Time and space, frequency and wavevector: or, what I talk about when I talk about propagation
Negative Frequency Waves?
Or: What I talk about when I talk about propagation
Paul Kinsler ∗ Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom. (Dated: Wednesday 29 th August, 2018)The existence of waves with negative frequency is a surprising and perhaps controversial claim which hasrecently been revisted in optics and for water waves. Here I explain a context within which to understand themeaning of the “negative frequency” conception, and why it appears in some cases.
Introduction
Claims or analyses involving negative frequency waves oc-cur in a wide range of fields: quantum mechanics, optics &EM [1, 2], and acoustics and water waves [3]. However,some regard the notion of negative frequencies as unphysical.Here I describe four alternative views of the universe with aview to clarifying why negative frequencies might appear, andwhether or not we should be worried: the omniscient , the tra-ditional , the directed , and the causal .The “traditional” and the “directed” pictures, involvingpropagation along time axis and one selected spatial axis re-spectively, have already been compared for a range of acousticwave equations [4]. In particular, the convenience of spatialpropagation in optics, and the additional utility of the unidi-rectional approximation has been widely studied (see e.g. [5]and references therein). Here I am most interested in compar-ing the traditional & directed views, and restrict notation toCartesian coordinates for simplicity; although extension to al-ternate systems is of course possible [6]. It is important to notethat in practical terms, utilising spatial propagation amounts tomaking an approximation, since not only the initial conditionsbut the ongoing propagation require a knowledge of the futurewhich we rarely have.It is useful to mention two ideas before we start: first, thestandard notion of the casual past of a point in time and space,generally called the past light cone; and second, the computa-tional past of that point. This notion of computational past ap-plies primarily to numerical simulations, but can also be use-ful from more abstract perspective. It consists of both the setof initial conditions and/or computed data which might haveaffected the calculated state of that chosen point.In what follows I will denote the profile of the wave fieldunder discussion as F , but rather than festooning it with aplethora of typographical tics (bars, tildes, primes, and thelike) to indicate what arguments may or may not have beentransformed, I use its arguments alone to define F ’s meaning– whether time t or space r = ( x , y , z ) , whether it is a spectrawith frequency ω or wavevector k = ( k x , k y , k z ) , and so on.Further, I will not consider any specifics as to why a specifiedwave field might develop a certain character or characteristics– pulses, oscillations, spectral features and the like – just whatwe can say about the wave given the available knowledge. ∗ Electronic address: [email protected]
In addition to the four sections addressing each of the omni-scient, traditional, directed, and causal pictures in turn, alongwith a brief conclusion, there are appendices which addresssome issues relevant in particular to the field of nonlinear op-tics: the handling of time-response models, and the appear-ance of negative frequency terms.
I. OMNISCIENT: ALL TIME AND ALL SPACE
The omniscient position is taken when we claim all possibleknowledge about how some wave has and will move throughits environment. Here we might express our knowledge ofsome wave-field F by writing it with full space and time ar-guments: i.e. F ( t , r ) . Most likely, this view is a result of ushaving an analytic solution to some specific case of the phys-ical model we are interested in.Given this fully known wave field solution F ( t , r ) , we are atliberty to Fourier transform in either time or space (or both) aswe see fit. In the resulting double spectrum F ( ω , k ) , we willquite naturally see not only positive and negative frequencycomponents, but also forward and backward wavevector com-ponents along each spatial coordinate axes.Since the spectrum of some function is closely related to thecomplex conjugate of that spectrum for negative frequencies,we know for the doubly transformed F ( ω , k ) that F ( ω , k ) = F ( − ω , − k ) . Notably, in this case there is no obvious dis-tinction between a positive frequency disturbance evolving inone direction and a negative frequency disturbance evolvingin the opposite direction. We therefore need not be troubledby the presence of negative frequencies, since we can reinter-pret them as oppositely directed positive ones – we might, forexample, refer to the idea that a positron is only an electrontravelling backwards in time .It is also possible to consider a “limited omniscience” po-sition, where we claim complete knowledge of the wave be-haviour in some defined region of space and time. This mightbe from some (past) experimental results, or be a numericalor analytic solution valid over that limited extent. In such acase, both time and space can be Fourier transformed, andthe remarks above about handling and/or interpreting nega-tive frequencies still hold. However, whilst such after-the-fact Of course the handling of most particles is complicated by their mass,whereas photons, being massless, can be their own antiparticle. a r X i v : . [ phy s i c s . op ti c s ] S e p t t t transmittedinput reflection z t ev o l u t i on propagation F I N I S H I N G S T A T ES T AR T I N G S T A T E FIG. 1: Temporal propagation of waves, where disturbances (orpulses) evolve either forward or backward in space. At any pointin the propagation, we know the spatial behaviour of our wave fieldat all points in space, as indicated by the pale green vertical lines.The pale green shaded triangles indicate the past light cones (i.e.the causal past) of a wave element (black circles) at selected timesalong the path of the disturbance. In a temporally propagated numer-ical simulation which has a maximum wave speed, the causal pastmatches the computational past. A notional interface has been addedto the diagram to show how a reflection would behave. analysis is typically very useful, caution should be taken whenapplying the conclusions to an on-going propagation: in suchinescapably dynamical situations, the alternatives discussed inthe next three sections are more applicable.
II. TRADITIONAL: TEMPORAL PROPAGATION
The traditional position is taken when we claim all knowl-edge about how the wave has moved through its environmentup until now. This picture is shown on fig. 1, and assumes thenormal preferred direction of propagation along the time axis,which is usually from t = − ∞ , through t =
0, and towards t = + ∞ .Here we might express our knowledge of the wave-field F by writing it at the current time t with full space arguments:i.e. F t ( r ) ; and where is is implied that we also knew F t ( r ) atall past times t < t also. The starting state of our propagationmatches the early-time boundary conditions – i.e. what wouldnormally be considered to be the initial conditions. An elec-tromagnetic FDTD simulation [7, 8], along with many otherfinite element approaches, uses this traditional approach. Inthis sense the starting state for some temporally propagatedsimulation – the “computational initial conditions” – are thesame as the traditional physical initial conditions . However, if we were to decide to propagate backwards in time from t = + ∞ to t = − ∞ , as is sometimes done when we have preferred final-time bound-ary conditions, note that those become the computational initial conditions, Given some wave field state F t ( r ) known over all space r atsome specified time t = t , we are at liberty to Fourier trans-form in space, but not time. The spatial transform will giveus the wavevector or momentum-like properties of the wavefield, in a spectrum F t ( k ) . This will contain both forward andbackward wavevector components along each spatial coordi-nate axis, and we can calculate it not only at the current time t , but also all past times t < t .For simplicity, imagine we are following some trajectorythrough time and space, while counting interesting events thatoccur locally. Time is always increasing but we might travelany direction through space, although it is easier to think ofthe case where our position is fixed. Naturally, our count ofinteresting events will only ever increase, thus any estimate ofthe time period T between events would be a positive number;as would any frequency estimation f in events-per-second.However, the events themselves may have different spatialcharacteristics: notably, we may see objects that pass us trav-elling left, or perhaps travelling right. The differing charac-ter of events, accessible to us through our spatial knowledge F t ( r ) , allows us to impose a sign (or signs) when adjusting ourcounting total – perhaps +1 for left-going objects, and -1 forright-going ones.Having discussed how a counting/timing argument thatgives us a strictly positive frequency estimate can be con-verted into a signed count using spatial information, let usrevist the temporally propagated spatial spectrum F t ( k ) ofour notional wave field. As already noted, the spatial spec-trum of the wave has both positive and negative wavevectorcomponents. Starting with the well known velocity relationfor waves v = ω / k , we can use this to claim that the positivewavevector part of the spectrum represents forward evolvingwaves (with v > v < F t ( z ) , F t ( − k ) = F ∗ t ( k ) . It is the changing complex phase(s)of F t ( k ) that represents the shift in F t ( z ) profile from one po-sition to another.Note that I have so far only discussed frequency estimates .Strictly speaking, in this picture, any system state exists onlyat an instant in time, so of itself that state has no frequencycontent at all. Further, even given our knowledge of paststates, we cannot calculate a true frequency dependence, be-cause we do not have the entire wave history to hand: we havethe past but not the future. The best we might do in getting anup-to-date idea of the spectrum is apply a Laplace transform over the known past information, converting t into a Laplace-spectral s , and get a hybrid spectrum F (cid:48) t ( s , k ) . Nevertheless,it may be possible to characterise the dominant frequency-like properties of the propagation as a function of the know and that the computational past of any point in the propagation will be its future light cone. The interested reader might try drawing a suitable coun-terpart to fig. 1 themselves. Although any one-sided transform that seemed appropriate, if applied overpast (known) data, would also be fine. Ω ( k ) [4] as a basis on which to simplify the propagation – perhapsby assuming it to be unidirectional.However, if we judge that omitting the future behaviour willhave a negligible effect, or that it is irrelevant since we onlycare about an experiment or computation that has already fin-ished, we can simply take data from the appropriate past timeinterval, and calculate the frequency spectrum of that. Thisafter-the-fact analysis of past data returns us to a version ofthe omniscient position, albeit one limited in scope, so thatthe same view of negative frequencies as there can be applied.This is what is usually done (either implicitly or explicitly),which is of course the reason why so many frequency re-sponses & spectra appear in the literature, textbooks, and ondatasheets.So in this traditional picture, we need not be overly trou-bled by negative frequencies. Either we are being very rig-orous, and so deny that a true frequency spectrum exists atall, or we have a spectrum calculated from known histori-cal data, which has negative frequencies reinterpretable aspositive ones, just as in the omniscient picture. As a finalnote, and as a result of the considerations above, the spectro-scopists habit of giving spatial (wavevector, or momentum-like) spectra rather than temporal (frequency, or energy-like)ones makes sense from a theoretical perspective, as well asfrom a practical experimental one. III. DIRECTED: SPATIAL PROPAGATION
The directed position is taken when we claim all knowledgeabout how the wave the wave has moved in a preferred direc-tion along some chosen path through its environment. Thispicture is shown on fig. 2, where the path is along the z axisfrom z = − ∞ , through z =
0, and towards z = + ∞ . Here wemight express our knowledge of the wave-field F by writingit at the current position (e.g. z = z ) with full time and trans-verse space arguments: i.e. F z ( t , x , y ) ; and where it is im-plied that we also knew F z ( t , x , y ) at all prior positions z < z along that path also. This often seems a very natural thing todo, especially when considering unidirectional beam propa-gation in waveguides or optical fibres. In particular, this posi-tion is made most explicit in the PSSD (pseudospectral spatialdomain) method for propagating electromagnetic pulses, butalso in many others [5, 9–12]. Here, the starting state at z = z i for some spatially propagated simulation – the “computationalinitial conditions” – could be written F z i ( t , x , y ) , and are em-phatically not the same as the traditional physical initial con-ditions at a time t = t i (written as e.g. F t i ( x , y , z ) ). Contrast, forexample, the starting states on figs. 1 and 2.Given some wave field state F z ( t , x , y ) known at some spec-ified path position z , we are at liberty to Fourier transformin x and y space, and over time t , but not along the propgationaxis z . The result will be a mixed spectrum F z ( ω , k x , k y ) as cal-culable for the current position z , as well as prior locations z < z . The spatial transform of this will give us the trans-verse wavevector spectrum of the wave field, which will con-tain both positive and negative wavevector components along z t z z z reverse−reflection transmittedinput p r op a g a t i on evolution FINISHING STATESTARTING STATE
FIG. 2: Spatial propagation of waves, where disturbances (or pulses)evolve either forward or backward in time. At any point in the prop-agation, we know the full time behaviour of our wave field – bothhistory and future, as indicated by the pale pink horizontal lines..The light pink shaded triangles indicates the computational past of awave element at particular points (black circles) along the path of thedisturbance. Note that unlike the temporally propagated case shownin fig. 1, the computational past of a spatially propagated system isnot the same as the causal past. A notional interface has been addedto the diagram to show how reflections behave – i.e. in an unexpectedway [4]. This is because a reflection should have been put in the ini-tial conditions, but was not due to an (assumed) lack of knowledgeof that future behaviour. each transverse coordinate axes. The temporal transform, us-ing the known full past-and-future history of that transversebehaviour, will contain both positive and negative frequencycomponents. This outcome is particularly convenient because(assumed) knowledge of the full time behaviour enables a fre-quency spectrum to be calculated, and arbitrary temporal ma-terial response (dispersion) to be implemented in a numeri-cally efficient way using (see commentary in e.g. [4, 5, 13]and references therein).For simplicity, and in analogy to the comparable discus-sion for the traditional picture, imagine we are following sometrajectory through space and time, while counting interestingevents that occur locally along that path. In this picture, thespatial coordinate along our propagation axis (e.g. z ) is alwaysincreasing but our trajectory can move forward or back alongthe other spatial axes ( x , y ), and even forward of back in time;although it is easier to think of the case where our x , y positionis fixed, as is time t . Naturally, our count of interesting eventswill only ever increase, but rather than measuring second be-tween events (or events per second), it would be in metersbetween events (or events per meter) – not a temporal period(or frequency), but a spatial interval λ or spatial recurrencerate (wavevector) k z = π / λ . Consquently, any λ (or k z ) esti-mation we might make would be a positive number. However,the events themselves may have different characteristics: no-tably, we may see objects that pass us travelling along a trans-verse spatial axis in either direction. Further, since F z ( t , x , y ) F z ( t , x , y ) , allows us to impose a sign(or signs) when adjusting our counting total – perhaps +1 forfuture-going objects, and -1 for past-going ones.Having discussed how a counting/distance argument thatgives us a strictly positive wavevector estimate can be con-verted into a signed count using temporal information, let usrevist the spatially propagated frequency spectrum F z ( ω ) ofour notional wave field. As already noted, the frequency spec-trum of the wave has both positive and negative frequencycomponents. Again using the well known velocity relation forwaves v = ω / k , we can now claim that the positive frequencypart of the spectrum represents forward (in time) evolvingwaves (with v > v < F ( t ) , F ( − ω ) = F ∗ ( ω ) . It is the changing complex phase of F ( ω ) that represents the shift in F ( t ) profile from one time toanother.Note that I have so far only discussed estimates of the prop-agation axis wavevector k z . In this picture, any system stateexists only at a specific location (e.g. z = z ) along its path,and so of itself that state provides no information about a k z at all. Further, even given our knowledge of previous stateson the path, we cannot calculate a true k z , because we onlyknow about where we have been ( z ≤ z ), not where we areyet to go ( z > z ). The best we might do is apply a Laplacetransform over the data from behind us along our path, con-verting z into a Laplace-spectral q z , and get hybrid spectrum F (cid:48) z ( ω , k x , k y , q z ) . Nevertheless, it is usually extremely advan-tageous to characterise the dominant k z -like properties of thepropagation as a function of the known frequency and so beable to use a reference k z ( ω ) [4, 5] as a basis on which tosimplify the propagation – perhaps by assuming it to be uni-directional.So in the directed picture, we can directly obtain a truefrequency spectrum at any point in our propagation, and thatspectrum will have negative components. These negative fre-quencies have a precise and well-defined meaning, but thatmeaning results from the convenient (but physically approxi-mate) decision to treat propagation as if it were along a spatialpath, rather than forward in time. IV. CAUSAL: THE PAST LIGHT CONE
The causal position taken when we claim all knowledge al-lowed by causal signalling about how the wave has movedthrough its environment. Here we might express our knowl-edge of some wave-field F by writing it F ( τ , B ) ; where τ and B are the proper time interval into the past and B the vector“rapidity” needed for a signal to travel from the past to ourcurrent location. These τ and B are constructed in a simi-lar way to Rindler coordinates, and quite naturally respect thelight cone. This has been discussed in more detail elsewhere[14].Since here our frequency-like quantity relates to the proper transmitted r e f l ec t e d input τ B i on ev o l u t propagation F I N I S H I N G S T A T E I S T AR T I N G S T A T E FIG. 3: Causal propagation, where our knowledge only advanceswith the edge of a single point’s (observer’s) lightcone; here I showthe point travelling at slightly less than the maximum speed (i.e. ofthat of light) for clarity. At any point in the propagation, we knowonly the behaviour in our past lightcone; we show three such pastlightcones, one being inscribed with curves showing how the rapidity B varies at a selection of fixed proper time τ intervals. The initialconditions are a single point, and the final state the lightcone border.A notional interface has been added to the diagram to show howreflections behave. time τ and not t , and because our wavevector-like quantityrelates to B and not r , I leave any more systematic analysis tolater work. However, note that τ is like time t in the sense thatit is one sided – we only know the past; and that B is like r V. SUMMARY
The above discussion shows that negative frequencies canhave a well grounded and physical basis – as long as we eitherare sufficiently omniscient and have a complete knowledge ofthe behaviour we consider relevant, or if we start from thepremise that choosing spatial propagation is reasonable. Insuch cases we cannot object to the appearance of negative fre-quency components, but it is worth noting that that we are notomniscient, and that spatial propagation – however useful –is (in practice) an approximation of reality. This means that the concept of negative frequencies must be treated with somecaution in any kind of dynamical situation. Acknowledgments
I would like to acknowledge the role of the recent “TheNonlinear Meeting 2014” in Edinburgh (May 2014), fundedby the Max Planck Society, in helping crystallize some of mythoughts on this topic, and motivating me to write them up. Ialso acknowledge financial support from EPSRC [16]. [1] E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno,D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt,F. König, and D. Faccio,Phys. Rev. Lett. , 253901 (2012),doi:10.1103/PhysRevLett.108.253901.[2] M. Conforti, N. Westerberg, F. Baronio, S. Trillo, D. Faccio,Phys. Rev. A , 013829 (2013),doi:10.1103/PhysRevA.88.013829.[3] G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, U. Leon-hardt,New J. Phys. , 053015 (2008),doi:10.1088/1367-2630/10/5/053015.[4] P. Kinsler,(2012),arXiv:1202.0714.[5] P. Kinsler,Phys. Rev. A , 013819 (2010),doi:10.1103/PhysRevA.81.013819,arXiv:0810.5689.[6] P. Kinsler,(2012),arXiv:1210.6794.[7] K. S. Yee,IEEE Trans. Antennas Propagat. , 302 (1966),doi:10.1109/TAP.1966.1138693[8] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D.Joannopoulos, and S. G. Johnson,Comput. Phys. Commun. , 687 (2010),doi:10.1016/j.cpc.2009.11.008.[9] K. J. Blow and D. Wood,IEEE J. Quantum Electronics , 2665 (1989),doi:10.1109/3.40655.[10] M. Kolesik, J. V. Moloney, and M. Mlejnek,Phys. Rev. Lett. , 283902 (2002),doi:10.1103/PhysRevLett.89.283902.[11] G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley,Opt. Express , 5382 (2007),doi:10.1364/OE.15.005382.[12] P. Kinsler, S. B. P. Radnor, and G. H. C. New,Phys. Rev. A , 063807 (2005),doi:10.1103/PhysRevE.75.066603,arXiv:physics/0611215.[13] J. C. A. Tyrrell, P. Kinsler, G. H. C. New,J. Mod. Opt. , 973 (2005),doi:10.1080/09500340512331334086.[14] P. Kinsler,Eur. J. Phys. , 1687 (2011), doi:10.1088/0143-0807/32/6/022,arXiv:1106.1792.[15] P. Kinsler, G. H. C. New, and J. C. A. Tyrrell,(2006),arXiv:physics/0611213.[16] M. W. McCall, P. Kinsler,grant number EP/K003305/1.[17] J. R. Gulley and W. M. Dennis,Phys. Rev. A , 033818 (2010),doi:10.1103/PhysRevA.81.033818.[18] P. Kinsler and G. H. C. New,Phys. Rev. A , 033804 (2005),doi:10.1103/PhysRevA.72.033804,arXiv:physics/0606111.[19] R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus,J. Opt. Soc. Am. B , 1159 (1989),doi:10.1364/JOSAB.6.001159.[20] P. Kinsler,Phys. Rev. A , 055804 (2010),doi:10.1103/PhysRevA.81.013819,arXiv:1008.2088. Appendix: Time response models
The time response of a propagation medium is handledrather differently in the temporal and spatial propagation ap-proaches. Consider a set of material properties n i that followsome kind of dynamical (temporal) response models definedby a differential equation. The differential equations for each n i will depends on both the local material states and that of thelocal field: ∂ kt n i = r i ( { n j } , F ) . (5.1)Typical examples might be nonlinear response delays [9], afree carrier density model [17], a Raman response [18, 19],or even just a Drude or Lorentz oscillator [14] giving rise todispersion. The requirement here for the model to be properlycausal is that the order of the time derivative k is greater thanany other time derivative parts present in the response function(or operator) r i [20]. (cid:73) In the traditional time propagated picture, each spatial lo-cation needs to not only know its field state F ( r ) , but also itsmaterial property state(s) n i ( r ) F or n i from earlier times – i.e. values from the computationalpast of the simulation – may need to be stored for use in sub-sequent computations. Further, as the propagation proceedsstep-by-step in time, both the field F ( r ) and current materialproperties n i ( r ) need to be updated. (cid:73) In the (directed) space propagated picture, each currentlyheld state of the field holds the time history of each point.This means that it is never necessary to store the computa-tional past of the simulation, since the values for earlier timesare already incorporated into the current computational state F ( t , x , y ) . We can simply solve equations such as eqn. (5.1)by directly integrating them from the distant past up to the de-sired time, using only that current state information. And, asalready mentioned, if the response model is linear and has aknown frequency response, as in the case of typical dispersivebehaviours, it can be applied quickly and efficiently as a sim-ple pahse shift applied to the frequency spectrum – a processrequiring only two Fourier transforms and a multiplication. Appendix: Nonlinear optics and negative frequencies
There are some specific issues relating to nonlinear optics(NLO) and negative frequencies which bear additional exam-ination. Consider the nonlinear Schrödinger equation (NLSE)as derived from Maxwell’s equations in the 1+1D regime of ( t , z ) , with some sources of dispersion and a perturbative thirdorder (Kerr) nonlinearity. Generally in NLO it is written inthe convenient (i.e. directed) spatially propagated picture, in aunidirectional approximation with fields evolving forward intime only, as the zNLSE [5] ∂ z E + ( t ) = ıK E + ( t ) + ∑ j β j ∂ jt E + ( t ) + ı χ z (cid:12)(cid:12) E + ( t ) (cid:12)(cid:12) E + ( t ) , (5.2)but note that it is also possible to derive a temporally propa-gated version, for the case of a unidirectional field evolvingforward in space only, as the tNLSE ∂ t D + ( z ) = ı Ω D + ( z ) + ∑ j γ j ∂ jz D + ( z ) + ı χ t | D + ( z ) | D + ( z ) . (5.3)Note that in the “directed” spatial zNLSE eqn. (5.2) allsources of dispersion – whether due to time-dependent ma-terial response or the geometric properties of a confiningwaveguide – are treated as if they were solely due to a time-dependent response (by means of the power series in timederivatives). In contrast, in the “traditional” temporal zNLSEeqn. (5.3), all sources of dispersion in the tNLSE are treated asif they were solely due to the local geometric properties of thatmaterial (by means of the power series in spatial derivatives).However, given that most dispersions are rather weak in ap-plications NLSE propagation, either picture can plausibly be Author’s derivation, unpublished assumed to be able to treat dispersion accurately enough forsuch purposes.In both these cases we can see that it would be useful toFourier transform the field to get temporal and spatial spectra˜ E + ( ω ) and ¯ D + ( k ) respectively, because then the dispersionmodel can be directly applied as a simple polynomial phaseshift, i.e. either of β ( ω ) = ∑ j β j . ( − ı ω ) j (5.4) γ ( k ) = ∑ j γ j . ( ık ) j (5.5)It often stated that a Kerr nonlinearity causes third harmonicgeneration, but we now also need to consider what this gener-ation is a “third harmonic” of. In the usual zNLSE model ofeqn. (5.2), an existing harmonic field E + ( t ) (cid:39) E + e − ı ω t + c.c.leads to not only some self phase modulation (SPM) butalso third harmonic frequency generation at ω = ω . Inthe tNLSE model of eqn. (5.3), an existing harmonic field D + ( z ) (cid:39) D + e ık z + c.c. leads to not only some self phase mod-ulation (SPM) but also third harmonic wavevector generationat k = k .For clarity, let us write this out as carefully as possible,in the case where we spilt the real-valued fields E + or D + into complex conjugate halves, in the style of [15]. With E + ( t ) = E (cid:48) ( t ) e − ı ω t + E (cid:48)∗ ( t ) e + ı ω t , the zNLSE equation canbe partitioned into two complex conjugate halves. We can alsodispense with the exponential oscillation by setting ω = ∂ z E (cid:48) ( t ) e − ı ω t = ıK E (cid:48) ( t ) e − ı ω t + ∑ j β j ∂ jt E (cid:48) ( t ) e − ı ω t + ı χ z (cid:2) E (cid:48) ( t ) e − ı ω t + E (cid:48) ( t ) (cid:2) E (cid:48) ( t ) (cid:3) ∗ e − ı ω t (cid:3) , (5.6) ∂ z E (cid:48)∗ ( t ) e + ı ω t = ıK E (cid:48)∗ ( t ) e + ı ω t + ∑ j β j ∂ jt E (cid:48)∗ ( t ) e + ı ω t + ı χ z (cid:2) E (cid:48)∗ ( t ) e + ı ω t + E (cid:48)∗ ( t ) (cid:2) E (cid:48)∗ ( t ) (cid:3) ∗ e + ı ω t (cid:3) . (5.7)The sum of these two equations is just eqn. (5.2). Further,since they are exact complex conjugates of one another, wecan solve just one version, which automatically gives us a so-lution to the other, and hence the solution to the real valuedfield E + ( t ) .Note in particular that the nonlinearity in this exact mathe-matical re-expression of the zNLSE equation drives only res-onant or third-harmonic frequencies. There is no explicit cou-pling to negative frequencies, which can be seen most clearlywhen assuming finite ω and a constant E (cid:48) ; then, eqn. (5.6)does not drive negative frequencies of itself. Nevertheless,although the complex E (cid:48) ( ω ) ∂ z E (cid:48) ( t ) e − ı ω t = ıK E (cid:48) ( t ) e − ı ω t + ∑ j β j ∂ jt E (cid:48) ( t ) e − ı ω t + ı χ z (cid:20) E (cid:48) ( t ) e − ı ω t + E (cid:48) ( t ) (cid:2) E (cid:48) ( t ) (cid:3) ∗ e − ı ω t + (cid:2) E (cid:48) ( t ) (cid:3) ∗ E (cid:48) ( t ) e + ı ω t (cid:21) , (5.8)and there is of course also a matching complex congugatecounterpart of eqn. (5.7), and the sum of both will be equalto the original zNLSE equation. Whether or not this repre-sentation containing explicit driving of negative frequenciesmight be useful is another matter, but it is certainly possi-ble to (re)express the mathematical model in order to con- struct them. But, apart from specific numerical difficulties thatmight occur when solving these propagation equations, the so-lutions gained from either form should be identical: both arejust different ways of representing the same physical model .Naturally one can apply the same method to the tNLSE aswell, setting D + ( z ) = D (cid:48) ( z ) e ık t + D (cid:48)∗ ( z ) e − ık z , and arriving attwo complex conjugate equations ∂ z D (cid:48) ( z ) e + ık z = ıK D (cid:48) ( z ) e + ık z + ∑ j γ j ∂ jz D (cid:48) ( z ) e + ık z + ı χ t (cid:104) D (cid:48) ( z ) e + ık z + D (cid:48) ( z ) (cid:2) D (cid:48) ( z ) (cid:3) ∗ e + ık z (cid:105) , (5.9) ∂ z D (cid:48)∗ ( z ) e − ık z = ıK D (cid:48)∗ ( z ) e − ık z + ∑ j γ j ∂ jz D (cid:48)∗ ( z ) e − ık z + ı χ t (cid:104) D (cid:48)∗ ( z ) e − ık z + D (cid:48)∗ ( z ) (cid:2) D (cid:48)∗ ( z ) (cid:3) ∗ e − ık z (cid:105) . (5.10)As we should expect, these two equations sum to eqn. (5.3).Also, here there again is no explicit coupling to opposite partsof the wavevector spectra D (cid:48) ( k ))