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Title Nonlinear photonic crystal fibres: pushing the zero-dispersion towards the visible

Author(s) Saitoh, Kunimasa; Koshiba, Masanori; Mortensen, Niels Asger

Citation New Journal of Physics, 8, 207https://doi.org/10.1088/1367-2630/8/9/207

Issue Date 2006-09-22

Doc URL http://hdl.handle.net/2115/14820

Rights Copyright © 2006 IOP Publishing Ltd.

Type article (author version)

File Information NJP8.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Nonlinear photonic crystal fibres: pushing the zero-dispersion toward the visible

Kunimasa Saitoh and Masanori Koshiba Hokk;:ti do UT1 i v~r~ i ty, North 14 \~r~st 9 , K it<'l -kll, Sapporo, 060-08 14 , Japan

Niels Asger IVlortensen§ I\U C D epartment of !vlicro and Nanotech nolog::y, NanoD TlJ, Technical Uni vl?r~it,y of Denmark, BId . 345 1?<'I.;:;t, DK-2800 K()ngo?n~ Lyngby, Denmark

Abs tra ct. The strong waveguide d ispersion in photonic crystal fibres provides unique opport unities for nonlinear optics with a zero-dispersion 'wavelengt h .\0 far below the limit o/",-...; 1.3/un set by the material d isper::don of siH ~a . Ry tlIning the a ir-hole d iameter d, the p itch A, and t he number of rings of air holes N , the strong waveguide d isper sion can in princip le be used t o e.x tend >'0 \vell into t he \'hliblp., (l lbeit to some extend at. the C;O>lt. of muit.imode op eration. \,\'e study iTl detail the in terplay of t he 7.ero-d ispersion wavelength, the cut-ofT wavelength Ac, and the leakage loss in the parameter space spanned by d, A , and N . As a particular resu lt 'we identify values of d ('"'-' fiOO nm) and A ("-' 700 nm) which I"ad lita t e t he shnrt.e>lt. pO>lsib le :t.ero-disper>lioTl wl'lvelengt.h ('" 700 nm) whi le the fibre is st ill sinp;le-mode for lonp;er wavelen p;ths .

PACS numbers : 42 .70.Qs, 42 .8 1.Dp , 42 .8 I.-i

Submitted to: New J. Phys. (Focus on Nanophotonics)

~ Corresponding aut hor : nam@mic.dtu.d k

Saitoh, Koshiba fj Mortensen: Nonlinear photonic crystal fibres ... 2

1. Introduction

Photonic crystal fibres (PCFs) [1, 2] have led to an enormous renewed interest in nonlinear fibre optics [3, 4, 5] . In particular, the strong anomalous group-velocity dispersion has facilitated visible super-continuum generation [6], which apart from being a fascinating and rich non-linear phenomenon [7] also has promising applications including frequency-comb metrology [8, 9] and optical coherence tomography [10].

The regular triangular arrangement of sub-micron air-holes running along the full length of the fibre [11, 12], see Fig. 1, is a key concept for the realization of strong anomalous chromatic dispersion which arises in a competition between material dispersion and wave-guide dispersion originating from the strong transverse confinement of light. At the same time the strong transverse confinement of light also serves to dramatically decrease the effective area and increase the non-linear coefficient [13] so that very high optical intensities may be achieved relative to the input power.

The zero-dispersion wavelength may be tuned down to the visible [14, 15] by carefully increasing the normalized air-hole diameter d/ A while at the same time decreasing the pitch A. However, in many cases the very short zero-dispersion wavelength is achieved at the cost of multi-mode operation. Furthermore, the reduction of the pitch may require a large number N of rings of air holes in order to circumvent leakage loss.

In this paper we in detail study the complicated interplay of the zero-dispersion wavelength Ao, the cut-off wavelength Ae, and the leakage loss in the parameter space spanned by d, A, and N . As a particular result we identify values of d and A which facilitate the shortest possible zero-dispersion wavelength while the fibre is still single­mode for longer wavelengths, i.e. Ae ~ Ao.

Figure 1. Cross section of a photonic crystal fibre w ith N = 4 rings of a ir holes in the photonic crystal cladding surrounding the core defect which is formed by the omission of a single air hole in the otherwise periodic structure. The photonic crystal cladding comprises air holes of diameter d arranged in a triangular array of pitch A.

Saitoh, Koshiba fj Mortensen: Nonlinear photonic crystal fibres ... 3

2. Competition between material and wave-guide dispersion

The linear dynamics and response of a waveguide is typically studied within the framework of temporal harmonic modes, E(r, t) = E(rJ.)e-'iwt = E(rJ.)e'i((3 z- wt) , where E(r') is a solution to the vectorial wave equation

(1)

Here, dr, w) = n 2 (r, w) is the spatially dependent dielectric function of the composite air-silica dielectric medium, see Fig. 1, and n(r, w) is the corresponding refractive index.

Solving the wave equation, Eq. (1), provides us with the dispersion relation w(/3) which contains all information on the spatial-temporal linear dynamics of wave packets and it is furthermore important input for studies of non-linear dynamics. However, quite often the dynamics and evolution of pulses are quantified by the derived dispersion parameters /3n = (onw / on /3) -1 with /31 = V;; 1 being recognized as the inverse group velocity. The group-velocity dispersion /32 is in fiber optics commonly quantified by the dispersion parameter D (typically in units of ps/nm/km) which can be written in a variety of ways including

02 /3 0/31 w2

D = o>..ow = 0>.. = - 27rl2 (2)

where>.. = 27rc/w is the free-space wavelength. The dispersion in the group velocity makes different components in a pulse propagate at different speeds and thus pulses

r--'\

~ t:: --. C/1

~ t:: 0 ..... C/1 I-< (!) 0, C/1 .....

Q

106

~ 103 ~

~ C/1 C/1 0 -400 -(!) OJ)

-600 1 ro ~ ro

high-order mode (!)

.....l -800

-1000 L.......!-'---'----'-----'-----'--'--'_'---'-----'-----'-----'----' 10-3

0.6 0.8 1.0 1.2 1.4 1.6

Wavelength [11m]

Figure 2. Left axis: Dispersion parameter versus wavelength for a photonic crystal fibre w ith d/A = 0.7, A = 700nm, and N = 10. The solid curve shows the total dispersion, Eq. (2), obtained through a self-consistent numerical solution of the wave equation, Eq. (1), while the dashed line shows the approximate addition of waveguide and material dispersion, Eq. (7). The dotted curve shows the pure waveguide contribution, Eq. (5), while the dot-dashed curve shows the silica material dispersion, Eq. (6). Right axis: Leakage loss versus wavelength for the fundamental and the first high-order mode. The rapid increase in the leakage loss of the high-order mode marks the cut-off wavelength.

Saitoh, Koshiba fj Mortensen: Nonlinear photonic crystal fibres ... 4

2.0,-------------,---,-------,

1.5 , ,

E , , ~ , , , ,

0 « 1.0

0.5 1.0 1.5 2.0

Fiber diameter [/lm]

Figure 3. Zero-dispersion wavelength versus fibre diameter for a strand of silica in air. The dashed line indicates the cut-off wavelength and the blue shading indicates the multi-mode phase.

will compress or broaden in time depending on the sign of D. The zero-dispersion wavelength Ao, defined by D(Ao) = 0, is thus of particular relevance to pulse dynamics in general and non-linear super-continuum generation in particular. By pumping the fiber with high-intensive ultra-short nano and femto-second pulses near Ao the pump pulses will loosely speaking maintain their high intensity for a longer time (or propagation distance) thus allowing for pronounced non-linear interactions.

The finite group-velocity dispersion is a consequence of both the dispersive properties of the host material itself as well as the strong transverse localization of the light caused by the pronounced spatial variations in the dielectric function. Since the variation with frequency of the refractive index of silica is modest (at least in the transparent part of the spectrum) the dielectric function satisfies

c(rJ.,w) = co(rJ.,w) + Oc(rJ.,w),

Oc(rJ.,w) = c(rJ.,w) - co(rJ.,w)« co(rJ.,w)

(3)

(4)

where W is some arbitrary, but fixed frequency where consequently Oc = O. The high attention to the telecommunication band has often made 5. = 1550 nm a typical choice, though this is by no means a unique choice. From co one may define a pure waveguide contribution Dw to the dispersion parameter by the definition

D - [P!3o w - [)A[}w (5)

where !3o(w) is the solution to the wave equation with c(rJ.,w) = co(rJ.,w), i.e. the frequency dependence of the dielectric function is ignored. Furthermore, this has the consequence that the wave equation, Eq. (1), becomes scale invariant which is very convenient from a numerical point of view since results for one characteristic length scale can easily be scaled to a different length scale.

Saitoh. K osltiba f!.1 !\1ortcns(;.n: }\'onlincaT llhotonic crystal fibTCS ... 5

libr the dielectric material itself one likewise define~ a mal,erial dh.;persion J)m

by solving the \\'ave equation ,>,:ith £(i-:'"..L, u.:) = £n1 (:/J) = n;n c, •. :). .From the simple homogeneous-space dispersion relation it readily follO\\'s that

>. fPnm D,n = -- 'J") .

C (A-(6)

Intuitiw-:lYl one might speculate that the hvo kindf-l of sources of dispersion simply add 11 p ,lno. ad.llally tlw approxirnati(lIl

D::::: D·Il , +Dm (7)

is Ilsnd 'viddy in the litmaturn. \Vhil(! th(~ approximation is Ilsnflll in flualitativdy under~tanding the zero-dispersion properties it is however ah;;o clear (see e.g. the work of Ferranoo ct al. [Hi]) that quantitative correct results requires eithC'I a. self-consistent solution of the vvave equation or some accurate perturbative method [17; lR].

In this papf'r we use a fully self-consistent solution of the WHVf' equationl Eq. (1). For the dielecLric function we use the frequency-independent yalue c: = I in the air­hole regions while we ror silica employ the usual three-term Sell meier polYl1omial description,

:i 'J

2 , (1.j"'\-Cm(A) = nm(A) = 1 + ~ V _ A2

)=1 j

where the absorption lines Aj and the corresponding strengths aj are given by

,\, = O.06841J43 I'm,

~\2 = 0.1162414 p.m,

A" = 9.896161 Jl111,

Q., = 0.6961663,

(J2 = 0.4079426,

(J:; = 0.8974794.

(8)

('l' )

(10)

(11)

Figure 2 illustrates the typical dispersion properties of a. photonic cr:/stal fiLrc. The' st.rongl~r Ilcgativ(' I1l8terial dispersioIl Dm of silica hdow ",\ ..... 1.3 fun tend t.o make the total dispersion D of Rtandard fibres negative for A :S 1.31lHl: simply hecause of t.he very "veak \vaveguide coniributlon TJ 11 ,. However. pholonic crystal fibres are contrary to this since the composite air-sHica cladding b seen to provide the guided mode with a strongly positive \vaveguidC" dispersion Dw which tends to shift the zcro-oispcrsion vvavc!ength >'0 far below 1.;1 fun towards the visible. \~'~hil(' thC' material dispcn::ion is fixeo. the waveguide diHperslon varief-i Htrongly in the phi-:lse-space f-ipauned hy d ano. A and in this \-yay the COmpel-1UOn betvveen \vaveguide a.nd material dispersion becomes a powerful mechanism in engineering tIle ~ero-dispersion wavelength.

3. A strand of silica in air - the ultimate limit?

AR mentiolled ill the introductioll the zero-dispersioll wavelength may he pushed to lower values by simply inerc~asing the air ho1c~ diaJllN(~r and decrea .. «:ing· the pitch. For d/:\ --+ 1 this will to Rome extend en·ectively leave us with a single strand or silica surrounoed L,Y air. This limiting case has becn emphasized previously in the literature and in Fig. :1 ,ve reproduce the zero-dispersion vVllvf'iength results reported by Knight et al. [1·1]_ Generally, the strand of silica vvi11 have eitlwr t\vo dispersion zeros or none. \vith the exception of the special case \-vhere it only supports a single dispersion zero. PCFs turn out to foHow the same overall pattern and the existence of t,vo dispersion L.;cros t.urns out to have interesting applications in super-continuum generation [19].

Saitoh, Koshiba €1 Mortensen: Nonhnear' photanic cTystal jib1'es ... 6

The results in Fi~. 3 leave promises for a zero-dispersion \vavelen~ih dO\vn io below 500 nm which will eventually also be the ultimate limit [or silica based PCFs. However, as also indicated by t he dashed line t he large index contrast bchvccn air and silica in general prevents single-IIlode operat ioll at the zero-dispersion wavelength. In the follmving \ve wi ll study io which degree the photollic crystal cladding concept of PCFs can be used to circumvent this problem.

4. Photonic crystal cladding as a modal sieve

As demoIlstr ated already by Birks fd al. [12] t he photonie crystal cladding of it P CF ads a ... '" modal sieve \vhieh ma:r prevent localization of high-order lUodes to t he core region. T his so-called endlessly single-mode property has later been studied in great detail [1:l, 20, 21, 22] and it was recently ar~ued that the endlessly sin~le-mode

phenomena is a pure geometrical effect and t hat the P CF is eIHllessly single mode for d/A ~ 0.42 irn'!spedively of t he fihre material refradive index [23] . The photonic crystal cladding t hus serves t o limit t he number of guided modes and at t he same time the ~uided modes will to some extend inherit the chromatic dispersion properties observed [or t he strand o[ silica in air [14] .

The problem of zero-dispersion wavelength versus pitch has previously been studied for P CFs v.rith an infinite photonie crystal cladding [18] demonstrating curves quaHtatively resembling the curve in Fig. 3. Here~ v.re extend that v.rork to PCFs \vith a photonic crystal cladding or finite spatial extent. Tn particular, we study the erreci of a varying nUlnber Iv' of rings of air holes surroundin~ the core region. \Ve also study the cut-off and leakage properties to explore the possibility for a single-mode PCF with a zero-dispersion wavelength in t he visible.

Our numerical solutions of the v.rave equation, Eq. ( I ), are based on a fin ite­element approach which is described in detail in Rer. [24]. Por the calculation of the cut-alI wavelen~th and the leakage loss we refer to Refs. [25, 26] and references therein.

The results of extensive numerical simulat ions are summarized in Fig. 4. First of all \ve notice t hat the zero-dispersion v.ravelength versus pitch has a curve-shape quaHtatively resembling the result in Fig. 3 fo r a strand of silica in air. Furthermore~ we see that the number lV or rin~s or air holes has Ht tle innuence on the zero-dispersion \vavelength. In particular, t he results for 1V = 10 are in full quantitative agreement \vith t hose reported in Ref. [1 8]. On t he other hand, t he spatial extent .!.V X A of t he photonic er:ystal cladding is as expected seen to hav"f! a huge impad on t he leakage loss [27, 26] as seen from the red shadin g indicati ng the region with a leakage loss exceeding 0.1 dB/km. Furthermore, N has as expected little elIect on the cut-alI \vavelength since the cut -off and the modal sieving is governed by t he width (A - d) of t he silica regions hehveen t he air holes [23] rather than the spatial extent .N X A of t he photonie (:rystal cladding.

Finally, we note tllat by c1loosin~ dj " rv 0.7 we may reali:;l,e a peF wiLlI a single ~ero-dispersion \ .... 'avclength dmvn to rv 700 nm with the fibre being sin~le-mocle for longer wavelengths. \Ve believe this to be the ultimate limit for silica-based P CFs having a photonie crystal cladding comprising a triangular arrangement of circular air holes. Such results have been demonstrated experimenta lly by e.g. Knight et al. [14] . Tn praciice1 the lin-.it mi~llt be pushed sH~llt1y ruril ler toward the visible since real PCFs tend to have a sli~htly shorter cut -off wavclen~ih compared to the expectations based on the ideal fibre structure [22] . l\1ost likely~ this tendency originates in the presence of s(:attering loss in real fibres ,vhich also ads in suppressing the high-

Saitoh, Koshiba fj Mortensen: Nonlinear photonic crystal fibres ...

o ~

o ~

A [Jlm]

7

Figure 4 . Zero-dispersion wavelength '>"0 versus pitch A for different values of the normalized air-hole diameter d/ A. Panels a), b), and c) are for N = 6,8, and 10 rings of air holes, respectively. Regions with a leakage loss larger than 0.1 dB/ km are indicated by red shading. Similarly, the multi-mode regime is indicated by blue shading.

Saitoh, Koshiba €1 Mortensen: Nonhnear' photanic cTysta{ jib1'es ... 8

order modes even though they are \veakly ~uided by the phoionic crystal c1addin~. In order to push the ~cro-dispcrsion wavclcn~;th further into the visible one \vould have to tolerate guidance of high-order modes or alternatively employ smIlc'"vliat morc complicated designs involving a varying air-hole dianletf'!I throughout the cla<i(ling [28].

5. Conclusion

Tn conclusion "ve have studied the :;;ero-dispersioll wavelength AO in silica based photonic crystal fibres \~'ith special emphasis OIl the interplay with the cut-off \vavelength aml leakage loss. In the large parameter space spannp.cl 1)), the air-hole diameter d and the pitch A 'l,re have identified the valnes facilitating the shortest possible "7;ero-dispersion \yavelength (.-v 700 nm) \yhile the fibre is siill single-mode for longer wavelengths.

\Ve believe that our An-maps arc an important input for the efforts in designing nonliIH'!ar photonie crystal fibres with still shorter zero-dispersion \yavelengths for super-continuum generation in the visible.

6. Acknowledgments

N. A . N1. acknO\yledges discussions with J. Lcegsgaard as well as the collaboration on the zero-dispersion results in Ref. [18] v,rhich strongly stimulated the present \york.

Saitoh, Koshiba €1 Mortensen: Nonhnear' photanic cTysta{ jib1'es ...

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