Spectral compression of optical parabolic

Ann.Phys.Fr.32(2-3),71–74(2007)

c EDP Sciences,2008

DOI:10.1051/anphys:2008010

Spectral compression of optical parabolic similaritons

C.Finot1,a,A.Guenot1and P.Dupriez2

1Institut Carnot de Bourgogne,UMR5209CNRS-Universit′e de Bourgogne,

21000Dijon,France

2Fianium UK,20Compass Point,Ensign Way,Southampton,UK

Abstract

We numerically investigate the spectral recompression experienced by a self-

similar parabolic pulse with an anormal initial chirp.Spectral compression

factors above10and high-quality output pulses can be predicted.

1INTRODUCION

For various applications,ultrashort optical pulses with a broad spectrum are not

bene?cial.The solution usually used to narrow the spectrum is to spectrally?l-

ter the pulse.Unfortunately,this process is poorly energy-e?cient.A much more

elegant approach takes advantage of the Kerr nonlinearity of silica?bers[1].It be-

comes then possible to recompress,without any energy loss,an initial femtosecond

pulse into a transform-limited picosecond pulse.

We numerically investigate in this context the potentiality of self-similar

pulses[2].Those optical pulses with a parabolic intensity pro?le combined with

a linear frequency chirp progressively appear during the?ber ampli?cation of ul-

trashort pulses in presence of gain,normal dispersion and nonlinearity.Successful

experimental demonstrations have been carried out both in rare-earth doped am-

pli?ers[3]and Raman ampli?ers[4].

We will outline that,compared to initial pulses with hyperbolic secant(sech)

or Gaussian intensity pro?les,parabolic pulses enable a recompressed spectrum ex-

hibiting noticeably reduced sidelobes.Some analytical guidelines will be proposed

and discussed to predict the requirements and performance of the setup.

a e-mail:christophe.finot@u-bourgogne.fr

Article published by EDP Sciences and available at https://www.360docs.net/doc/dd18044471.html, or https://www.360docs.net/doc/dd18044471.html,/10.1051/anphys:2008010

72ANNALES DE PHYSIQUE

2INFLUENCE OF THE INITIAL PULSE SHAPE

During its propagation in an optical?ber,a high peak-power pulse undergoes self-phase modulation(SPM)[5].The resulting frequency chirp is in?rst approx-imation linear and positive over the central part of the pulse.This chirp can thus partly compensate an initial chirp with an opposite slope(applied for example by a pair of di?raction gratings).Consequently,the pulse will undergo a spectral compression[1].

In order to illustrate our study,we simulate the evolution of optical pulses in a photonic crystal?ber(second order dispersionβ2=1.4×10?3ps2m?1and Kerr-nonlinearityγ=40×10?3W?1m?1)at1060nm which is typical of ytterbium doped?ber lasers or ampli?ers[3].The electrical?eldψ(z,T)of the pulse can be modelled by the NonLinear Schr¨o dinger Equation(NLSE)[5]:

i ?ψ

?z=

β2

?2ψ

?T2

?γ|ψ|2ψ.

We?rst compare the results obtained for initial pulses with Gaussian,sech and parabolic intensity pro?les with an initial full-width at half maximum(FWHM) temporal duration?T=4.75ps and a linear frequency chirp C T=4THz ps?1 (Fig.1a1).Those pulses have a FWHM spectral bandwidth?F=3THz(inset Fig.1b).

Figure1.In?uence of the initial pulse shape.Three intensity pro?les are compared (Gaussian,sech and parabolic,grey dotted line,dashed line and black solid line respec-tively).(a)Temporal chirp and intensity pro?les before(a1)and after recompression(a2).

(b)Spectral intensity pro?le before(inset)and after recompression.

Spectrally recompressed pulses are plotted Figure1(a2,b).Gaussian and sech pulses both lead to a non-zero frequency chirp in the central part of the pulse.On the contrary,as the chirp induced by the SPM of similaritons is strictly linear[6], the chirp is perfectly cancelled for parabolic pulses.Similar compression ratios are observed(?F =0.15THz can be demonstrated in all cases,i.e.a compression factor of20.)The di?erence lies mainly in the amount of energy contained in the spectral sidelobes.Indeed,parabolic pulses lead to a spectral intensity pro?le exhibiting notably reduced substructures(with the shape of a Bessel function of the?rst kind[6]).

COLOQ’10

3IMPACT OF THE INITIAL FREQUENCY CHIRP We will now focus our attention on parabolic pulses and more speci?cally on the impact of their initial chirp.Let us consider an initial pulse with ?F =3THz and a given initial energy E =500pJ.We will vary the linear chirp coe?cient C F (in the case of a parabolic pulse ,C T =4π2/C F ).The resulting parabolic pulse has a temporal width ?T =?F C F /2π(Fig.2a1),with a peak power P given by P =3π√2E/2?F C F (Fig.2a2).After spectral compression,we obtain a Fourier transform-limited parabolic pulse with a temporal width ?T and a spectral width ?F =0.73/?T (Fig.2b1)[6].The optimum recompression distance L depends on the product γE and can be estimated by L =?F 3C 2F /6π√2γE .

Figure 2.(a)Evolution of ?T and P according to C F .(b)Evolution of ?F and L versus C F .Analytical results (solid line ),are compared to a ?ber with a dispersion D =?2.33ps km ?1nm ?1(circles )or D =?23ps km ?1nm ?1(crosses ).(c)Spectral intensity pro?les.Results obtained by numerical integration of NLSE with normal dispersion (solid line ),of GNLSE with normal dispersion (crosses ),and GNLSE with anomalous dispersion (circles ).

We can check Figure 2b the agreement between the analytical predictions and results based on the numerical integration of NLSE.The mis?t observed regard-ing L can be explained by the in?uence of the chromatic dispersion which has been neglected in the preceding analysis.The normal dispersion leads to an additional chirp and thus reduces the required ?ber length.Simulations carried out with a ?ber exhibiting a higher normal dispersion con?rm this qualitative explanation.The dispersion value does not seem to heavily a?ect the compression ratio or the spectral recompressed shape.

We ?nally compare Figure 2c numerical results obtained for various conditions.Results obtained by integration of the NLSE are in qualitative agreement with the results relying on the integration of the generalized NLSE which includes higher dispersive e?ects,Raman e?ect,linear loss,etc.Similar simulations carried out with an anomalous ?ber (D =+2.33ps km ?1nm ?1),show that this dispersion

74ANNALES DE PHYSIQUE

regime is not more suitable:some instabilities can appear and potentially degrade the recompressed pulses.

4CONCLUSION

We have demonstrated the potential bene?ts of the use of parabolic self-similar pulses in the?eld of spectral recompression.Analytical approach is con?rmed by numerical results.This technique can also be combined with optical?ber ampli?cation and thus lead to an elegant way to produce ps transform limited parabolic pulses with high energy[6].

References

[1]M.Oberthaler,R.A.H¨o pfel,Appl.Phys.Lett.63,1017(1993).

[2]J.M.Dudley et al.,Nature Phys.3,597(2007).

[3]P.Dupriez et al.,Opt.Expr.14,9611(2006).

[4] C.Finot et al.,IEEE J.Select.Top.Quant.Electron.10,1211(2004).

[5]G.P.Agrawal,Nonlinear Fiber Optic s,3rd edn.2001.

[6] C.Finot et al.,Opt.Expr.14,3161(2006).