Relaxed phase-matching constraints in zero-index waveguides
Justin R. Gagnon, Orad Reshef, Daniel H. G. Espinosa, M. Zahirul Alam, Daryl I. Vulis, Erik N. Knall, Jeremy Upham, Yang Li, Ksenia Dolgaleva, Eric Mazur, Robert W. Boyd
RRelaxed phase-matching constraints in zero-index waveguides
Justin R. Gagnon † , Orad Reshef † , Daniel H. G. Espinosa , M. Zahirul Alam , Daryl I. Vulis ,Erik N. Knall , Jeremy Upham , Yang Li , Ksenia Dolgaleva , Eric Mazur , andRobert W. Boyd Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5, Canada School of Electrical Engineering and Computer Science, University of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5,Canada John A. Paulson School of Engineering and Applied Sciences, Harvard University, 9 Oxford Street, Cambridge, Massachusetts02138, USA State Key Laboratory of Precision Measurement Technology and Instrument, Department of Precision Instrument, TsinghuaUniversity, 100084 Beijing, China Institute of Optics and Department of Physics and Astronomy, University of Rochester, 500 Wilson Blvd, Rochester, NewYork 14627, USA † J. Gagnon and O. Reshef contributed equally to this work. * Corresponding author: [email protected]
Abstract
The nonlinear optical response of materials is the foundation upon which applications such as frequency conversion, all-optical signal processing, molecular spectroscopy, and nonlinear microscopy are built [1–4]. However, the utility of allsuch parametric nonlinear optical processes is hampered by phase-matching requirements [5]. Quasi-phase-matching [6,7], birefringent phase matching [8], and higher-order-mode phase matching [9, 10] have all been developed to addressthis constraint, but the methods demonstrated to date suffer from the inconvenience of only being phase-matched for asingle, specific arrangement of beams, typically co-propagating, resulting in cumbersome experimental configurationsand large footprints for integrated devices [11]. Here, we experimentally demonstrate that these phase-matchingrequirements may be satisfied in a parametric nonlinear optical process for multiple, if not all, configurations of inputand output beams when using low-index media. Our measurement constitutes the first experimental observationof direction-independent phase matching for a medium sufficiently long for phase matching concerns to be relevant.We demonstrate four-wave mixing from spectrally distinct co- and counter-propagating pump and probe beams, thebackward-generation of a nonlinear signal, and excitation by an out-of-plane probe beam. These results explicitly showthat the unique properties of low-index media relax traditional phase-matching constraints, which can be exploited tofacilitate nonlinear interactions and miniaturize nonlinear devices, thus adding to the established exceptional propertiesof low-index materials [12].
When light is generated by a parametric nonlinear in-teraction ( e.g., harmonic generation [13]), the propaga-tion direction of the generated output light is dictatedby the properties of the input beams [5, 11]. This depen-dence is due to conservation of momentum, also known asphase-matching [5, 14]. The amount by which the phase-matching condition is not satisfied is quantified by thephase mismatch, ∆ k , the difference in the momentumof the constituent beams. Approaches such as quasi-phase-matching [6, 7], birefringent phase matching [8],and higher-order-mode phase matching [9, 10] have beendemonstrated as means to achieve phase matching. How-ever, these methods suffer from the inconvenience of only being phase-matched for one specific configuration ofthe participating beams, which is typically collinear andalong the direction of propagation [11], and only for anarrow range of wavelengths [15]. These constraints posesevere limitations on potential applications in nonlinearoptics, where flexibility and compactness are highly de-sired.There has been significant interest in using metama-terials to lift such constraints and explore the resultingnovel behavior [11, 15–20]. Metamaterials provide ulti-mate flexibility in the engineering of optical materials,enabling many unusual and interesting properties, in-cluding negative indices of refraction [21–23]. Materi-1 a r X i v : . [ phy s i c s . op ti c s ] F e b ls with a negative refractive index have been used todemonstrate the second-harmonic generation of a non-linear signal wave propagating against the pump wave,known as backward phase matching [15,24]. This uniquebehavior may be further explored when considering zero-index media [25, 26].As the magnitude of the momentum wave-vector k isproportional to the refractive index n ( k = 2 πn/λ , where λ is the free-space wavelength), it vanishes for light prop-agating in a zero-index medium. Consequently, light in azero-index mode does not contribute any momentum tophase-matching considerations, and its propagation di-rection becomes inconsequential to the phase mismatch(Figs. 1a – b). By virtue of this unique quality, many oth-erwise forbidden phenomena, such as the simultaneousgeneration of both forward and backward-propagatinglight, become possible [11].In our experiment, we explore these phenomena usingDirac-cone metamaterials that achieve an effective refrac-tive index of zero via the simultaneous zero-crossing ofthe permittivity and permeability while maintaining afinite impedance [26, 27]. These metamaterials consistof a pair of silicon-based, corrugated ridge waveguideswhose dispersion profiles have zero-crossings at 1600 nmor 1620 nm. Figures 1c,d show an image of a fabricatedwaveguide and its measured refractive index profile. Bysampling five distinct configurations of pump, signal,and idler waves, our experimental results support theexistence of direction-independent phase matching (SeeSec. S1: Idler power predictions and theoretical supportfor phase matching free of directional restriction).In the four-wave mixing (FWM) interaction under in-vestigation, a powerful pump beam interacts with a sig-nal (probe) beam, converting two pump photons of fre-quency ω p into one signal photon of frequency ω s , andone idler photon of frequency ω i = 2 ω p − ω s [28]. Asis usually studied, all the beams of a FWM processare co-propagating, and the phase mismatch is given by∆ k fw = 2 k p − k s − k i , where suffixes p , s , and i repre-sent the pump, signal, and idler, respectively. In a stan-dard silicon ridge waveguide, this phase-matching con-dition may be satisfied (∆ k ≈ i.e., counter-propagating with respect to the pump beam), becausethen ∆ k bw = 2 k p − k s + k i = ∆ k fw + 2 k i ≈ k i . Simi-larly, phase mismatch would prevent efficient FWM if thesignal wave was counter-propagating against the pumpwave.To explore the impact of a low-index response on phasematching, we consider the special case of co-propagatinginput beams when the idler wave is generated at the zero- Figure 1: Phase matching in a low-index medium.a)
In a conventional medium ( n > b) In a low-index medium ( n ≈ c) Scanning electronmicroscope image of a Dirac-cone zero-index waveguidesurrounded by photonic band gap materials (triangularlattice of holes). d) Refractive index profile of one ofthe waveguides used in the experiment, crossing zero at λ = 1600 nm. The shaded region indicates a refractiveindex below the measurement threshold of n < . k i . Indeed, simula-tions predict that the backward-propagating idler waveis strongest when the idler is located at the zero-indexwavelength (See Sec. S2: Phase-matching nonlinear scat-tering theory). Results
As a first step towards demonstrating directionally un-restricted phase matching, we show the simultaneousgeneration of forward and backward-propagating idlerwaves when considering the pump and signal beamsco-propagating in a waveguide (Fig. 2). Through thecareful simultaneous adjustment of the pump and sig-nal beams, this measurement produces idler waves forwavelengths of λ i ranging from 1570 to 1630 nm, cross-ing through the zero-index wavelength at λ = 1600 nm2Fig. 2c – d). The backward-propagating light peaks at λ i = 1606 nm, while the forward-propagating light has adip centered at 1596 nm. We also plot our theoretical pre-dictions alongside our experimental results (black curvesin Figs. 2c – d). The forward and backward-generatedspectra show almost perfect agreement with the theoryin terms of both peak wavelength and rate of drop off.The forward-generated idler wave dips in power shortlybefore the zero-index wavelength at 1596 nm. This dipis caused by dispersive propagation loss and permeabil-ity values (See Sec. S3: Generation of the theoreticalcurves and loss profile). Such effects are less prominentin the backward-generated light, where phase match-ing is shown to be the dominant factor [27]. Beyondthe strong theoretical agreement, the fact that the mostpowerful backward-generated idler wave is not locatedat the same wavelength as the least powerful forward-generated idler wave constitutes additional proof that thebackward-propagating idler wave is independently gener-ated, and does not merely consist of back-scattering ofthe forward-propagating light due to possible reflectionsat the zero-index wavelength.We next consider the phase-matching condition forother phase-matching configurations not possible in con-ventional waveguides. For counter-propagating pumpand signal beams, simulations predict that the bright-est forward-propagating idler wave will occur when thesignal wave is at the zero-index wavelength (here at λ = 1620 nm), while for the backward-propagating idlerwave it is predicted when the pump wave is at the zero-index wavelength. We perform measurements with apump beam at 1600 nm and a signal beam at 1565 nm,where both requirements are best satisfied given exper-imental limitations. The resulting spectra are shown inFig. 3a. The simultaneous generation of forward andbackward-propagating idler waves is again clearly visi-ble, here at λ i = 1630 nm.We further establish phase matching without direc-tional restriction by coupling the pump beam intothe waveguide as before and shining the signal beamonto the waveguide from out of the plane of the de-vice. A backward-propagating idler wave is observed at λ i = 1605 nm as shown in Fig. 3b. In addition to con-firming our theoretical predictions, observing the FWMprocess from a signal beam coupling from outside theplane of the device layer provides further proof that low-index waveguides significantly ease restrictions on para-metric nonlinear effects by relaxing the phase-matchingcondition. Figure 2: Collinear phase-matching measurements.a)
Spectra of the pump and signal waves when measuredafter propagating independently through a 15- µ m-longlow-index waveguide. b) When these same pump andsignal beams are simultaneously applied to the waveg-uide, an idler wave is generated in the forward directionat ω i = 2 ω p − ω s (1600 nm). The spectrum of the idlerwave closely follows that of the pump wave because ofthe narrowness of the signal-beam spectrum. Generatedidler wave spectra in the c) forward and d) backwarddirections in a FWM process with co-propagating pumpand signal beams. The red curves show the spectra ofthe idler beams for ten different values of the pump andsignal wavelengths (See Methods). For each wavelengthpair, the spectral gap between the pump and signal fre-quencies is held constant. The black curves show thepeak power of the pulses predicted by phase-matchingconstraints (See Sec. S3: Generation of the theoreticalcurves and loss profile), while the vertical dotted blacklines in (c) and (d) indicate the n = 0 wavelength. Discussion
The simultaneous generation of forward and backward-propagating idler light has been previously observedin a fishnet metamaterial with a total thickness of800 nm [11]. However, the thickness of that metama-terial was smaller than the free-space optical wavelength( λ = 1510 nm) and phase mismatch is not a concernover such small propagation lengths [29]. Our demon-stration uses similar wavelengths but a 14.8 µ m longwaveguide, corresponding to almost 10 free-space optical3igure 3: Counter-propagating and out-of-planephase-matching measurements. a)
Spectra show-ing FWM for counter-propagating pump and signalbeams with an idler wave generated in the forward (co-propagating with the pump wave, orange) and backward(co-propagating with signal wave, red) directions. Thesignal and pump beams are at 1565 nm and 1600 nm,respectively, while the idler wave appears at 1635 nm. b) Generated idler wave spectrum resulting from a sig-nal beam coupling from out of the plane of the waveguide.An idler wave is generated in the backward direction onlywhen the pump and signal beams are simultaneously ap-plied (red curve compared to the blue curve). The verti-cal dotted black lines in both figures indicate the n = 0wavelength.wavelengths and consistent with a lower bound estimateof the coherence length at 7.8 µ m (See Sec. S4: Lowerbound estimate on the coherence length). Therefore, low-index waveguides address the phase mismatch challengerather than side-stepping it, as would be the case in athin metasurface configuration. In addition, while thisearlier demonstration used intra-pulse FWM, our demon-stration uses multiple spectrally-distinct beams, enablingthe clean isolation of the generated nonlinear pulses fromthe inputs, resulting in an unambiguous demonstration.These factors support the conclusion that the process isstrongly phase-matched. While our current zero-indexplatform exhibits radiative losses, some methods havebeen proposed to reduce loss in similar zero-index plat-forms [30–33].Observation of the idler wave generated in the waveg-uide (Fig. 3b) when excited from outside the device layercould be explained in two ways: 1) The signal beam,which is incident on the waveguide from outside the de-vice and from a direction very different to that of theguided modes, can generate an idler wave because thephase-matching condition has been so relaxed by thewave vectors vanishing at at low refractive indices, as weclaim; 2) The signal beam couples into the guiding modeof the waveguide from free space, and subsequently gen-erate FWM at the low refractive indices. The present experiment cannot distinguish between these two expla-nations for the observed idler waves. However, it is clearthat the vanishing k -vector, and therefore, a near-zero re-fractive index, is the key to enabling FWM in the waveg-uide when excited from outside the device layer.In summary, we have experimentally demonstratedthat a low-index medium enables phase-matching free ofdirectional restriction for the constituent beams, whichgreatly relaxes conventional nonlinear optical constraintsand potentially enables all input and output beams totake on any desired configuration. While low-index ma-terials still require conventional phase matching throughthe careful engineering of its dispersion parameter (SeeSec. S5: Waveguide dispersion), they provide great flexi-bility in terms of propagation direction. We believe thatsuch structured low-index media have the potential tofacilitate the realization of nonlinear optical interactionsdue to the relaxation of this constraint and thus serveinnumerable roles in the field of nonlinear optics. Author contributions
JRG carried out the nonlinear measurements. OR con-ceived the basic idea for this work. JRG, OR, and DHGEdesigned the experiment. DIV and EK carried out thelinear measurements. OR and YL carried out the simula-tions. JRG, OR, JU, and ZA analyzed the experimentalresults. RWB, JU, EM, and KD supervised the researchand the development of the manuscript. JRG and ORwrote the first draft of the manuscript, and all authorssubsequently took part in the revision process and ap-proved the final copy of the manuscript.
Acknowledgements
Fabrication in this work was performed in part at theCenter for Nanoscale Systems (CNS), a member ofthe National Nanotechnology Coordinated InfrastructureNetwork (NNCI), which is supported by the National Sci-ence Foundation under NSF award no. 1541959. CNS ispart of Harvard University.The authors thank Kevin P. O’Brien for fruitful discus-sions. The authors gratefully acknowledge support fromthe Canada First Research Excellence Fund, the CanadaResearch Chairs Program, and the Natural Sciences andEngineering Research Council of Canada (NSERC [fund-ing reference number RGPIN/2017-06880]). R.W.B. andE.M. acknowledge support from the Defense AdvancedResearch Projects Agency (DARPA) Defense SciencesOffice (DSO) Nascent program and the US Army Re-search Office. O.R. acknowledges the support of theBanting Postdoctoral Fellowship of the Natural Sciences.4ortions of this work were presented at the 2016 Confer-ence on Lasers and Electro-Optics (CLEO) in San Jose,CA [34].
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Methods
The waveguides used in the experiment were fabricatedby writing a pattern into a negative-tone resist usingelectron-beam lithography, and subsequently transferringit to a silicon substrate using inductively-coupled plasmareactive ion etching [27]. To facilitate coupling intothe waveguides, polymer coupling pads with large cross-sectional areas were constructed on either end of thewaveguide. The waveguides consist of a row of zero-indexDirac cone metamaterial with a lattice constant of a =760 nm and a cylindrical hole of radius r = 212 nm [27].Two zero-index waveguides are used: waveguide A witha length of 14.8 µ m and a zero-index wavelength of1600 nm, and waveguide B with a length of 11.1 µ m with5 zero-index wavelength of 1625 nm. Their propagationloss has been previously determined to be wavelength-dependent, with values of up to 1 dB/ µ m [27]. Waveg-uide A is used for the co-propagating and out-of-planemeasurements (Figs. 2 and 3b). However, in a setupfeaturing counter-propagating beams, there is less poweroverlap between the pump and signal beams as a result ofpropagation losses in the waveguide. As a result, waveg-uide B is used for the counter-propagating measurements(Fig. 3a) due to its shorter length which allows for alarger power overlap.In this experiment, a pulsed laser provides the pumpbeam, and an amplified continuous-wave laser providesthe signal seed beam. The full setup can be seen in de-tail in Sec. S6: Experimental setup. The pulsed laserconsists of a Ti:Sapphire and optical parametric oscil-lator pumped by a 532 nm continuous-wave laser. Thissetup is capable of generating infrared pulses with a peakpower of 1300 W, a pulse width of 3 ps, and a repetitionrate of 76 MHz. The signal laser consists of a continuous-wave laser amplified by an erbium-doped fiber amplifiercapable of accessing wavelengths between 1535 nm and1565 nm with a peak power of 2 W. In measurementswith co-propagating beams involving a signal beam above1565 nm, a weaker erbium-doped fiber amplifier capableof generating up to 100 µ W was used. The spectra ex-iting the waveguide are measured using an optical spec-trum analyzer set to a resolution of 2 nm.In our measurement with co-propagating pump andsignal beams, we sweep the pump wavelength in incre-ments of 5 nm from 1555 nm to 1600 nm while main-taining a constant spectral separation between the pumpand signal waves (∆ f = c/λ p − c/λ s = 2 . λ = 1585 nm and the signal wave at 1565 nm. As thewaveguide will only accept light coming in at an incidentangle defined by Snell’s law and the refractive index at1565 nm is slightly positive (n ≈ upplementary Information: Relaxed phase-matching constraints inzero-index waveguides Justin R. Gagnon † , Orad Reshef † , Daniel H. G. Espinosa , M. Zahirul Alam , Daryl I. Vulis ,Erik N. Knall , Jeremy Upham , Yang Li , Ksenia Dolgaleva , Eric Mazur , andRobert W. Boyd Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5, Canada School of Electrical Engineering and Computer Science, University of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5,Canada John A. Paulson School of Engineering and Applied Sciences, Harvard University, 9 Oxford Street, Cambridge, Massachusetts02138, USA State Key Laboratory of Precision Measurement Technology and Instrument, Department of Precision Instrument, TsinghuaUniversity, 100084 Beijing, China Institute of Optics and Department of Physics and Astronomy, University of Rochester, 500 Wilson Blvd, Rochester, NewYork 14627, USA † J. Gagnon and O. Reshef contributed equally to this work. * Corresponding author: [email protected]
This document provides Supplementary Information for “Relaxed phase-matching constraints in zero-index waveg-uides.” In Section S1, we demonstrate a theoretical treatment which provides support for phase matching free ofdimensional restriction. In Section S2, we show a simulation of the generated nonlinear signal as a function of propa-gation length for forward and backward-propagating idler waves using nonlinear scattering theory. In Section S3, weshow the model used to generate the theoretical curves in Fig. 2, as well as provide an explanation for the shape of theprofile of the forward-generated idler peaks. In Section S4, we provide our calculation of the lower bound estimate onthe coherence length. In Section S5, we show the simulated dispersion profile of our zero-index waveguide. In SectionS6, we show and provide an overview of the setup used to collect our data. In Section S7, we show the data for thepower measurement that was used to demonstrate that the nonlinearity is third-order FWM.
S1 Idler power predictions and theoretical support for phase matchingfree of directional restriction
It can be useful to think of phase matching in terms of the coherence length [1] given by L coh = 2 / ∆ k. (1)This parameter indicates the length over which a nonlinear interaction remains coherent, i.e., where there is construc-tive interference of a generated idler wave. The lower the phase mismatch, the longer the coherence length.To obtain the coherence length for an arbitrary beam configuration, we can make the reasonable assumption that,as in all collinear cases, the phase relationship between the constituent beams is the principle governing factor in thegeneration of a powerful idler wave. Therefore, to judge the phase-matching properties of this beam configuration,we can calculate the coherence length L coh by generalizing the phase-matching relation for all possible orientations ofpump, signal, and idler waves. To do this, we split up the phase-matching relation into its x , y , and z components | ∆ (cid:126)k | = (cid:113) ∆ k x + ∆ k y + ∆ k z , (2)where ∆ k x , ∆ k x , and ∆ k z are defined by using the angles of the pump, signal, and idler waves as in Fig. S1. Withoutloss of generality, we may define the pump wave as being at φ, θ = 0, where φ and θ are the azimuthal and polar angles,1espectively. spherical coordinates. We then define the signal wave as being on the xy -plane, and describe its positionwith respect to the pump wave with an angle φ s . Finally, the position of the idler wave can be described using twoangles: φ i , the angle of the idler wave with respect to the pump wave on the xy -plane, and θ i , the angle of the idlerwave with respect to the pump wave on the z -axis. With this definition, the components k x , k y , and k z are given by∆ k x = 2 k p − k s cos φ s − k i cos φ i cos θ i (3)∆ k y = k s sin θ s + k i sin φ i cos θ i (4)∆ k z = k i sin φ i , (5)where the signs are chosen in accordance with Fig. S1. x yz φ i θ i ω i ω p ω s φ s Figure S1:
Representation of the three angles φ s , φ i , and θ i for arbitrary signal and idler waves relativeto the pump wave. Here, the pump wave is located on the x-axis, and the signal wave is located along the xy -plane.Note that θ i is defined coming up from the xy -plane, in contrary to the regular definition with spherical coordinates.After substituting these terms into Eq. (2), we obtain an expression which simplifies to | (cid:126) ∆ k | = (cid:113) k p + k s + k i − k p ( k s cos φ s + k i cos φ i cos θ i ) + 2 k s k i cos θ i (cos φ s cos φ i + sin φ s sin φ i ) . (6)From this expression, we can determine the phase-matching constraint for an arbitrarily oriented set of beams. Thisphase-matching condition, as a result, is a generalization of the phase-matching conditions in the main text. To getsome intuition as to the results of this equation, we can plot a small subset of all the possible angles using the dispersionprofile of our zero-index waveguide. In Fig. S2, we plot L coh for the angles φ s , φ i = 0 , π/ , π, π/ θ i = 0, π/ n = 0. In the case where all angles are equal to zero, corresponding to co-propagation of allwaves, we have phase matching everywhere. If only φ s changes, we can achieve phase matching by simply placing thesignal wave at the zero-index wavelength, and thereby eliminate the directional dependence of the displaced signalwave. This is the case for a forward-propagating idler wave with counter-propagating pump and signal beams, as seenin Fig. 3a. We can analogously do the same when there is a variation in φ i or θ i for the idler wave (corresponding to abackward-propagating idler wave with co-propagating pump and signal beams, as seen in Fig. 2). If φ s and φ i are bothaltered equally, we can achieve phase matching by placing the pump wave at the zero-index wavelength correspondingto a backward-propagating idler wave with counter-propagating pump and signal beams (as seen in Fig. 3a).There are cases where simply placing one of the components of the FWM interaction at the zero-index wavelengthto enable phase matching does not work because all three constituent beams travel in seperate directions. This isthe case for our demonstration with a signal beam incident from outside the plane of the device layer (Fig. 3b). Insuch cases, we see that L coh is large when all of the waves are clustered near the zero-index wavelength. While thephase-matching condition does vary with beam configuration, a FWM interaction for an arbitrary beam configurationis always phase-matched under some condition. This constitutes a theoretical prediction of phase matching which isfree of directional restriction in low-index waveguides. S2 Phase-matching nonlinear scattering theory
Using nonlinear scattering theory [2], we qualitatively demonstrate both forward and backward-phase-matching in azero-index medium consisting of a 2D Dirac cone photonic crystal. This method can be used to estimate the magnitude2igure S2:
Numerical predictions of the coherence length.
The coherence lengths are calculated using L coh =2 / ∆ k , for 20 separate combinations of the angles φ s , φ i , and θ i . As the angle of φ i does not matter for θ i = π (when θ i = π , the idler wave is entirely in the z -direction), we only include φ i = 0 for this case to avoid repetition. L coh isplotted as a function of the pump wavelength λ p and the signal wavelength λ s . L coh is plotted from 0 µ m to 15 µ m,and the angles are given in radians. 3f the nonlinear signal generated in an interaction as a function of propagation length in, and thereby allowing for thedirect estimate of the phase mismatch or coherence length. A benefit of this approach is that a realistic structuredmedium can be incorporated, and the effects of propagation loss or dispersion can be neglected. Therefore, theseresults can help us infer the contribution of phase matching.The platform we investigated consists of a 2D square array of air holes in a silicon bulk ( a = 583 nm, 2 r = 364 nm).The low-index waveguide probed in our measurements is a descendent of this theoretical metamaterial (Figs. S3a-c)which is designed to exhibit a Dirac cone at the center of its Brillouin zone. The modes at the Γ-point consist of adipole and a quadrupole mode (Figs. S3d-e).Figure S3: Specifications of the low-index waveguide.
The design for the low-index waveguide a) is derived froma 2D photonic crystal b) . We model the unit cell from this array with periodic boundary conditions c) , and obtain e) a band structure that has a Dirac cone at the center of the Brillouin zone consisting of dipole and quadrupole modes d) .Nonlinear scattering theory predicts that the intensity of the nonlinear idler wave generated within a medium I NL is proportional to the modulus square of overlap between an electric field originating from the detector at the idlerfrequency (cid:126)E and the nonlinear polarization induced by the source (cid:126)P NL : I NL ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:126)E detector · (cid:126)P NL dV (cid:12)(cid:12)(cid:12)(cid:12) . (7)For a four-wave-mixing interaction, the nonlinear polarization (cid:126)P NL can be calculated using the mode distribution atthe pump and signal frequencies (cid:126)P NL = χ (3) (cid:126)E ( ω p ) (cid:126)E ∗ ( ω p ) (cid:126)E ∗ ( ω s ); Lorentz reciprocity dictates that the detector electricfield (cid:126)E detector must be at the idler frequency. For a forward-propagating idler wave, the distribution of the detectorfield is (cid:126)E detector = (cid:126)E Q + i (cid:126)E D , where (cid:126)E Q and (cid:126)E D represent the field distributions of the quadrupole and dipole modes,respectively. By contrast, a backward-propagating idler wave will have a field of (cid:126)E detector = (cid:126)E Q − i (cid:126)E D (Fig. S4a).Using the method outlined above, we calculate the nonlinear signal generated in the zero-index medium as a functionof propagation length (Fig. S4b). Our calculation simulates I NL for a 2D array, which eliminates any out-of-planeradiative losses. Additionally, the calculation is performed at degenerate frequencies ( ω p = ω s = ω i = ω ). The gen-erated intensities are observed to grow quadratically in all propagation directions, indicating perfect phase-matching,consistent with a refractive index of zero. Additionally, the conversion efficiency for the backward-phase-matchedsignal is observed to be smaller due to a reduction in the total mode overlap caused by the out-of-phase dipole mode. Propagation Length [um] N on li nea r S i gna l ( a . u . ) forward forward backward backward Q + DQ - D a b
Figure S4:
Nonlinear signal generated in the zero-index medium as a function of propagation length.
Fora perfectly phase-matched process, the generated signal for a forward-propagating idler wave is shown in blue, whilethe backward-propagating wave is shown in yellow. 4
While the coherence length provides us with some strong intuition as to when an interaction is phase-matched, itis alone insufficient to predict the generated power of a nonlinear interaction. To adequately analyze experimentalresults, we require a model that takes dispersion and radiative losses into account. We derive this model by solvingthe wave equation for each component of the field [3]. The wave equation is given by ∇ E n − (cid:15) n µ n c ∂ E n ∂t = µ n (cid:15) c ∂ P NLn ∂t . (8)where E n is the electric field of the incident light, P NLn is the nonlinear component of the polarization density, (cid:15) isthe vacuum permittivity, (cid:15) n is the relative permittivity, µ n is the relative permeability, and c is the speed of light.Here, the subscript n is used to denote an individual component of the field ( i.e., pump, signal or idler). Given thescalar field approximation, we have the dispersion relation k n = n n ω n c , (9) n n = (cid:15) n µ n . (10)where k n is the wave vector, ω n is the angular frequency, and n n is the refractive index. For each component of thefield, we can substitute a trial solution for the electric field and polarization densities E n ( z, t ) = A n e i ( k n z − ω n t ) + c.c. (11) P NL n ( z, t ) = P NL n e − iω n t + c.c. , (12)where A n is the scalar amplitude of the electric field, and P NL n is the scalar amplitude of the nonlinear component ofthe polarization density. Here, c.c. represents the complex conjugate. The values P NL n for the pump, signal, and idlerrespectively (denoted by the subscripts p , s , and i ) are given by P NL p = 3 (cid:15) χ (3) A ∗ p A s A i e i ( k s z + k i z − k p z ) + c.c. (13) P NL s = 3 (cid:15) χ (3) A p A ∗ i e i (2 k p z − k i z ) + c.c. (14) P NL i = 3 (cid:15) χ (3) A p A ∗ s e i (2 k p z − k s z ) + c.c. (15)Here, χ (3) is the third-order nonlinear susceptibility. Solving the wave equation for every component of the FWMinteraction yields the 3 coupled-amplitude equations dA p dz = 3 iµχ (3) ω p k p c A ∗ p A s A i e − i ∆ kz , (16) dA s dz = 3 iµχ (3) ω s k s c A p A ∗ i e i ∆ kz , (17) dA i dz = 3 iµχ (3) ω i k p c A p A ∗ s e i ∆ kz . (18)To account for an interaction where the constituent waves are depleted by loss, we add a loss term proportional tothe amplitude in the final result to represent the fields depleting as the waves propagate through the waveguide. Fora forward-propagating idler wave, the coupled-amplitude equations, therefore, take the form: dA p dz = 3 iµ p χ (3) ω p n p c A ∗ p A s A i e − i ∆ k f z − α p A p (19) dA s dz = 3 iµ s χ (3) ω s n s c A p A ∗ i e i ∆ k f z − α s A s (20) dA i dz = 3 iµ i χ (3) ω i n i c A p A ∗ s e i ∆ k f z − α i A i . (21)Here, α n is the propagation loss, and ∆ k f ≡ k p − k s − k i . Note that we have used the slowly varying amplitudeapproximation, and ignored the second derivative of A i . One can obtain the parameter α from the propagation lossin dB/ µ mby using [4] α / m = ln 1020 α dB /µ m . (22)5o solve these equations, we may use a relaxed version of the undepleted pump approximation. This relaxedapproximation states that the depletion of the the pump and signal waves is dominated by propagation loss and notby conversion to idler waves. Using this approximation, we Eqs. (19 – 21) to the form dA p dz = − α p A p , (23) dA s dz = − α s A s , (24) dA i dz = 3 iµ i χ (3) ω i n i c A p A ∗ s e i ∆ k f z − α i A i . (25)These equations do possess an analytical solution, as opposed to the previous ones. To solve them, we can first solveEqs. (23) and (24) to obtain an expression for the pump and signal wave amplitudes as a function of z . For the pumpand signal waves, we obtain A p ( z ) = A p0 e − α p z . (26) A s ( z ) = A s0 e − α s z . (27)where A p0 and A s0 are the initial amplitudes of the pump and signal waves. After obtaining these expressions for thepump and signal waves, we can thereafter substitute them into Eq. (25) to obtain dA i dz = 3 iµ i χ (3) ω i n i c A A ∗ s0 e i ∆ k f z e − ∆ αz − α i A i (28)where we have defined ∆ α ≡ a p + a s for notational convenience. We are now left with a single linear differentialequation which can be solved. By using the initial condition A i (0) = 0, we obtain the final expression for the idlerwave amplitude as a function of zA i ( z ) = 3 iµ i χ (3) ω i n i c A A ∗ s0 (cid:18) e ( i ∆ k f − ∆ α ) z − e − α i z i ∆ k f − ∆ α + α i (cid:19) . (29)This equation can be used to calculate the power of a forward-propagating idler wave as a function of the propagatedlength z in the waveguide. We can follow an analogous procedure to obtain the expression for a backward-propagatingidler wave. In the backward-propagating case, the phase-matching condition is ∆ k b = 2 k p − k s + k i . As the idlerwave is counter-propagating against the pump and signal, we can define our pump and signal waves to begin at z = L . This effectively inverts the frame of reference of the waveguide (See Fig. S5 for an illustration clarifying this).We also appropriately invert the signs of the pump, signal, and idler wave momentum terms. In cases where therelaxed undepleted pump approximation holds, this approach is theoretically valid. Under these assumptions, thecoupled-amplitude equations take the form 6igure S5: Schematic demonstrating the swap in the frame of reference in the waveguide.
A pumpwave propagating from z = 0 to z = L is represented by a wave with initial amplitude A p that suffers loss e − α p z as it propagates through the waveguide, exiting with an amplitude A p e − α p L . In the treatment for the backward-propagating idler wave, we represent this same wave as propagating from z = L to z = 0. We also appropriately invertthe momentum k p of the pump wave. dA p dz = α p A p , (30) dA s dz = α s A s , (31) dA i dz = 3 iµ i χ (3) ω i n i c A p A ∗ s e − i ∆ k b z − α i A i . (32)Using the initial conditions A p (0) = A p0 e − α p L and A s (0) = A s0 e − α s L , we can obtain the pump and signal amplitudesas a function of z , obtaining A p ( z ) = A p0 e − α p L e α p z , (33) A s ( z ) = A s0 e − α s L e α s z . (34)By substituting these expressions for the pump and signal wave amplitudes into Eq. (32) and subsequently solving theresulting linear differential equation, we obtain the result A i ( z ) = 3 iµ i χ (3) ω i n i c A A ∗ s0 e − (2 α p + α s ) L (cid:18) e (∆ α − i ∆ k b ) z − e − α i z ∆ α − i ∆ k b + α i (cid:19) . (35)We use Eqs. (29) and (35) to analyze our experimental results. The power P of a wave (not to be confused withthe polarization density P NL ), which we measure is proportional to the square of the modulus of the field amplitude: P ∝ n | A | . In the power equation, we use the refractive index of the surrounding silicon waveguide which couples intothe zero-index waveguide where the measurement is performed.When based solely on phase-matching constraints, our discussion in Section S1 suggests that the forward-propagatingidler peaks should have equal magnitudes at every wavelength (Fig. S6d). However, our measurements results indicatethat this is clearly not the case. This discrepancy can be explained by incorporating a dispersive loss and permeabilityto the model (Fig. S6a–b). These factors are included in the model that generated the theoretical curves plotted inFig. 2 of the main text.In Fig. S6c, we isolate the effects of these two quantities on the output idler power by plotting superimposing twocurves over the measurement results: a constant permeability and therefore a constant impedance with a variable loss(red curve); and a variable permeability and a constant loss (green curve). The propagation loss values were extractedfrom previous measurements on a similar set of waveguides, and the permeability was extracted from simulations. Whenthe impedance is held constant, the forward-propagating idler is attenuated for wavelengths longer than 1600 nm. When7he loss is held constant, the forward-propagating idler is attenuated for wavelengths shorter than 1600 nm. Whenboth curves are compared, it is clear that the spectrum of the forward-propagating idler is a result of the combinationof these two factors. As a result, we can conclude that the spectrum of the forward-propagating idler peaks is causedby wavelength-variable impedance and loss, while the spectrum of the backward-propagating idler peaks is primarilythe result of the phase-matching condition.Figure S6: Loss and permeability over n values and their effects of the produced idler peaks. a) Lossprofile of a similar waveguide in dB/ µ m. The loss peaks shortly below the zero-index wavelength (dashed black line). b) Simulated µ/n , a quantity proportional to the impedance, plotted as a function of wavelength. This quantity isincluded as a factor in Eqs. 29 and 35. c) Generated idler wave spectra in the forward-propagating direction for aFWM process with co-propagating pump and signal beams, compared to two theoretical predictions. All results arenormalized to unity. For the red curve, the impedance is made wavelength-invariable, allowing us to isolate the effects ofloss. For the green curve, the loss is made wavelength-invariable, allowing us to isolate the effects of impedance. Whilethe red curve predicts an attenuation beyond the dip in the forward-propagating idler peaks, the green curve predictsan attenuation prior to the dip in the forward-propagating idler peaks. d) The same experimental data plotted againstthe theoretical prediction (black curve) when all wavelength-dependant quantities bar the phase matching conditionare held constant.In summary, the fact that the simultaneously generated forward and backward-propagating idler peaks respondso differently to passing through the zero-index wavelength is overall strong proof that the idler wave both showdirectional independence, and that each direction is subject to its own phase-matching requirement.
S4 Lower bound estimate on the coherence length
For collinear beams, the amplitude of the generated signal wave may be shown to be a function of the coherence length L coh = 2 / ∆ k and interaction length L [5]: A i ∝ L sinc ( L/L coh ) , (36)where we assume the slowly varying amplitude approximation, and neglect the propagation losses. Here, we define thelower bound on the coherence length as the shortest length for which the factor sinc ( L/L coh ) (obtained by squaringEq. 36) yields a pulse at a quarter of the power of the peak power. As the power is proportional to the square of theamplitude, we may write P i ∝ sinc ( L/L coh ) . (37)8he solution to sinc ( x ) = 0 .
25 is x = 1 . µ m waveguide, the lower bound of the coherencelength is given by L coh = 14 . µ m / .
895 = 7 . µ m. If we, instead, use a lower power to define our lower bound,the lower bound on the coherence length is longer. 7 . µ m is dramatically longer than a free-space wavelength, andproves we have phase-matching beyond what is possible in a metasurface. S5 Waveguide dispersion
The waveguides used in the experiment possess anomalous dispersion in a bandwidth of roughly 80 nm surroundingthe zero-index wavelength (Fig. (S7)). This anomalous dispersion ensures that the desired FWM nonlinear opticalprocess is phase-matched in the spectral region of interest, as elaborated upon in [5].Figure S7:
Simulated dispersion parameter D for a Dirac-cone zero-index waveguide.
D is positive, corre-sponding to anomalous dispersion over the 80 nm surrounding the zero-index wavelength.
S6 Experimental setup
A schematic of the complete setup can be seen in Fig. S8. Both beams must be collimated with roughly the same spotsize (0.5 cm in diameter) so that co-propagating beams can be coupled into the zero-index waveguide via the samelens. We use a system of lenses to collimate the beams, and we use a telescope to adjust the spot size while retainingthe collimation. To polarize the lasers, we use the combination of a half-wave plate and polarizing beam-splitter foreach laser to both achieve the desired transverse-electric polarization and provide a means with which we can modulatethe power of the pump and signal beams.Following polarization, the pump and signal beams are coupled into the zero-index waveguide. In the case of co-propagating beams, we combine the pump and signal beams using a beamsplitter cube (beam cube). The portionof the pump and signal beams that is not used to couple into the waveguide is then directed towards a detector todetermine the power of the pump and signal beams when performing measurements. Once combined, the pump andsignal beams can be coupled into the waveguide through its coupling pads.To determine the generated output, we collect the light using multi-mode optical fibers on both sides on thewaveguide. On the side of the input facet, we set up a non-polarizing beam cube to allow the input light to travelthrough while redirecting the output light to our detector. On the opposite side of the waveguide, we focus the outputlight on our multi-mode fiber. Once the light has been collected, it is spectrally decomposed by an AQ-6315E opticalspectrum analyzer (OSA) for subsequent analysis. 9igure S8:
Schematic of the setup used to couple the pump and signal beams into the zero-indexwaveguide.
The pump and amplified signal beams pass through lens systems to collimate the beams (C1 and C2),and have their spot sizes matched using telescopes (L1, L2, L3, L4). The beam is subsequently polarized using ahalf-wave plate and polarizing beam-splitter (HWP1, HWP2, P1, P2), and then spectrally filtered (F1, F2). In thecase of co-propagating pump and signal beams, the beams are combined using a beamsplitter cube (BS1) and a lens(L5) subsequently focuses the beams into the zero-index waveguide (ZI). Optical fibers (Fi1, Fi2) on either side collectthe forward and backward-propagating generated light exiting the waveguide, and send it to the OSA for furtheranalysis. For counter-propagating beams, the flip mirror (FM) is flipped into position, and the signal light is focusedinto the zero-index waveguide via the lens L6. Mirrors which redirect light are represented by M.
S7 Idler wave power measurement
To confirm that the generated idler wave is the product of a third-order FWM interaction, we plot the peak powerof the idler wave as a function of the peak power of the pump beam for both the forward and backward-propagatinglight (Fig. S9). 10igure S9:
Peak power of the idler wave P i as a function of the peak power of the pump wave P p . In a) , we plot the powers for forward-propagating light, while in b) , we plot the powers for backward-propagatinglight. The error bars represent measurement uncertainties extracted from the standard deviation of a repeated set ofmeasurements.For the forward-propagating light, we observe a slope of 2 . ± .
3, while for the backward-propagating light, weobserve a slope of 1 . ± .
3. Both of these values are close to 2, confirming the prediction of a quadratic relationshipbetween pump and idler power. Deviations from 2 likely occur due to the background noise of the OSA and fluctuationsin the pump spectrum. It can summarily be concluded that the idler wave is produced by a FWM process [3].
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Nature Materials (2015).[3] Boyd, R. W. Nonlinear Optics (Academic Press, San Diego, California, 2020), 4 edn.[4] Agrawal, G. P.
Nonlinear Fiber Optics (Academic Press, Boston, MA, 2007), 4 edn.[5] Foster, M. A. et al.
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