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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Grover Swartzlander
  • Vol. 30, Iss. 3 — Mar. 1, 2013
  • pp: 505–511
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Resonantly enhanced third harmonic generation in microfiber loop resonators

Timothy Lee, Neil G. R. Broderick, and Gilberto Brambilla  »View Author Affiliations


JOSA B, Vol. 30, Issue 3, pp. 505-511 (2013)
http://dx.doi.org/10.1364/JOSAB.30.000505


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Abstract

We theoretically study third harmonic generation in silica microfiber loop resonators wherein the large resonant field strength is exploited to increase the efficiency and reduce the required pump power, with a focus on the influence of loop parameters such as loss and coupling. For a 3 mm length loop, the conversion can reach several percent, that is, 640 times greater than an equivalent straight microfiber, for input powers as low as 100 W. The harmonic signal can be toggled between a high- and low-output state due to hysteresis at higher powers, and the efficiency can be further enhanced if the harmonic light is recirculated and coresonant with the pump.

© 2013 Optical Society of America

1. INTRODUCTION

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

9. V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef]

]. While it is theoretically possible to attain efficiencies exceeding 50% over 5 cm in a straight silica OM at 1 kW power levels [9

9. V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef]

], 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

21. G. Brambilla, V. Finazzi, and D. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express 12, 2258–2263 (2004). [CrossRef]

,22

22. L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef]

]. 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

18. M. Sumetsky, Y. Dulashko, J. Fini, A. Hale, and D. DiGiovanni, “The microfiber loop resonator: theory, experiment, and application,” J. Lightwave Technol. 24, 242–250 (2006). [CrossRef]

], 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

23. J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express 19, 11415–11421 (2011). [CrossRef]

25

25. Z. Bi, A. W. Rodriguez, H. Hashemi, D. Duchesne, M. Loncar, K. Wang, and S. G. Johnson, “High-efficiency second-harmonic generation in doubly-resonant χ(2) microring resonators,” Opt. Express 20, 7526–7543 (2012). [CrossRef]

] and microtoroids [26

26. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3, 430–435 (2007). [CrossRef]

], and THG was noted as an additional nonlinear effect during pulse-shaping experiments [3

3. A. Coillet, G. Vienne, and P. Grelu, “Potentialities of glass air-clad micro-and nanofibers for nonlinear optics,” J. Opt. Soc. Am. B 27, 394–401 (2010). [CrossRef]

].

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 L0 and Lc, 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 κω,3ω falls exponentially with their separation, and so it can be assumed that the coupling is only significant over a small range LcL0.

Fig. 1. Schematic of the microfiber loop resonator. The amplitudes in the two arms of the coupling region are denoted A1(s) and A2(s), and the amplitudes in the loop are A0(s).

Table 1. Summary of Parameters Used in the Simulation of the Loop Resonator when Phase-Matching the HE11(ω) Mode with the HE12(3ω) Mode for λω=1.55μm

table-icon
View This Table

In the loop region, the coupled-mode differential equations describing the evolution of the copropagating pump and third-harmonic-mode amplitudes A0ω and A03ω are adapted from [9

9. V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef]

] to include loss:
dA0ωds=αωA0ω+in(2)kω[(J1|A0ω|2+2J2|A03ω|2)A0ω+J3(A0ω*)2A03ωeiδβs],
(1a)
dA03ωds=α3ωA03ω+in(2)kω[(6J2|A0ω|2+3J5|A03ω|2)A03ω+J3*(A0ω)3eiδβs],
(1b)
where the overlap integrals J1, J2, J3, and J5 correspond to the terms for pump self-phase modulation (SPM), cross-phase modulation (XPM), pump-harmonic overlap, and harmonic SPM, respectively. Because the analysis concentrates on the narrow range of wavelengths (<1nm) around a resonance, it is reasonable to approximate these overlaps as constants with the values given in Table 1. The detuning is given by δβ=β3ω3βω, while kω=2π/λω is the pump free-space propagation constant and n(2)=2.7×1020m2/W is the silica nonlinear refractive-index coefficient. In the coupling region, the equations for the amplitudes Aiω(s) and Ai3ω(s) are similar, albeit with the addition of the linear coupling terms:
dAiωds=αωAiω+iκωAjω+in(2)kω[(J1|Aiω|2+2J2|Ai3ω|2)Aiω+J3(Aiω*)2Ai3ωeiδβs],
(2a)
dAi3ωds=α3ωAi3ω+iκ3ωAj3ω+in(2)kω[(6J2|Aiω|2+3J5|Ai3ω|2)Ai3ω+J3*(Aiω)3eiδβs].
(2b)

Equations (2a) and (2b) give the differential equations for the modes A1ω and A13ω propagating in the first arm of the coupling region when i=1 and j=2, and likewise for the second arm A2ω and A23ω when i=2 and j=1. These equations are then solved iteratively with the boundary conditions for field continuity:
A0ω,3ω(0)=A1ω,3ω(Lc)exp(iβω,3ωLc),
(3a)
A2ω,3ω(0)=A0ω,3ω(L0)exp(iβω,3ωL0)
(3b)
as well as the input initial conditions:
A1ω(0)=P0,
(4a)
A13ω(0)=0
(4b)
for an input pump power P0. The output transmissions can then be evaluated as
Pω,3ωP0=|A2ω,3ω(Lc)A1ω(0)exp(iβω,3ωLc)|2.
(5)

The free spectral range (FSR) of the 3 mm loop is FSRλ2/(neffL0)=740pm 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 Kc=κcLc=π(2m+1)/2 for integer m1 [18

18. M. Sumetsky, Y. Dulashko, J. Fini, A. Hale, and D. DiGiovanni, “The microfiber loop resonator: theory, experiment, and application,” J. Lightwave Technol. 24, 242–250 (2006). [CrossRef]

] with the lowest value (m=1) of κc=9.4×104m1. 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 104W. 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 ΔK=(κcκω)Lc1 (rather than zero).

3. DISCUSSION

First, the transmitted pump Pω and harmonic P3ω output powers will be studied for different input powers, as shown in Fig. 2, where the pump is detuned from a resonance at λR1550nm (to avoid confusion with the phase-matching detuning δβ, this detuning is denoted as δλ). The coupling between pump modes is chosen to be κω=8×104m1, which is achievable with close contact between the two microfibers and corresponds to ΔK=0.71 (for reference, values of ΔK=0 and ΔK=3π/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, κ3ω is roughly an order of magnitude smaller than κω. Furthermore, in the most general case, the resulting third-harmonic signal does not necessarily coincide with a resonance, and so κ3ω is set to zero here (situations in which κ3ω>0 are considered later).

Fig. 2. Pump Pω and third harmonic P3ω output power from a loop resonator when input pump power P0 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: L0=3mm, Lc=50μm, αω,3ω=5m1, κ=8×104m1, and δβ1440m1.

We choose δβ1440m1, 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 δβ=0m1. 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

12. N. G. R. Broderick, M. A. Lohe, T. Lee, and S. Afshar V., “Analytic theory of two wave interactions in a waveguide with a χ(3)nonlinearity,” in Proceedings of the International Quantum Electronics Conference and Conference on Lasers and Electro-Optics Pacific Rim 2011, (Optical Society of America, 2011), p. I366.

]).

For an input power of P0=100W, Fig. 2(a) confirms that the pump’s nonlinear resonance spectrum appears asymmetrically skewed toward longer wavelengths by δλ=25pm 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 η=P3ω/Pω=0.0174 is attained. For comparison, the theoretical conversion for an equivalent 3 mm long microfiber (calculated by setting κω,3ω to zero) would be η0=2.7×105. The loop therefore provides a resonant enhancement of ζ=η/η0=640 times greater than that of the straight microfiber. This enhancement arises primarily due to a large field enhancement inside the loop of Pcirc=|A0ω(0)|2=8.6P0, corresponding to an internal power level of 860 W, which should be well tolerated by the OM (if pumping with nanosecond pulses and 1W average powers) and hence possible to demonstrate experimentally using current fabrication techniques. However, far from resonance (e.g., at δλ=200pm), η falls below η0 because a fraction of the light bypasses the loop in the coupling region, and thus experiences an effective path length shorter than L0.

Increasing P0 to 250 W further redshifts the resonance wavelength, as shown in Fig. 2(b). In addition, the greater pump power increases both η and η0 to 0.17 and 2.24×104, respectively, yielding a resonant enhancement of ζ=750, that is, 17% times larger than for P0=100W. Although ζ 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 1440m1 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 ζ. However, the same mechanism would also reduce the efficiency if δβ was positively offset; physically this depends on whether the OM diameter is slightly larger or smaller than the critical phase-matching diameter.

When P0=600W, as shown in Fig. 2(c), the peak efficiency of η=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 ζ=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 δλ100pm 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 δλ=50pm, the pump wavelength resides within the original linear resonance and so the output is monostable. Increasing δλ further, however, introduces hysteresis as expected, and at δλ=96pm the upper nonlinear switching power coincides with the input power of P0=600W. For δλ=100pm, 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 P3ω/P0 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 P0=600W 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.

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 ζ for different ΔK. For comparison purposes, the situation presented earlier in Fig. 2(a), when κω=8×104m1, is highlighted by the dotted line at ΔK=0.71. In addition, Fig. 5 shows the corresponding circulating power ratio on resonance Pcirc/P0 for the same range of ΔK.

Fig. 4. Enhancement ζ against detuning δλω for different proximities to critical coupling ΔK (equivalent to 6.5×104m1<κω<8.2×104m1. Regions where ζ>1 generate a greater third-harmonic conversion than an equivalent straight microfiber. The dotted line at ΔK=0.71 corresponds to the situation in Fig. 2(a), when κω=8×104m1. P0=100W, and other parameters are the same as in Fig. 2.
Fig. 5. Maximum internal pump power on resonance inside the loop, Pcirc/P0, for different proximities to critical coupling ΔK and loss. P0=100W, 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.

Near to critical coupling, with ΔK=0.6, Pcirc is ninefold greater than the input power, which results in large enhancements in excess of 103, but only over a narrow range of 20 pm near resonance. Furthermore, ζ 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 ΔK increases, the Q factor of the resonance and hence the enhancement both decrease dramatically, with ζ=2 at ΔK=1.4 because the recirculating power is only 1.3P0. On the other hand, the conversion bandwidth increases with the resonance linewidth. Note that the peak efficiency also occurs closer to δλ=0pm and the P3ω spectrum is more symmetric, mirroring the loop’s linear resonance spectra, because the lower recirculating power induces a weaker SPM.

A higher pump loss αω reduces the circulating power ratio and ζ as expected, but the reduction becomes more significant with lower values of ΔK 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 Pcirc. Nonetheless, from Fig. 5 it can be seen that even for very large losses of αω=30 the recirculating power ratio still exceeds five, which provides a corresponding enhancement of ζ=125.

For the range of ΔK and loss discussed above, the highest efficiency is 5% (when ΔK=0.6 and α=3m1), 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 Pcirc. The enhancement can therefore be estimated from the linear properties accounting for the loss, which are discussed in [18

18. M. Sumetsky, Y. Dulashko, J. Fini, A. Hale, and D. DiGiovanni, “The microfiber loop resonator: theory, experiment, and application,” J. Lightwave Technol. 24, 242–250 (2006). [CrossRef]

]. 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
|T|2=|A2ω(Lc)|2P0=eαωL0+sin(κωLc)1+sin(κωLc)eαωL0
(6)
and the power recirculation is given by
PcircP0=1|T|21exp(2αωL0).
(7)

From Eqs. (6) and (7) the maximum enhancement can be estimated as:
ζ[1sin2(κωLc)(1+sin(κωLc)eαωL0)2]3.
(8)

Interestingly, Eq. (7) also predicts that Pcirc/P0 would counter-intuitively increase with increasing loss if π/2<ΔK<3π/2 (where ΔK was previously defined as the difference from the critical coupling value at κcLc=3π/2). This can indeed be seen from Fig. 5, where the highest loss (α=30m1) does not correspond to the lowest circulating power for values of ΔK 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 (|ΔK| closer to zero) achieved at the expense of bandwidth, as shown in Fig. 6. For weak coupling at ΔK=1.4, the full width at half-maximum bandwidth of ζ (measured from a baseline of ζ=1) is 185 pm, or roughly 25% of the resonator’s FSR. On the other hand, nearer to critical coupling at ΔK=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 ζ of the third harmonic power and (b) bandwidth of the enhancement (taken as the full width measured halfway between ζ=1 and maximum ζ) for different proximities to critical coupling ΔK. P0=100W, and other parameters are the same as in Fig. 2.

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

1. R. R. Gattass, G. T. Svacha, L. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414 (2006). [CrossRef]

,2

2. S. Leon-Saval, T. Birks, W. Wadsworth, P. St. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef]

]). 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 BFSRpump linewidth, the overall conversion falls and tends toward the off-resonance level.

Fig. 7. Third-harmonic conversion against detuning, when the harmonic light is partially recirculated (i.e., with κ3ω>0m1). P0=100W, and other parameters are the same as in Fig. 2.

The behavior can be understood from Eq. (2b), which states that at the start of the loop |A03ω| grows if 0<θ3ω(s=0)3θω(s=0)<π (neglecting phase modulation), where θi represents the phase of A0i. For this particular example, the condition is satisfied not at zero detuning but rather at the detunings around δλ+30pm 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 δλ=40pm, the pump is off resonance, and hence η 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

19. X. Jiang, L. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, and D. Yang, “Demonstration of optical microfiber knot resonators,” Appl. Phys. Lett. 88, 223501 (2006). [CrossRef]

] or the microcoil [20

20. M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12, 2303–2316 (2004). [CrossRef]

], which offer improved stability while maintaining the advantage of straightforward fabrication.

ACKNOWLEDGMENT

G. Brambilla gratefully acknowledges the Royal Society (London, UK) for his University Research Fellowship.

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Z. Bi, A. W. Rodriguez, H. Hashemi, D. Duchesne, M. Loncar, K. Wang, and S. G. Johnson, “High-efficiency second-harmonic generation in doubly-resonant χ(2) microring resonators,” Opt. Express 20, 7526–7543 (2012). [CrossRef]

26.

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3, 430–435 (2007). [CrossRef]

27.

G. Vienne, Y. Li, L. Tong, and P. Grelu, “Observation of a nonlinear microfiber resonator,” Opt. Lett. 33, 1500–1502 (2008). [CrossRef]

28.

K. S. Lim, A. A. Jasim, S. S. A. Damanhuri, S. W. Harun, B. M. Azizur Rahman, and H. Ahmad, “Resonance condition of a microfiber knot resonator immersed in liquids,” Appl. Opt. 50, 5912–5916 (2011). [CrossRef]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(230.5750) Optical devices : Resonators
(230.7370) Optical devices : Waveguides

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 14, 2012
Revised Manuscript: December 21, 2012
Manuscript Accepted: December 23, 2012
Published: February 8, 2013

Virtual Issues
February 15, 2013 Spotlight on Optics

Citation
Timothy Lee, Neil G. R. Broderick, and Gilberto Brambilla, "Resonantly enhanced third harmonic generation in microfiber loop resonators," J. Opt. Soc. Am. B 30, 505-511 (2013)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-3-505


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References

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  26. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3, 430–435 (2007). [CrossRef]
  27. G. Vienne, Y. Li, L. Tong, and P. Grelu, “Observation of a nonlinear microfiber resonator,” Opt. Lett. 33, 1500–1502 (2008). [CrossRef]
  28. K. S. Lim, A. A. Jasim, S. S. A. Damanhuri, S. W. Harun, B. M. Azizur Rahman, and H. Ahmad, “Resonance condition of a microfiber knot resonator immersed in liquids,” Appl. Opt. 50, 5912–5916 (2011). [CrossRef]

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