designing microstructured polymer optical fibers for cascaded quadratic soliton compression of...

1MporomcafGam

fma(somp(

ctst

460 J. Opt. Soc. Am. B/Vol. 26, No. 3 /March 2009 Morten Bache

Designing microstructured polymer optical fibersfor cascaded quadratic soliton compression

of femtosecond pulses

Morten Bache

DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby,Denmark

([email protected])

Received November 17, 2008; accepted December 19, 2008;posted January 8, 2009 (Doc. ID 104222); published February 12, 2009

The dispersion of index-guiding microstructured polymer optical fibers is calculated for second-harmonic gen-eration. The quadratic nonlinearity is assumed to come from poling of the polymer, which is chosen to be thecyclic olefin copolymer Topas. We found a very large phase mismatch between the pump and the second-harmonic waves. Therefore the potential for cascaded quadratic second-harmonic generation is investigated inparticular or soliton compression of femtosecond pulses. We found that excitation of temporal solitons fromcascaded quadratic nonlinearities requires an effective quadratic nonlinearity of 5 pm/V or more. This mightbe reduced if a polymer with a lower Kerr nonlinear refractive index is used. We also found that the group-velocity mismatch could be minimized if the design parameters of the microstructured fiber are chosen so therelative hole size is large and the hole pitch is of the order of the pump wavelength. Almost all design-parameter combinations resulted in cascaded effects in the stationary regime, where efficient and clean solitoncompression can be found. We therefore did not see any benefit from choosing a fiber design where the group-velocity mismatch was minimized. Instead numerical simulations showed excellent compression of �=800 nm 120 fs pulses with nanojoule pulse energy to few-cycle duration using a standard endlessly single-mode design with a relative hole size of 0.4. © 2009 Optical Society of America

OCIS codes: 060.4005, 190.4370, 320.5520, 320.7110, 320.2250, 190.5530.

rhlcTita

epftsTcbightsCotp

t

. INTRODUCTIONicrostructured optical fibers allow for an elaborate dis-

ersion control. Usually they are exploited to gain controlver the group-velocity dispersion (GVD) [1] (see [2] for aecent literature overview over the dispersion propertiesf microstructured optical fibers). Also the group-velocityismatch (GVM) for three-wave mixing processes can be

ompletely removed [3], which is important in order to re-lize second-harmonic generation (SHG) of ultrashortemtosecond pulses. However, the prize for having a smallVM is that the phase mismatch becomes very large [3],nd efficient SHG would therefore rely on quasi-phase-atching (QPM) techniques.The large phase-mismatch �� can instead be exploited

or cascaded ��2� :��2� SHG processes [4]. The second har-onic (SH) is within a coherence length 2� / �� gener-

ted and then back-converted to the fundamental waveFW). This cascaded process generates a nonlinear phasehift �NL on the FW [5], which can become several unitsf � by propagating in just a few centimeters of nonlinearaterial, and is conceptually equivalent to the nonlinear

hase shift observed through self-phase modulationSPM) with cubic nonlinearities [6].

The large nonlinear phase shift can be exploited toompress ultrashort femtosecond pulses [7]. This is par-icularly advantageous when the compression occurs in-ide the nonlinear material due to the soliton effect [8,9];he cascaded SHG process can generate a negative �
NL
0740-3224/09/030460-11/$15.00 © 2

esulting in a negatively chirped FW, and if the materialas normal GVD a temporal soliton is generated. By

aunching a higher-order soliton, the input pulse can beompressed by exploiting the initial pulse narrowing.his cascaded quadratic soliton compressor (CQSC) can

n principle compress the FW pulse to single-cycle dura-ion, and is ultimately limited by higher-order dispersionnd competing Kerr nonlinear effects [10].One requirement for efficient compression is that GVM

ffects are not too strong: they tend to distort the com-ressed pulses through a Raman-like effect [9–12]. Inact, when GVM dominates over the cascaded effects fromhe phase mismatch the compression is nonstationary, re-ulting in inefficient compression and distorted pulses.herefore the possibility offered by microstructured opti-al fibers to control GVM is very intriguing for the CQSC,ecause the compression can become stationary, whichmplies efficient compression and clean pulses. The fibereometry can also help in overcoming the problem of in-omogeneous compression in the transverse direction ofhe beam found in a bulk geometry [13]. No comparisonhould otherwise be made between the bulk and the fiberQSC; the bulk version works with high-energy femtosec-nd pulses with microjoule to millijoule energies, whilehe fiber version works with low-energy femtosecondulses with picojoule to nanojoule energies.We have done a preliminary investigation of the poten-

ial for using silica microstructured optical fibers for

009 Optical Society of America

CsWatdtcabgnsfiimcb

ppgitmsndsn

gdfi[filpttn[twSvtiibmtbstddgpcte

2TpsalmecnTli(titnpceGsG[

ssodirwt

tcig

etcon

wbhhttac

Morten Bache Vol. 26, No. 3 /March 2009/J. Opt. Soc. Am. B 461

QSC [2,14], where the quadratic nonlinearity was as-umed to come from thermal poling of the silica fiber [15].e surprisingly found that zero GVM was not as such an

dvantage, since the compressed pulses became very dis-orted. Another main result was that a very large qua-ratic nonlinearity deff�3–5 pm/V was needed in ordero generate suitably large �NL [2]. This is because thehosen fiber designs had very a large phase mismatch,nd �NL�deff

2 /��. Realistically, thermal poling of silica fi-ers can generate deff�0.5 pm/V [16]. Therefore we sug-ested to lower the phase mismatch using QPM tech-iques, and in this way a much lower deff�1 pm/V wasufficient [14]. However, periodic thermal poling of silicabers has until now proved inefficient, and has resulted

n a much lower deff�0.01 pm/V than expected [17]. Weust therefore conclude that periodically poled silica (mi-

rostructured) fibers cannot generate large enough �NL toe interesting for CQSC.A large deff would compensate for a large ��. In this pa-

er, we therefore turn our attention to microstructuredolymer optical fibers (mPOFs). Poling of polymers canenerate extremely large quadratic nonlinearities (rang-ng from 1 pm/V to hundreds) [18], while poling still haso be shown in a fiber context. Alternatively, the lowPOF drawing temperature (a few hundred degrees Cel-

ius) implies that nanomaterials with large quadraticonlinear responses (such as nanotubes [19]) can berawn into the fiber core. Also this solution promises atrong deff. The potential for using mPOFs for quadraticonlinear optics is therefore large.Here we calculate the dispersion properties of an index-

uiding mPOF with three rings of air holes in the clad-ing. The polymer material is chosen to be the cyclic ole-n copolymer Topas due to its broad transparency window20]. We point out that a problem with using polymer asber material is that unlike silica the material Kerr non-

inearity is very large, and it generates an SPM-inducedositive nonlinear phase shift that has to be overcome byhe negative cascaded nonlinear phase shift. We will showhat for the considered polymer Topas deff�5 pm/V iseeded to do so (compared to deff�3–5 pm/V in silica2,14]). Such a value could be achieved with polymers ashe fiber material. We also show that the fiber designsith a dramatically reduced GVM are multimoded in theH, and thus one risks to have cascaded nonlinear con-ersion to several SH modes. Quite surprisingly we findhat the compression is always in the stationary regime,rrespective of the choice of fiber-design parameters. Thiss very positive since efficient and clean compression cane obtained. On the other hand, there is no longer anyotivation for reducing GVM in order to enter the sta-

ionary regime. Thus, we conclude that there are very fewenefits of reducing GVM through fiber design. Thistatement is underlined by performing numerical simula-ions of the propagation equations for an mPOF havingeff=10 pm/V: an endlessly single-moded mPOF designominated mainly by material dispersion turns out toive clean and efficient compression of femtosecondulses, and obtaining compressed pulses with durationslose to single-cycle duration is possible. So if the poten-ially large deff of polymer material can be exploited, thenxcellent pulse compression can be obtained in mPOFs.

. NONLOCAL THEORYhe scope of this theoretical part is to point out the im-ortant parameters when choosing the proper fiber de-ign. The main hypothesis is that an mPOF can be useds a CQSC. The challenges with creating a quadratic non-inearity in the fiber aside, there are other obstacles that

ust be faced. Understanding these issues can be greatlynhanced by realizing that in the cascading regime, theoupled FW and SH propagation equations reduce to aonlinear Schrödinger equation (NLSE) for the FW [6].he action of the cascaded SHG can be modeled as a Kerr-

ike nonlinearity with an equivalent nonlinear refractivendex nSHG

I �deff2 /��. Importantly, when ��=�2−2�1�0

which is usually the case) nSHGI is self-defocusing by na-

ure: it generates a negative nonlinear phase shift �NL0. The material Kerr nonlinear refractive index nKerr

I isnstead usually self-focusing, and therefore counteractshe effects of the cascaded nonlinearities. Achieving

SHGI �nKerr

I is crucial to obtain a large negative nonlinearhase shift. An important point here is that because theascaded SHG induces a self-defocusing Kerr-like nonlin-arity, temporal solitons exist in the presence of normalVD. In contrast, the usual cubic temporal solitons ob-

erved for instance in telecom fibers require anomalousVD due to the self-focusing Kerr nonlinearity of silica

21].GVM is also playing a decisive role. As we recently

howed the pulse compression is clean and efficient in thetationary regime, where the phase mismatch dominatesver GVM effects [10,12]. These results were obtained byeriving a more general NLSE, where dispersion includ-ng GVM imposes a Raman-like nonlocal (delayed) tempo-al cubic nonlinearity. Contrary to earlier reports [7,11]e showed that the regime where efficient compression

akes place is independent of the input pulse duration.In the remainder of this section we briefly show how

he nonlocal NLSE for the FW is derived and discuss theonsequences for soliton pulse compression. The equations derived in dimensionless form based on the SHG propa-ation equations derived in Appendix A.

The nonlocal theory was derived in [10,12] (for a gen-ral review on nonlocal effects, see [22]). Essentially inhe cascading limit (large phase mismatch), the SH be-omes slaved to the FW. The normalized FW U1 can thenn the dimensionless form be modeled by the followingonlocal NLSE [12]:

�i�

�−

sgn��1�2��

2

�2

��2�U1 + NKerr2 U1�U1�2

= NSHG2 U1

*�−�

�

dsR±�s�U12�,� − s�. �1�

here Kerr cross-phase modulation (XPM) effects haveeen neglected, and for simplicity self-steepening andigher-order dispersion are not considered. On the left-and side (LHS) an ordinary NLSE appears with an SPMerm from self-focusing material Kerr nonlinearities. Onhe right-hand side (RHS) the effects of the cascaded SHGppear: it is also SPM-like in nature, but it turns out to beontrolled by a temporal nonlocal response. The dimen-

spgtwf

aso�bci

tri

�

T

wtnpn

w

Nimiwfiberm

f

wd

o

wt[

Nwin�o

Ud

Scb

sRtc=s

Tv=tih

3Ips


ionless temporal nonlocal response function R±�� ap-ears in two distinct ways: in the stationary regime it isiven by R+, and �R+��e−��/�b is localized having a charac-eristic width �b of a few femtoseconds. This happenshen the phase mismatch is dominating over GVM ef-

ects or, more precisely, when ��sr, with

��sr =d12

2

2�2�2�

= −�d12

2

D2�22 . �2�

nd D2=−2��2�2� /�2

2 is the fiber GVD parameter. In thetationary regime clean and efficient compression can bebtained [9,10,12,23]. Instead in the nonstationary regime��sr� GVM effects dominate and the compressionecomes distorted and inefficient. In this case the nonlo-al response function is given by R−, and �R−��sin�� /�b�s oscillatory and never decays.

The fiber designs presented here all turn out to be inhe stationary regime. In this case, and when the nonlocalesponse function can be assumed to be quasi-nstantaneous, Eq. (1) can be written as [9,11,12]

i�

�−

sgn��1�2��

2

�2

��2�U1 − �sgn��NSHG2 − NKerr

2 U1�U1�2

= − iNSHG2 sa�R,SHG�U1�2

�U1

��. �3�

he soliton order for the Kerr fiber nonlinearity is [24]

NKerr2 =

LD,1

LKerr, �4�

here LD,1=Tin2 / ��1

�2�� is the characteristic GVD length ofhe FW, and LKerr= � KerrPin�−1 is the characteristic Kerronlinear length. Pin=�0neff,1ca1Ein

2 /2 is the input peakower, related to the peak input electric field Ein. The Kerronlinear coefficient is

Kerr = nKerrI

�1

cAeff,1, �5�

here Aeff,1 is the effective Kerr FW mode overlap area

Aeff,1 =�dx�F1�x��2�2

Pdx�F1�x��4. �6�

ote that Aeff,1� f11−1 from Eq. (A10), and that as explained

n Appendix A the subscript P indicates that integrationust be done only over the part of the fiber mode sitting

n the polymer part of the mPOF. We also encounter theell-known Kerr nonlinear refractive index nKerr

I : the re-ractive index change due to the Kerr self-focusing effects defined as �n=nKerr

I I, where I is the intensity of theeam. Although it is not known for the Topas polymer, westimate (based on similar polymer types) that it is in theange nKerr

I �10–15�10−20 m2/W, which is an order ofagnitude larger than for fused silica.In a similar way we have introduced a soliton order

rom the cascaded SHG process

NSHG2 =

LD,1

LSHG, �7�

here LSHG= � SHGPin�−1. Here the nonlinear coefficient isefined as for the Kerr case

SHG = �nSHGI �

�1

cASHG, �8�

nly it contains the SHG overlap area ASHG,

ASHG =a1

2a2

�Pdx�F1*�x�2F2�x��2

, �9�

here aj are the fiber mode areas [Eq. (A6)]. The “effec-ive” nonlinear refractive index from the cascaded process5]

nSHGI = −

4�deff2

c�0�neff,12 neff,2��

. �10�

ote the negative sign in front of nSHGI ; since the cases we

ill discuss always have ��0 the cascaded nonlinearitys therefore self-defocusing. Thus, in order to generate aegative nonlinear phase shift we must have SHG Kerr. We can express this through an effective soliton

rder

Neff2 = NSHG

2 − NKerr2 = PinLD,1� SHG − Kerr�. �11�

sing this in Eq. (3) we then get an NLSE with a self-efocusing SPM term;

�i�

�−

1

2

�2

��2�U1 − Neff2 U1�U1�2 = − iNSHG

2 sa�R,SHG�U1�2�U1

��.

�12�

elf-defocusing solitons with strength Neff2 can then be ex-

ited if the FW GVD is normal, i.e., if �1�2��0, which has

een assumed in Eq. (12).So the LHS of Eq. (12) is now a self-defocusing NLSE

upporting solitons if Neff�1. The RHS contains theaman-like perturbation due to GVM. It becomes impor-

ant for large SHG soliton orders NSHG [10] and when theompressed pulse duration is of the order of TR,SHG�R,SHGTin, where the characteristic dimensionless timecale of the Raman-like perturbation is

�R,SHG �2�d12�

��Tin. �13�

ypically TR,SHG is of the order of 1–5 fs, but can becomeery large in the nonstationary regime [10]. Finally, sa

sgn�d12�2�2��: the Raman-like effect will—depending on

his sign—give either a redshifting �sa= +1� or blueshift-ng �sa=−1� of the FW spectrum. In the cases we presentere sa=−1.

. NUMERICAL RESULTSn this section the transverse fiber modes and their dis-ersion properties are calculated for various mPOF de-igns (varying the air-hole diameter d and the hole pitch

�dtcve

AWastadtm

ppTfad

BWtcbf

mcrdnIC

piwat=t(at

t=amntrs−2=is

ppb�sccisvea q6

pr�bndbctttTpsgntnt

wpeTa

hsvta


). Then we show compression examples for selected fiberesigns. These numerical simulations were done usinghe full coupled propagation equations (A9), which in-lude Kerr XPM effects and steepening terms, and arealid down to single-cycle resolution using the slowlyvolving wave approximation [23,25–27].

. Microstructured Fibere consider an index-guiding mPOF with a triangular

ir-hole pattern in the cladding; see Fig. 1. The mPOF de-ign parameters are the pitch � between the air holes andhe relative air hole size D=d /�, where d is the physicalir-hole diameter. We assume that the core has a qua-ratic nonlinearity from thermal poling of the polymer af-er the fiber drawing, or from including nonlinear nano-aterial in the fiber preform.We consider an mPOF made from the cyclic olefin co-

olymer Topas [20]. The advantage of using Topas com-ared to, e.g., poly(methyl methacrylate) (PMMA), is thatopas has a larger transparency window: it is transparentrom 300–1700 nm, interrupted by two absorption peaksround 1200 and 1450 nm [20]; see Appendix C for moreetails.

. Fiber Design, Dispersion, and Compressione will here search for fiber design parameters, where

he requirement is an optimal dispersion profile for cas-aded quadratic soliton compression. Details about the fi-er mode calculations and the dispersion calculations areound in Appendix B.

Two types of designs are discussed: one is a realisticPOF with a large air-hole pitch of around �=7 �m that

an easily be drawn, while another takes � small (compa-able to the FW wavelength) as to significantly alter theispersion parameters. Such an mPOF would probablyeed to be done by tapering an mPOF with a larger pitch.n the calculations the Sellmeier equation from Appendix

with T=25 °C was used.It is fairly standard to draw mPOFs with a air-hole

itch of ��5 �m. With this as a starting point we shown Fig. 2 the GVM and phase mismatch for a Topas mPOFith �=7.0 �m. We notice that the choice of D does notffect the dispersion parameters much. This is becausehe size of the waveguide [the core diameter is dcore��2−D�] is significantly larger than the wavelength of

he guided light, and so material dispersion is dominatingsee also Fig. 3). The microstructured cladding thereforelters the dispersion only very little, and we cannot getailored GVM or GVD properties.

Fig. 1. Microstructured fiber design considered here.

Since Topas has a primary transparency window be-ween 290 and 1210 nm, we focus now on SHG with �1800 nm. We would like the fiber to be single mode botht the FW and SH wavelengths, because then phaseatching can only occur between these modes and we doot have problems with the typical higher-order mode in-eraction usually observed in waveguided SHG. The crite-ion for single-mode operation in the particular micro-tructured fiber, we investigate, is [28] � /�=2.8�D0.406�0.89. This criterion gives the lines shown in Fig.(a) for the FW (where �=�1) and the SH (where �=�2�1 /2). We therefore choose a design with D=0.4, which

s indicated with a cross; this design is actually endlesslyingle-moded [29].

For the chosen design, we show in Fig. 3 the dispersionarameters as a function of wavelength. In Fig. 3(a) thehase mismatch �� is shown. It is always above theoundary to the stationary regime as given by Eq. (2)�sr=d12

2 /2�2�2�. Thus, the chosen fiber design for all the

hown wavelengths is in the stationary regime for solitonompression, which is very good news for the possibility oflean and efficient compression. This might seem surpris-ng given the large GVM seem in Fig. 3(b), but it is a con-equence of having a very large phase mismatch and theery large GVD at the SH wavelength; see Fig. 3(c). How-ver, the large phase mismatch unfortunately also impliessmall cascaded nonlinear parameter SHG. To achieve

SHG� Kerr, as required for solitons to exist, the effectiveuadratic nonlinearity deff must be quite high, aroundpm/V for �1=800 nm as shown in Fig. 3(d).In Fig. 4 we show numerical simulations of soliton com-

ression using the chosen design. For the Kerr nonlinearefractive index of the polymer Topas we use nKerr

I =1510−20 m2/W, which is a large but realistic value. To out-

alance this Kerr self-focusing nonlinearity the quadraticonlinearity must be deff�6 pm/V, see Fig. 3(d), andeff=10.0 pm/V was chosen. The compression is shown foroth a low and a high soliton order, and both are verylean for several reasons: first, the compression occurs inhe stationary regime, and second, the Raman-like per-urbation is very small, TR,SHG=1.3 fs. Third, looking athe FW frequency spectra there are no dispersive waves.hese tend to induce trailing oscillations on the com-ressed pulse [10], but we checked that for this fiber de-ign they are not phase matched in the transparent re-ion of Topas. We should also mention that we haveeglected any cubic Raman effects of the material due tohe lack of material knowledge in this respect, and if sig-ificant Raman effects are present as for silica fibers thenhis would have an impact on the results.

So a standard mPOF, where the fiber simply serves as aaveguide but otherwise has little influence on the dis-ersion, can give decent compression (provided deff is highnough). What benefits can we get with a reduced GVM?o get a detailed understanding of this, let us investigatepossible fiber design having very small GVM.To significantly alter the dispersion the mPOF must

ave a lower hole pitch. As an extreme example Fig. 5hows the calculated fiber properties for �=0.65 �m (thisalue is unrealistic but serves the purpose of discussinghe conditions under which zero or low GVM can bechieved): the GVM can be very small or even zero along

tGc�t

=tGlsalct

rf3

[Ssggwdimm

Fen

FG


he black curve. Unfortunately for �1=800 nm this low-VM area is in an anomalous dispersion region, where

ascaded solitons do not exist, so let us instead work at1=1040 nm, which would be relevant for compressinghe output of Yb-doped fiber lasers.

The dispersion for �1=1040 nm of this mPOF with �0.65 �m is shown in Fig. 6(a) as function of D. Clearly

he dispersion changes dramatically as D is changed. TheVM goes from a normal value �d12�−200 fs/mm� for

ow D to zero at high D. Nonetheless, we remain in thetationary regime even for low D values since ��srlways. This is because both �� and the SH GVD D2 arearge. Notice also that the FW third-order dispersion �1

�3�

hanges dramatically: it is very large for high D and closeo zero for low D. Finally, Fig. 6(b) shows the critical deff

ig. 2. (Color online) GVM parameter d12 and phase mismatchration are indicated for the FW and the SH modes. The FW zormal. Finally the X indicates the chosen fiber design for single

(a)

(b)

ig. 3. Dispersion for �=7.0 �m and D=d /�=0.4. (a) The phasVD parameter D=−2�c��2� /�2 versus �. The material dispersion

� for nI =15�10−20 m2/W.
SHG Kerr Kerr
equired in order to overcome the material Kerr self-ocusing effects. It is around 2–3 times higher than in Fig., which is a consequence of a large ��.Zero GVM can therefore be achieved choosing D=0.95

at the cross, marked (1) in Fig. 5]. This choice leaves theH multimoded and is therefore not suitable for compres-ion: there would be different modes with different de-rees of phase mismatch interacting with the FW andaining control over the compression would be difficult. Ife want the SH to be single moded, we should choose aesign below D=0.6 (see Fig. 5), and the cross marked (2)n Fig. 5 has the advantage of being endlessly single

oded. In this regime the GVM is not too different fromaterial dispersion.So the penalty of choosing a small-pitched fiber in order

r an mPOF with �=7.0 �m. In (a) the lines for single-mode op-persion point is also indicated; below this line the FW GVD isoperation at �1=800 nm.

)

)

atch ��=�2−2�1 and (b) GVM d12=�1�1�−�2

�2� versus �1. (c) Theo shown. (d) shows the effective nonlinearity required to achieve

�� foero dis-mode

(c

(d

e mismis als

tvsm

tmta

F=TPao

F(


o reduce the GVM seems to be too high. The main moti-ation behind controlling GVM through the fiber micro-tructure was to be in the stationary regime. However, al-ost all mPOF designs we have investigated have been in

(b)

(c)

(a)

ig. 4. (Color online) Numerical simulation of soliton compr10 pm/V. Top left plot shows the FW compressing to 15 fs FWHop right shows the spectral broadening and development of SPin=4.6 kW, and the pulse energy was 0.6 nJ. The middle row shpulse energy of 5.7 nJ), which compresses to 4.4 fs after 6 mm o

rder dispersion was included �md=10�. 213 temporal points and

ig. 5. (Color online) As Fig. 2 but with �=0.65 �m. A zero GVMZDP). The two crosses indicate the chosen fiber designs at � =1
1
he stationary regime thanks to the large phase mis-atch. Another motivation was to reduce the characteris-

ic time of the Raman-like response TR,SHG= �d12� /2�� sos to get cleaner compressed pulses, but TR,SHG is low

in an mPOF with the design parameters of Fig. 3 and deffr 30 mm of propagation (at the dashed line) starting from 120 fs.uced sidebands of the FW. The pulse parameters were Neff=3,simulation for a higher soliton order (Neff=9, Pin=41.8 kW, andgation. A cut at z=6 mm is shown in the bottom plot. Up to 10thsteps per coherence length were used.

e is indicated with black as well as the FW zero dispersion point.

essionM afteM-indows af propa�15 z

curv040 nm

epsswhwbpoiop

nellss(c4=s

es=db=

4Ospw

FhTs

(

(

(

Fl�5TN


ven when GVM is not reduced again thanks to the largehase mismatch. What is more, choosing a zero GVM de-ign surprisingly has been shown not to benefit compres-ion: in our previous study [14] we performed simulationsith just the lowest SH transverse mode (neglecting theigher-order modes), and the fiber designs with zero GVMere giving quite distorted compressed pulses. This cane partially explained by the fact that higher-order dis-ersion is significant when the relative hole size is large;bserve for instance how large the third-order dispersions in Fig. 6 for D close to unity: this value is several ordersf magnitude stronger than the material third-order dis-ersion.An advantage of small-pitched fibers is instead that

onlinear effects occur with a much smaller power. As anxample, choosing �=1.0 �m and D=0.4 gives an end-essly single-mode mPOF, where GVM and GVD are simi-ar to the material dispersion (see the calculated disper-ion parameters in Fig. 7). This situation is thereforeimilar to the large-pitch case discussed previouslywhere �=7.0 �m). However, due to the much smallerore diameter (d=1.6 �m compared to d=11.2 �m in Fig.) and mode areas (a1=0.43 �m2 compared to a121.2 �m2 in Fig. 4) a compression similar to the onehown in Fig. 4 can be done with 10 times lower pulse en-

(a)

(b)

ig. 6. (Color online) (a) Dispersion as function of the relativeole size D as calculated for �1=1040 nm and �=0.65 �m. (b)he the effective nonlinearity required to achieve SHG� Kerr as-uming nKerr

I =15�10−20 m2/W.

rgy and power: one can compare the compressed pulsehown in Fig. 7 for �=1.0 �m with the one for �7.0 �m shown in Fig. 4. Both have the same soliton or-er and achieve essentially the same level of compression,ut the pump power required is Pin=3.1 kW when �1.0 �m and Pin=41.8 kW when �=7.0 �m.

. CONCLUSIONSur recent theory for cascaded quadratic soliton compres-

ion [10,12,23] has shed new light on the role of the dis-ersion in the compression of femtosecond near-IR pulsesith cascaded SHG. But what can be gained by having

a)

b)

c)

ig. 7. Dispersion calculated for D=0.4 nm and �=1.0 �m. Theower plot shows the results of a numerical simulation, where a1=800 nm Tin=120 fs FWHM input pulse is compressed to.5 fs. The pulse cut is shown after z=7.0 mm of propagation.he fiber had deff=10 pm/V as in Fig. 4. The input pulse hadeff=9, Pin=3.1 kW, and pulse energy 0.42 nJ.

tflbpsn

smresphmotmfiss

qschhlmsaSccSmaowSrrG

ttdib[dgr[

smclpw

w�eapoa

sfcicecbmd

wttpwo

APFagsrtwe

T=ivb

wt


he control over the GVM that microstructured fibers of-er [3]? An initial study on silica microstructured fibers il-ustrated that zero GVM is not a definite advantage [14]ecause it surprisingly gave a more distorted compressedulse. Moreover a very large phase mismatch was ob-erved [2], which required a very large effective quadraticonlinear response by thermal poling of the fiber.Therefore we here investigated index-guiding micro-

tructured polymer optical fibers, because poling of poly-ers or the inclusion in the polymer matrix of nanomate-

ials potentially can meet the requirement of a largeffective quadratic nonlinear response deff. A definite an-wer to the required nonlinearity cannot be given since atresent the investigated polymer type (Topas) has yet toave its material Kerr nonlinear refractive index nKerr

I

easured. However, based on nKerrI measurements of

ther similar polymer types such as PMMA, we estimatehat it is around 10 times that of silica. Consequently deffust exceed 5 pm/V and ideally be around 10 pm/V if ef-cient generation of large negative nonlinear phasehifts. This does pose a serious challenge for fabricatinguch an mPOF, but we believe it should be possible.

GVM is considered a major obstacle for obtaining high-uality [12] and short duration compressed pulses [10]. Toubstantially reduce GVM we found from dispersion cal-ulations of the transverse fiber modes that the relativeole size had to be quite large and at the same time theole pitch had to be close to the wavelength of the pump

ight; this is consistent with the results obtained in silicaicrostructured fibers [3]. The problem with choosing

uch a design is that the SH is no longer single moded,nd partial phase matching can take place to any of theH transverse modes, thereby giving rise to many cas-aded effects. This is clearly undesirable. (Note in thisonnection that the study in [3] in case of a multimodedH assumed that phase matching to the lowest order SHode was done using QPM, so a single-moded SH was not

s crucial as it is here.) On the other hand, the main goalf reducing GVM was to enter the stationary regime,here efficient and clean compression may take place.ince the fiber designs turned out to be in the stationaryegime almost irrespective of the size of the hole pitch orelative hole size, then the motivation for reducing theVM fades.The main reason why the fiber designs are always in

he stationary is the large phase mismatch. Thus, whilehe phase mismatch poses problems in demanding a largeeff, it has the benefit of leaving the cascaded interaction

n the stationary regime, which is exactly characterizedy the phase mismatch dominating over GVM effects10,12]. It is worth remarking that we here present a fiberesign where compression can occur in the stationary re-ime at �1=800 nm; most bulk quadratic nonlinear mate-ials are in the nonstationary regime at this wavelength12].

The surprising conclusion is therefore that a fiber de-ign, where the dispersion is not too different from theaterial dispersion, gave the best compression. Numeri-

al simulations evidenced that few-cycle pulses at a wave-ength of 800 nm can be generated from oscillator-levelulse energies (a few nanojoules). Such pulse compressionas demonstrated with an endlessly single-mode mPOF

ith a relative hole size of D=0.4 both with a large pitch=7.0 �m and a small pitch �=1.0 �m. In both cases an

ffective nonlinearity of deff=10 pm/V was used. The onlydvantage of the small-pitch design was that much lowerump powers and pulse energies are required, but on thether hand it is much more difficult to actually draw suchsmall-pitch mPOF.A natural question is whether the polymer material can

ustain the large intensities involved when dealing withemtosecond pulses; typically kilowatts of power are fo-used to mode areas of a few micrometers squared givingntensities ranging from 10–1000 GW/cm2. However, be-ause the femtosecond pulses are so short in time, the en-rgy fluences in the examples shown in this paper were10 mJ/cm2, a value that unlike the intensity does not

hange much during compression. Such fluences shoulde below the threshold for writing, e.g., gratings in poly-er materials [30], and we therefore expect that material

amage can be avoided.We can therefore state that if an mPOF can be drawn

ith large enough effective nonlinearities to outbalancehe material Kerr nonlinearity, then only a few centime-ers of fiber is needed in order to generate clean few-cycleulses in the near-IR with very low pulse energies. Thisould be a remarkable add-on to any femtosecond pulsedscillator.

PPENDIX A: GENERALIZEDROPAGATION EQUATIONSor SHG in a waveguiding medium with both quadraticnd cubic nonlinear material response, a general propa-ation equation for the electrical fields including self-teepening and cubic higher-order nonlinear terms caneadily be derived from [23,26,27]. There bulk propaga-ion is considered, but with some slight modifications thatill become apparent in what follows. The propagationquations for the electrical fields Ej�z ,�� read

L1E1 + �SHG,1S1E1*E2ei��z + �Kerr,1S1�E1�F11�E1�2 + 2F12�E2�2�

= 0, �A1a�

L2E2 + �SHG,2S2E12e−i��z + �Kerr,2S2�E2�F22�E2�2 + 2F12�E1�2�

= 0. �A1b�

he fields are taken scalar E�x ,z , t�12 x�j=1

2 Ej�x ,z , t�e−i�jt+c.c., by assuming they are polar-zed along the same polarization direction x. The trans-erse field is split from the longitudinal propagation fieldy looking for solutions of the form

Ej�x,z,t� = Ej�z,t�Fj�x�ei�jz, �A2�

here �j are the mode propagation wavenumbers, Fj arehe transverse mode profiles, and x= �x ,y�.

The linear propagation operators are

L1 � i�

�z+ i

�

2+ D1, �A3a�

w

Tctavgatp

w

ap�dtm

wli=

pKbmotcd

Fs

prdwanStcv�tISSSmfawbmp

p�lfiEc=

wi[mifia

FT


L2 � i�

�z+ i

�

2− id12

�

��+ D2,eff, �A3b�

here Dj are dispersion operators up to order md

Dj � �m=2

md

im�j

�m�

m!

�m

��m , �A4a�

D2,eff � D2 + S2−1

d122

2�2

�2

��2 , �A4b�

he unusual form of the SH dispersion, Eq. (A4b), is dis-ussed below. The mode effective indices neff,j are relatedo the propagation constants as �j=neff,j�j /c. The fieldsre in the frame of reference traveling with the FW groupelocity vg,1 by the transformation �= t−z /vg,1, whichives the GVM term d12=1/vg,1−1/vg,2, where vg,j=�j

�1�,nd �j

�m��m�j /��m��=�j. Linear losses are included

hrough the loss-parameter �. Finally, ��2−2�1 is thehase mismatch.The quadratic nonlinear coefficients are

�SHG,j ��1

2cneff,j

�dx��2��x��F1*�x�2F2�x��

aj

=�1deff

cneff,j

�Pdx�F1*�x�2F2�x��

aj, �A5�

here the mode overlap area is

aj �� dx�Fj�x��2, �A6�

nd ��2� is the quadratic nonlinear tensor value along theolarization direction of the interacting waves, and deff��2� /2 in the reduced Kleinman notation. The depen-

ence of ��2� on x implies that in an index-guiding mPOFhe nonlinearity is only present in the polymer, as re-inded by the subscript P in the integral of Eq. (A5).The cubic nonlinear coefficients are

�Kerr,j =3�j Re��3��

8cneff,jaj=

�j

cajnKerr,j, �A7�

here nKerr,j�3 Re��3�� /8neff,j, and ��3� is the cubic non-inear tensor in the polarization direction of the interact-ng waves. We neglect two-photon absorption so Im��3��0. The mode overlap integrals are defined as

Fjk ��P

dx�Fj�x��2�Fk�x��2. �A8�

To describe ultrashort pulses adequately, the usual ap-roach adopted in silica fibers is to divide the materialerr response must be divided into an electronic and a vi-rational (Raman) response [31]. However, for the poly-er materials used for optical fibers, the delayed nature

f the Raman response is unknown. Therefore we decidedo neglect the delayed vibrational Raman response. Thisan be also be justified by the fact that the propagationistances are very small, of the order of a few centimeters.

inally, we have included steepening terms through aelf-steepening operator Sj�1+ �i /�j�� /��.

The Eqs. (A1) are valid in the slowly evolving wave ap-roximation (SEWA)[25], which is a general spatiotempo-al model with space–time focusing terms important forescribing femtosecond spatiotemporal optical solitons. Itas recently extended to SHG by Moses and Wise [26],nd as a plane-wave model for SHG with competing cubiconlinearities by Bache et al. [23]. The advantage of theEWA model is that it does not pose any constriction onhe pulse bandwidth, and therefore holds to the single-ycle regime. Instead, the more commonly used slowlyarying envelope approximation (SVEA) only holds for� /�1/3 (and that only when including steepening

erms and the general Raman convolution response [31]).n the absence of diffraction, the difference between theHG SEWA model and the usual SVEA model is that theH has an effective dispersion term [Eq. (A4)]. In the SHGEWA model we must remember that one assumptionade when deriving Eqs. (A1) was that the spectra of the

undamental and SH do not overlap (substantially). Thisssumption allows us to separate the fields into twoaves. We chose �� /�j=0.9. This could give some overlapetween the fundamental and SH spectra, but we alwaysade sure that the spectral components in the overlap-

ing regions were negligible.We now rescale space and time (in our notation, a

rimed variable is always dimensionless) so z��z /LD,1,�=� /Tin, where LD,1�Tin

2 / ��1�2�� is the characteristic GVD

ength of the FW and Tin is the input pulse duration. Theelds are now normalized to the peak input electric fieldin=E�z=0, t=0�. Since the SH has no input field, wehoose to rescale it to the FW input field Ein, so U1

E1 /Ein and U2=E2 / nEin, and the equations become

L1�U1 + ��NSHGS1�U1*U2ei��z�

+ NKerr2 S1�U1��U1�2 +

2nf12

f11�U2�2� = 0, �A9a�

L2�U2 + ��NSHGS2U12e−i��z� +

2n2f22

f11NKerr

2 S2�U2

��U2�2 +2f12

nf22�U1�2� = 0, �A9b�

here the dimensionless SHG soliton number is definedn Eq. (7) (an extension of the bulk soliton number from9,23] to the waveguiding case). The dimensionless phase

ismatch is ��LD,1. The cubic soliton number NKerrs given by Eq. (4) and is well-known from the NLSE inber optics [24]. In Eq. (A9) the usual overlap integralsppear

fjk �Fjk

ajak=

Pdx�Fj�x��2�Fk�x��2

dx�Fj�x��2dx�Fk�x��2. �A10�

inally, n�neff,1 /neff,2, which is typically close to unity.he dimensionless propagation operators are

w=

Fs�s

wa

w

sbnawsttw

ATDTfifvHtq

B=cf

c−Suttmesn�stlpuns1

AEWtat

wmwpotp

llm1mw�it


L1� � i�

�z�+ i

��

2+ D1� , �A11a�

L2� � i�

�z�+ i

��

2− id12�

�

��+ D2,eff� , �A11b�

Dj� � �m=2

md

im�j�m�

�m

��m , �A11c�

here we have introduced the dimensionless loss ��LD,1 and the dimensionless dispersion coefficients

d12� � d12

LD,1

Tin, �j

�m� � LD,1

1

Tinmm!

�j�m�. �A12�

inally, the steepening operators working with dimen-ionless time are S1��1+ is�� /�� and S2��1+ i�s� /2�� /��, where s��1Tin�−1. The SH effective disper-

ion [Eq. (A4)] in dimensionless form is

D2,eff� � D2� + S2�−1

�

2

�2

��2 , �A13�

here the dimensionless factor ��cLD,1 /�2neff,2LGVM2 ,

nd LGVM=Tin/ �d12� is the GVM length. By using

S2�−1 = �

m=0

� �− is�

2 �m �m

��m ,

e get [23]

D2,eff� = �m=2

md

im��2�m� +

�

2� s�

2 �m−2� �m

��m . �A14�

The dimensionless propagation equations (A9) are thetarting point of the analysis. The difference between theulk equations we presented in [23] is that the solitonumbers and the coefficients for the SPM and XPM termsre modified to include the mode overlap areas, and thate are dealing with power and mode propagation con-

tants instead of intensity and wavenumbers. However,he dimensionless form is general, so the scaling laws andhe critical transition points to compression found in [23]ill still hold.

PPENDIX B: CALCULATION OF THERANSVERSE FIBER MODES ANDISPERSIONhe propagation equations describe the dynamics of theeld envelope in the z propagation direction and wereound by describing the field as in Eq. (A2). The trans-erse modes Fj�x� will from this analysis have to obey aelmholtz-type of equation, whose solution will give the

ransverse eigenmodes Fj�x� and corresponding eigenfre-uencies �j allowed by the fiber.We calculated the fiber modes with the MIT Photonic-

ands (MPB) package [32]. Each unit cell contained nC2

482 grid points, and the supercell contained nSC2 =72 unit

ells. For a given �jMPB=�j� /2�, the fundamental mode

requency �MPB=� � /c and group velocity were first cal-
1 1
ulated, followed by iterations of the SH until ��2MPB

2�1MPB�10−4. Material dispersion, parameterized by a

ellmeier equation (see Appendix C), was then includedsing a perturbative technique [33], whose advantage ishat many different � values can be calculated perturba-ively from the MPB data (where � is unity). From theseodified data we may then calculate the dispersion prop-

rties of the fiber including the effect of material disper-ion. We should mention that after the perturbative tech-ique is applied we get a modified set of eigenfrequencies

˜ jMPB, and therefore the requirement ��2

MPB−2�1MPB�

10−4 no longer holds. However, the ��jMPB, �j

MPB� dataets were afterwards converted to dimensional form, andhen fitted to a regular grid. This ensures that the calcu-ation of the SH dispersion was actually done at theroper frequency �2=2�1. The higher-order dispersionsed in the numerics was calculated with a robust poly-omial fitting routine, that gave proper convergent re-ults compared to the original �j values when fitting up to0 polynomial orders.

PPENDIX C: TOPAS SELLMEIERQUATION AND LOSSESe here make an accurate fit with a Sellmeier equation to

he refractive index data from [20] made in the visiblend near-IR (see Table 1). Using MATHEMATICA we fit tohe following single-resonance Sellmeier equation:

n2�� = 1 + B/�1 − A/�2�, �C1�

here � is the wavelength measured in micrometers. Theeasurements were made from 15 °C–75 °C and withavelengths between 435.8 and 1014 nm [20]. The fittingarameters presented here are more accurate than thenes reported in [20] (this could be because in [20] ahree-parameter fit was used, while we use a two-arameter fit).The linear losses were included in the numerical simu-

ations, but had only a little influence since the fiberengths considered here were of the order of a few centi-

eters. The absorption peaks of Topas around �=1.2 and.4 �m were modeled using a loss method: top-hat trans-ission profiles with a maximum transmission of unityere fitted to the three main spectral windows ��0.29,1.21, [1.25, 1.35], and �1.4,1.7 �m as measured

n [20] in a L=3.2 mm sample. The linear losses werehen calculated as �=−ln�T� /L+�0, where T is the top-

Table 1. Sellmeier Equation (C1) Fitting Param-eters for Refractive Index Data Points in Topas

Grade 5013 Measured in [20] at Various Tempera-tures and With Wavelengths Between 435.8 and

1014 nm

T(°C)

A��m2� B

15 1.08199�10−2 1.3121125 1.09177�10−2 1.3083550 1.08080�10−2 1.3013975 1.09199�10−2 1.29348

h(l

ATogN0

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

3

FlT


at transmission profile, and �0 are the base linear lossesestimated to 0.5 dB/cm [20], i.e., �0=0.115cm−1). Theoss profile of the simulations is shown in Fig. 8.

CKNOWLEDGMENTShe work presented could not have been completed with-ut fruitful discussion with J. Moses, F. W. Wise, J. Lægs-aard, and O. Bang. Financial support from The Danishatural Science Research Council (FNU, grant 21-04-506) is acknowledged.

EFERENCES1. T. Birks, D. Mogilevtsev, J. Knight, and P. St. J. Russell,

“Dispersion compensation using single-material fibers,”IEEE Photon. Technol. Lett. 11, 674–676 (1999).

2. J. Lægsgaard, P. J. Roberts, and M. Bache, “Tailoring thedispersion properties of photonic crystal fibers,” Opt.Quantum Electron. 39, 995–1008 (2007).

3. M. Bache, H. Nielsen, J. Lægsgaard, and O. Bang, “Tuningquadratic nonlinear photonic crystal fibers for zero group-velocity mismatch,” Opt. Lett. 31, 1612–1614 (2006).

4. G. I. Stegeman, D. J. Hagan, and L. Torner, “��2�, cascadingphenomena and their applications to all-optical signalprocessing, mode-locking, pulse compression and solitons,”Opt. Quantum Electron. 28, 1691–1740 (1996).

5. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W.Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt.Lett. 17, 28–30 (1992).

6. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary wavesdue to ��2� :��2� cascading,” J. Opt. Soc. Am. B 11, 2434–2443(1994).

7. X. Liu, L. Qian, and F. W. Wise, “High-energy pulsecompression by use of negative phase shifts produced bythe cascaded ��2� :��2� nonlinearity,” Opt. Lett. 24,1777–1779 (1999).

8. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda,“Soliton compression of femtosecond pulses in quadraticmedia,” J. Opt. Soc. Am. B 19, 2505–2510 (2002).

9. J. Moses and F. W. Wise, “Soliton compression in quadraticmedia: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. 31, 1881–1883 (2006).

0. M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. W.Wise, “Limits to compression with cascaded quadraticsoliton compressors,” Opt. Express 16, 3273–3287 (2008).

1. F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise,“Controllable Raman-like nonlinearities fromnonstationary, cascaded quadratic processes,” J. Opt. Soc.Am. B 21, 376–383 (2004).

ig. 8. Loss-parameter � used in the numerics: both linearosses (estimated to 0.5 dB/cm [20]) and the absorption peaks ofopas, see [20], are modeled.

2. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocalexplanation of stationary and nonstationary regimes incascaded soliton pulse compression,” Opt. Lett. 32,2490–2492 (2007).

3. J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise,“Efficient high-energy femtosecond pulse compression inquadratic media with flattop beams,” Opt. Lett. 32,2469–2471 (2007).

4. M. Bache, J. Lægsgaard, O. Bang, J. Moses, and F. W. Wise,“Soliton compression to ultra-short pulses using cascadedquadratic nonlinearities in silica photonic crystal fibers,”Proc. SPIE 6588, 65880P (2007).

5. D. Faccio, A. Busacca, W. Belardi, V. Pruneri, P. Kazansky,T. Monro, D. Richardson, B. Grappe, M. Cooper, and C.Pannell, “Demonstration of thermal poling in holey fibres,”Electron. Lett. 37, 107–108 (2001).

6. P. G. Kazansky, Optoelectronics Research Centre,University of Southampton, Highfield, Southampton, S0171BJ, UK (personal communication, 2007).

7. P. G. Kazansky and V. Pruneri, “Electric-field poling ofquasi-phase-matched optical fibers,” J. Opt. Soc. Am. B 14,3170–3179 (1997).

8. C. Chang, C. Chen, C. C. Chou, W. J. Kuo, and J. J. Jeng,“Polymers for electro-optical modulation,” J. Macromol.Sci., Polym. Rev. 45, 125–170 (2005).

9. G. Y. Guo and J. C. Lin, “Second-harmonic generation andlinear electro-optical coefficients of BN nanotubes,” Phys.Rev. B 72, 075416 (2005).

0. G. Khanarian and H. Celanese, “Optical properties of cyclicolefin copolymers,” Opt. Eng. (Bellingham) 40, 1024–1029(2001).

1. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon,“Experimental observation of picosecond pulse narrowingand solitons in optical fibers,” Phys. Rev. Lett. 45,1095–1098 (1980).

2. W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller,J. Rasmussen, and D. Edmundson, “Modulationalinstability, solitons and beam propagation in spatiallynonlocal nonlinear media,” J. Opt. B: QuantumSemiclassical Opt. 6, s288–s294 (2004).

3. M. Bache, J. Moses, and F. W. Wise, “Scaling laws forsoliton pulse compression by cascaded quadraticnonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007).

4. G. P. Agrawal, Nonlinear Fiber Optics, 3ed. (Academic,2001).

5. T. Brabec and F. Krausz, “Nonlinear optical pulsepropagation in the single-cycle regime,” Phys. Rev. Lett. 78,3282–3285 (1997).

6. J. Moses and F. W. Wise, “Controllable self-steepening ofultrashort pulses in quadratic nonlinear media,” Phys. Rev.Lett. 97, 073903 (2006).

7. J. Moses and F. W. Wise, “Derivation of nonlinear evolutionequations for coupled and single fields in a quadraticmedium,” arXiv:physics/0604170.

8. B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke,“Modal cutoff in microstructured optical fibers,” Opt. Lett.27, 1684–1686 (2002).

9. T. A. Birks, J. C. Knight, and P. S. Russell, “Endlesslysingle-mode photonic crystal fiber,” Opt. Lett. 22, 961–963(1997).

0. A. Baum, P. J. Scully, W. Perrie, D. Jones, R. Issac, and D.A. Jaroszynski, “Pulse-duration dependency of femtosecondlaser refractive index modification in poly(methylmethacrylate),” Opt. Lett. 33, 651–653 (2008).

1. K. J. Blow and D. Wood, “Theoretical description oftransient stimulated Raman scattering in optical fibers,”IEEE J. Quantum Electron. 25, 2665–2673 (1989).

2. S. Johnson and J. Joannopoulos, “Block-iterativefrequency-domain methods for Maxwell’s equations in aplanewave basis,” Opt. Express 8, 173–190 (2001).

3. J. Lægsgaard, A. Bjarklev, and S. Libori, “Chromaticdispersion in photonic crystal fibers: fast and accuratescheme for calculation,” J. Opt. Soc. Am. B 20, 443–448(2003).

designing microstructured polymer optical fibers for cascaded quadratic soliton compression of...

Documents