Appl. Phys. B 73, 181–184 (2001) / Digital Object Identifier (DOI) 10.1007/s003400100629 Applied Physics BLasersand Optics

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Enhanced spectral broadening of short laser pulses inhigh-numerical-aperture holey fibersA.B. Fedotov1, A.M. Zheltikov1,∗, A.P. Tarasevitch2, D. von der Linde2

1 Physics Faculty, International Laser Center, M.V. Lomonosov Moscow State University, 119899 Moscow, Russia(Fax: +7-095/939-39-59, E-mail: zheltikov@top.phys.msu.su)2 Institut für Laser- und Plasmaphysik, Universität Essen, 45117 Essen, Germany(Fax: +49-201/183-2120, E-mail: tar@ilp.physik.uni-essen.de)

Received: 23 April 2001/Revised version: 18 June 2001/Published online: 18 July 2001 – Springer-Verlag 2001

Abstract. The influence of the structure of holey-fibercladding on the spectral broadening of femtosecond laserpulses is experimentally studied. These experiments demon-strate that the spectral broadening of 70-fs pulses of 800-nmTi:sapphire laser radiation transmitted through 2- and 3-µm-pitch holey fibers can be enhanced by a factor of about 1.5 byincreasing the air-filling fraction of the fiber cladding from 16up to 65%.

PACS: 42.70.Qs; 42.65.Wi

Remarkable properties of holey fibers (HFs), i.e. fibers wherea cladding has the form of a two-dimensional (often periodic)array of closely packed glass capillaries drawn at a high tem-perature, have attracted much attention during the last fiveyears, and many exciting applications of these fibers havebeen discussed and demonstrated [1–12]. The fabrication offibers with a composite cladding where air holes were ar-ranged in a two-dimensional periodic structure in glass wasfirst reported by Knight et al. [1]. One of the main advan-tages of fibers of this type is that they support single-modewaveguiding within a remarkably broad spectral range, allow-ing radiation-energy losses to be considerably reduced in thesingle-mode regime [2]. It was soon realized that the highlight localization degree attainable with HFs and their disper-sion properties also hold much promise for nonlinear opticalapplications [8–11], including frequency conversion, spectralbroadening of short pulses, and supercontinuum generation.Recent experiments [10] have revealed the ability of HFs tospectrally broaden femtosecond pulses to more than one op-tical octave (see also [13]). This property of HFs has beenemployed by Holzwarth et al. [13] and Diddams et al. [14]to implement a frequency chain linking a 10-MHz radio-frequency reference to the optical region in one step and ina phase-coherent way.

Thus, holey fibers seem to offer new elegant solutions tomany problems of fiber optics [2, 3], nonlinear optics [8–11],atomic optics [6, 15], the physics of photonic crystals [3, 12],

∗Corresponding author.

high-precision optical frequency measurements [13], biomed-ical optics [16], and optical data transmission [10, 17]. Theprogress achieved in the last few years in the technologyof fabricating holey fibers allowed the structure of thesefibers to be modified in such a way as to provide desir-able waveguiding properties. In particular, a strong wave-guiding with zero group velocity dispersion around 565 nmhas been recently demonstrated in HFs with a very high air-filling fraction [17, 18]. Other significant recent achievementsin holey-fiber optics include periodical poling demonstratedfor such fibers [19], as well as lasing [20], harmonic gener-ation [21], and birefringence effects [22] observed in holey-fiber experiments.

In this paper, we investigate the influence of the structureof the HF cladding on the spectral broadening of femtosecondlaser pulses. Our experiments performed with 70-fs pulses ofTi:sapphire laser radiation demonstrate that the efficiency ofthis process can be noticeably improved by increasing the air-filling fraction of the HF cladding, due to a higher degree oflight localization in the core of such a holey fiber.

1 The general idea

The idea of enhancing nonlinear optical processes in a HF byincreasing the diameter of holes in the cladding (and, there-fore, increasing the numerical aperture of the fiber) is basedon the fact that the effective size of a waveguide mode de-pends on the difference between the refractive indices of thefiber and the cladding. In the case of a conventional step-index fiber, the size of the waveguide mode is given by [23]

a = w+1/p , (1)

where w is the fiber core radius, p2 = k2(n2c cos2 ϕ−n2

cl) isthe transverse component of the wave vector in the fiber core,k = 2π/λ, λ is the radiation wavelength, nc and ncl are the re-fractive indices of the core and the cladding, respectively, andϕ is the incidence angle characterizing the mode in the core ofthe fiber. In the case of a holey fiber, when the cladding is nolonger solid, but includes a large number of holes (Fig. 1a, b),

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Fig. 1a–c. Cross-sectional images of holey fiberswith a pitch of the cladding equal to 3 µm. Theair-filling fraction is a 16% and b 65%. c Thegeneralized unit cell of the periodic structure ofa HF cladding

(1) can be used only as a rough estimate, and we will employit here just to provide a qualitative illustration of the idea. Wewill modify (1) to include the structure of the HF cladding byreplacing ncl by the effective refractive index of the claddingneff, introduced in accordance with [2]

neff = βcl/k , (2)

where βcl is the propagation constant of the fundamentalspace-filling mode, i.e. the fundamental mode of an infinitestructure obtained by periodically translating a unit cell of theHF cladding. Then, (1) for the effective size of the waveguidemode can be rewritten as

a = w+ λ

2π

√n2

c cos ϕ−n2eff

. (3)

Although the angle ϕ involved in (3) is still to be calculatedfor each particular waveguide mode, the refractive index ofthe core, nc, and the effective refractive index of the cladding,neff, explicitly appear in (3), thus showing the dependenceof the waveguide mode size on these parameters. The re-fractive index neff is defined by (2) through the propagationconstant of the fundamental space-filling mode, which canbe calculated by solving the dispersion equation for an infi-nite periodic two-dimensional structure of the HF claddingwithout a defect of this structure introduced by the fiber core.A recipe of an approximate solution of this problem involv-ing the replacement of the hexagonal unit cell of the HFcladding (Fig. 1c) by an annular unit cell has been proposedby Birks et al. [2], who also provided an illuminating discus-sion of the results obtained by this method.

A light pulse propagating through a HF experiences spec-tral broadening due to self-phase modulation (SPM). Therelative frequency deviation induced by this effect is givenby [24]

∆ω

ω= n2

c

P0

SτL , (4)

where n2 is the nonlinear refractive index, c is the speed oflight, P0 is the peak power of the laser pulse, S = πa2 is theeffective waveguide mode area, τ is the pulse duration, andL is the fiber length.

When studying the spectral broadening of light pulses inoptical fibers, one generally has to take into consideration dis-persion effects, which may considerably modify the spectraof short pulses emerging from the fiber [25]. To avoid the in-fluence of these effects in our experiments, we used very short

HF samples (with lengths not exceeding 3 cm). No significantinfluence of dispersion effects was seen from autocorrelationtraces under these conditions. This is why we neglect disper-sion effects in our simple qualitative analysis of self-phasemodulation in a holey fiber.

An obvious result that follows from (4) is that the effi-ciency of SPM spectral broadening can be improved by re-ducing the effective waveguide mode area S. In fact, not onlySPM, but many other nonlinear optical processes, includingwave mixing, harmonic generation, stimulated Raman scat-tering, and cross-phase modulation, could be also enhancedwith such an approach.

A straightforward method of increasing the numericalaperture of HFs is to increase the air-filling fraction f of thecladding by making the holes in the cladding larger. Pursuingthis plan, one can achieve numerical apertures much higherthan those typical of conventional fibers. However, it wouldbe a gross overestimate to think that mode localization can beimproved to the same degree as the air-filling fraction of thefiber cladding. In fact, the increase in the degree of light local-ization is much lower, of course, since light is mainly guidedalong a high-refractive material in the cladding, and muchless light is left to fill the air. These factors are included in theeffective refractive index introduced in (2). An illuminatingdiscussion of the properties of the fundamental space-fillingmode for HF claddings was provided by Birks et al. [2], whodemonstrated that the increase in the diameter of air holesin the cladding still allows a noticeable increase in the nu-merical aperture of the fiber to be achieved. By looking at(2)–(4), we easily recognize that this conclusion implies thatnonlinear optical processes can be also enhanced under theseconditions. These expectations were verified by our nonlinearoptical measurements on HFs described below.

2 Experimental

The details of the technology employed to fabricate HFs usedin our experiments, which was similar to the process de-veloped by Knight et al. [1], were described in our earlier pa-pers [11, 12]. Briefly, the fabrication process involved draw-ing identical glass capillaries stacked into a periodic preformat an elevated temperature, cutting the resulting structure intosegments, and repeating the technological cycle again. Thisprocedure allowed the fabrication of HFs with a pitch ofthe cladding ranging from 400 nm up to 32 µm, as reportedin [12].

To demonstrate the idea of enhancing nonlinear opticalprocesses in high-aperture HFs, we employed several HF

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samples with different air-filling fractions of the cladding(Fig. 1a, b). While HFs shown in Fig. 1a have small air holesin the cladding and are characterized by the air-filling fractionf equal to approximately 16%, the holes in the cladding ofHF samples in Fig. 1b are much larger, corresponding to anair-filling fraction f of about 65%. The generalized unit cellof the periodic structure of a HF cladding, shown in Fig. 1c, ischaracterized by the ratio of the air hole radius R to the pitchof the structure Λ. Although the structure of the HF claddingwith f = 16% is more complicated (see Fig. 1a), includingtwo subsets of holes, we will formally characterize it witha ratio R/Λ = 0.2, which corresponds to the air-filling frac-tion of this structure. For high-numerical-aperture HFs shownin Fig. 1b, the R/Λ ratio is equal to 0.4. The pitch of thecladding of holey fibers used in our experiments ranged from1.7 up to 5 µm.

A Ti:sapphire laser consisting of an oscillator and a mul-tipass amplifier pumped by a pulsed Nd:YAG laser was usedas a source of radiation in our experiments. Light pulses pro-duced by this laser at a repetition rate of 1 kHz had a dura-tion of 70 fs. The energy of these pulses may be as high as1 mJ, but we never used more than 1 µJ of this energy in ourexperiments.

A micro-objective used to couple laser radiation into a ho-ley fiber provided a coupling efficiency of 10–25%, depend-ing on the size of the fiber core and the type of the fiber. Thespectra of light pulses emerging from the fiber were analyzedwith the use of a spectrometer and a CCD camera.

3 Results and discussion

To study the influence of the structure of HFs on nonlinearoptical processes occurring in the fiber core, we employedshort pieces of HF samples, typically with a length of 3 cm,in order to avoid the spectral superbroadening of laser pulsesdue to a combination of several nonlinear optical processesin a HF (this effect was earlier observed in holey fibers byRanka et al. [10]) and to reduce the influence of dispersioneffects. The spectra of 70-fs Ti:sapphire laser pulses at theinput and at the output of a holey fiber with a length of

Fig. 2. The spectra of 70-fs Ti:sapphire laser pulses (bold line) at the inputand (1–7) at the output of a holey fiber with a length of 3 cm, the pitch ofthe cladding equal to 3 µm, and the air-filling fraction f = 16% for pulseenergies of (1) 0.5 nJ, (2) 5 nJ, (3) 10 nJ, (4) 20 nJ, (5) 30 nJ, (6) 40 nJ, and(7) 50 nJ

3 cm and the pitch of the cladding equal to 3 µm are shownin Fig. 2. The bold line in this figure shows the spectrumof the pulse going into the fiber. The output-pulse spectra1–7 correspond to input-pulse energies of 0.5 nJ (1), 5 nJ (2),10 nJ (3), 20 nJ (4), 30 nJ (5), 40 nJ (6), and 50 nJ (7). Muchmore efficient spectral broadening was achieved in experi-ments [10, 13, 14] where subnanojoule femtosecond pulseswere shown to allow supercontinuum generation under cer-tain experimental conditions. A lower efficiency of spectralbroadening in our experiments was due to shorter lengthsof HF samples, which allowed us to study the influence ofthe HF cladding geometry on self-phase modulation in theHF core in the regime when dispersion and superbroaden-ing effects were negligible, and due to a lower quality of thefibers, which resulted in considerable optical losses in ourexperiments.

Figure 3a and 3b present the spectral broadening (definedas the spectral width of a light pulse at the output of the fiberminus the spectral width of the same pulse at the input ofthe fiber) of 70-fs pulses of 800-nm Ti:sapphire laser radia-tion at the output of 2- and 3-µm-pitch HF samples (whosecross-sectional views are shown in Fig. 1a, b) with a lengthof 3 cm and air-filling fractions equal to 65% (dependences 1)and 16% (dependences 2) as functions of radiation energy

a

bFig. 3a,b. The spectral width ∆ω of 70-fs pulses of 800-nm Ti:sapphirelaser radiation transmitted through a 2- and b 3-µm-pitch HF samples witha length of 3 cm and the air-filling fraction equal to (1) 65% and (2) 16% asfunctions of radiation energy coupled into the fiber

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coupled into the fiber. While the initial phase of supercon-tinuum generation was observed for laser-pulse powers onthe order of 10 kW in our experiments with 3-cm HF sam-ples, the increase in radiation energy for lower powers of laserpulses resulted in a linear growth in the spectral widths oflaser pulses transmitted through a HF (see Fig. 3a, b). Theenhancement in the SPM efficiency, defined as η = ∆ω ( f =65%)/∆ω ( f = 16%), can be deduced from the ratio of theslopes of the dependences shown in Fig. 3a and b. Such anestimate yields an enhancement factor η of about 1.46 in thecase of 2-µm-pitch HFs and approximately 1.45 for 3-µm-pitch HF samples. These estimates agree well with waveguidemode diameters estimated directly from images of the out-put ends of holey fibers for the studied HF samples (5 and3.4 µm for 3-µm and 2-µm-pitch HFs with f = 16%, respec-tively, and 4 and 2.7 µm for 3-µm and 2-µm-pitch HFs withf = 65%).

Holey fibers with air-filling fractions of 16 and 65% em-ployed in our experiments may support more than one wave-guide mode in accordance with the 2πw

λ

(n2

c −n2eff

)1/2< 2.405

criterion. The fact that the SPM-induced spectral broadeningcan be approximated with a linear function of the pulse in-tensity within some range of input-radiation energies (Fig. 3aand b) is also important in this context, as it indicates thatthe effective mode area S introduced in (4) is not just a for-mal quantity, which may change, depending on the energycoupled into the fiber due to the excitation of higher-ordermodes, but a parameter that adequately describes the influ-ence of light localization in a HF core on spectral broaden-ing of short pulses in our experimental geometry within thestudied range of pulse energies.

4 Conclusion

Experimental investigation of the influence of the structure ofthe holey-fiber cladding on the spectral broadening of fem-tosecond pulses performed in this paper has demonstratedthat the efficiency of self-phase modulation in such a fiber canbe noticeably improved due to the decrease in the effectivearea of the waveguide mode. By increasing the air-filling frac-tion for 2- and 3-µm-pitch holey fibers from 16 up to 65%,we were able to improve the efficiency of spectral broad-ening of 70-fs pulses of 800-nm radiation of a Ti:sapphirelaser by a factor of about 1.5. This possibility of enhanc-ing nonlinear optical processes in holey fibers by modify-ing the HF cladding geometry, demonstrated in this paper,seems to be more than a purely technical fiber-optic issue,as it provides additional means of controlling nonlinear opti-cal processes in optical fibers, which may be useful for pulsecompression, soliton formation, and stimulated Raman scat-tering experiments.

Although our analysis presented in this paper was re-stricted to two different air-filling fractions and two configu-rations of hole patterns in the HF cladding, we believe that theresults of these studies are quite representative due to the factthat one of the f values taken for our experiments was ratherlow (16%), while the other was very high (65%). The experi-mental results presented in this paper are consistent with ourexpectations based on the qualitative examination of nonlin-ear optical processes in holey fibers. This agreement betweenexperimental data and theoretical analysis allows predictions

for intermediate values of the air-filling fraction to be madeby interpolating the results of measurements presented in thispaper. Our experiments performed with holey fibers of othercladding geometries and air-filling fractions have confirmedthe general tendencies outlined in this paper.

Acknowledgements. We are grateful to V.I. Beloglazov, N.B. Skibina, andYu.S. Skibina for fabricating the fiber samples. We would also like to ex-press our gratitude to Prof. M. Aeschlimann and the members of his groupM. Bauer, M. Scharte, and M. Wessendorf for help with the Ti:sapphirelaser.

This study was supported in part by the Volkswagen Foundation(project I/76 869). The work of A.B.F. and A.M.Zh. was also supportedin part by a President of the Russian Federation Grant No. 00-15-99304,the Russian Foundation for Basic Research Project No. 00-02-17567, andAward Nos. RP2-2266 and RP2-2275 of the US Civilian Research and De-velopment Foundation for the Independent States of the former SovietUnion (CRDF).

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