Dispersion in time and space: what propagating optical pulses in time (& not space) forces us to confront

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by  Paul Kinsler

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Dispersionintimeand space:
whatpropagating optical pulses intime(& not space)forces us to confront
P. Kinsler∗
BlackettLaboratory, ImperialCollegeLondon, PrinceConsortRoad, LondonSW72AZ,UnitedKingdom.
(Dated:Friday20thFebruary,2015)
Ideriveatemporallypropagateduni-directionalopticalpulseequationvalidinthefewcyclelimit.Temporal
propagationisadvantageousbecauseitnaturallypreservescausality,unlikethecompetingspatiallypropagated
models. Theapproachgeneratesexactcoupledbi-directionalequations,whichcanbeefficientlyapproximated
downtoauni-directionalformincaseswhereanopticalpulsechangeslittleoveroneopticalcycle. Italsoalso
allowsadirectterm-to-termcomparisonofanexactbi-directionaltheorywithanapproximateuni-directional
theory. Notably, temporal propagation handles dispersion in a different way, and this difference serves to
highlight existing approximations inherent in spatially propagated treatments of dispersion. Accordingly,
I emphasise the need for future work in clarifying the limitations of the dispersion conversion required by
thesetypesofapproaches; sincetheonlyalternativeinthefewcyclelimitmaybetoresorttothemuchmore
5
computationallyintensivefullMaxwellequationsolvers.
1
0
2 NOTE: despite the implication of the claimed “substantial text overlap” flagged up by arXiv, the phys-
ical meaning of this work differs from that of arXiv:0810.5689 – it merely uses the same technique for
b decomposing waves into directional pieces, and as a consequence can have similar phrasing even when the
e
physicsisdistinct.
F

9
1 I. INTRODUCTION torspectrum,aswellasacausallyappropriatetimeevolution
[17]. This requiresan alternate approachto the temporalre-

] sponse of the propagationmedium, which highlightsthe ex-
s The importance of having simple and robust methods for
c istingandsomewhatpoorlycharacterisedapproximationsin-
thepropagationofopticalpulseshasundergoneincreasingat-
i herentinspatiallypropagatedtreatmentsofdispersion.
t tentioninrecentyears. Thisisduetothemultitudeofappli-
p I start in section II by deriving a customized second or-
cations[1]inwhichincreasingshortpulsesarerelieduponto
o
der wave equation from Maxwell’s equations, and reorga-
. actasakindofstrobe-lamptoimageultrafastprocesses[2,3],
s nize itto definethe materialpropertiesappropriatelyand set
orevendesignerelectricfieldprofiles[4–6]toexcitespecific
c
up the factorization stage. In section III, I use the method
i atomic or molecular responses. Alternatively, strongnonlin-
s of factorization that allows us to construct an explicitly bi-
y earityisusedtoconstructequallywide-bandbutalsotempo-
directional model, and which is then reduced to the popular
h rallyextendedpulses–i.e. (whitelight)supercontinnua[7–9]
uni-directionallimitinsectionIV.SectionIVDdiscussestyp-
p – or even come full circle and use the strong nonlinearityto
[ generatesub-structurethatisagaintemporallyconfined,asin ical modificationsthat can be applied to the equationsgiven
opticalroguewaves[10];oreventhetemporallyandspatially in sections III and IV in order to and simplify them appro-
2
priately and compare them to existing models. Having de-
v localizedfilamentationprocesses[11–13].
rived this main result – the propagation equations and their
9 It is clear, therefore, that progress towards diminishing
various specialisations – I turn in V to the consequences of
6 pulse durations as well as their increasing spectral band-
5 temporalpropagationand the new perspectiveit provideson
widths, and higher pulse intensities are all either stretching
5 the handling of dispersive effects in this and standard (spa-
existingpulsepropagationmodelstotheirlimits,orbreaking
0 tiallypropagated)models. ThisisthenfollowedinsectionVI
them. In such regimes, we need to be sure that our numeri-
1. byaspecificcomparisonbetweentheordinaryspatiallyprop-
calmodelsstillwork,andhaveaclearideaofwhathasbeen
0 agated nonlinear Schrödinger (NLS) equation – perhaps the
neglected,and whatthe side-effectsof those approximations
5 most widely investigated equation in nonlinear optics – and
are.InthispaperIuseamoderndirectionalapproachtoshow
1 itstemporallypropagatedcounterpart.Finally,insectionVII,
: howastraightforwardandrelativelysimplederivationallows
v Ipresentmyconclusions.
aside-by-sidecomparisonofexactandapproximatepropaga-
i
X tion equations, whilst still providing the numerical and ana-
lyticalconvenienceofafirst-orderwaveequation.
r
a However, unlike the typical approachin nonlinaroptics, I II. SECONDORDERWAVEEQUATION
consider propagationin time rather than along some chosen
spatial axis [14, 15]. The resulting evolution equations for Mostopticalpulseproblemsconsiderauniformandsource
the wave profiles in space no longer give us access to the freedielectricmedium. Insuchcasesagoodstartingpointis
frequency spectrum [16], but instead we have the wavevec- thesecondorderwaveequationfortheelectricfield,whichre-
sultsfromthesubstitutionofthe(cid:209) ×H=¶ D+JMaxwell’s
t
equationinto (cid:209) ×E=−¶ B (see e.g. [18]). Here, however,
t
IwanttofollowthedisplacementfieldD(notE)sinceitnat-
∗Electronicaddress:[email protected] urallyoccursinconjuctionwitha time derivative– likewise,
1
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D2OWE Timepropagatedopticsanddispersion http://www.kinsler.org/physics/
theotherimportantfieldisBratherthanH. Thuswerexpress Nowtakethetimederivativeofthisequation,usethefactthat
thecurlMaxwell’sequationsas a m isindependentoftime, defineam =1−a m andae =1−
a e ;andthensubstitutefor¶ tB,sothat
m ¶ D=m (cid:209) ×H−m J =(cid:209) ×B−m (cid:209) ×M−m J.
0 t 0 0 0 0
(2.1) e0m 0¶ t2D=e0(cid:209) ×am ¶ tB−e0m 0(cid:209) ׶ tMB−e0m 0¶ tJ (2.11)
e ¶ B=−e (cid:209) ×E−e K =−(cid:209) ×D+(cid:209) ×P−e K.
0 t 0 0 0 (2.2) c−2¶ t2D=−(cid:209) ×am [(cid:209) ×D−(cid:209) ×P+e0K]
−c−2(cid:209) ׶ M −c−2¶ J (2.12)
t B t
Notethatforbothoftheseequations,wewillalso (asusual)
want to specify the material response, whether dielectric or ¶ t2D=−c2(cid:209) ×am [(cid:209) ×D−(cid:209) ×a eD−(cid:209) ×PD+e0K]
magnetic, in terms of a dielectric polarization P=D−e E −(cid:209) ׶ tMB−¶ tJ (2.13)
0
andamagnetizationM=B−m 0H. Eitheroftheseequations =−c2(cid:209) ×am [(cid:209) ×(1−a e)D−(cid:209) ×PD+e0K]
given above might be either in the spatial domain (with ar- −(cid:209) ׶ M −¶ J (2.14)
t B t
gumentr=(r ,r ,r )),orthewavevectordomain(withargu-
x y z
mentk=(kx,ky,kz)). =−c2(cid:209) ×am (cid:209) ×aeD+c2(cid:209) ×am (cid:209) ×PD
Now we define the total magnetization M in terms of a −c2(cid:209) ×am e0K−(cid:209) ׶ tMB−¶ tJ
component M linearly dependent on B, and another com-
L (2.15)
ponentM containingtherest. Asaresult,theexcesspolar-
B
izationMm ≡MB,whichwilltypicallyincludeanynonlinear-
ity, isalsoafunctionofBandnotH. IntraditionalH-based Rememberthatinthispicture,allfieldpropertiesarespec-
spatially-propagatedapproaches,thetemporalresponse(“dis- ifiedasfunctionsofourchosenprimaryfieldD; notably,the
persion”) can be incorporated naturally, here it cannot. We magneticfieldBwhichisusuallythemainargumentforMB
havethat mustbe determinedfromthecalculatedD usingthe relevant
Maxwell’sequation(i.e. eqn.(2.2)).
M(r)=ML(r)+MB(r) =a m (r)B(r)+MB(r), (2.3)
whichinthespatialfrequency(wavevectork)domainis
A. Simplecase:isotropic&homogeneous
M(k)=ML(k)+MB(k) =a m (k)⋆B(k)+MB(k), (2.4)
sinceafteraFouriertransformfromrintok,theproductsbe- Sincetryingtotreatallthedetailsofeqn. (2.15)correctly
comespatialconvolutions(“⋆”); whichemphasisesthatany- leadstoexcessivelycomplicatedexpressions,Iwillfirstsim-
thing wavevector dependent is necessarily a non-local con- plifyitdowninto thecase where(a)the material’sreference
cept.Notethatthisistheconverseoftheusualsituationwhere properties am , ae do not vary in space, and (b) the effective
atemporalresponseisdescribedusingtime-domainconvolu- monopole current is zero (K=0). Since (magnetic) mono-
tions,andthesebecomeproductsinthefrequencydomain. ples do not exist, and all non reference properties can still
Now we definethe totalpolarizationP in termsof a com- be encodedin PD, MB, or J, this simplification doesnot (of
ponent P linearly dependenton D, and another component itself) impose any approximation. However, the first simpli-
L
P containing the rest. As a result, the excess polarization fication means that our reference properties cannot incorpo-
D
Pe ≡PD,whichwilltypicallyincludeanynonlinearity,isalso rate any exact knowledge about the spatial structure of the
afunctionofD. Wehavethat propagationmedium1. This restriction on obtainingthe best
possible match between the reference behaviour and the ex-
P(r)=PL(r)+PD(r) =a e (r)D(r)+PD(r), (2.5) actbehaviourofthepropagationmediumwilltypicallyaffect
later approximations, where we assume deviations from the
whichinthespatialfrequency(wavevectork)domainis reference behaviour are small. One additional advantage of
selectingconstantreferenceparametersa m anda e isthatwe
P(k)=a e (k)⋆D(k)+PD(k). (2.6) canreplacethemwithmatricestoallowfore.g. birefringence
andcross-polarizationcouplings.
With these definitions for ¶ D, ¶ B, and for M and P set
t t The simplified secondorderwave equationforchangesin
up,wecannowproceedwiththederivationofasecondorder
the displacement field D(r) as it propagatesforward in time
waveequationforD–
m ¶ D=(cid:209) ×B−m (cid:209) ×M −m (cid:209) ×M −m J (2.7)
0 t 0 L 0 B 0
=(cid:209) ×B−(cid:209) ×a m B−m 0(cid:209) ×MB−m 0J (2.8) 1Thisdiffersfromthesituationpresentinthecomplementaryspatiallyprop-
=(cid:209) ×B−a m (cid:209) ×B− (cid:209) a m ×B−m 0(cid:209) ×MB−m 0J acgalaltyedcaanppbreoasuchbs[u1m4]e,dwinhteorethteheremfeerdeinucme’bselhinaevaioruter.mNpeovraelrtrheeslpeossn,seintynpeii--
(cid:0) (cid:1) (2.9) ther case canthe spatial properties beincorporated exactly into theref-
=(cid:209) × 1−a m B−m 0(cid:209) ×MB−m 0J. (2.10) e“rsepnactiealbdeihsapveirosuiorn;”al–thcoaungbheamppadroex.imations –notably thatofaso-called
(cid:2) (cid:3)
2
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is ourdefinitionofthedirectionalfields,sinceitisincompatible
withourtemporalpropagation.Toincludethiswemustmodel
¶ t2D=−c2am ae (cid:209) ×(cid:209) ×D +c2am (cid:209) ×(cid:209) ×PD itexplicitlyasisdoneinMaxwellsolverssuchasFDTD[20,
−(cid:209) ׶ M −¶ J (2.1) 21]; or approximate it as if it were a spatial effect (see e.g.
t B t
[22],orthediscussionsinthelatersectionV.
=+c2am ae (cid:209) 2D−c2am ae(cid:209)(cid:209) ·D
−c2am (cid:209) 2PD+c2am (cid:209)(cid:209) ·PD
−(cid:209) ׶ M −¶ J. (2.2)
t B t
Notethattheretentionofthe(cid:209) ·Dand(cid:209) ·P termsallowsus
D
toincludechargeeffects,asisneededwhendiscussinghigh-
powernonlinearsituationssuchasfilamentation[19].
InthespatialFourier“wavevector”regime,wereplace(cid:209) → III. FACTORIZATION
ık, so that k2 =k·k, and reorganize slightly. The equation
for changes in the displacement field D(k) as it propagates
forwardintimeis
Inowfactorizethewaveequation[14,23],aprocesswhich
¶ t2D−c2am ae k2D=−c2am k2PD−¶ tJ−ık׶ tMB tnieoantalylaapvporiodascshoems.eFoafctthoerizaaptpioronxtiamkaetsioitnssnnaemceesfsraormyitnhetrafadci-t
−c2am ae kk·D+c2am kk·PD (2.3) that the LHS of eqn. (2.2) or (2.4) is a simple difference of
¶ t2D−W 2D=+W 2ae−1PD−¶ tJ−ık׶ tMB squares which might be factorized, indeed this is what was
done in a somewhat ad hoc fashion by Blow and Wood in
−W 2k−2kk·D+W 2k−2a−e 1kk·PD, 1989[24].Sincethefactorsarejustoftheform(¶ ∓ık),each
t
(2.4)
by itself looks like a forward (or backward) directed wave
wherewehavedefinedareferencefrequencyW fortheprop- equation. It therefore allows us to define a pair of counter-
propagatingGreensfunctions,andsodividethesecondorder
agation,basedonthereferencematerialproperties,i.e.
waveequationintoabi-directionalpairofcoupledfirstorder
W 2(k)=c2am ae k2. (2.5) waveequations. Inaddition,itallowsustodirectlycompare
theexactbi-directionalandapproximateuni-directionaltheo-
We could– if we believedwe understoodthe spatial proper- riestermforterm,whereasinotherapproachesthebackward
tiesofthepropagationmediumsufficientlywell–simplyalter partssimplyvanishandarenotdirectlyavailableforcompar-
thisdefinitionbyincorporatinganalternatekdependencethat ison. Ashortsummaryofthefactorizationmethodisgivenin
mimics an assumed spatial dispersion. However, it needs to appendixA.
be understood that such a step is distinctly ad hoc, since it
lackstheconvolutionsoverknecessaryforanyaccuraterep- To re-iteratean importantpoint– the choiceof ae ,am and
resentation of the material structure. But, that said, such an thereforeW (k)ineqn. (2.4)definesthespecificGreensfunc-
approachmaystillhaveconsiderablepracticalutility;andin- tionsused; it thereforealso definesthe basisuponwhichwe
deed is functionallyequivalentto how other methodshandle willthenpropagatethedisplacementfieldD.
spatial dispersion, as a non-local (and hence, strictly speak-
ing, a non-causal) process: e.g. those that import a waveg-
uide dispersioncalculatedfromthe (spatial) modesgivenby
the guide’stransverse cross-section. To allow for this possi-
bility, in the following I will replace the physically rigorous
magnitude k argumentfor W with k, its vector counterpart2.
This allows for not only more complicated assumed spatial
dispersions,butalsoadditionalorientationdependence.How- A. Bi-directionalwaveequations
ever, these benefits(only)arise as a result ofapproximation.
Strictly speaking, both on the grounds covered above, and
onestakingstricterviewofcausality[16],anynotionofspa-
A pair of bi-directionalwave equationssuggests similarly
tialdispersionisincompatiblewithlocality.
bi-directionalfields, so I split the electric displacement field
Unlikeforthespatiallypropagatedschemein[14],wehave into forward (D+) and backward (D−) directed parts, with
notincludedtemporalresponseofthepropagationmediumin D=D++D−. Inthefollowingequations,rememberthatP ,
D
M , etcalldependonthe fullfieldD, andnot onlyonepar-
B
tialfield(e.g. onlyD+). Thisisanimportantpointsincewe
seetheydrivebothforwardandbackwardevolutionequations
2Insomecases,dependingontheapproximatemodelchosen,thisnewW (k) equally.
definitionmayalsoneedtobeconvolvedwithotherpartsoftheexpression,
e.g.W 2(k)PD(k)→W 2(k)⋆PD(k),althoughsuchanadditionalcomplica-
The coupled bi-directional first order wave equations for
tionislikelytopreventusobtainingmuchconceptualormathematicalben-
efit.Consequently,inwhatfollowsIdonotallowforsuchconvolutions. thedirectedfieldsD± are propagatedforwardin timewhilst
3
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beingevolvedforwardandbackwardinspaceare fieldoscillationsinD±(r)wouldmoveforward(+)orback-
wards (−) in space. This is analogous to the choice of ref-
¶ D±(k)=±ıW (k)D±(k) ±ıW 2(k)a−e 1P (D,t;k) erence when constructing directional fields [26], but here is
t 2W (k) D donefortemporalratherthanspatialpropagation[27].
J˙(t;k) ı Residual evolution accounts for the discrepancy between
∓ ∓ k×M˙ (D,t;k) thetrueevolutionandtheunderlying/referenceevolution,and
2W (k) 2W (k) B
is every part of the material response not included in ae ,am
ıW 2(k) ıW 2(k)a−e 1 (and hence W (k)). i.e. it is all the remaining terms on the
∓ kk·D(k) ± kk·P (D,t;k)
2W (k)k2 2W (k)k2 D RHS of eqn. (3.2). These typically include any non-linear
(3.1) polarization, or spatially or orientationally dependent linear
ıW (k) terms; they are analogousto the correctiontermsused in di-
¶ D±(k)=±ıW (k)D±(k) ± P (D,t;k) rectionalfields models. In the languageused by Ferrandoet
t D
2ae al. [23], these residuals are “source” terms. Although we
∓J˙(t;k) ∓ ı k×M˙ (D,t;k) mighthopetheywillbeonlyaweakperturbation,sothatwe
2W (k) 2W (k) B mightmakethe(desirable)uni-directionalapproximationdis-
ıW (k)kk ıW (k)kk cussedlater, thefactorizationprocedureis neverthelessvalid
∓ 2 k2 ·D(k) ± 2ae k2 ·PD(D,t;k) foranystrength.
(3.2)
Here I havereplacedthe partialtime derivativesonthe RHS C. Referenceevolution:choiceofW andtheresultingD±
withover-dots,thisbeingtoemphasisethatifourexpression
is to remain strictly causal [17], then our models for current InowexaminehowthechoiceofW affectstherelativesizes
JandmagnetizationMB mustreturntheirtime-derivativesas oftheforwardandbackwarddirectedD+andD−. TodothisI
explicitfunctionsofknownquantities(here,typicallyDort). considerthesimpleexampleofamediumforwhichthefieldis
Note that factorized equations can be rebuilt into a single knowntopropagatewithfrequencyw ;butfordemonstration
second-orderequationbytakingthesumanddifference,then purposeswechooseareferenceevolutiondeterminedbyafre-
substitutingoneintoanotherwiththeassistanceofaderivative quencyW thatisdifferentfromw . E.g.,foralinearisotropic
w.r.t.t (seesectionIV.Bof[23]). medium we could exactly define w 2 =W 2+D 2; but in gen-
A remaining clarification required is to understand the eralwewouldjusthavesomeresidual(source)termQ. This
meaningof“forward”and“backward”inthecontextofthree meansthatourdefinitionsofforwardandbackwarddirected
dimensionalspace–thisbeinglessstraightforwardthaninthe fields do not exactly correspond to what the wave equation
spatiallypropagatedcase,whereonespatialaxisisselected3, will actually evolve forward and backward as we propagate
andforwardandbackwardrefersto timet. Here, we should towardslargert.
understand “forward”and “backward”as being with respect The second order wave equation is (¶ 2+W 2)D = −Q,
t
to a particularchoice of k, and those individualvector com- whichinthelinearcasehasQ=D 2D,sothat(¶ 2+w 2)D=0.
t
ponents,thatweareinterestedin. ThefactorizationintermsofW isthen
ıQ
¶ D±=±ıW D±± . (3.1)
B. Propagation,evolution,anddirectedfields t 2W
Now if we select the case where our field D only evolves
Notethatsinceoursolutionsofthewaveequationsenforce forward, we know that D=D exp[ıw t]. Consequently D±
0
propagationtowardslatertimest,thefieldsD±(t)aredirected musthavematchingoscillations:i.e. D±=D±exp[ıw t],even
0
forwardsandbackwardsinspace;thesefieldsthenevolvefor- thoughD−isdirectedbackwards.Substitutingtheseintoeqn.
wards and/or backwards in space as t increases. Note that I (3.1)gives
usethisterminology(propagated,directed,evolved)through-
outthispapertomeanthesethreespecificthings. W −w
D−= D+, (3.2)
When examining the wave equation eqn. (3.2) which 0 W +w 0
evolves the directed fields D± as they propagate forward in
t, weseethattheRHShastwotypesofterms: whichIlabel whichspecifieshowmuchD−weneedtocombinewithD+so
the“reference”and“residual”parts[25]. thatourpulseevolvesforward;sincetheD− willbedragged
Reference evolution (or “underlying evolution”) is that forward by its coupling to D+. This interdependentD± be-
given by ±ıW (k)D± term, and is determined by our chosen haviourisgeneric–nomatterwhattheoriginofthediscrep-
ae andam , and anyadditionalad hocrefinements. By itself, ancy between W and the true evolution of the field (i.e. the
itwoulddescribeanordinaryoscilliatoryevolutionwherethe residual/source terms such as mismatched linear behaviour,
nonlinearity,etc): somenon-zerobackwarddirectedfieldD−
must exist but still evolve forwards with D+. Analogous
behaviour can be seen in the directional fields approach of
3Typicallyacartesianaxis,althoughotherchoicesarepossible[15]. Kinsleretal. [26].
4
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UsuallywehopethatthisresidualD−contributionissmall ing and backwardevolving componentsis ∼2W ; in essence
enoughsothatitcanbeneglected.IfweassumeD−≃0,then itiscomparabletothecommonrotatingwaveapproximation
we find thatw ≃W +D 2/2W , which isjust theexpansionof (RWA). This rapid relative oscillation means that backward
w =(W 2+D 2)1/2 tofirstorderinD 2/W 2. Followingthis,we evolvingcomponentsneveraccumulate,aseachnewaddition
findthateqn. (3.2)thensaysthatD−≃(D 2/4W 2)D+,which willbeoutofphasewiththepreviousone;itisnotquitea“no
0 0
hascomefullcircleandprovideduswiththescaleonwhich reflection”approximation,butone thatasserts that the many
D− canbe considerednegligible. Outsidetherestricted(lin- micro-reflectionsdo not combine to produce something sig-
ear) case where we knowD 2, the true frequencyw mightbe nificant. An estimate of the terms required to break this ap-
difficulttodetermine,andinnonlinearpropagationanylocal proximationaregiveninappendixB;generallyspeakingthis
estimateofthefrequencycanchangeasthepulsepropagates. is a much more robust approximation than the slow evolu-
There is a further important point to notice: if we choose tion one. Of course, a periodic temporal modulation of the
W =W (k) with a wave-vector dependence, then we see that medium will give periodic residual terms, and these can be
thesource-liketermsinheritthatdispersion;andthisisalmost engineeredto force phase matching. In spatially propagated
inevitable,since typicallyW ∼ck. Thismeans, forexample, analyses,wherewetalkofspatial(wavevector)phasematch-
thatevenifwestartedwithnonlinearpolarizationmodelwith- ing and not this temporal (frequency) phase matching [28],
outanyk dependence,ourfactorizedequationswillhavea k thiswouldbe a periodicitybasedona relativelysmallphase
dependenceon the nonlinearterms. This is the complemen- mismatch (see e.g. quasi phase matchingin Boyd [29]); but
taryprocesstowhathappensinthespatiallypropagateddirec- mightevengoasfarasmatchingthebackwardwave(seee.g.
tionalfieldsapproachofKinsleretal. [26],wherechoosinga [30]).
dispersivereferencehasanequivalenteffectonthecorrection Itisalsoimportanttonotethatthesamesmallsizeofpertur-
terms. bationfromtheresidualtermscanaccumulateontheforward
evolving field components (or, indeed, the backward pertur-
bationonthebackwardevolvingfieldcomponents).Although
themagnitudeoftheresidualtermsactingontheforwardand
IV. UNI-DIRECTIONALWAVEEQUATIONS
backwardfieldevolutionareidentical,forwardevolvingcom-
ponentsoftheresidualscanaccumulateontheforwardevolv-
Makingonlyasingle,welldefinedclassofapproximation
ingfieldbecausetheyarephasematched;whereasbackward
I can now reduce the exact coupled bi-directional evolution
residualsarenot,andrapidlyaveragetozero.
of D down to a single uni-directionalfirst order wave equa-
tion. Aswiththecomplementaryspatiallypropagatedtheory
[14],Idonotrequireamovingframe,asmoothenvelope,or
A. Residualterms
toassumeinconvenientsecondorderderivativesaresomehow
negligible,aswasnecessaryintraditionaltreatments.Theap-
proximationis that the residualterms are weak comparedto The handling and criteria for different types of residual
the (reference) ±ıW D term – e.g. weak nonlinearity or ori- termshasbeendiscussed already[14] forthe spatially prop-
entationdependence. Thisenablesmeto assertthatif Istart agatedcase. Whileitwouldbestraightforwardtorepeatthat
withD−=0,thenD− willremainnegligible. Inthiscontext, levelofdetailhere,infactthebasicprocessisvirtuallyiden-
ticalexceptforthedifferentrolesplayedbythet andrargu-
“weak”meansthatnosignificantchangeinthebackwardfield
ments. AccordinglyIwillconsideronlythemostcommonly
isgeneratedinadistanceshorterthanonewavelength(“slow
considered residuals – the dielectric scalar and vector terms
evolution”); and that small effects do not build up gradually
aswellascurrents. Thoseinterestedinmagneticormagneto-
overpropagationtimesofmanyperiods(“noaccumulation”).
electriceffectscanadapttheprocesstotheirownsystemsina
Slow evolution is where the size of the residual terms is
similarmanner.
much smaller than that of the underlying linear evolution –
i.e.smallerthanW D.Thisallowsustowritedownstraightfor- TherelevanttotalpolarizationPD canthenbedecomposed
intopiecesbeforeseeinghoweachmightsatisfytheslowevo-
wardinequalitieswhichneedtobesatisfied. Itisimportantto
lutioncriteria. Thuswewrite
notethecloserelationshipbetweentheseandagoodchoiceof
W ,asdiscussedinsubsectionIIIC.IfW isnotagoodenough
match,therealwaysbesignificantcontributionsfrombothfor- PD(D,r,t)=ae (D,r,t)D(r,t)+UD(D,r,t)
ward and backwarddirected fields; and evenif nothingends =aL(r,t)∗D(r,t)+aN(D,r,t)∗D(r,t)
up evolving backwards, an ignored backward directed field +U (r,t)∗D(r,t)+U (D,r,t). (4.1)
L N
willresultin(e.g.) miscalculatednonlineareffects,sincethe
totalfieldD=D++D− willbewrong. The part which is scalar in nature is represented by ae, and
No accumulation occurs when the evolution of any back- mightcontain(non-reference)linear partsandtime response
ward directed field D− is dominated by its coupling via the (a ), perhaps a convolution(“∗”) over the past, or angle de-
L
residualtermstotheforwarddirectedfieldD+;andnotbyits pendence; it might also contain nonlinear contributions a
N
preferred underlying backward evolution. No accumulation suchasthethirdorderKerrnonlinearitywitha D(cid:181) (D·D)D.
N
meansthatforwardevolvingfield componentsdonotcouple The vector part U , if present, could be from e.g. a sec-
D
to field components that evolve backwards; this the typical ondordernonlinearity,whichcouplestheordinaryandextra-
behavioursince the phase mismatch betweenforwardevolv- ordinaryfield polarizations. Note thatthis descriptionof the
5
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materialparametersdoesnotrestricttheminanyway,andcan withthesame strength,so thatitisfarfromobviousthatwe
e.g. includeanyorderofnonlinearity. cansetD− tozero,butstillkeeptheD+ withoutbeingincon-
y y
sistent. However,asdescribedabove,itisthephasematching
which ensures that forward residuals accumulate, whilst the
B. Slowevolution? non-matchedbackwardresidualsaresubjecttotheRWA,and
becomenegligible:hencewecanstillrelyoneqn.(4.2),albeit
NowIwilltreateachpossibleresidualterminorder,where undercaution.
theoppositelydirectedfieldisnegligible:i.e.,forD±,wehave
thatD∓≃0. wherethescalare containsthelinearresponse Thirdly,wehavetheresidualtermsindicatedbythepresence
L
ofthematerialthatisbothisotropicandlossless(orgain-less); oftime-varyingcurrents.Theseare
sincehereitisatime-responsefunction,itisconvolvedwith J˙ ≪2W 2 D± . (4.3)
theelectricdisplacementfieldD. NotethatthefieldvectorD, i i
andindeedthematerialparametereLareallfunctionsoftimet (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
andspacer=(x,y,z);thepolarizationPe anditscomponents Fourthly,wehavethedivergencetermskk·D,kk·PD,which
f e ,Ve areafunctionsoftimet,spacer,andthefieldD. amountto chargedensity contributionseither fromD or PD,
Below I will refer to field componentsDi and D±i , where with s P(k) = a−e 1k·PD and s D(k) = k·D. Thus if the
D++D− =D≡(D ,D ,D ) and i∈{x,y,z}; also to wave wavevectorisorientedalongaunitvectoru,then
x y z
vectorcomponentskifromk=(kx,ky,kz),withk⊥2 =kx2+ky2. ıuW |s |/2k≪ıW D±
However,notethatintheconstraintsbelow,thatk isalsoused i i
j
asasubstitutesymboltorepresentanyoneofkx,ky,orkz. s ui≪2k(cid:12)(cid:12)D±i (cid:12)(cid:12). (4.4)
Any correctionsdue to imperfect choice of W will appear (cid:12) (cid:12)
istherelevantcriterion. (cid:12) (cid:12)
intheP polarizationcorrectionterm.However,unlikeinthe
D
spatially propagated case, there is no diffraction-like term –
To summarize, these various criteria assert that modulations
here,thatisquitenaturallypartofthereferencebehaviour.
away from the reference evolution must be weak. Notably,
weak nonlinearityis invariablyguaranteedby material dam-
Firstly,scalarpolarizationtermsf ,whichcanbeeitherlinear
age thresholds[31]. However, and in contrastto the usually
(f (k))ornonlinear(f (D,k)). Thesemightencodee.g. the
L N favoured (but strictly speaking less causal) spatially propa-
time-response of the medium (and hence its dispersion), or
gated methods, the lack of any rigorous way to incorporate
be some nonlinearity response such as the third-order Kerr dispersionintothereferencefrequencyW (k)makesitharder
nonlinearity.Sincethesearelocallycorrelatedinspace,inthe
tosatisfythecriteriafortheordinarylinearresponse.
wavevector picture they include convolutions over k, and so
giveusthecriterion
ıW f ⋆ D++D− /2 f ⋆ D± /2 f C. Uni-directionalequationforD+
i i ≃ i ≃ ≪1. (4.1)
ı(cid:12)W D± (cid:12) (cid:12)D±(cid:12) 2
(cid:12) i (cid:12) (cid:12) i (cid:12) In the case where all of the slow-evolution criteria listed
(cid:12) (cid:12) (cid:12) (cid:12)
Inthelinear(cid:12)case(cid:12),f ≡f isind(cid:12)epe(cid:12)ndentofD,soonlythema- above hold, we can be sure that the backward directed field
L
D− is negligible, and if the no-accumulation condition also
terialparametersareconstrainedandthepulsepropertiesplay
holds,thenneitherwilltherebeanynon-negligiblebackward
norole. Inthenonlinearcase,thecriteriamakesdemandson
evolvingcontributionsto the field. Consequently,we can be
the peak intensity of the pulse – but does notapply smooth-
surethataninitiallynegligibleD− remainsso, andeqn.(3.2)
nessassumptionsorbandwidthrestrictions.
becomes
Secondly, linear and nonlinear vector terms from U. These ıW (k)
¶ D+(k)=+ıW (k)D+(k) + P (D+,t;k)
will have a criterion broadly the same as the scalar cases in t 2ae D
eqn. (4.1),butwithUreplacingf D. Thusfori∈{x,y,z},we J˙(t;k) ı
canwritedownconstraintsforeachcomponentUi,whichare −2W (k) −2W (k)k×M˙ B(D+,t;k)
ıW |Ui|/2≪ıW D±i =⇒ |Ui|≪2 D±i . (4.2) −ıW (k) kk·D+(k) +ıW (k) kk·P (D+,t;k).
In the linear case, U(cid:12)(cid:12) ≡(cid:12)(cid:12)U , and since U and(cid:12)(cid:12)D h(cid:12)(cid:12)ave some 2 k2 2ae k2 D
L L (4.1)
linear relationship, thiscriteriononly constrainsthe material
parameterscontainedinU ,notthefieldprofileoramplitude. wherenowthepolarizationP ,diffraction,anddivergenceare
L D
The nonlinearcase is similar to the scalar one, and the peak solelydependentontheforwarddirectedfieldD+.
pulse intensity is restricted; e.g. for a c (2) medium, |U |∼
N
c (2)|D|.
However,onecomplicationofthevectorcasesisthatafield D. Modifications
consistingofonlyonefieldpolarizationcomponent(e.g.D+)
x
may induce a driving in the orthogonal (and initially zero) Just as for the comparable spatially propagatedderivation
components(e.g. D±). Hence both D± fields will be driven [14]wemightalsoapplysomeofthestrategiesusedinother
y y
6
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approaches to get a more tractable and/or simpler evolution picture,wecanstillfindthesamewaveguidemodesandtheir
equation.Unlikein“traditional”derivations[32–35],noneof dispersions,butthistimegivenasfunctionsofk , notw . So
z
thesearerequired,buttheyneverthelessmaybeuseful. forsomen-thmodeofthewaveguide,asexpandedaboutsome
Briefly, a co-moving frame might be added, using z′ = centralevolutionwavelengthk ,whichcorrrespondstoaref-
zn
z+tv . This is a simple linear process that causes no ex- erencefrequencyW =W (k ),wemighthave
f n zn
tracomplications;theleadingRHSıcD+ termisreplacedby
ı(c∓v)D±. Note thatsetting c=v willfreezethe phase ve- W (k )=W +(cid:229) g (k −k )m, (5.5)
z n m z zn
locity of the pulse, not the group velocity. Also, a carrier-
m
envelope separation could be implemented using D(z) =
A(z)exp[ı(w 1t−k1z)]+A∗(z)exp[−ı(w 1t−k1z)]definingthe whereg1=c+g¯1hastwocontributions–thatduetothevac-
envelopeA(z)withrespecttowavevectork1 andcarrierfre- uum (just c), and the estimated modifications due to spatial
quencyw 1. Multipleenvelopescentredatdifferentwavevec- effects(g¯1).
torsandcarrierfrequencies(k,w )couldalsobeusedtosplit
i i We may also wish to add the effects of whatever tempo-
the field up. This is typically done when the field princi-
ral response the material might have to our propagationcal-
pallycomprisesanumberofdistinctnarrowbandcomponents
culation; but also decide that an approximate model where
(e.g. [29,36,37]). Thewaveequationcanthenbeseparated
we modify the expression above in eqn. (5.5) is sufficent.
into one equation for each piece, coupledby the appropriate
This is the converse of the traditional procedure, where the
wavevector-matchedpolarizationterms(cf[28]). waveguidedispersionisaddedtowhateverb (w )isappropri-
ate for the medium’s time response. Consequently we need
onlymakethesameassumptions: (a)thatthetwosourcesof
V. DISPERSION
dispersionareweakandsocansimplybeadded,(b)thatthe
temporalresponseisassumedtohavenegligibleeffectinthe
Consider a dielectric slab waveguideof thicknessd in the
waveguide’spropagatingmodes,and(c)thatthetemporaldis-
perpendicularx direction,propagationin thezdirection,and
persionisthesameatallpointsinthewaveguide.
with core and cladding permittivities e and e . In the CW
1 0 For simplicity we consider the usual situation where our
limit, the spatialstructureofthe waveguidegivesrise to fre-
dispersionhasalreadybeencharacterisedand representedas
quency dependent properties that are usually called “spatial
a truncated power series in frequency. In line with the rec-
dispersion”. Evenforthis simple slab design, the dispersion
ommendationsof[25],Iexcludeimaginarycomponents(e.g.
relation hasa non trivialform; in this case it is givenby the
lossorgainterms)ineitherthewavevectorreferenceb orthe
solution to a transcendental equation [38]. Typically, this is
frequencyreferenceW . Consequently,thereferencewavevec-
writteninawayimplyingwewanttocalculatek andb =k
x z torconsideredherehasthesimpleform
from w , i.e. for the transverse electric (TE) field modes we
have
b (w )=b +k (w −w )+k (w −w )2, (5.6)
0 1 0 2 0
T(k d)=k−1 w 2m (e −e )−k2, (5.1)
x x 0 1 2 x
q
where b 2(w )=w 2m e −k2, (5.2) where k 1 =c−1+k¯1 has two contributions– that due to the
0 1 x
vacuum(just 1/c), and the modificationdue to the temporal
andT(kxd)iseithertan(kxd)or−cot(kxd). response(k¯1).
However,wecannowseethatthisisanimplicitlyspatially I have chosenthis formfor b (w ) becauseit is possible to
propagatedrepresentation, since in time propagatedsystems calculate its inverse, although in general – if there are more
wedonot(cannot)knoww ,butwedoknowk. Thereforeeqn. termsintheexpansion,orb hassomespecificandnon-trivial
(5.2) should be rewritten accordingly, using kz for b (since analyticform–theproblemismorecomplicated.Forthiswe
wavevector is no longer a propagationreference), and W for againreplacew withW todenotethenewroleasareference
w (sincefrequencynowisapropagationreference);thus frequency,andreplaceb withk . WithW =ck ,thisgives
z n zn
e
c2W 2(k )=k2 1 T2(k d)+1 . (5.3) k
where kx2=cx2W(e21(−k )e2−)k(cid:2)2. x (cid:3) (5.4) W (kz)=W n−2k12(cid:20)1∓q1+4(kz−kzn)k 2/k 12(cid:21), (5.7)
z x x
Hereweseethatthenatural(reference)propagationfrequency whichwecouldexpandintoapowerseriesforsmallkzn−kz.
is defined solely by the transverse wavevector kx, as is the Since k 1 includesthe vacuum,and dispersionk 2 is assumed
matchingevolutionwavevectork . Whileapparentlysimpler tobesmall,weshouldmakethetopsignchoicesinceitgives
z
thanthe traditionalform,since W is directlygivenby k , we the physicallyrelevantsmallfrequencycorrections. The lin-
x
usuallywanttoneglectanydirectknowledgeofthetransverse ear term in the series has the coeffient g1T = 1/k 1 and the
properties. ThismeansthatwhatwereallywantisnotW (kx), quadratictermg2T =k 2/k 13.
butW (k ),theinverseofthetraditionallypreferredb (w ). We now combine the two dispersive effects – the spa-
z
Despite these complications, which are in any case rather tial/geometric/waveguide ones based on parameters g , and
i
similartothoseweseeinthetraditionalspatiallypropagated convertedtemporalonesbasedonk ,togiveatotalreference
i
7
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D2OWE Timepropagatedopticsanddispersion http://www.kinsler.org/physics/
frequencybasedonk of astemporaldispersionworks–evenifthislatterprocessisso
z
ubiquitousin opticsas to be effectivelyinvisible. Removing
W ′(kz)=W n+g¯1(kz−kzn)+ (cid:229) gm[(kz−kzn)]m, eitherofthemismatchesdiscussedabove–eitherinreference
m>1 behaviourorconvolution–requiresanarrowband(CW)limit,
k
− 1 1− 1+4(k −k )k /k 2 (5.8) a process which necessarily obscures the parameter regime
2k 2(cid:20) q z zn 2 1(cid:21) where interesting tests and comparisonscan (and should) be
1 k made–i.e. theshortpulse,largebandwidthlimit. Indeed,the
≃W n+(cid:18)g¯1+k (cid:19)(kz−kzn)+(cid:18)g2−k 23(cid:19)(kz−kzn)2. stringent nature of the CW approximation would defeat the
1 1 entirepointofthecomparison.
(5.9)
Nevertheless, it may be possible to design numericalsim-
ulationscontainingstructuresand temporalresponseswhose
Note that to avoid incorporating the vacuum contribution
scalesarecarefullygraduatedsothatthemundanebandwidth
twice when summing the dispersive effects, I have replaced
g withg¯ . issues can be kept separate from the interesting propagation
1 1
ones. Suchadesignwould,asdesired,allowthepropagation
Inthecomplementarycase, usefulforspatialpropagation,
approximations(only)tobecomparedandcontrastedbynu-
wecanfollowtheconverse–andindeedtheusualprocedure
mericalsimulation.However,thisisanon-trivaltask,andone
–toconverttheCWeffectofthewaveguide’sspatialstructure
W (k ) into a frequencydomainform[38, 39]. Thisthen can whichIleaveforlaterwork.
z
becombinedwithtemporalresponseofb (w )toget
1 g VI. PULSEPROPAGATION
b ′(w )=b + k¯ + (w −w )+ k − 2 (w −w )2.
0 (cid:18) 1 g (cid:19) 0 (cid:18) 2 g 3(cid:19) 0
1 1
(5.10) Thirdorder nonlinearitiesare commonin manymaterials,
e.g.inthesilicausedtomakeopticalfibres[18,38,39]. Here
I compare 1+1D propagation such a waveguide, in order to
Discussion elucidatethedifferencesbetweenthestandardspatiallyprop-
agatedpictureandthetemporallypropagatedpicture.
Theassumptionsmadearethoseoftransversefields,weak
We mighthavehopedto finda less approximatecompari-
dispersivecorrections, and weakly nonlinearresponse; these
sonbetweenthespatiallypropagatedandthetemporallyprop-
allallowustodecoupletheforwardandbackwardwaveequa-
agatedapproachestodispersionwhathasjustbeendiscussed.
tions. Thisdecouplingthen allowsus, withoutany extraap-
Further, although qualitative comparisons can also be made,
proximation,toreduceourdescriptiontooneofforwardonly
ideallywe would like a specific andquantitativecomparison
pulsepropagation.Thespecificexamplechosenhereisforan
asdonefore.g. testingtheunidirectionalapproximation[31].
instantaneouscubicnonlinearity,butitiseasilygeneralizedto
Thatthisisnotassimpleasitmightsoundcanbedemon-
non-instantaneouscasesorevenotherscalarnonlinearities.
strated as follows: write two mathematically identical but
physically and notationally distinct propagation equations,
i.e.
A. SpatiallypropagatedNLSE
¶ E(w )=+ıK(w )E(w ) (5.11)
z
¶ tD(z)=+ıW (z)D(z). (5.12) ThespatiallypropagatedNLSforEx+(w )basedonadirec-
tionalfactorization[14]canbewritten
Forsimplicity,thiscomparisonisdoneinthe1+1Dlimitand
forunidirectionalequations;theresidualw orzbehaviourhas ¶ zEx+=+ıb Ex+ +(cid:229) k m.(w −w 0)mEx+
beenmergedwiththereferencebehaviourtogivesingleprop- m
agationparametersK(w )orW (z). ık2
+ 0F c (3)E2(t)E+(t) , (6.1)
Superficially, the comparison between eqn. (5.11) and 2b x x
h i
(5.12) looks promising. However, the vacuum part of K(w )
is w /c, but the vaccum part of W (z) is in contrast a spatial whereF[...]istheFouriertransformthatconvertsthenonlin-
derivative, i.e. c¶ . Even if it were possible to match up earpolarizationtermintoitsfrequencydomainform.Herewe
z
thefunctionalformsofthenon-vacuumtemporalresponseof have chosen a fixed, non-dispersivereferencewavevector b ,
K(w )−w /c with a spatial structure giving W (z)−c¶ , the thetemporalresponseofthepropagationmediumisencoded
z
vacuum parts would remain unmatched. Alternatively, we inthecoefficientsk . Forthetemporal(timeresponse)ofthe
j
couldcomparethez-propagatedw -domaintothet-propagated material,thisisanin-principleexactre-representationofthat
k-domain. However, although in this case the reference be- responseasaTaylorseriesexpansion;althoughinpractisethe
havioursmatch, butthek-domainmodelnowhasa convolu- seriesisusuallytruncatedafterafewtermsandthepropagat-
tionthatdoesnotappearinthew -domainone. ing fields restricted to within a bandwidth where only those
Asaresult,itisunclearhowwellapproximatingtemporal termsaresignificant.
dispersion as spatial propertiesworks in practise; but then it The important point to remember is that in this spatially
is equally unclearhow wellapproximatingspatialproperties propagatedpicturewe do knowthe fulltime history, at least
8
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D2OWE Timepropagatedopticsanddispersion http://www.kinsler.org/physics/
inacomputationalsense–i.e.aswecomputeasolutiontothe it explicitly, then we can use the approach described in the
propagationequation. This “computationalpast” [16], com- previoussection,wherethetemporalresponseofthematerial
prisingallpreviouslycomputedvaluesofthefieldproperties was (approximately)represented as if it were instead due to
andmaterialstatemeansthatafullfrequencyspectrumandre- spatialproperties.
sponseisknown.Asaresult,thetimeresponseofthematerial
Finally, note that in ordinary temporally propagated sim-
canbe directly encodedin an accurateand usefuldispersion
ulations, the computationalpast and the physical (temporal)
relation,andthenbeapproximatedasaTaylorseriesinw .
pastareidentical;thisisnotthecaseforspatiallypropagated
Since this propagation is assumed to be taking place in a
simulations,asmentionedabove.
waveguide,wealsohaveitsgeometric–wavelengthsensitive
VII. CONCLUSION
– propertiestoconsider. However,as discussedin theprevi-
oussectionwetypicallyjustre-representthoseasiftheywere
insteadgeneratedbya temporalresponse(see e.g. [38, 39]) I have deriveda generalfirst orderwave equationforuni-
Thus the coefficients k usually combine both the effect of directionalpulsepropagationintimethatallowsforarbitrary
j
timeresponseandgeometricpropertiesintoasingletotaldis- dielectricpolarization,diffraction,andfreeelectricchargeand
persion; this is the very basis of dispersion compensation in currents. This propagation equation has a utility all of its
opticalfibres,sincethe material(temporal)dispersionisoff- own,allowingefficientunidirectionalpropagationinastrictly
setbythewaveguidedesign(spatialdispersion)[38]. causalmanner(i.e. timedirected). However,itrequirescom-
parable approximations to those used in deriving spatially
propagatedunidirectionalpropagationequations,althoughthe
B. TemporallypropagatedNLSE specifictradeoffsaredifferent.
The propagation equation was derived by first factoriz-
Usingtheapproachderivedandpresentedinthispaper,we
ing the second order wave equation for D into an exact bi-
can developa time propagatedcounterpartto eqn. (6.1). In
directional model, it applies the same type of well defined
a purelymathematicalsense, itis, apartfromnotation, iden-
type of approximationto all non-trivialeffects (e.g. nonlin-
tical to eqn. (6.1). However, in physical terms, the change
earity,diffraction),andso reducesthe propagationequations
isnottrivial,sincethephysicalmeaningandboundarycondi-
to a first order uni-directionalwave equation. One nice fea-
tionsdiffersignificantly. Startingfromeqn. (4.1)wecanget
tureofthisfactorizationmethodisthatitprovidesforaterm-
apropagationequationinwavevectorspaceforD+(k)which
x to-termcomparisonoftheexactbi-directionaltheorywithits
reads
approximateuni-directionalcounterpart.
¶ D+=+ıW D+ +(cid:229) g .(k −k )mD+ Inadditiontoitsusewherethecausalsimulationofaprop-
t x x m z z0 x
m agation problem is of strict concern, it also illuminates the
ıW rathershadowyprocessofhandlingmaterialresponseandspa-
+ G c (3)D2(z)D+(z) , (6.1)
2ae h x x i tial structure in waveguides by constructing a single “com-
bined dispersion” parameter. As discussed in section V, we
whereG[...]istheFouriertransformthatconvertsthespatially cannowseethatthissimpleadditionoftemporalandspatial
localized nonlinear polarization into its wavevector domain dispersionsin an ad hoc mannerstands on rather poorfoun-
form. Here, alldispersivebehaviourhasbeencombinedinto dations, particularly in the few-cycle pulse limit. Further, it
thecoefficientsg ,whichmimicthecommonnotionofspatial is hard to see any straightforward way of testing the limita-
j
dispersion. Such terms can arise directly from a Taylor ex- tions of this common procedure, which poses an interesting
pansioninkofarelevantCWdispersionrelationaboutsome challengefor the future. That said, in most applications, we
suitablereferencewavevector;however,althoughwecanuse expectthatacombineddispersionwillremainaperfectlyad-
theexpansiontorepresentthespatialstructureofthepropaga- equateapproximation,sincethepulsesarerarelysoshort,and
tionmedium(e.g. a waveguide),the conversionfromspatial itseemsreasonableto assumethatthe characterofthe pulse
structure to k expansion is only relatively straightforward in evolution should remain similar, even if details might differ.
theCWlimit. Lastly, the lessons learnt from this investigation could also
If the propagation medium also has a temporal response be applied to other types of wave propagation,most notably
(i.e. temporal dispersion), and if we do not want to model acousticsystems(seee.g.[27]).
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