic generation [4–6], and third harmonic generation (THG) [3,6–11], as well as the potential for photon-triplet generation and downconversion [12–14], due to their strong modal confinement. OMs are typically fabricated by heating and tapering optical fiber to a diameter comparable to the wavelength, which in conjunction with the large glass–air refractive-index contrast offers modal areas down to a few square micrometers. For OMs fabricated from standard silica single-mode fiber, the effective nonlinearity $\gamma$ can be enhanced by approximately two orders of magnitude [15,16]. Furthermore, the large evanescent field in the surrounding air may be easily accessed and exploited to self-couple light between adjacent segments of the OM and thus form loop [17,18], knot [19], and microcoil [20] resonators in which the high internal amplitudes near resonance would be ideal for reducing the input threshold powers required to observe nonlinear effects.

In this work, we study resonantly enhanced THG in loop resonators, which are the simplest of the resonator geometries. Harmonic generation in microfibers relies on intermodal phase matching of the fundamental pump mode to a higher-order harmonic mode possessing a similar effective index, a condition that is satisfied at certain critical OM diameters [9]. While it is theoretically possible to attain efficiencies exceeding 50% over 5 cm in a straight silica OM at 1 kW power levels [9], in practice the reported conversion rates have often been limited by the fabrication difficulties in maintaining the required diameter and uniformity over such extended lengths [21,22]. In particular, the tolerance of the OM diameter must be of the order of 1 nm in order to achieve conversion rates of several percent. Whether the phase matching occurs at the waist of a parabolic taper profile or in the transition regions, this constraint greatly restricts the effective interaction length over which the third harmonic signal can grow. For this reason, the loop resonator [18], which is typically only a few millimeters in length, provides a convenient technique to improve the efficiency without the need for more expensive, higher-power sources. The effects limiting the efficiency of THG in the microfiber do not limit the resonant enhancement, and so the problem of fabricating the fiber diameter to a high tolerance is thus traded for the far less challenging problem of fabricating a high-$Q$ resonator. An equivalent resonantly enhanced harmonic-generation technique has also been reported in ring resonators [23–25] and microtoroids [26], and THG was noted as an additional nonlinear effect during pulse-shaping experiments [3].

The following sections discuss the effect of the pump light resonance on the conversion efficiency, focusing on the influence of the coupling and loss parameters (which are both known to dictate the resonance characteristics) for a silica loop resonator. At higher powers, the transmission is modified by hysteresis, and the effect of bistability on the THG is analyzed, as well as the coresonant case in which the harmonic signal is partially recirculated.

2. THEORETICAL MODELING DETAILS

The ideal loop-resonator geometry is illustrated in Fig. 1, formed from a microfiber arranged into loop and coupling regions with lengths $L_0$ and $L_c$, respectively. Nonlinear phase-modulation effects and THG are assumed to take place throughout the entirety of the resonator. Typically, the diameter of a loop varies between several hundred micrometers to a centimeter, while the coupling length can vary considerably according to the tightness of the loop. The coupling coefficient between the adjacent OMs ${\kappa }_{\omega ,3\omega }$ falls exponentially with their separation, and so it can be assumed that the coupling is only significant over a small range $L_c\ll L_0$.

 

Fig. 1. Schematic of the microfiber loop resonator. The amplitudes in the two arms of the coupling region are denoted $A_1(s)$ and $A_2(s)$, and the amplitudes in the loop are $A_0(s)$.

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Here, we focus on simulations using the loop parameters listed in Table 1, which can be readily realized using manual stages without the need for specialist equipment after the microfiber is fabricated, with $L_0=3\mathrm{mm}$ and $L_c=50\mathrm{\mu m}$. Because the ${\mathrm{HE}}_{11}$ mode at the harmonic wavelength has a higher effective index than that at the pump wavelength, the two fundamental modes always experience a phase mismatch, and so the ${\mathrm{HE}}_{11}(\omega )$ mode must instead be intermodally phase matched to a higher-order harmonic mode. Of all such higher-order harmonic modes, the ${\mathrm{HE}}_{12}(3\omega )$ mode experiences the largest modal overlap with the ${\mathrm{HE}}_{11}(\omega )$ pump mode, and in order to ensure their phase matching the OM diameter is therefore chosen to be 767 nm, at which both modes have an effective index of $n_{\text{eff}}=1.08$ for a pump wavelength of ${\lambda }_{\omega }=1.55\mathrm{\mu m}$. Only these two modes experience any significant power exchange because the microfiber is single moded at $\lambda ={\lambda }_{\omega }$ with a $V$ number of 1.6, and all other harmonic modes are far from phase matched.

Table 1. Summary of Parameters Used in the Simulation of the Loop Resonator when Phase-Matching the ${\mathrm{HE}}_{11}(\omega )$ Mode with the ${\mathrm{HE}}_{12}(3\omega )$ Mode for ${\lambda }_{\omega }=1.55\mathrm{\mu m}$

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In the loop region, the coupled-mode differential equations describing the evolution of the copropagating pump and third-harmonic-mode amplitudes $A_0^{\omega }$ and $A_0^{3\omega }$ are adapted from [9] to include loss:

Equations (2a) and (2b) give the differential equations for the modes $A_1^{\omega }$ and $A_1^{3\omega }$ propagating in the first arm of the coupling region when $i=1$ and $j=2$, and likewise for the second arm $A_2^{\omega }$ and $A_2^{3\omega }$ when $i=2$ and $j=1$. These equations are then solved iteratively with the boundary conditions for field continuity:

The free spectral range (FSR) of the 3 mm loop is $\mathrm{FSR}\approx {\lambda }^2/(n_{\text{eff}}{L}_0)=740\mathrm{pm}$ near the 1.55 μm pump wavelength. A larger resonant enhancement would be expected if the loop resonator were nearer to critical coupling, which occurs when $K_c={\kappa }_c{L}_c=\pi (2m+1)/2$ for integer $m\ge 1$ [18] with the lowest value ($m=1$) of ${\kappa }_c=9.4\times {10}^4{\mathrm{m}}^{-1}$. However, the power within a critically coupled loop may be several orders of magnitude larger than the original input power, which would damage the microfiber—in particular, problems associated with hotspots typically become problematic if the peak power of nanosecond pulses in an OM exceeds ${10}^4\mathrm{W}$. Furthermore, the loop will often be slightly undercoupled or overcoupled due to fabrication limitations in controlling the spacing of the microfibers, and for these reasons we study the dependency of the resonant enhancement on proximity to critical coupling for cases where $\mathrm{\Delta }K=({\kappa }_c-{\kappa }_{\omega })L_c\approx 1$ (rather than zero).

Powers in excess of 100 W were studied, which can be straightforwardly achieved in experiments using pulsed sources. It is worth noting from previous reports that such high powers can induce thermal phase shifts arising from the temperature-dependent refractive index and thermal expansion, which introduce their own nonlinearities and hysteresis into the power-transfer function of a microfiber resonator [27]. However, provided that the repetition rate of the source is greater than the inverse of the thermal response time (typically of the order of 0.1–1 ms, depending on the OM diameter and surrounding environment), we may assume here that the resonator exists in a dynamic thermal equilibrium in which the loop geometry and linear/nonlinear ($n_2$) index are steady state, which along with the loss and coupling coefficients would collectively determine the resonance characteristics [18,28].

3. DISCUSSION

First, the transmitted pump $P_{\omega }$ and harmonic $P_{3\omega }$ output powers will be studied for different input powers, as shown in Fig. 2, where the pump is detuned from a resonance at ${\lambda }_R\approx 1550\mathrm{nm}$ (to avoid confusion with the phase-matching detuning $\delta \beta$, this detuning is denoted as $\delta \lambda$). The coupling between pump modes is chosen to be ${\kappa }_{\omega }=8\times {10}^4{\mathrm{m}}^{-1}$, which is achievable with close contact between the two microfibers and corresponds to $\mathrm{\Delta }K=0.71$ (for reference, values of $\mathrm{\Delta }K=0$ and $\mathrm{\Delta }K=3\pi /2$ would correspond to the critically coupled and totally uncoupled extremes, respectively). Because the higher-order harmonic mode’s transverse profile contains more oscillations/zeroes, ${\kappa }_{3\omega }$ is roughly an order of magnitude smaller than ${\kappa }_{\omega }$. Furthermore, in the most general case, the resulting third-harmonic signal does not necessarily coincide with a resonance, and so ${\kappa }_{3\omega }$ is set to zero here (situations in which ${\kappa }_{3\omega }>0$ are considered later).

 

Fig. 2. Pump $P_{\omega }$ and third harmonic $P_{3\omega }$ output power from a loop resonator when input pump power $P_0$ is (a) 100, (b) 250, and (c) 600 W. The labeled dots indicate the position of the detunings for the transfer characteristics shown in Fig. 3. Parameters: $L_0=3\mathrm{mm}$, $L_c=50\mathrm{\mu m}$, ${\alpha }_{\omega ,3\omega }=5{\mathrm{m}}^{-1}$, $\kappa =8\times {10}^4{\mathrm{m}}^{-1}$, and $\delta \beta \approx 1440{\mathrm{m}}^{-1}$.

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We choose $\delta \beta \approx -1440{\mathrm{m}}^{-1}$, which is close to the optimum detuning for THG in the loop resonator. The value is negatively offset from zero to compensate for the nonlinear phase shifts, although a smaller but nonetheless significant third-harmonic signal would still be detectable if $\delta \beta =0{\mathrm{m}}^{-1}$. It should be noted that the magnitude of the optimum detuning for THG in a loop resonator, which experiences stronger SPM/XPM on resonance, is generally larger than that of the straight microfiber (for the OM, the ideal detuning and input parameters can be deduced by finding soliton solutions to the third-harmonic interaction that offer the greatest conversion [12]).

For an input power of $P_0=100\mathrm{W}$, Fig. 2(a) confirms that the pump’s nonlinear resonance spectrum appears asymmetrically skewed toward longer wavelengths by $\delta \lambda =25\mathrm{pm}$ due to the accumulated phase shift from SPM. The extinction ratio also exceeds that of the linear resonance due to exchange of power into the harmonic mode—indeed, a peak conversion of $\eta =P_{3\omega }/P_{\omega }=0.0174$ is attained. For comparison, the theoretical conversion for an equivalent 3 mm long microfiber (calculated by setting ${\kappa }_{\omega ,3\omega }$ to zero) would be ${\eta }_0=2.7\times {10}^{-5}$. The loop therefore provides a resonant enhancement of $\zeta =\eta /{\eta }_0=640$ times greater than that of the straight microfiber. This enhancement arises primarily due to a large field enhancement inside the loop of $P_{\text{circ}}={|A_0^{\omega }(0)|}^2=8.6P_0$, corresponding to an internal power level of 860 W, which should be well tolerated by the OM (if pumping with nanosecond pulses and $\approx 1\mathrm{W}$ average powers) and hence possible to demonstrate experimentally using current fabrication techniques. However, far from resonance (e.g., at $\delta \lambda =200\mathrm{pm}$), $\eta$ falls below ${\eta }_0$ because a fraction of the light bypasses the loop in the coupling region, and thus experiences an effective path length shorter than $L_0$.

Increasing $P_0$ to 250 W further redshifts the resonance wavelength, as shown in Fig. 2(b). In addition, the greater pump power increases both $\eta$ and ${\eta }_0$ to 0.17 and $2.24\times {10}^{-4}$, respectively, yielding a resonant enhancement of $\zeta =750$, that is, 17% times larger than for $P_0=100\mathrm{W}$. Although $\zeta$ is to a large extent dictated by the intrinsic resonance characteristics (namely the proximity to critical coupling, as is discussed later), the growth of the third-harmonic signal remains nonetheless highly sensitive to any changes to the phase-matching conditions. In this case, the detuning of $-1440{\mathrm{m}}^{-1}$ is slightly lower than the optimum detuning, so at the higher power levels experienced inside the loop, the greater nonlinear phase modulation serves to compensate for this offset and thus increase the conversion and $\zeta$. However, the same mechanism would also reduce the efficiency if $\delta \beta$ was positively offset; physically this depends on whether the OM diameter is slightly larger or smaller than the critical phase-matching diameter.

When $P_0=600\mathrm{W}$, as shown in Fig. 2(c), the peak efficiency of $\eta =0.50$ is sufficiently high that the maximum enhancement becomes limited by pump depletion, with the pump extinction ratio being 2.3 times greater than the linear case. For this reason, the enhancement of $\zeta =240$ is smaller than that predicted for the lower input powers. Furthermore, the output spectra from the resonator becomes multivalued for a band of red-detunings near $\delta \lambda \approx 100\mathrm{pm}$ due to bistability. To explain this behavior, Figs. 3(a) through 3(d) provide the power-transfer characteristics at several positive detunings for a loop resonator of the same parameters. When $\delta \lambda =50\mathrm{pm}$, the pump wavelength resides within the original linear resonance and so the output is monostable. Increasing $\delta \lambda$ further, however, introduces hysteresis as expected, and at $\delta \lambda =96\mathrm{pm}$ the upper nonlinear switching power coincides with the input power of $P_0=600\mathrm{W}$. For $\delta \lambda =100\mathrm{pm}$, the same input power resides on the bistable region, with the upper and lower branches corresponding to the two values in Fig. 2(c). Note that only the upper branch for $P_{3\omega }/P_0$ generates any significant harmonic power because the pump light in the other branch is in an off-resonance, high-transmission state. Finally, when the detuning is increased to 120 pm, the output for a $P_0=600\mathrm{W}$ pump becomes single valued again because both the upper and lower switching powers exceed 600 W.

 

Fig. 3. Transfer characteristics when the pump is detuned from resonance by (a) 50, (b) 96, (c) 100, and (d) 120 pm. Other parameters are the same as in Fig. 2.

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As mentioned earlier, the maximum possible enhancement is greater if the loop resonator is closer to critical coupling, as can be seen in Fig. 4, which shows the expected $\zeta$ for different $\mathrm{\Delta }K$. For comparison purposes, the situation presented earlier in Fig. 2(a), when ${\kappa }_{\omega }=8\times {10}^4{\mathrm{m}}^{-1}$, is highlighted by the dotted line at $\mathrm{\Delta }K=0.71$. In addition, Fig. 5 shows the corresponding circulating power ratio on resonance $P_{\text{circ}}/P_0$ for the same range of $\mathrm{\Delta }K$.

 

Fig. 4. Enhancement $\zeta$ against detuning $\delta {\lambda }_{\omega }$ for different proximities to critical coupling $\mathrm{\Delta }K$ (equivalent to $6.5\times {10}^4{\mathrm{m}}^{-1}<{\kappa }_{\omega }<8.2\times {10}^4{\mathrm{m}}^{-1}$. Regions where $\zeta >1$ generate a greater third-harmonic conversion than an equivalent straight microfiber. The dotted line at $\mathrm{\Delta }K=0.71$ corresponds to the situation in Fig. 2(a), when ${\kappa }_{\omega }=8\times {10}^4{\mathrm{m}}^{-1}$. $P_0=100\mathrm{W}$, and other parameters are the same as in Fig. 2.

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Fig. 5. Maximum internal pump power on resonance inside the loop, $P_{\text{circ}}/P_0$, for different proximities to critical coupling $\mathrm{\Delta }K$ and loss. $P_0=100\mathrm{W}$, and other parameters are the same as in Fig. 2. The inset confirms the cubic pump dependency of the enhancement over the range of values simulated.

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Near to critical coupling, with $\mathrm{\Delta }K=0.6$, $P_{\text{circ}}$ is ninefold greater than the input power, which results in large enhancements in excess of ${10}^3$, but only over a narrow range of 20 pm near resonance. Furthermore, $\zeta$ only exceeds unity across a 150 pm span—outside of this range, the harmonic signal can become two orders of magnitude weaker than that of the original OM. As $\mathrm{\Delta }K$ increases, the $Q$ factor of the resonance and hence the enhancement both decrease dramatically, with $\zeta =2$ at $\mathrm{\Delta }K=1.4$ because the recirculating power is only $1.3P_0$. On the other hand, the conversion bandwidth increases with the resonance linewidth. Note that the peak efficiency also occurs closer to $\delta \lambda =0\mathrm{pm}$ and the $P_{3\omega }$ spectrum is more symmetric, mirroring the loop’s linear resonance spectra, because the lower recirculating power induces a weaker SPM.

A higher pump loss ${\alpha }_{\omega }$ reduces the circulating power ratio and $\zeta$ as expected, but the reduction becomes more significant with lower values of $\mathrm{\Delta }K$ because the light traverses a longer effective path length within the loop on resonance. Although the bend losses of loop resonators with a few millimeters diameter are negligible for submicrometer OM diameters, the microfiber surface may become contaminated by dust or moisture from the atmosphere, which increases the surface scattering and absorption losses to reduce $P_{\text{circ}}$. Nonetheless, from Fig. 5 it can be seen that even for very large losses of ${\alpha }_{\omega }=30$ the recirculating power ratio still exceeds five, which provides a corresponding enhancement of $\zeta =125$.

For the range of $\mathrm{\Delta }K$ and loss discussed above, the highest efficiency is 5% (when $\mathrm{\Delta }K=0.6$ and $\alpha =3{\mathrm{m}}^{-1}$), and so the pump can be approximated to be undepleted such that the pump distribution inside the loop is similar to what would be observed in the absence of THG. Indeed, the inset in Fig. 5 confirms that the enhancement increases cubically with $P_{\text{circ}}$. The enhancement can therefore be estimated from the linear properties accounting for the loss, which are discussed in [18]. On resonance, when $m=1$ (where $m$ is the integer eigenvalue index for the critical coupling condition as mentioned previously), the power transmission simplifies to

From Eqs. (6) and (7) the maximum enhancement can be estimated as:

Interestingly, Eq. (7) also predicts that $P_{\text{circ}}/P_0$ would counter-intuitively increase with increasing loss if $\pi /2<\mathrm{\Delta }K<3\pi /2$ (where $\mathrm{\Delta }K$ was previously defined as the difference from the critical coupling value at ${\kappa }_c{L}_c=3\pi /2$). This can indeed be seen from Fig. 5, where the highest loss ($\alpha =30{\mathrm{m}}^{-1}$) does not correspond to the lowest circulating power for values of $\mathrm{\Delta }K$ near 1.5 [the behavior shown in the numerically calculated graph differs somewhat from the aforementioned analytical prediction based on Eq. (7), due to the phase-modulation effects]. This phenomenon is however unlikely to be of practical use because it occurs far from critical coupling, where the enhancement is poor, and the larger loss values would detriment the outcoupling of the third harmonic signal.

In general, the bandwidth $B$ of the enhancement is dictated by the resonance linewidth, with a greater resonant enhancement from a higher-$Q$ resonance ($|\mathrm{\Delta }K|$ closer to zero) achieved at the expense of bandwidth, as shown in Fig. 6. For weak coupling at $\mathrm{\Delta }K=1.4$, the full width at half-maximum bandwidth of $\zeta$ (measured from a baseline of $\zeta =1$) is 185 pm, or roughly 25% of the resonator’s FSR. On the other hand, nearer to critical coupling at $\mathrm{\Delta }K=0.8$, the $Q$ factor is approximately 2.8 times greater and limits $B$ to only 40 pm, which may however be useful for applications requiring a narrow linewidth at the harmonic wavelength.

 

Fig. 6. (a) Resonant enhancement factor $\zeta$ of the third harmonic power and (b) bandwidth of the enhancement (taken as the full width measured halfway between $\zeta =1$ and maximum $\zeta$) for different proximities to critical coupling $\mathrm{\Delta }K$. $P_0=100\mathrm{W}$, and other parameters are the same as in Fig. 2.

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A relatively wide bandwidth is advantageous for converting a wider range of the pump light’s nonlinear broadened wavelength components, which may have been generated even before reaching the taper-waist region by SPM (additional broadening mechanisms may become significant depending on the pulse duration, power, and the pump wavelength’s position on the OM dispersion curve, and indeed their influence has been discussed in the context of continuum generation—see for example [1,2]). The broadened pump components each experience a different enhancement depending on their proximity to the resonance as explained from Fig. 2, and it should thus be noted that in the limit when $B\ll \mathrm{FSR}\ll \mathrm{pump}$ linewidth, the overall conversion falls and tends toward the off-resonance level.

Finally, it is interesting to consider cases with ${\kappa }_{3\omega }>0$ in which the harmonic is also near resonance because previous studies on resonantly enhanced harmonic generation in ring resonators [23,25] suggest that coresonance of the pump and harmonic can further increase the conversion. For values of ${\kappa }_{3\omega }$ up to $2\times {10}^4{\mathrm{m}}^{-1}$, Fig. 7 shows the expected harmonic conversion spectrum for a loop with the same parameters as Fig. 2(a) and $P_0=100\mathrm{W}$ input. Both ${\kappa }_{\omega }$ and ${\kappa }_{3\omega }$ can be altered in practice by adjusting the OM index, surrounding index, OM separation, and pump wavelength. Altering one of these parameters inevitably affects both couplings, but by changing two or more parameters simultaneously it is possible to keep constant one of the couplings and change the other (or alternatively, tailor their ratio). As ${\kappa }_{3\omega }$ increases from zero, the conversion generally falls because the third-harmonic signal is being recirculated rather than coupled out. However, for detunings in the range $+20$ to $+30\mathrm{pm}$, the efficiency can be enhanced significantly (by up to 50% greater when ${\kappa }_{3\omega }=2\times {10}^4{\mathrm{m}}^{-1}$).

 

Fig. 7. Third-harmonic conversion against detuning, when the harmonic light is partially recirculated (i.e., with ${\kappa }_{3\omega }>0{\mathrm{m}}^{-1}$). $P_0=100\mathrm{W}$, and other parameters are the same as in Fig. 2.

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The behavior can be understood from Eq. (2b), which states that at the start of the loop $|A_0^{3\omega }|$ grows if $0<{\theta }_{3\omega }(s=0)-3{\theta }_{\omega }(s=0)<\pi$ (neglecting phase modulation), where ${\theta }_i$ represents the phase of $A_0^i$. For this particular example, the condition is satisfied not at zero detuning but rather at the detunings around $\delta \lambda \approx +30\mathrm{pm}$ where the power of the recirculated harmonic seed grows with distance as it propagates around the loop. Note that although this phase condition is also met at $\delta \lambda =40\mathrm{pm}$, the pump is off resonance, and hence $\eta$ is low. For this reason, the coresonant enhancement is only apparent over a relatively narrow 10 pm range.

4. CONCLUSION

We have studied the use of microfiber loop resonators for intermodally phase-matched THG. The recirculation of the pump light near resonance can greatly enhance the efficiency by several orders of magnitude greater than that of the straight microfiber, which allows a significant conversion of several percent to be attained even at low pump powers down to 100 W. Importantly, this is achievable over microfiber loop lengths of only a few millimeters, which can realistically be fabricated with a high diameter uniformity. A range of interesting characteristics arise from the interplay of the nonlinear properties of the fiber and the resonant behavior. For example, at higher powers, the hysteresis of the resonator is reflected in the nonlinear switching of both the pump and third-harmonic power levels. Furthermore, when the harmonic light is coresonant with the pump, it is possible to enhance the conversion further.

Although the simulations here have focused on loop resonators, it should also be possible to observe similar effects in other microfiber resonators such as the knot [19] or the microcoil [20], which offer improved stability while maintaining the advantage of straightforward fabrication.