We present a scheme to generate the third harmonic in the fundamental mode using microfibres, by introducing a counter-propagating pump field to partly compensate the phase mismatch and thus enhance the harmonic conversion efficiency

Brambilla, Gilberto

Jiang, Xiujuan

Lee, Timothy

Southampton (e-Prints Soton)

Enhancement of Fundamental Mode ThirdHarmonic Generation Efficiency in MicrofibresXiujuan Jiang1;2, Timothy Lee2, and Gilberto Brambilla21School of Electro-mechanical Engineering, Guangdong University of Technology, Guangzhou, 510006, China2 Optoelectronics Research Centre, University of Southampton, Southampton, SO17 1BJ, United Kingdomjiangxj@gdut.edu.cnAbstract: We present a scheme to generate the third harmonic in the fundamental modeusing microfibres, by introducing a counter-propagating pump field to partly compensate thephase mismatch and thus enhance the harmonic conversion efficiency.OCIS codes: 190.2620,190.4370.1. IntroductionThird harmonic generation (THG) remains an important area of research, with numerous applications. For example,with high-power laser drivers, such as the Omega Facility in Rochester University, USA and Shenguang II Facility atthe Shanghai Institute of Optics and Fine Mechanics, China, THG is used to convert an infra-red pump into ultra-violetwavelengths, as the latter have a higher energy coupling efficiency in laser-plasma interactions. The systems for THGutilizing crystals are usually complicated because many optical components are required. In recent years however,harmonic generation in microfibres has been reported [1], thus opening the possibility for an all-fibre system for THGas a more robust alternative to the existing nonlinear-crystal technologies.As with other nonlinear harmonic conversion processes, phase matching should be satisfied to attain efficient THG inmicrofibres. However, because of material and waveguide dispersion, only intermodal phase matching can be achieveddirectly in glass microfibres (i.e. between the fundamental pump mode and a higher order harmonic mode [2, 3]) andno efforts on generating the fundamental third harmonic mode in microfibres have been reported to date. Nonetheless,a fundamental mode output is often desired when developing a laser, because its nearly Gaussian field distributionswould be easily compatible with other laser sources and equipment, and better for irradiation smoothing. In thispaper, we therefore present a scheme in which the phase mismatch is partly compensated by introducing a counter-propagating pump field into the microfibre, thereby allowing the harmonic to be generated in the fundamental mode.2. Basic principlesIn the process of THG, the total electric field can be represented as a sum of the mode fields at each frequency w jE(x;y;z; t) =åE j(x;y;z; t) =åA j(z)F j(x;y)exp [i(b jz w jt)]+ c:c:; (1)where j = 1 for the pump and j = 3 for the third harmonic; F j(x;y) are the electric mode field distributions and A j(z)are the amplitudes of the modes normalised to their powers, that is, jA j(z)j2 = Pj(z). b j are the propagation constantsof the modes in the fibre defined as b j = (w j=c)ne f fj (w j), and ne f fj are the effective indices of the modes.In step-index glass-air circular fibres, the following coupled-mode equations can be used to describe the THGprocess [1],¶A1¶ z= in(2)k1(J1jA1j2+2J2jA3j2)A1+ J3A21 A3 exp(idb z); (2a)¶A3¶ z= in(2)k1(6J2jA1j2+3J5jA3j2)A3+ J3A31 exp( idb z); (2b)where db = b3 3b1 is the phase mismatch, k1 = w1=c= 2p=l1 is the propagation constant of the pump in vacuum,n(2) is the nonlinear refractive index coefficient, and Ji are nonlinear overlap integrals (given in detail in Ref. [1]).Fig.1 shows the effective index curves for the fundamental pump mode HE11(w1) and fundamental third harmonicmode HE11(3w1) in a silica microfibre for l1 = 1:55mm. At the microfibre diameter of 550 nm where ne f f1 = 1:012 andne f f3 = 1:334, the phase mismatch is db = 3:9106m 1. As this phase mismatch is large (comparable to k1), the termexp( idb z) in Eq.(2b) oscillates very fast and prevents the fundamental mode harmonic from growing. Thus, efficientTHG between the fundamental mode of the pump and the fundamental mode of the harmonic would be impossiblewithout employing special methods to overcome such a large phase mismatch, as described next.Microfibre diameter (µm)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Effective Index11.051.11.151.21.251.31.351.41.45HE11(3ω1)HE11(ω1)Fig. 1. Effective index curves for the fundamental pump mode HE11(w1) (solid red) and the funda-mental third harmonic mode HE11(3w1) (dashed blue) in a silica microfibre for l1 = 1:55mm.If we introduce a weak counter-propagating light field [4, 5] at the fundamental frequency w1 into the fibre, the su-perposition of the forward-travelling and weak counter-propagating beams can be described by a forward-propagatingfield that is modulated in both amplitude and phase along the direction of propagation with a periodicity of half thelaser wavelength [4], that is,E1(x;y;z; t) = A1t(z)F1(x;y)exp[i(b1tz w1t)]+ c:c:; (3)where b1t = b1+b 0(z), and b 0(z) has the formb 0 = 2rb1 cos(2b1z)+2r2b11+2r cos(2b1z)+ r2: (4)Thus the actual total phase mismatch is dbt = b3 3b1t = b3 3b1 3b 0(z) and hence b 0(z) acts as a compensativedetuning which we can use to enhance the THG conversion. Importantly, the variation of b 0(z) along the propagationdistance can be adjusted by changing the ratio of the backward-forward field amplitudes r = A 1 =A+1 .3. Simulation resultsFigure 2 shows the compensative and total detuning against propagating distance when the backward-forward ampli-tudes ratio is r = 0:14. The total detuning periodically approaches and even reaches zero, and at these positions, theharmonic power can be built up.We calculate the third harmonic power in the presence of such a counter-propagating field by solving Eq.(2) nu-merically, and the results are given in Fig. 3. For small r, it is assumed that the counter-propagating pump generates anegligible counter-propagating harmonic signal. In the situation r = 0, i.e. with only the forward pump, the harmonicpower remains very low, but when the counter-propagating field is introduced (though only with r = 0:14, or about2% power of the forward pump) the harmonic power grows along the propagating direction, and in a microfibre tensof micrometres in length, the conversion efficiency obtains an enhancement of about one order of magnitude.4. ConclusionWe have presented a scheme to enhance the fundamental-mode third harmonic generation efficiency in microfibres. Byintroducing an appropriate weak counter-propagating beam, the phase mismatch between the fundamental modes ofthe pump and third harmonic is partly compensated, enabling the harmonic power to grow along the propagation direc-tion. This scheme offers an alternative idea for efficient harmonic generation by modifying the pump beam, and couldpotentially be applied to other small core waveguides such as wagon-wheel holey fibres or integrated waveguides.z (µm)0 1 2 3 4 5Compensative detuning (m-1 )×106-202Total detuning (m-1 )×107-101Fig. 2. The compensative detuning b 0(z) introduced by the backward beam (black dashed curve) andthe total detuning dbt (red solid curve) against propagating distance z, when the backward-forwardamplitude ratio is r = 0:14.z (µm)0 5 10 15 20 25 30P 3 / P0×10-900.511.52r = 0.14r = 0Fig. 3. The third harmonic generation power against the propagating distance z. The pump powerP0 = 1kW, nonlinear overlap integral J3 = 0:31mm 2, l1 = 1:55mm, and n(2) = 2:710 20m2=W.AcknowledgmentsThis work is supported by the National Natural Science Foundation of China (Grant No.11204043) and the ScholarshipFund of China (Grant No.201308440325).References1. V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13,6798–6806 (2005).2. T. Lee, Y. Jung, C. A. Codemard, M. Ding, N. G. R. Broderick and G. Brambilla, “Broadband third harmonicgeneration in tapered silica fibres,” Opt. Express 20, 8503–8511 (2012).3. T. Wu, Y. Sun, X. Shao, P. Shum and T. Huang, “Efficient phase-matched third harmonic generation in anasymmetric plasmonic slot waveguide,” Opt. Express 22, 18612–18624 (2014).4. X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterparopagating light,” Na-ture Phys. 3, 270–275 (2007).5. J. Peatross, S. Voronov and I. Prokopovich, “Selective zoning of high harmonic emission using counter-propagating light,” Opt. Express 1, 114–125 (1997).

Enhancement of fundamental mode third harmonic generation efficiency in microfibres

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