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Broadband third harmonic generation in tapered silica fibres |
Optics Express, Vol. 20, Issue 8, pp. 8503-8511 (2012)
http://dx.doi.org/10.1364/OE.20.008503
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– Abstract
1. Introduction
2. Experimental details
3. Simulations
4. Discussion
5. Conclusion
– References and links
– Abstract
1. Introduction
2. Experimental details
3. Simulations
4. Discussion
5. Conclusion
– References and links
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Abstract
Optical microfibres have recently attracted much attention for nonlinear applications, due to their tight modal confinement. Here, we report broadband third harmonic generation based on the intermodal phase matching technique in silica microfibres of several centimetres. The third harmonic signal is predominantly generated from the taper transition regions (rather than the waist), wherein the range of diameters permits phase matching over a wide bandwidth. Microfibres up to 4.5 cm long were fabricated with waist diameters below 2.5 μm to allow a λ = 1.55 μm pump to phase match with several higher order third harmonic modes; conversion rates up to 3 × 10−4 were recorded when pumped with 4 ns pulses at a peak power of 1.25 kW. Analysis of the third harmonic frequencies generated from the nonlinearly broadened pump components indicate a 5 dB conversion bandwidth of at least 36 nm, with harmonic power detected over a 150 nm range.
© 2012 OSA
1. Introduction
Third harmonic generation (THG) has attracted considerable attention because of its applications ranging from all-optical full logic units for signal processing [1] to scanning microscopy [2] and material processing [3]. Although THG has been demonstrated in a number of different media [4–6], the third harmonic signal has been generated predominantly in the fundamental mode and with a bandwidth of only few nanometers, largely because the pump laser has a linewidth limited to few nanometers and the phase matching conditions in materials occur over a small range of wavelengths, typically several nanometers. In this manuscript, we demonstrate THG in higher order modes detected over ranges up to 150 nm by exploiting long optical microfibre tapers, which simultaneously favour: (1) pump broadening because of nonlinear effects such as self-phase modulation, and (2) phase matching for the various pump wavelengths along different parts of the taper.
R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun. 207, 155–160 (2002). [CrossRef]
Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]
H. Endert, M. Scaggs, D. Basting, and U. Stamm, “New ultraviolet lasers for material processing in industrial applications,” J. Laser Appl. 11(1), 1–6 (1999), [CrossRef]
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]
Research interest in optical microfibres for several nonlinear applications arises from their wavelength-scale diameter D and strong refractive index contrast, which together provide tight modal confinement. For an air-clad silica microfibre, the effective area is of the order of (λ/nsilica)2 which enhances the nonlinearity by over 100 times compared to standard single mode fibre (SMF) [7], hence allowing diverse nonlinear effects to be investigated not only at low powers but also over propagation lengths as short as several millimetres.
S. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwave-length structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef]
In particular, efficient third harmonic generation (THG) is possible by compensating the material and waveguide dispersion with intermodal dispersion to phase match the fundamental pump HE11(ω) mode with a higher order third harmonic mode which experiences the same effective index [8–12], i.e. neff(ω) ≈ neff(3ω), or equivalently Δβ ≡ β (3ω) – 3β (ω) ≈ 0 (optimum conversion efficiency requires a small offset from Δβ = 0 to compensate for the intensity dependent change in refractive index): this condition is only satisfied for certain microfibre diameters and provides extremely small bandwidths [8].
V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef] [PubMed]
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]
V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef] [PubMed]
Here, we discuss the fabrication and characterisation of long tapers with lengths L > 3 cm wherein the potential of intermodal dispersion to compensate the inherent material and waveguide dispersions utilises the spread of diameters in the transition regions to collectively allow phase matching over a wider range of pump wavelengths. This technique can be considered analogous to chirping the poling period of quasi phase matched devices to extend their bandwidth [13], and will allow broad third harmonic continua to be generated in the green down to the UV wavelengths for uses such as biological spectroscopy. The influence of the input power and pump wavelength detuning are investigated for several different silica tapers with diameters near 2 μm.
M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997). [CrossRef]
2. Experimental details
The modified flame brushing method [14] was used to fabricate the tapers from standard SMF (SMF-1300/1500-9-125-0.25-L, OZ Optics, Canada) using a rig in which the fibre is tensioned by two computer controlled stages whilst being translated back and forth through a stationary ceramic heater at 1320 °C until the desired microfibre diameter is reached. Compared to the more commonly used flame burner, the heater’s wide hot zone of 4 mm offers more uniform heating. The discussion will analyse results obtained using three tapers: (A)D = 2.1 μm, L = 45 mm; (B) D = 1.8 μm, L = 45 mm; and (C)D = 2.4 μm, L = 30 mm, where D is the waist diameter and length L includes the transition regions.
G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. Sessions, E. Koukharenko, X. Feng, G. Murugan, J. Wilkinson, and D. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1, 107–161 (2009). [CrossRef]
From scanning electron microscope (SEM) images, the reconstructed profile of microfibre A’s downtaper is summarised in Fig. 1(a). The uptaper is identical and the transition profiles for tapers B and C are similar. The diameter falls exponentially (∼ exp(−0.31 mm−1z)) to provide a gentle gradient close to the phase matching diameter which ensures a sufficient interaction length for the harmonic to grow. At the waist, the diameter is uniform over approximately 3 mm. The SEM images taken along the taper, examples of which are shown in Figs. 1(b)–1(d), also indicate a low surface roughness < 10 nm.
Fig. 1 (a) The diameter profile for the transition region of taper A (D = 2.1 μm and L = 45 mm) as characterised through SEM images, showing an exponential decrease in diameter with taper distance. (b–d) SEM images of the taper at the start, transition region and waist.
Whilst pulling the taper, the fibre was pumped with 4 ns 1.55 μm pulses from a fully fiberised 100 kHz source as shown in Fig. 2. The taper output was spliced onto a fibre shortpass filter with a 1300 nm cutoff wavelength to attenuate the pump signal so the optical output can be monitored in-situ using a spectrum analyser (Yokogawa AQ-6315A). In this way, it is possible to determine when the critical diameter has been reached by observing the onset of THG, so the rig can be manually stopped accordingly. For taper C, the rig was stopped just after the third harmonic signal peaked, whereas tapers A and B were pulled for a slightly longer time to provide a narrower waist. The shortpass filter is a straightforward bend-loss filter formed by coiling the collecting fibre and exhibits a loss of approximately 5 dB at the third harmonic wavelength. Using nanosecond pulses minimises walkoff issues, whilst maintaining high average and peak powers of Pav = 0.5 W and P = 1.25 kW which were well tolerated by the microfibre. Prior to entering the taper, the pump pulses’ full-width at half-maximum linewidth is typically less than 0.5 nm since the preceding SMF is limited to 2 m at most.
3. Simulations
To find the required phase matching diameters, the rigourous modal eigenvalue equations for a step index profile were solved [15]. We only consider phase matching to TH hybrid modes with odd azimuthal order ν, since the overlap for the other modes (including transverse electric and magnetic modes) can be shown to be identically zero due to symmetry considerations.
Figure 3 plots the predicted phase matching wavelength at different diameters for ν = 1,3 hybrid TH modes. When compared against the profile of taper A, it is clear that a 1.55 μm pump will phase match to several modes, including the EH31(3ω), EH12(3ω), HE32(3ω), and HE13(3ω) modes, albeit at different points along the transition region. Indeed, since any taper with a waist narrower than D=2.5 μm will permit phase matching with these four modes at various positions, we shall refer to this as the critical diameter. To assess the relative contribution of the modes, the modal overlaps J3 between the pump and each mode were calculated as follows [8]:
where F1(x, y) and F3(x, y) denote the power-normalised electric field distributions of the fundamental pump and third harmonic mode respectively. Since the overlaps for the four modes are similar and reside within the range 10−4 < J3 < 10−3 μm −2, it is expected that the third harmonic signal at the taper output will have a significant fraction of power in each mode. Although the overlap with the HE12(3ω) mode at D = 0.77 μm is greater, our experiments primarily aim to increase the bandwidth and therefore focus on the phase match points with larger diameter microfibres, which can be more consistently fabricated to longer lengths and are capable of tolerating higher pump powers.
V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef] [PubMed]
Fig. 3 The phase matched pump wavelength (left) against microfibre diameter, when matching to different third harmonic modes. The corresponding distance z along taper A (right) is also shown, and the red dots indicate positions where a λ1 = 1.55 μm pump is phase matched. At each point, the J3 overlap integral is given in italics in units of [μm−2].
4. Discussion
When the long tapers’ waist neared the phase matching diameter during the tapering process, the THG signal took several minutes attain its peak conversion, in contrast to previous experiments with short tapers where the onset of THG was sudden [10]. Firstly, when fabricating longer tapers, the microfibre is drawn slower (i.e. smaller dD/dt) when the diameter approaches a few microns because the stages move at a fixed speed and hence take a longer time to physically translate the length of the microfibre through the heater. Secondly, it should be noted that when the fibre waist decreases, the lower limit of the range of pump wavelengths which can undergo phase matching will also decrease. Hence, as this limit approaches the 1550 nm pump peak, the THG peak is expected to gradually rise. Indeed, as the waist narrowed from 3.3 μm to 2.4 μm, the TH peak grew by 25 dB whilst simultaneously falling from 600 nm to 520 nm as expected from Fig. 3. Furthermore, the TH signal remains even after the waist diameter falls several hundred nanometres below the critical value (although the peak is reduced by over 10 dB), implying that the locale of THG has shifted from the waist to the transition regions.
V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007). [CrossRef]
For taper A, the output spectrum at the pump and third harmonic wavelengths are shown in Figs. 4(a)–4(b) and 4(c) respectively, with the former characterised via a −25 dB coupler. The pump pulse is initially narrowband with a full-width half-maximum linewidth < 0.5 nm, but becomes nonlinearly broadened to over several hundred nanometres as it propagates through the taper waist. These broadened components are then phase matched along the uptaper resulting in the broadband THG spectrum as observed. Comparison of the pump spectrum before and after the taper in Figs. 4(a) and 4(b) indicates that the pump broadening becomes more apparent for higher powers - for example, at a peak power of 1.3 kW, the power at 1525 nm was measured from the source as −29 dBm/nm (being comprised largely of ASE) but increases to −24 dBm/nm after the taper. Additionally, the generation of side peaks in Fig. 4(b) was noticed, which did not exist in the original source spectrum, due to nonlinear effects such as self phase modulation and modulation instability. Note that these are partially masked by the source’s background amplified spontaneous emission peaks, which are continuous wave and hence do not contribute to the nonlinear processes.
Fig. 4 (a) The pump spectrum measured via a −25dB coupler from the source output and also (b) at the taper output with an input pump wavelength of λ1 = 1.55 μm. (c) the third harmonic spectrum, measured after a shortpass filter with approximately 5 dB loss. (d) The estimated experimental conversion efficiency at each of the pump wavelengths. Taper A parameters: D = 2.1 μm and L = 45 mm.
The TH spectrum shows not only the distinct peak at 522 nm, but also the third harmonic of the secondary peaks and broadened components at longer wavelengths. To some extent, the shape of the TH signal is also be influenced by the generation of multiple third harmonic modes, which individually have non-identical conversion spectra. As the pump power increases from 0.9 kW up to 1.3 kW, these extend from 540 nm up to 570 nm. Whereas the components of the pump which broadened to longer wavelengths are able to phase match within the transition regions, the shorter wavelengths require a phase matching diameter narrower than the taper waist and hence cannot generate a detectable TH signal. The TH spectrum therefore appears asymmetric and skewed towards longer wavelengths.
Calculating the efficiency over moving 5 nm steps at maximum pump power (10 μJ pulse energy) results in the conversion rates shown in Fig. 4(d). At λ = 1550 nm, the value of η = 10−6 is low, although still comparable to the efficiency obtained in previous experiments with uniform diameter waveguides [10]. For shorter pump wavelengths, the conversion falls to 10−7 because THG can only occur in the non-phase matched regime. On the other hand, for λ > 1560 nm, η rises by over a hundred-fold to more than 0.3 × 10−3 with a 5 dB bandwidth of at least 36 nm. It is likely that the true bandwidth may be greater yet, but this would require a broadband source and longer wavelength spectrum analyser to confirm.
V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007). [CrossRef]
To study the dependency upon the taper profile further, taper B was fabricated with a narrower waist of 1.8 μm. From its output spectrum in Fig. 5, the third harmonic signal is over twice as broad as that of taper A and covers a wider range of 150 nm at P = 1.2 kW, because the shorter wavelength components of the pump are also able to contribute to the third harmonic signal. The TH spectrum therefore appears much more symmetrical and extends down to 450 nm (60 nm lower than for taper A). Compared to the THG in uniform waveguides in [9], this spectral width is an order of magnitude broader, despite pumping at a tenfold lower peak power. An important consequence of the narrower waist is that the efficiency of η = 5 × 10−5 is notably lower than that of taper A, since the third harmonic generation is now occurring in a steeper section of the transition region.
Fig. 5 Third harmonic spectrum for different pump powers, showing broad TH signal detectable over 150 nm. Pump wavelength is λ1 = 1.55 μm. Results obtained using taper B (D = 1.8 μm and L = 45 mm).
For powers less than 0.7 kW, the third harmonic peak is distinct as shown in Fig. 5(a) and 5(b) - in the latter, the secondary peaks are also visible. On the other hand, at higher powers the nonlinear broadening is severe enough to almost mask the peak as in (d) when P = 1.2 kW. This was not observed in the spectra of 4 (c) and can be attributed to a combination of the higher field intensity in the narrower taper B, as well as differences in their dispersion properties which vary strongly with fibre diameter.
To investigate the effect of tuning the pump wavelength λ1 in further detail, taper C was fabricated with a shorter length of 30 mm with a diameter of 2.4 μm and characterised at approximately P = 0.6 kW. These two changes intend to limit the nonlinear effects so that a distinct TH peak can be identified, as shown in Fig. 6. As λ1 is tuned from 1535 nm up to 1560 nm, the TH peak wavelength varies as λ1/3. For this taper, the THG from phase matching to the EH31(3ω) third harmonic mode is likely to be dominant near the waist since the 2.4 μm diameter is very close to the critical diameter of 2.5 μm. However, due to the slight difference, the largest conversion occurs at the shorter wavelength of λ1 = 1545 nm rather than 1550 nm. Across the 25 nm pump wavelength range, the TH peak level only varies by 6 dB which suggests the distribution of the corresponding phase matching diameters is fairly even in the transition region as expected from the SEM characterisation. The TH peak does however become narrower when pumped at the longer wavelength of 1560 nm, which may be partially due to a fall in the gain spectrum of the source amplifier as well as the steeper gradient at the corresponding section of the taper.
5. Conclusion
We have demonstrated that the third harmonic signal generated from tapered optical microfibres several centimetres long can provide a much wider bandwidth than bulk or uniform waveguides. Silica tapers were fabricated with diameters close to the phase matching value of D = 2.5 μm and lengths up to 45 mm, and characterisation using a 1.55 μm pump revealed third harmonic signals over a range of wavelengths exceeding 150 nm, conversion rates in the range 10−6 < η < 0.3×10−3 and a 5 dB bandwidth in excess of 36 nm. Efficiency can be improved by optimising the taper profile design, by improving phase matching conditions for the HE12(3ω) mode using narrower tapers, and also by using soft glasses with nonlinear coefficients significantly greater than that of silica.
We note that extending the analogy with chirped quasi-phase matched devices means that we should be able to simultaneously compress and convert broadband chirped femtosecond pulses [13]. Finally, optimising the process of broadband THG will also help in optimising the inverse process of triple photon production from a single pump photon as discussed in [16] and so such tapers could find important applications in quantum information processing. In particular the broadband generation of entangled photons at different wavelengths jointly with adequate wavelength multiplexing/demultiplexing devices could prove important for the realisation of multichannel quantum optical communication.
M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997). [CrossRef]
S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011). [CrossRef] [PubMed]
Acknowledgments
The authors gratefully acknowledge insightful discussions with Konstantin Karapetyan, Prof. D. Meschede and Prof. A. Rauschenbeutel. G. Brambilla gratefully acknowledges the Royal Society (London, U.K.) for his University Research Fellowship.
References and links
R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun. 207, 155–160 (2002). [CrossRef] | |
Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef] | |
H. Endert, M. Scaggs, D. Basting, and U. Stamm, “New ultraviolet lasers for material processing in industrial applications,” J. Laser Appl. 11(1), 1–6 (1999), [CrossRef] | |
D. Nikogosyan, Properties of Optical and Laser-Related Materials: A Handbook (Wiley, 1998). | |
D. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005). | |
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] | |
S. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwave-length structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef] | |
V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13, 6798–6806 (2005). [CrossRef] [PubMed] | |
D. Akimov, A. Ivanov, A. Naumov, O. Kolevatova, M. Alfimov, T. Birks, W. Wadsworth, P. Russell, A. Podshivalov, and A. Zheltikov, “Generation of a spectrally asymmetric third harmonic with unamplified 30-fs Cr: forsterite laser pulses in a tapered fiber,” Appl. Phys. B 76, 515–519 (2003). | |
V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007). [CrossRef] | |
U. Wiedemann, K. Karapetyan, C. Dan, D. Pritzkau, W. Alt, S. Irsen, and D. Meschede, “Measurement of submicrometre diameters of tapered optical fibres using harmonic generation,” Opt. Express 18, 7693–7704 (2010). [CrossRef] [PubMed] | |
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] | |
M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997). [CrossRef] | |
G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. Sessions, E. Koukharenko, X. Feng, G. Murugan, J. Wilkinson, and D. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1, 107–161 (2009). [CrossRef] | |
A. Snyder and J. Love, Optical Waveguide Theory , 1st ed. (Springer, 1983). | |
S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett. 36, 3000–3002 (2011). [CrossRef] [PubMed] |
OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4160) Nonlinear optics : Multiharmonic generation
(190.4370) Nonlinear optics : Nonlinear optics, fibers
ToC Category:
Nonlinear Optics
History
Original Manuscript: November 7, 2011
Revised Manuscript: January 16, 2012
Manuscript Accepted: January 16, 2012
Published: March 28, 2012
Citation
Timothy Lee, Yongmin Jung, Christophe A. Codemard, Ming Ding, Neil G. R. Broderick, and Gilberto Brambilla, "Broadband third harmonic generation in tapered silica fibres," Opt. Express 20, 8503-8511 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8503
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References
- R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun.207, 155–160 (2002). [CrossRef]
- Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett.70, 922–924 (1997). [CrossRef]
- H. Endert, M. Scaggs, D. Basting, and U. Stamm, “New ultraviolet lasers for material processing in industrial applications,” J. Laser Appl.11(1), 1–6 (1999), [CrossRef]
- D. Nikogosyan, Properties of Optical and Laser-Related Materials: A Handbook (Wiley, 1998).
- D. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).
- 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]
- S. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwave-length structures part I: Kerr nonlinearity,” Opt. Express17, 2298–2318 (2009). [CrossRef]
- V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express13, 6798–6806 (2005). [CrossRef] [PubMed]
- D. Akimov, A. Ivanov, A. Naumov, O. Kolevatova, M. Alfimov, T. Birks, W. Wadsworth, P. Russell, A. Podshivalov, and A. Zheltikov, “Generation of a spectrally asymmetric third harmonic with unamplified 30-fs Cr: forsterite laser pulses in a tapered fiber,” Appl. Phys. B76, 515–519 (2003).
- V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass micro-fiber,” Opt. Commun.274, 447–450 (2007). [CrossRef]
- U. Wiedemann, K. Karapetyan, C. Dan, D. Pritzkau, W. Alt, S. Irsen, and D. Meschede, “Measurement of submicrometre diameters of tapered optical fibres using harmonic generation,” Opt. Express18, 7693–7704 (2010). [CrossRef] [PubMed]
- A. Coillet, G. Vienne, and P. Grelu, “Potentialities of glass air-clad micro-and nanofibers for nonlinear optics,” J. Opt. Soc. Am. B27, 394–401 (2010). [CrossRef]
- M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett.22, 1341–1343 (1997). [CrossRef]
- G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. Sessions, E. Koukharenko, X. Feng, G. Murugan, J. Wilkinson, and D. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon.1, 107–161 (2009). [CrossRef]
- A. Snyder and J. Love, Optical Waveguide Theory, 1st ed. (Springer, 1983).
- S. Richard, K. Bencheikh, B. Boulanger, and J. A. Levenson, “Semiclassical model of triple photons generation in optical fibers,” Opt. Lett.36, 3000–3002 (2011). [CrossRef] [PubMed]
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