Experimental characterization of domain walls dynamics in a photorefractive oscillator

0.3 MB·English

by  Adolfo Esteban

Experimental characterization of domain walls dynamics in a photorefractive oscillator
Adolfo Esteban–Mart´ın1, Victor B. Taranenko1,2, Javier Garc´ıa1, Eugenio Rolda´n1, and Germ´an J. de Valca´rcel1
1Departament d’O`ptica, Universitat de Val`encia,
Dr. Moliner 50, 46100–Burjassot, Spain and
2Institute of Physics, National Academy of Sciences of the Ukraine, Kiev, Ukraine
Abstract
We report the experimental characterization of domain walls dynamics in a photorefractive resonator in a degenerate four
wavemixingconfiguration. Weshowhowthenonflatprofileoftheemittedfieldaffectsthevelocityofdomain walls aswellas
thevariationsofintensityandphasegradientduringtheirmotion. Wefindaclearcorrelationbetweenthesetwolastquantities
that allows theexperimental determination of thechirality that governs the domain walls dynamics.
6
PACSnumbers: 42.65.Sf,47.54.+r,42.65.Hw
0
0
2
INTRODUCTION been predicted to occur in type I [11] and type II [12]

n
DOPOs, in intracavity type II second–harmonic genera-
a
tion[13],andinvectorialKerrcavities[14]. Recently,we
J Extended nonlinear systems with brokenphase invari-
have reported the first observation of this phenomenon,
2 ance (e.g., systems with only two possible phase values
1 for a givenstate), are commonin nature. These systems the NIBT, in an optical system, namely a photorefrac-
tive oscillator [7]. Moreover, our observation is rare in
may exhibit different types of patterns but, importantly,
] the sense that we observed a hysteretic NIBT [7, 15].
S the broken phase invariance lies at the origin of the ap-
P pearance, in particular, of domain walls (DWs) which The aim of the present work is to study in detail the
. arethe interfacesthatappearatthe boundariesbetween dynamics of the DWs we reported in [7] by means of
n
the measurement of the different DWs characteristics,
i twospatialregionsoccupiedbytwodifferentphasestates
l namely intensity, phase gradient and velocity, establish-
n [1, 2, 3].
[ ing relations among them. In particular, we consider
In nonlinear optics there are several examples of spa-

whether the chiralityparameter,which will be described
1 tially extended bistable systems that can show solutions
later on, is appropriate for characterizing the DW.
v forthe emittedfieldwithagivenamplitudebutopposite
6 phases (that is, phases differing by π), such as degener-
2
ate optical parametric oscillators (DOPOs) or intracav-
0 EXPERIMENTAL SETUP
itydegeneratefour–wavemixing[4]. Theinterfacewhich
1
0 connects both solutions, the DW, can be of either one
6 of two different types: On the one hand, there are Ising Our experimental setup, Fig.1, is a single-longitudinal
0 walls in which the light intensity of the field at the core modephotorefractiveoscillator(PRO)formedbyaFabry
/
n of the interface is zero and the phase changes abruptly Perot resonator in a near self-imaging arrangement [16]
i fromφtoφ+π;ontheotherhand,thereareBlochwalls choseninordertoachieveahighFresnelnumber[17,18].
l
n in which the light intensity never reaches zero and the The nonlinear material, a BaTiO3 crystal, is pumped
:
v change of phase is smooth across the DW [5]. In addi- by two counterpropagating laser beams of the same fre-
i tion to this, Ising walls are always static whereas Bloch quency. In this way a degenerate four wave mixing pro-
X
wallsareusuallymovingfronts(theyarestaticonlywhen cess occurs within the cavity. The degeneracy implies
r
a the systemis variationalwhatisanuncommonsituation thatthe fieldexitingthe nonlinearcavityisphase-locked
for dissipative systems). It is important to remark that and only two values of the phase (differing by π) are al-
Blochwallsarechiral(theyarenotidenticaltotheirmir- lowed [19]. Hence DW formation is allowed.
ror images) as in the Bloch wall the phase angle rotates The system performance is ruled by different parame-
continuously through π and two directions of rotation terssuchasdetuning(whichisthedifferencebetweenthe
are possible. This fact has important dynamical conse- frequency of the pump and the frequency of the cavity
quences as Bloch walls with opposite chirality move in longitudinalmodeinwhichemissionoccurs),gain,losses
opposite directions [5]. Both Ising and Bloch walls have and diffraction. All these parameters can be controlled
beenfoundinnonlinearopticalcavityexperiments[6,7]. up to some extent. We choose as in [7, 18] cavity detun-
When a control parameter is varied a bifurcation that ing as the control parameter as it can be finely tuned in
changes the nature of the DW may happen. This is the an actively stabilized system [20].
nonequilibrium Ising–Bloch transition (NIBT) that has Regardingdiffraction,thesystemisintentionallymade
been investigated theoretically in [5, 8] and has been re- quasi–one dimensional in the transverse dimension (1D
peatidly observedin liquid crystals (see, e.g., [9, 10]). In system) in order to avoid the influence of DW curvature
the context of nonlinear optical systems, the NIBT has in the observed dynamics: Curvature induces a move-
1
ment in the DW [4, 12] that contaminates that due to phase values (say 0 and π, modulo 2π) can be ampli-
the nature of the DW (i.e., its Ising or Bloch character). fied. Thus, the regions of the field whose phase lies in
This is achievedby properly placing slits inside the non- the domain ]−π/2,+π/2[ will be attracted towards the
linear cavity (D in Fig. 1), in particular, at the Fourier 0 phase value, whilst those with a phase lying in the do-
planes (FP in Fig. 1). The width of the slits is adjusted main ]+π/2,+3π/2[will be attracted towards the phase
to the size of the diffraction spot in these planes. In this value π.
way beams with too large inclination (such that their Asaninmediateconsequence,theappearanceofaDW
transverse wavevector falls outside the plane defined by is forced as the points at which φ= ±π/2 (modulo 2π).
the center line of the slit) are not compatible with the TheseDWsaredarkbecausethosephasevaluescannnot
diffraction constraints of the cavity. This Fourier filter- be amplified but also, and this is a more fundamental
ing allows the use of finite width slits and still gets rid reason, because these points separate adjacent domains
of most 2D effects. It is also by using a diafragm that with opposite phases. These points, the DWs, are thus
spatialfrequenciesbelongingtootherlongitudinalmodes topological defects [26] what makes them very robust as
than the one of interest are removed [18]. they cannot be removed via continous changes.
Detuning, our control parameter, can be changed by In Fig. 2 we show several snapshots of the interfero-
means of a piezo-mirror. Depending on detuning, differ- gram of the output field during the writing process. In
ent types of structures can be found [17, 18, 21, 22] but Fig. 2a the initial homogeneous field is shown, and the
forourpresentpurposesitsufficestosaythatDWs exist interferogram shows the homogeneity of the field phase.
in resonance or for positive cavity detuning (i.e., when Then, in Fig. 2b the tilted beam is injected. While the
the frequency of the pumping field is smaller than the injection is being applied, a blurred DW appears on the
frequency of the nearest cavity mode): At zero (or small left, Fig. 2c, and moves to the right, Figs. 2d-2e. When
positive) cavity detuning DWs are static (Ising type), the wall is located at the centre of the crystal, Fig. 2f,
whilst they start moving (Bloch type) if detuning is in- the injection is blocked and the wall is generated, Figs.
creased enough [7]. 2g and 2h. In this way a single DW is written at the
desired location along the transverse dimension. Notice
that more than one DW can be written if larger angles
INJECTION OF DOMAIN WALLS in the writing beam are used.
DWs can form spontaneously from noise when the de-
CHARACTERIZATION OF THE DWS
tuningvalueliesintheappropriatedomain,asitwasthe
case with the DWs reported in [17, 18]. But waiting for
the appearance of DWs from noise is not the most ade- In order to fully characterize the features of the dif-
quatestrategyfortheirstudyforseveralreasons. Onthe ferent DWs that are being observed, it is necessary to
one hand one must wait until the formation of the DW measure the complex field. This is achieved by means
occursanditisverylikelythatnotasingleDWbutsome of the interferometric technique already used in [6]: A
of them appear, which complicates the analysis because CCD video camera records the interference pattern be-
of DW interaction. On the other hand even when a sin- tweenthenearfield(i.e.,theintracavityfieldatthecrys-
gle structure is formed from noise, there is the problem tal plane) anda homogeneousreference beam. From the
that its position on the transverse plane will not always interferometric pattern, like those of Fig. 2, a Fourier
bethesame. Owingtotheseandotherreasonsitismost transformtechniqueallowsthereconstructionofboththe
convenient to find a way for injecting, or writing, DWs amplitude and phase of the optical field, see [6] for full
at will. details. Some comments are in order regarding the way
An injection technique to remove the initial structure data must be extracted and processed.
and write a single DW has been devised [7, 23]. It Animportantfeaturethatmustbe pointedoutisthat
consists in injecting, for a short time (the shutter in thespatialprofileoftheemittedfieldisnotflat,ascanbe
Fig. 1 remains open for a few seconds), a laser beam appreciatedinFig. 3. Ofcourseonecannotexpectaper-
into the photorefractive oscillator that is tilted with re- fectlyflatprofileasthepumping fieldsareGaussian,but
spect to the resonator axis. This tilted beam has a the interesting point is that the more positive cavity de-
phase profile in the transverse dimension x given by tuningisthelessflatandlessextendedtheoutputprofile
φ(x) = φ0 +(2π/λsinα)x, being φ0 a constant, λ the is , i.e., there is a noticeable self–focusing effect [24, 25].
lightwavelength,andαthetiltanglewithrespecttothe This is particularly relevant in the ”free” transverse di-
resonator axis. mension (x direction in Fig. 3, which is the direction
Now, this injected tilted beam forces a phase vari- that is not limited by the intracavity diafragms).
tion, along the transverse dimension, of the intracav- This fact is at the origin of one effect that has to be
ity field phase, but given the phase sensitive nature of taken into account when characterizing the DW dynam-
the degenerate four-wave mixing process [19], only two ics: The variation of the intensity profile along the x di-
2
mensionisquitestrong,strongerthanfornegativedetun- DYNAMICS OF DWS
ing, and no simple experimental arrangement can cope
withthis. Weshallcomebacktothispointinthefollow-
InthissectionwedealwiththedynamicsofDWsinour
ing section when studying the dynamics of Bloch walls
particular non flat background profile. Fig. 6 shows the
as x is the direction along which Bloch walls move.
trajectories followed by both a static wall and a moving
Apart from these nonuniformities along the x direc-
one that were created at the same spatial position (the
tion, one has to take into account also the nonuniformi-
centre of the nonlinear crystal). As commented below,
ties along the direction of the DW, i.e., the y direction,
the static walls shows some slow dynamics.
see Fig. 3. The finite extension of the pump as well as
The static wall is an Ising wall, as its intensity and
thealreadycommentedself–focusingeffectmakethatthe
phase profiles for different intants of time, shown in Fig.
output intensity departs noticeably from a top hat pro-
7, clearly show. One can observe that after an initial
file. This fact affects the nature of the DW as it cannot
transient in which the intensity does not reach zero and
be strictly uniformin the y directionthus making neces-
the phase jumps are relatively smooth, the intensity and
sary to discard the borders of the illuminated region in
phase profiles progresively approach those of an Ising-
order to characterize DWs properly.
type: After the transient, the change of phase through
Let us consider, for example, the interferogram cor-
theinterfaceisquiteabruptandtheintensityatthe cen-
responding to a static wall shown in Fig. 4a. For the
ter of the wall is very close to zero. It must also be
reconstruction of the field amplitude and phase first we
noticed that the amplitude of the field on the right and
must choose between taking data for a particular value
on the left of the DW are similar. The initial move-
of y or making an average for different values of y. We
ment can be easily attributed to the transiet behaviour
choose the second option in order to smooth the effect
that occurs until the DW reaches its final profile. The
of any local imperfection. Then we average the field ob-
remaining movement must be attributed to noise as the
tained from the interferometric technique [6] and obtain
trajectoryshowsmoreawanderingbehaviourthanasys-
the average field hA(x)i . But before we must decide
y tematic one. The above completely characterizes static
if we consider all data or we take into account only the
DWs that are clearly identified as Ising DWs.
central part (in the y direction): If only the central part
of the interferogram is considered for reconstructing the Letusnowconsidermovingwalls. InFig. 8wedisplay
field amplitude and phase, one obtains the profile of an theirintensityandphaseprofiles,againfordifferenttime
Ising wall, Fig. 4b, with null intensity at the center and values. The behaviour is very different from that of the
a sharp phase jump. Contrarily, if one does not discard static wall of Fig. 7: The moving DW moves continu-
the borders and takes all data into account, one obtains ously to the left, its velocity changes with time, and the
a profile that does not correspond so clearly to an Ising intensity and phase also change during the motion. We
wall, as can be seen in Fig. 4c (the intensity does not can conclude that it is a Bloch wall (smooth change of
reach zero and the phase jump is smoother). Of course phaseandnonzerointensityatthecoreofthewall). The
this isnothing butthe influence ofabordereffectthatis factthatthevelocityisnotconstantcanbeattributedto
duetothealreadycommentednonuniformityofthefields the factthat for the different positions the DW is placed
alongtheydirection,andindicatesthewayinwhichdata on a different background as the intensity decreases to-
must be treated because the Ising character of the DW wards the borders of the illuminated region, Fig. 3. In
is clear from its dynamics: They are static. other words, the acceleration shown by the DW can be
In the case of moving walls the border effect is not attributedtotheexistenceofanintensitygradientinthe
so noticeable: Fig. 5a shows a typical interferogramof a field. It is also to be noted that the intensity in the left
movingwallandthecorrespondingaveragesoftherecon- side of the DW is smaller than that in the right side,
structedfield(withoutbordersinFig. 5bandwiththem and also that this effect increases as the wallapproaches
in Fig. 5c). In this case the wallmovesfromrightto left the border of the illuminated region. We have checked
andit is interesting to notice that the field amplitude on thatallthesefeaturesarecapturedbyasimplemodelfor
the left side of the Bloch wall is smaller than that of the DWdynamics(aGinzburg–Landauequationwithbroken
right side. This occurs always: the intensity is larger on phasesymmetry,see[5])whenpumpisspatiallyinhomo-
the back side of the wall with respect the direction of geneous.
movement, a feature that is reproduced with the simple Lookingathowintensity andphase changeduring the
model used in [15] for reproducing a hysteretic NIBT. accelerated motion, one can qualitatively conclude that
Then the experimental procedure consists in injecting the larger the wall velocity is, the larger (smaller) the
a single DW, recordingthe interferogram,extracting the intensity (the phase gradient) at the centre of the wall
information about intensity, phase gradient and spatial is. These facts make it difficult the determination of the
positionofasingleDWfordifferentinstantsoftime,and DW velocity for a given set of parameters: Our proce-
repeatingthis procedurefordifferentvaluesofthe cavity dure has consisted in taking an averagevelocity in those
detuning. cases in which it is more or less constant, discarding al-
3
ways the velocities corresponding to the regions located move in opposite directions [5]. In [8] a chirality param-
close to the border of the illuminated region (where the eter defined as
acceleration is more obvious).
The measurements we have just commented corre- χ=I0∇φ0, (2)
spond to a particular value of the cavity detuning. In
order to see how a change in this parameter affects the wasusedforcharacterizingtheDWdynamics(otherdef-
moving DW, we show in Fig. 9 the intensity and phase initions of the chirality can be proposed). For Ising
profiles for three different values of the cavity detuning wallsthechiralityparameteriszeroandthewallremains
butcorrespondingtoapproximatelythesamepositionon static,whilstforBlochwallsthechiralityisnonzeroand
the transverse direction, i.e., to the same intensity back- the wall moves with a velocity proportional to the value
ground. We observe again that for increasing detuning of χ.
the velocity increases and, again, the more velocity the From our experimental results we can now justify the
wall gets, the more intensity and less phase gradient it use of (2) as a quantity that characterizes the DW dy-
has. namics. In Fig. 11 a linear fit between the velocity and
Tosumup,wecanconcludethatthereisexperimental the chiralityparameterofa DW, asit evolveswith time,
evidence ofthe connectionbetweenthe DW velocityand is shown for a particular value of the detuning. It is
the characteristicsofthe opticalfieldatthecentreofthe worth stressing that the measurement of the velocities
interface: Thesmootherthephasejumpisandthelarger is not so reliable as the measurement of intensity and
the intensity at the wall core is, the faster the DW is. phase, for the reasons commented above, and for this
reason the linear fit is not so good as that of Fig. 10.
Wehavealsotriedusingotherquantitiesasthechirality,
CHIRALITY the results inFig. 11being the best (e.g., if one uses the
intensity or the inverse of the phase gradient as chirality
In the previous section we have shown how the inten- parameters, the correlations one obtains are 0.9540 and
sity and phase gradient of the DW change with detun- 0.9813,respectively,whichareconsiderablysmallerthan
ing and background conditions. We pass now to a more the 0.9967 obtained in Fig. 11).
quantitativestudyoftherelationexistingbetweenthein-
tensityatthecoreoftheDW,I0,andthephasegradient,
∇φ0. THE ISING–BLOCH TRANSITION
In Fig. 10 we represent, in a log–log plot, ∇φ0 versus
I0 forfixedparameters,thedifferentpointscorreponding ThereremainstoexplainhowthetransitionfromIsing
to different time instants (it is a parametric plot). It is walls to Bloch walls (the nonequilibriium Ising–Bloch
clear that a linear fit is well suited, which leads to the transition,NIBT)occursascavitydetuning isincreased.
relation In fact this was the subject of Ref. [7]. We reproduce
I0m∇φ0 =10−n, (1) here a brief explanation of this for the sake of complete-
ness.
where the typical values for the slope and crossing point In Fig. 12 a typical plot of the DW velocity as a func-
are m =−0.56 and n= 2.12, respectively, and the coef- tion of cavity detuning is shown. The rare aspect of this
ficientofdeterminationR–squaredhasavalueof0.9989. NIBTisthatitexhibitshysteresis: Westartwithastatic
A similarly good correlation is also found for other de- wall at resonance and increase detuning until the DW
tuningvalues,althoughmandnchangeslightlywithde- starts moving, signaling an Ising–Bloch transition (this
tuning (m variesbetween −0.67and −0.56,andn varies ismarkedasIBTinFig. 12). Forlargerdetuningvalues,
between 2.32 and 2.12). It is obvious that these fits do DWsalwaysmove. Then,fromalargepositivedetuning,
notallowsayingthatmandnareconstant,butwethink we inject a DW and check whether it moves or not, and
that their changewith detuning couldbe attributed to a repite this operationfor decreasingdetuning values until
correponding change with detuning in the shape and in- we obtain an static (Ising) wall (this is marked as BIT
tensityofthebackground(duetothealreadycommented in Fig. 12). The interesting thing is that the two NIBTs
self–focusingeffect). Inanycase,thisresultinvitestotry occur at different detuning values. In other words, there
todescribethe dynamicsintermsofonlyonemagnitude isafinite detuningrange(betweenBITandIBT)where,
of the field at the interface, either the phase gradient or for fixed detuning, we obtain either Ising or Bloch walls
the intensity or a combination of both. in subsequent injections.
As already commented, Bloch walls are chiral struc- Noticethatthe bistabilityrangeanthevelocityvalues
turesasthecomplexfieldcanrotateinthecomplexplane in Fig. 12 are smaller, roughly by a factor of 50%, than
when passing from one side to the other of the DW with thosereportedin[7]. Thereasonisthatthetwocasescor-
two different directions [5]. This has an important phys- respondtotwodifferentexperiments withdifferentgains
ical consequence, as Bloch walls with different chiralities what obviously affects the DW dynamics quantitatively.
4
Let us finally comment that a simple model that pre- [8] G.J.deValc´arcel,I.P´erez–Arjona,andE.Rold´an,Phys.
dicts a hysteretic NIBT like the one we have just com- Rev. Lett.89, 164101 (2002).
mented can be found in [15]. In this model one finds [9] T.Frisch,S.Rica,P.Coullet,andJ.M.Gilli,Phys.Rev.
Lett. 72, 1471 (1994).
that the origin of the hysteresys lies on the existence of
[10] T. Kawagishi, T. Mizuguchi, and M. Sano, Phys. Rev.
bistability in the homogeneous solutions of the system,
Lett. 75, 3768 (1995).
a bistabillity that we have also found in the experiment.
[11] I. P´erez-Arjona, F. Silva, G. J. de Valc´arcel, E. Rold´an
We refer the interested reader to [7, 15] for further de- and V. J. S´anchez-Morcillo, J. Opt. B: Quantum Semi-
tails. class. Opt.6, S361 (2004).
[12] G.Izu´s,M.SanMiguelandM.Santagiustina,Opt.Lett.
25, 1454 (2000).
[13] D.Michaelis,U.Peschel,F.Lederer,D.V.Skryabin,and
CONCLUSIONS W. J. Firth, Phys.Rev.E 66, 066602 (2001).
[14] V.J. S´anchez–Morcillo, V. Espinosa, I. P´erez–Arjona, F.
Silva,G.J.deValc´arcelandE.Rold´an,Phys.Rev.E71,
In conclusion, we have reported a detailed experimen- 066209 (2005)
tal characterization of domain walls dynamics in a non- [15] V.B. Taranenko, A. Esteban–Mart´ın, G.J. de Valc´arcel,
linear optical cavity, namely a photorefractive oscillator and E. Rold´an, e–print nlin.PS/0510018.
[16] J. A.Arnaud,Appl.Opt. 8, 189 (1969).
workingin adegeneratefour–wavemixing configuration.
[17] V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys.
An injection technique to remove any spontaneous ini- Rev. Lett.81, 2236 (1998).
tial structure and to write a single DW at any desired
[18] A. Esteban–Mart´ın, J. Garc´ıa, E. Rold´an, V.B. Tara-
location has been shown. The dynamics of the DWs has nenko, G.J. de Valc´arcel, and C.O. Weiss, Phys. Rev. A
been characterized in terms of the intensity and phase 69, 033816 (2004).
gradient of the optical field. We have also shown the re- [19] P.Yeh,Introduction to Photorefractive Nonlinear Optics
lation existing between these two quantities that leads (Wiley & Sons,New York,1993)
[20] M. Vaupel and C. O. Weiss, Phys. Rev. A 51, 4078
naturallyto the identification ofthe chiralityparameter.
(1995).
Finally, the hysteretic nonequilibrium Ising–Bloch tran-
[21] K. Staliunas, G. Slekys, and C. O. Weiss, Phys. Rev.
sition, which turns out to be hysteretic in our case, has Lett. 79, 2658 (1997).
been shown. [22] A.V.MamaevandM.Saffman,Opt.Commun.128,281
Acknowledgements (1996).
[23] A.Esteban-Mart´ın,V.B.Taranenko,E.Rold´an,andG.J.
This work has been financially supported by the de Valc´arcel, Opt.Express 13, 3631 (2005)
Spanish Ministerio de Ciencia y Tecnolog´ıa and Euro-
[24] Wenotethatfornegativecavitydetuning(whereastripe
pean Union FEDER (Projects BFM2002-04369-C04-01, pattern is observed, see [18]) a wide output beam is ob-
FIS2004-06947-C02-01 and FIS2005-07931-C03-01), and tained in the x direction. As detuning is made positive
by the Ag`encia Valenciana de Ci`encia i Tecnologia of andincreasedtheoutputbeambecomesincreasinglynar-
the Valencian Government (Project GRUPOS03/117). rower in the ”free” transverse dimension. This observa-
tion is compatible with theobservations in [25].
V.B.T. was financially supported by the Spanish Minis-
[25] M.Vaupel,O.Mandel,andN.R.Heckenberg;J.Opt.B:
terio de Educaci´on, Cultura y Deporte (grant SAB2002- Quantum Semiclass. Opt. 1, 96 (1999)
0240).
[26] S. Trillo, M. Haelterman, and A. Sheppard, Opt. Lett.
22, 970 (1997).
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5
FIGURE CAPTIONS
erated at the same spatial position. The detuning for
Fig. 1.- Scheme of the experimental setup. M: cavity the static and moving wall are zero and 20% of the free
mirrors; l: effective cavity length; D: diaphragm located spectral range, respectively.
in the Fourier plane (FP) that makes the system quasi- Fig. 7.- Reconstructed amplitudes (top) and phases
1D; PM: piezo–mirrorfor controlof the cavity detuning; (bottom)correspondingtoastaticdomainwallatdiffer-
CCD: cameras to take pictures from near field; and L: ent instants (the time interval between different profiles
lenses. is5s). Thedifferentlocationofthephasegradients(bot-
Fig. 2.-Experimentalinterferometricsnapshotsofthe tom) is artificial and has been introduced for the sake of
domain wall injection process. a) Initial homogeneous clarity (this applies also to Fig. 8).
field, b) injection is applied, c)-e) a domainwall appears Fig. 8.-AsFig. 7butforamovingdomainwall. The
and moves, f) the injected wall is located at the centre time interval between different profiles is 2s.
of the crystal, g) the injection beam is blocked, and h) Fig. 9.- Reconstructed amplitudes and phases of do-
the domain wall is finally written. Time runs from top main walls located at the same spatial position but ob-
to bottom in steps of 5 s. The transverse dimension is tained for different detunings. Detuning increases from
1.6 mm. lefttorigth(inpercentageofthefree–spectralrange): a)
Fig. 3.- Experimental snapshot of a 3D view of the 0%,b)10%andc)20%. Velocityincreasesasdetuning
emitted field containing a domain wall. does: a) 0 µm.s−1, b) 7 µm.s−1, and c) 12 µm.s−1.
Fig. 4.- Experimental interferometric snapshot of Fig. 10.- Logarithm of the intensity at the core of
a static domain wall (a), reconstructed amplitude and the DW, I0, versus the logarithm of the phase gradient,
phase neglecting the boundaries (b), and reconstructed ∇φ0,fordifferentinstantsandfixeddetuning(20%ofthe
amplitude and phase taking into accountthe boundaries free–spectralrange). Squarescorrespondtoexperimental
(c). The cavity detuning is zero. data and the straight line correspondto the fit.
Fig. 5.- Experimental interferometric snapshot of Fig. 11.- Relation between the velocity of a moving
a static domain wall (a), reconstructed amplitude and domain wall and the chirality parameter measured dur-
phase neglecting the boundaries (b), and reconstructed ing its motion in a non flat background field. Squares
amplitude and phase taking into accountthe boundaries correspond to the experimental data and straight lines
(c). Thecavitydetuningis12%ofthefreespectralrange correspond to the linear fit.
(FSR). The FSR of the cavity is 120Mhz. Fig. 12.- Velocity of the DWs as a function of cavity
Fig. 6.- Plot of the trajectories of a static (full line) detuning. BIT and IBT mark the two nonequilibrium
and a moving (dashed line) domain wall that were gen- Ising–Bloch transitions. See text for more details.
6
Reference PUMP 1
Shutter
PM L L L L M
FP c+ FP
CCD
D D M
Output BaTiO3
l f f f 2f f f f
PUMP 2
Injection

Figure 1

A. Esteban-Martín et. al, Experimental characterization of domain walls…
a

b

c

d

e

f

g

h

Figure 2

A. Esteban-Martín et. al, Experimental characterization of domain walls…
Figure 3

A. Esteban-Martín et. al, Experimental characterization of domain walls…
1 p
phase
a
b
amplitude
0 0

75 0 -75
1 p
phase
c
amplitude
0 0

75 0 -75
X (m m)

Figure 4

A. Esteban-Martín et. al, Experimental characterization of domain walls…