Table Of ContentCounterpropagating dipole-mode vector soliton
Jochen Schro¨der and Philip Jander and Cornelia Denz
Institute of Applied Physics, Westf¨alische Wilhelms-Universit¨at Mu¨nster,
D-48149 Mu¨nster, Germany
Tobias Richter and Kristian Motzek and Friedemann Kaiser
Institute of Applied Physics, Darmstadt University of Technology,
D-64289 Darmstadt, Germany
5 CompiledFebruary6,2008
0 We experimentally observe a counterpropagating dipole-mode vector soliton in a photorefractive SBN:60Ce
0 crystal. We investigate the transient formation dynamics and show that the formation process differs
2 significantly from the copropagating geometry. The experimental results are compared with fully anisotropic
numericalsimulationswhichshowgoodqualitativeagreement. 2008 OpticalSocietyofAmerica
n OCIS codes: 190.5530,190.5330 ©
a
J
0
1 Spatial optical solitons in a counterpropagating (CP) theywillcrossinsidethecrystal,resultinginachangeof
geometry have been proposed some time ago.1 It has the soliton interaction length. Therefore, beam-bending
]
not been until recently however, that they have drawn hasasignificanteffectonthetransientformationdynam-
S
P widerattention2–5 andwereinvestigatedtheoreticallyin ics as will become apparent later. Although at a first,
. saturable Kerr and local photorefractive media in one roughglancethecounterpropagatinggeometrymightbe
n
transversedimension(1D). In addition, Cohenet al. ex- very similar to configurations for e.g. double phasecon-
i
l perimentally demonstrated the existence of stable CP jugate (DPC) mirrors, the actual experiments are quite
n
[ solitons with narrow stripe beams.3 distinct. Both interacting beams are mutually incoher-
In a previous publication, we numerically predicted ent, thereby not interfering in the material. The lack
2
the existence of stable two-dimensional higher-order of an interference grating is obvious due to the insensi-
v
2 counterpropagatingvectorsolitonsinsaturableKerrme- tivity of the beams to slight angular changes once the
1 dia,6 commonly used as an approximation for the pho- solitons are formed. This effect can directly be seen
0 torefractivenonlinearity.Inthepresentpaperwewillex- when observing the dynamics of the soliton formation.
1
perimentally and numericallydemonstrate the existence In contrast, the existence of a grating would result in
0
of a counterpropagating dipole-mode vector soliton in a highly selective Bragg-condition. Such a behaviour is
5
0 a photorefractive SBN:60Ce crystal and investigate the typical for double phase conjugation configurations,but
/ transient formation dynamics. does not appear in our case. Moreover, the scales are
n
A dipole-mode vectorsolitonconsistsof twomutually much smaller than for DPC experiments (input beams
i
l incoherent beams: an optical dipole and a fundamental- areintherangeof20micrometers,interactionanglesare
n
◦
: mode(FM) beam. The individually propagating dipole smallerthan1 ),andeffectsthataretypicallyassociated
v
does not form a spatial soliton due to repulsion of the withfour-wavemixingandphase-conjugationdonotap-
i
X dipole components.7,8 However, if an FM beam which pear on that scale. Nevertheless, in all our experiments
r is incoherent to the dipole is launched in between the we checkedthoroughly that the effects we are observing
a
dipole components, they will be “trapped” via incoher- are indeed caused by counterpropagatingsolitons.
ent attraction.7,9 The experimental setup is depicted in Fig. 1. A
The counterpropagating geometry poses some addi- frequency-doubled Nd:YAG-laser beam is divided into
tional challenges to the experimental setup. A soliton two arms. The beams are rendered mutually incoher-
propagating inside a photorefractive crystal will be dis- ent by a means of a piezo-driven mirror (PM) oscillat-
placedinthepositivedirectionofthecrystalc-axis.This ing much faster than the response time of the crystal
effect is well-known as the so-called “beam-bending”- and are then focused onto the two faces of the pho-
effect caused by charge-carrier diffusion. This effect has torefractive SBN-crystal via two beamsplitters. Due to
been extensively investigated in the case of single soli- the mutual incoherence of the beams, they do not inter-
ton formation. For the case of interaction copropating fere in the materialnor produce an interference grating,
solitons or copropagatingvector solitons,the effect does thereby only interacting via their common phase modu-
notchangethefundamentalinteractionbehaviourasthe lation and induced refractive index channel. The crystal
beams are displaced by the same amount.11 Two coun- is biasedwith anexternalDC fieldalongthe x-axispar-
terpropagatingsolitonshoweverarebendinthesamedi- allelto the crystallographicc-axis.To take advantageof
rection. Thus when launched in a head-on configuration the large r electro-optic coefficient of SBN, the beams
33
1
PH l/2 PBS GP PM (a) (b) (c)
Nd:YAG
532nm y
F F l/2
M
x,c t=0s t=25s 60mm t=128s
F F
M (d) (e) (f)
BS
M
SBN CCD
BS BS
t=500s t=35min t=35+5min
M F M
Fig.2.Imagesofthedipoleexitfaceofthecrystal(beam
Fig. 1. Experimentalsetup for the realizationof a coun-
from FM input face to camera is blocked). The spot in
terpropagatingdipole-modevectorsoliton.(PH:pinhole,
the middle is the reflection of the FM input serving as
F: lens, λ/2: waveplate, (P)BS: (polarizing) beamsplit-
a reference. (a) linear propagation.Soliton formation at
ter,GP:glass-plate,M:mirror,PM:piezo-drivenmirror)
times t = 25 s (b), t = 128 s (c) and t = 500 s(d) af-
ter the nonlinearity has been turned on. (e) stable state
aftert≈35min.(f)individuallypropagatingdipoleap-
are polarized along the same direction. Background il- proximately t=5 min after the FM beam was blocked.
lumination is realized with a white light source, which
allows adjustment of the dark-intensity. To be able to
observe the two crystal faces simultaneously, both are dipole draws intensity from the non-trapped part and
imaged onto the same CCD-camera. The optical dipole the non-trapped part is attracted horizontally toward
is generated by entering a tilted glass-plate into one of the FM input (d). After approximately 35 minutes, the
the arms,a well-knowntechnique for the creationof op- systemreachesequilibrium,withalmostallpowerofthe
tical dipoles, and is oriented perpendicular to the c-axis dipole trapped (e). When the FM beam is blocked, the
to minimize effects of the photorefractive anisotropy.10 dipole components quickly repel and move to the right
In order to compensate for the above-mentioned beam- due to beam-bending. Image (f) shows the dipole after
bending,thebeamsarelaunchedintothecrystalinsuch propagating individually for approximately 5 minutes.
awaythattheinputfaceofonebeamcoincideswiththe The separationof the dipole components of the trapped
exitfaceofthecounterpropagatingbeampropagatingas andnon-trappedpartinimage(e)wasd ≈38µm
trapped
◦
anindividualsoliton.Thisresultsinasmallangle(<1 ) and dnon−trapped ≈ 87 µm respectively, compared to
between the two incident beams. The power of the FM d ≈ 57 µm for the individually propagting
individual
beam is PFM−beam ≈ 1.00 µW and the power of the dipole.
dipole P ≈1.35 µW. The approximate diameter of The transient formation dynamics differ significantly
dipole
the beams at the input face is d ≈ 20 µm FWHM. The from the copropagating case, which makes them partic-
geometry of the SBN crystal is 23(a)×5(b)×5(c) mm ularly interesting.The most striking fact is the splitting
and propagation is along the a-axis of the crystal for of the dipole into two parts, which is not observed in
maximumpropagationlength,correspondingtoapproxi- the copropagatinggeometry.Althoughata quickglance
mately4diffractionlengths.Theexternalfieldwassetto suchaneffectmightbeattributedtophenomenaascon-
E ≈1.2 kV/cm andthe background-illuminationwas ical scattering, a thourough analysis shows immediately
ext
adjustedinordertoallowstableself-focusingandsoliton that those effects can be excluded: the two interacting
formation of the individually propagating FM beam. solitonsaremutuallyincoherent,andtherebynotableto
The formation of the dipole-mode vector soliton is interfere with each other.Moreover,the angles at which
shown in Fig. 2. The images are taken at the exit face these new beams appear are in the range of one degree,
of the dipole beam, the less intense spot in the mid- much smaller than the scattering cone which should be
◦
dle of the images is the reflection of the FM input at well above 10 . Instead, the splitting can be explained
the crystal face. It serves as a reference and does not by the angle between the beams which is introduced to
influence the soliton formation process. When the non- compensate effects of beam-bending. Soliton formation
linearity is turned on, the dipole initially self-focuses to inphotorefractivecrystalsisatwo-stepprocess,inwhich
the left of the FM beam (Image (b)). This is due to the beam self-focuses initially and is then displaced by
the initial angle between the beams. The dipole is then beam-bending.11 After self-focusing of the counterprop-
stronglydeflected to the right,as beam-bending and at- agatingbeams self-focus but before beam-bending takes
traction from the FM soliton act in the same direction. place,thebeamsonlycrossatonepointinsidethecrystal
The dipole splits up and two spots of low intensity with duetotheangularadjustment.Atthispointthecolliding
a much smaller separationappear on the left side of the beams will interact strongly, which causes the observed
FM beam (c). As time elapses, this trapped part of the splitting. The non-trappedpartof the dipole is then de-
2
(a) (b) varying envelopes F and B. The total optical field can
thenbe writtenasF exp(ikz)+Bexp(−ikz),withk be-
y ing the wavevector in the undisturbed crystal. Since we
are considering F and B to be mutually incoherent, the
x,c total intensity is just given by I = |F|2 + |B|2. The
Kukhtarevmodelofthephotorefractivecrystalleadsun-
der simplifying but welljustified assumptions to the fol-
Fig. 3. Images of the dipole input face at steady state lowing equation
≈35 min after the nonlinearity has been turned on. (a)
τ
The output of the FM beam is displaced to the left of ∂t∆φ+∆φ+∇ln(1+I)∇φ=
1+I
the dipole input. (b) Output of the FM when the dipole
E ∂ ln(1+I)
is shortly blocked, a small fraction of the FM beam has ext x
coupled into the dipole waveguide. +κ(cid:2)∆ln(1+I)+(∇ln(1+I))2(cid:3) (1)
forthepotentialφoftheelectricalscreeningfield,which
(a) (b) causes the nonlinear change of the refractive index via
the Pockels effect. E denotes the externally applied
ext
electrical field, κ = −k T/e is the diffusive coupling
B
y strength, τ is the effective time constant of the crystal,
and I is measured in units of the background intensity.
x,c t=100t t=280t
The propagation of the beams is in paraxial approxi-
mation described by the set of equations
i i
Fig. 4. Numerical simulations of the dipole-mode vector ∂zF − ∇2⊥F + γ(EextF −∂xφF)=0 (2a)
soliton. The images show the exit face of the dipole at 2 2
i i
(a) t = 100 τ and (b)t = 280 τ after the start of the −∂zB− ∇2⊥B+ γ(EextB−∂xφB)=0 (2b)
simulation.The locationof the input beams is indicated 2 2
by the dashed lines. with the nonlinear coupling constant γ = k2x2n2r ,
0 0 eff
which contains the refractive index of the unperturbed
crystaln , the effective element of the electro-optic ten-
0
flected by the combined forces of beam-bending and in- sor r , and the transverse scaling constant x . ∇2 de-
eff 0 ⊥
coherent attraction of the dipole, causing the dipole to notesthetransverseLaplacian.Thepropagation(z-)axis
“overshoot”. This explains why the non-trapped dipole is scaled to the diffraction length L =kx2. The values
D 0
part is initially displaced further then the individually used in the simulations are x = 10µm (typical beam
0
propagatingdipole,whichcanbeseenbycomparingim- widths are about 1...2x ), r =280pm/V, n =2.35,
0 eff 0
ages(c)and(d) withimage (f) inFig.2.The secondre- E = 2.5kV/cm, and k = 2πn /λ with λ = 532nm.
ext 0
markablefeatureofthecounterpropagatingdipoleisthe The total propagation distance was chosen to be 4L
D
longtimescaleofthe formationprocess.Itdiffers signifi- (11.1 mm).
cantlyfromcopropagatingsolitons,withformationtimes TheresultsofthesimulationsareshowninFig.4.Im-
intherangeof20−50secondsorevenshorterforhigher age (a) and (b) show the exit face of the dipole beam
intensities,11 compared to around 30 minutes for the at times t = 100 τ and t = 280 τ after the start of
counterpropagating dipole-mode vector soliton to reach the simulation. Although the numerical and experimen-
astablestateforsimilarparameters.Ascanbeseenfrom tal results differ in detail, the main features of the in-
Fig. 2(c)-(e), the trapped dipole is not aligned with the teraction are very similar. The splitting of the dipole
input of the counterpropagating beam, but is displaced into a trapped and non-trapped part can clearly be ob-
slightly to the left. The output of the FM beam, which servedinFig.4(a),althoughitisnotaspronouncedasin
can be seen in Fig. 3 shows a similar picture. The FM the experiment. As time elapses the trapped part grows
soliton is displaced to the left of the dipole input. This in intensity drawing energy from the non-trapped part
isinagreementwithearliersimulations6 whichpredicta (Fig. 4(b)). The second main feature, the exceptionally
thresholdpropagationlengthwherethe symmetryalong long timescale is reproduced as well, with the time con-
the dipole axis is broken. Thus our experimental results stant τ corresponding to approximatelyτ ≈15 s.
areabovethis thresholdlength.Additionallyweobserve To summarize, we have shown the existence of a sta-
that the the FM beam splits as well. Fig. 3(b) shows ble counterpropagating dipole-mode vector soliton in a
the exit face of the FM beam when the dipole is shortly photorefractive SBN crystal. The vector soliton differs
blocked. A fraction of the beam has split from the main considerablyinmanyaspectsfromitscounterpartinco-
part and has coupled into the dipole waveguide. propagatinggeometry.Thetimescaleofthetransientdy-
Toverifyourexperimentalresults,wecarriedoutsim- namicsaresignificantlylarger.Duringtheformationthe
ulations in the anisotropic, nonlocal model.12 The two beams split up, and a trapped and non-trapped part of
counterpropagating beams are denoted by their slowly the dipole can be observed.
3
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