(2009)Observation of nonlinear Airy-like beam evolution in lithium niobate

Observation of nonlinear Airy-like

beam evolution in lithium niobate Jacopo Parravicini,1,*Paolo Minzioni,1Vittorio Degiorgio,1and Eugenio DelRe2,3 1CNISM and Dipartimento di Elettronica,Universitàdegli Studi di Pavia Via Ferrata1,27100Pavia,Italy

2Dipartimento di Ingegneria Elettrica e dell’Informazione,Universitàdegli Studi de L’Aquila,

67100Monteluco di Roio,L’Aquila,Italy

3SOFT-CNR Research Center,Universtitàdi Roma“La Sapienza”,00185Rome,Italy

*Corresponding author:jacopo.parravicini@unipv.it

Received September4,2009;revised November6,2009;accepted November10,2009;

posted November24,2009(Doc.ID116743);published December15,2009

We report the observation of Gaussian beam fragmentation into Airy-like waveforms during nonlinear propagation.The effect is supported by the high-intensity photovoltaic nonlinearity arising in unbiased pure congruent lithium niobate.The process is found to occur when the nonlinear response is dominated by the nonlocal effects associated with the charge-displacement process.?2009Optical Society of America OCIS codes:050.1940,350.5500,190.5330.

When visible light propagates in congruent LiNbO3a space-charge?eld is created,producing,through the electro-optic effect,a refractive index pattern that gives rise to photorefraction[1],one of the basic ef-fects to optically manipulate beams.It is worth re-membering that when such light is a micrometer-sized diffracting beam,the nonlinearity combines with diffraction so as to provide a playground for soli-ton formation;a mechanism has evolved into a con-solidated?eld[1–3].Without an external bias, LiNbO3manifests a self-defocusing nonlinearity sup-porting dark solitons[4].This is a direct consequence of the nature of the photovoltaic(PV)effect that drives photorefractive charge separation,and in par-ticular of the sign of the Glass coef?cient.The non-linearity can be switched to a self-focusing one using various techniques[5,6]allowing also the observa-tion of bright solitons.

In this Letter we report the?rst observation(to our knowledge)of nonlinear asymmetric Airy-like beam formation in unbiased LiNbO3,a qualitatively new phenomenon that involves the asymmetric frag-mentation of a Gaussian beam into multiple self-focused fragments.This process was?rst predicted in the time domain as a consequence of temporal de-layed response, e.g.,in Raman-dominated pulse propagation in?bers[7,8],and in unbiased non-PV photorefractive crystals as a consequence of thermal diffusion[9].The effect is caused by a strong nonlocal nonlinearity that depends on the transverse spatial derivative of the optical intensity,and it emerges as a consequence of the high-intensity PV response. Whereas nonlocal nonlinear corrections are known in relation to solitons in biased photorefractive crystals, where charge diffusion can produce self-bending[10], the present study involves a fundamentally different effect for which no soliton-supporting nonlinearity is present.Indeed the nonlocal component is a speci?c product of the high-intensity PV response amounting to a nonlinear component that is almost3orders of magnitude stronger than the well-studied nonlinear component mediated by thermal charge diffusion (equivalent to a?eld of tens of volts per centimeter), in combination with a weak self-defocusing.

Our results con?rm the qualitative validity of pre-dictions,proving the possibility of asymmetric quasi-localized beam propagation(the single fragments)in unbiased LiNbO3.The formation of asymmetric Airy-like beams that bend along a parabolic trajectory vests moreover a more general role in the under-standing of the basic paradigm where a nonlocal non-linearity governed by a term having a?rst derivative in the intensity is coupled to Kerr nonlinearity[7] and comes in a timely fashion with renewed interest into quasi-nondiffracting linear Airy beams.These, recently observed in air[11,12],are nonspreading so-lutions of the Schr?dinger propagation equation in a parabolic reference system.

In our experiments we have observed the fragmen-tation of a monochromatic cw TEM00Gaussian beam (probe beam,PB)of typical peak intensity I p ?5kW/cm2and input width w0?25?m,?=532nm,

polarized along the crystal optical axis(say;x?), launched along its orthogonal axis(say;z?),for a propagation length in the LiNbO3sample of L z ?4mm.The beam is superimposed on a wide-area copropagating beam(background beam,BB)of equal ?,orthogonally polarized,of intensity I b,so that I p/I b is in the range1–10.

The physical origin of the phenomenon lies in the predominant role of nonlocal nonlinear effects.We consider the model of the photorefractive effect in a PV crystal in an open circuit condition without exter-nal bias at steady state,limiting the analysis to a tractable one transverse?x?and one propagation axis ?z??1+1D case).In our simulations we consider beams propagating in the z direction and con?ned in the x direction,having an in?nite extension along y. We call E the static electric?eld induced inside the crystal by light.We derive the equation describing the x dependence of E starting from the Gauss law,

d

d x

E+

q

?

?N+N a?N d i?=0,?1?

where N a is the concentration of acceptor impurities,?is the static dielectric constant,N d i is the ionized do-

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nor density,N is the free electrons density,and q is the electron charge.In distinction to previous studies [5,13],here the used intensities invalidate the simpli-

fying condition that N?N d i,and N must be included in Eq.(1).

We use a model,developed,e.g.,in[5],which in-cludes the charge-transport equation and the rate

equation s??A?2+I b??N d?N d i???NN d i=0,describing charge excitation,and the transport relationship

q?NE+?eff s?N d?N d i???A?2+I b?=0,where thermal dif-fusion of charges is neglected.Here,s is the photoion-ization cross section,?A?x,z??2is the intensity of the PB of amplitude A?x,z?,I b is the BB intensity,N d is the donor density,?is the recombination coef?cient,?is the electron mobility,and?eff is the Glass con-stant(assuming?eff=?iij??iii).An analogous situa-tion was described in[13]for non-PV materials.The PV response of our material in the high-intensity re-gime,when N?N d i is invalid,activates a strong non-linearity that would otherwise be latent,as discussed in[5].Through these equations we can connect the charge densities to the intensity of the incident light. Introducing the dimensionless variables Q=Is/?N a;?=N d/N a(??1in LiNbO3);Y=E/E PV;x0 =??E PV?/qN a;?=x/x0(where E PV=?eff?N a/q?is the photovoltaic?eld as in[4,5];and I=?A?x,z??2+I b),the equation for the?eld is given by

Y=1

2

?Y?+1+Q??Q2+2Q?2??1?Y???,?2?

where we denote with a prime the derivative with re-spect to?.For typical LiNbO3material constants, 25-?m-wide beam,and for the intensities used(peak intensity of about 5.3kW/cm2),it turns out that ?Y???1??10?3?whereas Q?0.1?10(depending on the actual intensity used).This means that the equa-

tion can be approximately solved with a recursive

method[14].We start from the zero-order solution Y?0?=1/2?1+Q??Q2+2Q?2??1??.The?rst-order so-lution Y?1?is obtained by substituting Y?0?in Eq.(2) and keeping the?rst-order terms in Y?,i.e.,

Y?1?=1

2?1+Q??Q2+2Q?2??1?

?

+

1

4?2Q+2??1

?Q?Q+4??2?

2??1

Q+4??2?Q?.?3?

As experiments were carried out using high-intensity values,let us evaluate the previous equation under the high-intensity approximation.We note that Eq.

(3)has a?rst term that is a local Kerr-like response with an asymptotic behavior Y L??4??2?2/16Q,a nonlocal term that has the asymptotic behavior Y NL ?Q?/2,and a constant term that,being independent of?,has no direct effect on the nonlinear optical propagation and can be dropped.Once the electric ?eld is calculated,its impact on the beam propagation can be predicted through?n =?1/2n03r33E?I?,where?n is the index of refraction modulation,n0is the unperturbed index of refraction, r33is the relevant electro-optic coef?cient,and E?I?=E PV Y from Eq.(3).Finally,the effects on the propa-gating beam can be described in the scalar paraxial approximation by considering that the amplitude A obeys the equation??z??i/2k??xx?A?x,z?=?ik/n0??nA?x,z?,where k=2?n0/?.

In Fig.1we report the results of beam-propagation-method simulations for the intensity pattern of the PB,for a peak intensity of I p =5.3kW/cm2,w0=25?m,L z=4mm,?n PV =?1/2?n03r33E PV=6.85?10?4(the scale of the nonlin-ear?n),for I b=I p/3[cases in(A)and(C)]and I b=0 [cases in(B)and(D)].In Figs.1(C)and1(D)are re-ported the relative output intensity x pro?les.In the top case,the beam clearly suffers asymmetric split-ting and bending,as well as self-focusing,a signature of Airy-like nonlinear effects.This all occurs in a glo-bally diffracting beam,as expected in the presence of self-defocusing.The effect is strongly weakened in the bottom case,where quasi-symmetric spreading is dominant.As shown in Fig.1(C),the output beam is characterized by sharp intensity peaks with a maxi-mum width of20?m.

The physics becomes clear considering the asymptotic high-intensity nonlinear propagation equation,cast in the dimensionless form,with?=A?s/?N a and z=kx02?,i.e.,i??+1/2??=?k2/n0?n?, that is,

i??+

1

2

??=???2??1?2???2+????2????,?4?

where?=k2?n PV/2n0.As described in[7–9],the asymptotic Eq.(4)falls in that class of nonlinear equations that supports quasi-stationary solutions in the form of nonlinear Airy-like

beams[15]but with a predominantly saturated Kerr-like term[the?rst

Fig.1.(Color online)Nonlinear Airy-like effects and frag-mentation of the PB.Numerical prediction of the intensity in the xz plane in cases of strong(A)and weak(B)back-ground illumination and(C),(D)respective output inten-sity x pro?les.The black dashed lines are the calculated trajectories of the average beam center(see text).

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term on the right side of Eq.(4)].In such conditions the nonlinearity produces a parabolic (accelerated)trajectory of the light whose tails obey an Airy equa-tion and hence gives rise to the fragmentation [7,8].Experiments were carried out using a Mach–Zehnder-interferometer-like scheme,the two beams being the PB and the BB.Several lenses are imple-mented to obtain the suitable beam sizes,and ?/2waveplates in tandem with polarization selectors are used to tune the optical powers.In distinction to the tractable theoretical picture described above,the PB is a 2D circular Gaussian beam with w x =w y =w 0=25?m and is focused onto the input facet of the sample.The wide-area BB has w 0x ?=250?m in the x direction,while w 0y is several millimeters.We car-ried out measurements launching both beams onto the 8x ?14y ?4z mm congruent LiNbO 3z -cut crystal.Effects are observed to reach a steady state after ap-proximately 5min,in agreement with [16].The PB transverse intensity distribution in the xy plane is imaged onto the camera.A typical measurement is reported in Fig.2(top).The situation is analogous to that simulated in Figs.1(A)and 1(C),i.e.,I p =5.3kW/cm 2,I b =I p /3.In qualitative agreement with the simulations,we observe the formation of sharp intensity peaks of widths between 10and 20?m [Fig.2(top right)],the signature of asymmetric splitting and narrowing.The results in Fig.2(bottom)are relative to the weaker nonlinear regime,i.e.,for I p =5.3kW/cm 2,I b ?0,and once again are in agree-ment with the model predictions [Figs.1(B)and 1(D)],underlining the predominant role of the overall intensity I =?A ?2+I b .

In conclusion we have presented the ?rst experi-mental and numerical evidence of the formation of optical nonlinear Airy-like beams in a medium in the spatial domain.Experiments have been carried out in unbiased bulk congruent LiNbO 3crystals using an intense cw Gaussian beam copropagating with a plane-wave-like beam.The phenomenon consists of an asymmetric beam fragmentation,an effective lo-cal self-focusing,and an overall beam bending.We associate this effect with a saturated nonlocal nonlin-ear response that arises through the photorefractive PV response in LiNbO 3and forms the spatial em-bodiment of phenomena that were previously ob-served only in the temporal domain,such as Raman-dominated pulse propagation in optical ?bers.The different parameters that appear to play a role,i.e.,beam width,intensity,and background intensity,suggest a rich variety of effects that may lead to fur-ther developments and understanding into nonlinear Airy-like beams.

This work was partially supported by the FIRB project RBIN04NYLH and by grants from Fondazi-one C ARIPLO and from the FILAS initiative.References

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Fig. 2.(Color online)Experimental observations.Top,Airy-like beam fragmentation:(left)output intensity distri-bution in xy plane and (right)horizontal pro?le.Bottom,weak nonlinear regime:(left)output intensity distribution in xy plane and (right)horizontal pro?le (see text).

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