Table Of ContentExperimental Confirmation of the General Solution to the Multiple
Phase Matching Problem
Alon Bahabad,† Noa Voloch,‡ Ady Arie,† and Ron Lifshitz‡
7 †School of Electrical Engineering, Wolfson Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
0 ‡School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv
0 69978, Israel
2
n CompiledFebruary6,2008
a We recently described a general solution to the phase matching problem that arises when one wishes to
J perform an arbitrary number of nonlinear optical processes in a single medium [Phys. Rev. Lett. 95, 133901
6 (2005)].Hereweoutlineindetailtheimplementationofthesolutionforaonedimensionalphotonicquasicrystal
1 whichacts asasimultaneous frequencydoubler forthree independent optical beams.We confirmthis solution
experimentally using an electric field poled KTiOPO4 crystal. In optimizing the device, we find—contrary
] to common practice—that simple duty cycles of 100% and 0% may yield the highest efficiencies, and show
i that our device is moreefficient than a comparable device based on periodicquasi-phase-matching. (cid:13)c 2008
c
OpticalSocietyofAmerica
s
- OCIS codes: 190.2620, 190.4160,190.4360.
l
r
t
m
1. Introduction tors. Here we present the first experimental realization
.
t ofadeviceusingthisgeneralsolution:aone-dimensional
a Three-wave mixing is a nonlinear optical process that
m three-wave doubler.
can take place within dielectric materials having a non-
Note that other schemes for multiple harmonic gener-
- linear χ(2) coupling coefficient. Such processes are used
d ationhavebeen demonstratedbefore.12–14 Nevertheless,
foravarietyofopticalfrequencyconversionapplications.
n we choose this relatively simple application of the LAB
Usually, due to dispersion, the three interacting beams
o solution,asitallowsustoprovideadetailedpedagogical
c donotpropagateinphaseandsoefficientenergytransfer
description of the approach. Other than demonstrating
[ between them is prevented.1 One of the common meth-
that the LAB solution indeed works, we wish to clarify
ods to solve this problem, called quasi-phase-matching
1
allthe steps inthe design process,so that others will be
v (QPM), is to periodically modulate the sign of the rel-
able to implement it as well.We stress that the solution
8 evant component of the nonlinear dielectric tensor at a
5 periodcorrespondingtothephasemismatch.1,2Thisap- is generalandis notlimited to suchsimple applications.
3
proach is very successful, but unless one is extremely
1 2. Simultaneous Phase Matching of Several In-
lucky it is limited to the phase matching of a single
0 teractions
7 optical process. In recent years, the need to simultane-
0 ously phase match several different processes arose in We consider second order nonlinear optical interactions
t/ numerousapplicationssuchascreationofmultipleradia- in which three beams couple through the nonlinear sus-
a tion sources,3 of multi-coloredsolitons,4 of multi-partite ceptibility χ(2). For a planar process in which two con-
m
entanglement sources,5 and for all-optical processing.6 stant undepleted beams, E1 and E2, give rise to a third
d- This need was addressed by developing ad hoc general- output beam E3, its integrated field amplitude is given
n izations for the quasi-phase-matching procedure, based by
o eitheronperiodicstructuresinonedimension7 (fornon- E (∆k)=Γ g(r)exp(i∆k·r)d2r, (1)
c collinear processes) and two dimensions8–10 or specific 3 ZA
:
v quasiperiodic structures in one3,11–14 and two15 dimen- where A is the interaction area, and Γ is a parameter
i sions. In a recent publication [16, henceforth LAB] we depending on the amplitudes of the incoming waves,
X
explainedhowtosolvethemostgeneralproblemofmul- on the indices of refraction of all three waves, on the
r
a tiplephasematching—designingadevicetophasematch strengthofthe relevantcomponentofthe nonlinearsus-
anarbitrarysetof processes,definedby anygivensetof ceptibility tensor d , and on the interaction width W.
ij
phase-mismatch values. The LAB solution is based on For example, for sum frequency generation in MKS,
the general observation that the phase matching prob- Γ=ω2d E E /ic2k W, where c is the speed of light in
3 ij 1 2 3
lem is a consequence of momentum conservation, and vacuum. The function g(r) gives the spatial dependence
thatincrystallinematter,i.e.matterwithlong-rangeor- of the relevant nonlinear coupling coefficient, and ∆k is
der,17 momentum conservation is replaced with crystal- the phase mismatch vector of the interacting waves.For
momentum conservation. Thus, all that one needs to sum-frequency generation ∆k would be k +k −k .
1 2 3
do is to design a nonlinear photonic crystal (NPC)— It is clear from Eq. (1) that the intensity of the out-
whetherperiodicorquasiperiodic—whoseFouriertrans- put beam is proportionalto the Fourier spectrum of the
form is peaked at all the required mismatch wave vec- function g(r), evaluated at the mismatch vector ∆k.
1
an NPC by fabricating whole strips normal to the tiles
along the line.
Before starting we wish to point out that in special
cases, the D phase mismatch vectors ∆k(j) may be in-
tegrally dependent. This means that one can use fewer
thanD wavevectorsto generate the NPC and stillhave
Bragg peaks at all D points. It is then a matter of
choicewhethertousethefullsetofdependentvectors—
although, as pointed out by LAB, it may be difficult
in this case to control the intensities of the peaks—or
to prefer a smaller set of independent vectors. Here we
keepallmismatchvectorsandtreatthemasiftheywere
integrally independent.
3. Designing a One-Dimensional Three-Wave
Fig.1.(Coloronline)IllustrationoftheLABsolutionfor
Doubler
designing a one-dimensional NPC for multiple collinear
optical processes, using the dual grid method. (a) The
We wish to design a device that will simultaneously
required mismatch vectors. (b) The dual grid, in which
phase-matchthreecollinearsecond-harmonic-generation
each family of lines is shown with a different color. (c)
processes, for three different wavelengths in the fiber
Tiling of the real-space line according to the order in
telecom C-band: 1530nm, 1550nm, and 1570nm. We
which lines of different families appear in the dual grid.
choose to use the nonlinear crystal KTiOP and oper-
4
(d) Associating a given duty cycle with each tiling vec- ate at a temperature of 100◦C. At these conditions, the
tor. Positively-poled segment are shown in blue, and phase-mismatch values for the three processes are:22,23
negatively-poled segments are shown in white. ∆k(1) = .263µm−1, ∆k(2) = .256µm−1, and ∆k(3) =
.249µm−1 respectively.Thus,weneedtodesignanNPC
whose Fourier spectrum contains three collinear wave
Thus, if we wish to simultaneously phase-match a set of
vectors with these dimensions, as shown schematically
D three-wave optical precesses, characterized by phase
mismatch vectors ∆k(j), j = 1,...D, we should design in Fig. 1a.
Inwhatfollowswedescribethedesignofsuchastruc-
the spatial structure of g(r) so that its Fourier spec-
ture, as a particular example of the LAB solution for D
trum is peaked at all the D mismatch vectors. For a
single process, the standard QPM solution1,2 is to de- collinearprocesses.GeneralizingfromD =3processesto
an arbitrary number D of processes, follows directly by
sign a one-dimensional NPC with a period of 2π/|∆k|,
replacing all 3-component and 2-component vectors be-
for which there is a first order Bragg peak in the spec-
lowbyD-componentand(D−1)-componentvectors,re-
trum at ∆k. The LAB solution shows how to design
spectively. Generalizing to higher dimensional processes
an appropriate nonlinear photonic crystal—whether pe-
requires use of the full solution, as described by LAB.
riodic orquasiperiodic—suchthat its spectrumcontains
Bragg peaks at any given set of D mismatch vectors.
The approachthat LAB adopt for this purpose is based A. Finding the tiling vectors
on the so-called dual-grid method, originally developed
Tocalculatethe correspondingthreecollineartilingvec-
by de Bruijn18 and later generalized19–21 to become one tors, a(i), i = 1,...3, we first construct a single three-
of the standard methods for creating tiling models of component vector k = (∆k(1),∆k(2),∆k(3)) from the
1
quasicrystals.
three given mismatch values. This vector spans a one-
ThereaderinencouragedtoconsultLAB16 foracom-
dimensional subspace of an abstract three-dimensional
plete and rigorous treatment of the most general two-
vector space. We complete it to an orthogonal basis of
dimensionalmultiplephase-matchingproblem,whichwe
the three-dimensional space by adding two orthogonal
do not repeat here. Instead, we give a detailed demon-
vectorsq andq . These are,ofcourse,notunique, and
2 3
strationofthe LABsolutioninonedimension,whereall
we choose them to be q = (0.6483,−0.3421,−0.3331)
2
the optical processes are chosen to be collinear. In this
andq =(−0.3421,0.6672,−0.3240).Weusethesethree
3
casethe implementationofthe dual-gridmethodforthe
vectors as the columns of a 3×3 non-singular matrix,
design of the NPC, as well as the experimental fabrica-
tionofthedevice,arerelativelysimple.Nevertheless,we
K(1) ∆k(1) q(1) q(1)
can still design nontrivial and interesting devices, such 2 3
K(2) ≡ ∆k(2) q(2) q(2) , (2)
as the three-wave doubler, implemented here. The ba- 2 3
sic idea is to find a set of one-dimensional tiling vectors K(3) ∆k(3) q2(3) q3(3)
a(i), i = 1,...D, with which we can generate a one-
dimensional tiling of the line, whereby a tile is simply whoserowsK(j),j =1,...3,spanthethree-dimensional
an interval on the line. We then convert the tiling into vector space as well. We then find the three dual basis
2
The rest follows immediately, as the required order of
the tiles in the real-spacestructure is givenby the order
inwhichlinesofdifferentfamiliesappearinthedualgrid.
This is illustrated in Fig. 1c. This is the sense in which
the topology of the dual grid determines the real-space
tilinginthistrivialone-dimensionalsetting.Theduality
is a statement that each line in the grid is associated
with a tile, or interval, in the tiling; and each interval
in the grid with a vertex of the tiling. In our example,
approximately 800 lines are required in each family to
generatea1cmlongone-dimensionalnonlinearphotonic
quasicrystal.
Fig.2.Anopticalmicroscopeimageofthedemonstrated
NPC. The prominent elements correspond to the a(1) C. Buildinganonlinearphotoniccrystalfromthetiling
tiling vector,poledwith a100%duty cycle.Their width
To create an actual nonlinear photonic quasicrystal we
is 8.5µm. The distances between these elements are
modulate the relevant component of the nonlinear sus-
quasiperiodically ordered along with the a(2) and a(3) ceptibilitytensorχ(2)accordingtotheconstructedtiling.
tiling vectors, whose widths are 8.1µm and 7.9µm, re-
Technology usually permits us to use a binary modula-
spectively, and which are poled with a 0% duty cycle. tionofχ(2) sothatthe actualcrystalcanbe represented
by a normalized function g(r) = ±1. The simplest rep-
resentation from a theoretical standpoint would be to
vectors, denoted
attach a thin strip of value g(r) = 1 to every vertex
A(1) a(1) b(1) b(1) of the tiling, while assigning the background a value of
2 3
A(2) ≡ a(2) b(2) b(2) , (3) g(r)=−1. This is equivalent to a simple convolution of
A(3) a(3) b(223) b(333) tahnedstthreiprewfoitrhegdievletsatfhuencstimionpsleasttathnealvyetritciacleesxopfrtehsesitoinlinfogr,
the Fourier transform of the function g(r).16 Neverthe-
by solving the three-dimensional orthogonality relations
less, it does not necessarily produce an optimal NPC—
A(i)·K(j) =2πδ . (4) one in which the strongest Bragg peaks are associated
ij
withthethreemismatchvectors.Ingeneral,onecanuse
Eachrowofthematrix(3)isadual-basisvectorofthe a numerical procedure in order to optimize the required
form A(j) = (a(j),b(j)). The a(j) are the three required Bragg peaks. See for example the treatment by Norton
collineartilingvectors,whosevaluesarecalculatedtobe and de Sterke.24 Here we want to give a few quick-and-
a(1) = 8.3946µm, a(2) = 8.1652µm, and a(3) = 7.95µm. simplesolutions—inadditiontothethinstrips—thatare
The2-dimensionalvectorsb(j) canbeused,asexplained worthtryingifonedoesnotwishto dealwithnumerical
by LAB, to analytically calculate the Fourier transform optimization.
of the NPC in order to determine the expected efficien- A second option would be to use strips whose widths
cies for the different nonlinear processes. are equal to the tile vectors and simply to change the
sign of g(r) from one strip to the next. In this way, ex-
B. Constructing the tiling
actly half the tiles will give strips of value g(r)=1, and
If we were now asked to generate all points at integral the other half strips of value g(r) = −1. The generated
linear combinations of the three tiling vectors we would setofstrips wouldbe analogousto anantiferromagnetic
get the unwanted outcome of a dense filling of the real quasicrystal,25,26 whose Fourier transformcould also be
line. To avoid this situation we construct the dual grid, calculated analytically. We have found that this option
whose topology determines which of the integral linear does not yield an optimal NPC for this application.
combination of the tiling vectors are to be included in A third option, and the one which we actually imple-
the one-dimensional tiling. mented, is again to use strips whose widths are equal
The dual grid is constructed by associating with each to the tile vectors, but this time associate a so-called
mismatch vector ∆k(j), j = 1,...3, an infinite family dutycyclewitheachtiling vector.Thisisdonebydivid-
of parallel lines separated by a distance L =2π/∆k(j), ing each tiling vector into two segments, and assigning
j
as illustrated in Fig. 1b. The set of all families together a value of g(r) = 1 to one segment, and g(r) = −1 to
constitutes the dual grid. We use the freedom to shift the other. The duty cycle is the fraction of each strip
eachfamilyfromtheoriginbyanarbitraryvalueoff L , withg(r)=1.Thisgeneralprocedureisshownschemat-
j j
where 0 ≤ f < 1, so that lines from different families icallyinFig.1d.Itgivesustheabilitytoperformsimple
j
neverexactlycoincide.Becausethemismatchvectorsare spectral shaping. By varying the three duty cycles, as-
independentovertheintegers,suchashiftproducesaso- sociated with the three tiling vectors, we can engineer
called gauge-transformation20,21 which has no effect on the magnitudes of the Fourier coefficients of the three
the resulting NPC. required Bragg peaks. What we actually find—contrary
3
Fig. 4. (Color online) Normalized pump wavelength re-
sponse.Experimentandsimulationresultsofsecondhar-
monicgenerationasafunctionofpumpwavelength.This
figure corresponds to panel (a) of Fig. 3.
to common practice3,11,27—is that the optimal NPC’s
are obtained when we use duty cycles of either 100% or
0%. These are also easiest to fabricate in terms of the
required resolution as we associate a value of g(r) = 1
or g(r)=−1 to tiles as a whole.
IntheexperimentalimageoftheNPC,showninFig.2,
the tiling vector a(1) is associated with strips of value
g(r) = 1, or a duty cycle of 100%, while the other two
have a value of g(r) = −1, or a duty cycle of 0%. In
Fig. 3 we show numerical calculations of the magnitude
of the Fourier transform of g(r), for the three possible
assignments of a value of g(r) = 1 to one tiling vector,
and a value of g(r) = −1 to the other two. One clearly
sees that in all cases there are pronounced Bragg peaks
exactly where we want them to be, but the distribution
of intensities changes as we vary the tiling vector that
is assigned a 100% duty cycle. The magnitudes of the
Fourier coefficients are comparable to the 2/π ≃ 0.6366
figure of merit,which is the magnitude of the firstorder
Fouriercomponentfor a one-dimensionalperiodic NPC.
In fact, because the efficiency is measured in terms of
energy transfer it depends on the Fourier intensity and
on the square of the interaction length. Thus, the real
comparison should be with the efficiency per process of
Fig.3.(Coloronline)Spectralshaping.Eachpanelshows
a sequence of three periodic NPC’s, each of length L/3,
the magnitude of the Fourier transform for a 1cm long
whichisproportionalto 1 · 2 2 =0.045.Forthehomo-
NPCmadetophasematchthethreecollinearprocesses, 3 π
geneous nonlinear photo(cid:0)nic qu(cid:1)asicrystal that we imple-
describedinthetext.Ineachpaneloneofthetilingvec-
mentherethelowestprocessefficiencyisproportionalto
tors is given a duty cycle of 100%, denoted as 1, and
(1·0.23)2 =0.053andthehighestto(1·0.365)2 =0.133,
the remaining two a duty cycle of 0%, or 0. Each panel
both better than the composite periodic structure.
alsoshowsapiece ofthe correspondingreal-spacerepre-
The ability to shape the spectrum using different rep-
sentationofthe NPC,where the smallestelement sizeis
resentation of the function g(r) can be very useful for
8µm.
cascaded processes that to some extent are better per-
formedinsequence.Forexample,inmulti-harmonicgen-
eration,wheretheoutputsofsomeoftheprocessesserve
as inputs for others,it mightbe beneficial to give prece-
4
Table 1. Conversion efficiencies. Maximum conversion efficiencies are obtained for the indicated wavelengths. In the
simulation, the maximum efficiency of the third process was shifted by +0.4nm.
Wavelength Experimental efficiency Theoretical efficiency Simulated efficiency
[1/Watt] [1/Watt] [1/Watt]
1531nm 1.19e-3 1.5e-3 1.46e-3
1551nm 3.37e-4 6.13e-4 6.83e-4
1571nm 3.18e-4 5.49e-4 6.43e-4
dence to the efficiencies of the initial processes at the 5. Summary
beginning of the NPC and gradually, along the inter-
Wepresentthefirstexperimentalrealizationofanonlin-
action length, change the balance in favor of the latter
ear optical device employing the general solution of the
processes.
phase matching problem, given by LAB. The demon-
strated device is a one-dimensional three-wave doubler.
We show that by simply using 100% or 0% duty cy-
4. Experimental Results
cles, each of the three wave doubling processes exhibits
The NPC was fabricated by electric field poling28 of high efficiency. Moreover, the efficiencies are all higher
KTiOPO , and is shown in Fig. 2. The spatial modula- than for a device employing a periodic modulation with
4
tionwasperformedalong1cmwitha2×1mm2crosssec- the sameoverallinteractionlength.We demonstratethe
tion.TotesttheNPCweusedapumpbeamfromatun- ability to perform spectral shaping of the response by
able external cavity diode laser,followed by an Erbium- changing the duty cycles, associated with the different
doped fiber amplifier, and a fiber polarizationcontroller tiles.Thisallowsustostrengthencertainprocessesatthe
whose purpose is to polarize the beam to be perpendic- expense of others. It should be stressed once more that
ular to the plane of modulation of the NPC. The pump the LAB solution is general and can support any set of
beam was then chopped at a frequency of 1kHz and fo- nonlinear processes, not only for one-dimensional prob-
cusedtoawaistof20µminthemiddleofthecrystal.Its lems but also in two or three dimensions, without any
power was varied during the experiment in the range of symmetry restrictions. In addition, this general method
4−40mW. The NPC was kept at a constant tempera- canbe appliedto abroadrangeofproblemsfromgener-
ture of 100◦C. The input pump wavelength was varied ation of radiation sources through all-optical processing
inthe range1528nm−1577nmandthe resultingsecond to generation of entangled photons in quantum optics
harmonic power was measured using a calibrated Sili- applications.
conphoto-diode,followedby alock-inamplifier.The re-
sults werecomparedwithasimulationemployingasplit 6. Acknowledgments
stepFouriermethod,29inwhichanon-depletedGaussian
ThisresearchissupportedbytheIsraelScienceFounda-
beamwasusedasapump.Bothresultsexhibitexcellent
tion through grants no. 960/05and 684/06.
agreement, as can be seen in Fig. 4, and show that this
device is indeed most efficient for second harmonic gen-
eration of the three designated pump wavelengths. References
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6
