An interferometric method for determining the losses of spatially multi-mode nonlinear waveguides based on second harmonic generation
Matteo Santandrea, Michael Stefszky, Ganaël Roeland, Christine Silberhorn
AAn interferometric method for determining thelosses of spatially multi-mode nonlinear waveguidesbased on second harmonic generation.
Matteo Santandrea, Michael Stefszky, Gana¨el Roeland, Christine Silberhorn Integrated Quantum Optics, Paderborn University, WarburgerStraße 100, 33098 Paderborn, Germany Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS, ENS-PSLResearch University, Coll`ege de France, 4 place Jussieu, F-75252Paris, France ∗ Corresponding author: [email protected]
Abstract
The characterisation of loss in optical waveguides is essential inunderstanding the performance of these devices and their limitations.Whilst interferometric-based methods generally provide the best re-sults for low-loss waveguides, they are almost exclusively used to pro-vide characterization in cases where the waveguide is spatially single-mode. Here, we introduce a Fabry-P´erot-based scheme to estimate thelosses of a nonlinear (birefringent or quasi-phase matched) waveguideat a wavelength where it is multi-mode. The method involves mea-suring the generated second harmonic power as the pump wavelengthis scanned over the phasematching region. Furthermore, it is shownthat this method allows one to infer the losses of different second har-monic spatial modes by scanning the pump field over the separatedphase matching spectra. By fitting the measured phasematching spec-tra from a titanium indiffused lithium niobate waveguide sample to themodel presented in this paper, it is shown that one can estimate thesecond harmonic losses of a single spatial-mode, at wavelengths wherethe waveguides are spatially multi-mode.
Optical waveguides have enabled the expansion of optical networks in a veryshort time. Naturally, this technology is advancing in order to include, forexample, high power, high efficiency and/or quantum applications [1, 2].For the most demanding applications, such as squeezing in fibre networks[3] or on-chip entanglement [4], the losses of the waveguide are of critical im-1 a r X i v : . [ phy s i c s . op ti c s ] O c t ortance. Therefore, the reliable characterisation of these losses is a criticalissue.A number of methods for loss characterization and variants of thesemethods exist. These methods can be categorised into a few broad schemes:cut-back methods, fluorescence/scatter imaging, resonance techniques andoptical transmission measures. These various methods perform differentlyunder given circumstances. Interferometric methods tend to have greaterprecision as the losses decrease and so are more suited for characterisationof low loss waveguides [5, 6, 7].Unfortunately, such resonance-based methods are generally unsuitablein waveguides that are spatially multi-mode for the probe field. This isdue to the fact that it is experimentally very difficult to couple light intothe waveguide such that only a single propagation-mode of the waveguideis excited. These different spatial modes have disparate dispersion proper-ties, leading to different free spectral ranges (FSRs) for the various spatialmodes. The resulting transmitted power will consist of multiple resonanceconditions with unknown magnitude and phase, generally making the prob-lem intractable. Under certain conditions the losses can still be obtainedfrom such a measurement, but this requires the fulfillment of a number ofconditions which are, in general, not satisfied [8].For this reason, devices implementing multi-colour processes, such asdifference or sum-frequency generation, typically have their losses charac-terised at the longest wavelength, where the waveguide is single-mode. Thisvalue is often used to estimate or bound the losses at shorter wavelengths.However, one cannot know a priori the exact relationship between the lossesat different wavelengths. This is problematic when the losses at the shorterwavelengths are critical, for example in frequency converters that aim to pro-duce a field close to the transparency cut-off region of a particular material[9]. Here we present a method for loss characterisation in such systems bymeasuring the phase matching spectrum of the second harmonic (SH) pro-cess as the pump (fundamental) wavelength is varied over the phasematchingspectrum. The resulting phase matching spectrum is compared to theoryin order to estimate the losses of the second harmonic field. Additionally,this method allows one to probe the second harmonic losses for the spatialmode of one’s choosing . The approach is quite general and can be appliedto both birefringent and quasi-phase matched systems. One could extendthe theory to other processes such as sum frequency generation and type IIsecond-harmonic generation processes.2 Measurement Strategy and Theory
In the standard low-finesse Fabry-Perot loss measurement the power trans-mitted through a waveguide is recorded when scanning a probe field overwavelengths where the system is single-mode [5]. Given that one knows thereflectance of the end facets to a high precision, the interference effects ob-served in the transmitted power can be used to determine the losses insidethe resonator at the same wavelength as the probe field.The general strategy employed in the method presented here is that,in addition to first determining the losses at the fundamental wavelengthusing the standard method, we also measure the interferometric fringingthat one observes in the generated second harmonic field when scanning thepump over the wavelengths where phasematching occurs. One can then fitthe obtained phasematching spectrum to a model of this system and gaininformation about the losses of the second harmonic field. A single spatial-mode for the second harmonic field is guaranteed due to the fact that thesingle-mode pump field is phasematched to only a single second harmonicspatial-mode over the wavelength region of interest. The unique dispersionproperties of different spatial modes generally ensures that this is the case.The system is modeled using an extension of the second harmonic gen-eration theory presented by Berger [10]. In this method, the internal secondharmonic fields are first described and thereafter solved simultaneously inorder to find a self-consistent cavity solution. In order to arrive at an an-alytic expression it is assumed that the pump field, at the fundamentalfrequency, is not depleted by the nonlinear process. This assumption is triv-ial to establish experimentally by correctly choosing the power in the pumpfield. It may be possible to remove this restriction by considering a numer-ically based iterative approach [11]. However, this will further complicatethe treatment and will not provide an analytic expression.The circulating fundamental field amplitude travelling in the cavity inthe forward direction E fω (0) is given by the usual Fabry-Perot resonancecondition E fω (0) = E in τ ω, − ρ ω, ρ ω,L · e − i k ω L · e − α ω L (1)where k ω [ m − ] is the wavevector of the fundamental field, α ω [ m − ] are theintensity losses for the fundamental field, ρ ω, /L is the complex reflectivityfor the input/output facet at ω and τ ω, is the complex transmission of theinput facet at ω . Energy conservation ensures that | ρ ω | + | τ ω | = 1. Notethat from these definitions one can also express the circulating fundamentalfield amplitude travelling in the backwards direction, E bω ( L ) = ρ ω,L E ω e ik ω L .With the non-pump depletion approximation, the generated second har-monic field amplitude can be calculated from [11] asd E ω dz = iγ [ E ω e − α ω z/ ] e i ∆ kz − α ω E ω , (2)3here E ω/ ω ( z ) is the fundamental/second harmonic field amplitude at po-sition z , α ω [ m − ] represents the (intensity) losses of the second harmonicfield, γ [ m/V ] is the nonlinear coupling coefficient determining the strengthof the nonlinear process, the wave vector mismatch between the fundamen-tal and second harmonic field is defined by ∆ k = 2 k ω − k ω + k QP M [ m − ],and k ω [ m − ] is the wave vector of the second harmonic field. The term k QP M = 2 π/ Λ is required only when analysing periodically poled struc-tures, with period Λ . Note that the effect of losses in the fundamentalfield in eq. (2) have been included by considering a spatially dependentfundamental amplitude in the form E ω e − α ω z/ .Integration of eq. 2 with initial conditions ( E ω ( z ) , E ω ( z )) over a crys-tal length L yields the component of the second harmonic amplitude afterpassing through the length z due to the nonlinear interaction E ω ( z ) = SH z ( E ω ( z ) , E ω ( z )) , (3)= 2 iγ e ( α ω − α ω +2 i ∆ k ) z/ − α ω − α ω + 2 i ∆ k E ω e − α ω z/ + E ω e − α ω z/ . (4) left mirror right mirror Figure 1: Sketch detailing the forward and backward propagating wavesused for the theoretical treatment of the waveguide resonator.To derive an expression for the circulating second harmonic field ampli-tude, one defines the second harmonic field amplitudes travelling in the for-ward direction at the left and right sides of the sample, E f ω (0) and E f ω ( L ),and in the backwards direction at the left and right sides of the sample, E b ω (0) and E b ω ( L ), respectively, as illustrated in Figure 1. The relation4etween these four amplitudes can be described by the following system: E f ω ( L ) = SH L (cid:16) E fω (0) , E f ω (0) (cid:17) e ik ω L , (5) E b ω ( L ) = ρ ω, E f ω , (6) E b ω (0) = SH L (cid:16) E bω ( L ) , E b ω ( L ) (cid:17) e ik ω L , (7) E f ω (0) = ρ ω,L E b ω , (8)where E bω ( L ) = ρ ω,L E fω (0)e − ik ω L e − α ω L/ . The total circulating second har-monic field at steady-state can be found by simultaneously solving theseequations, thereby ensuring self-consistency of the SH field amplitude.Solving this set of equations, propagating through the right side mirrorin order to find the second harmonic field exiting the cavity and substituting(1) we find the output second harmonic field amplitude as E out ω = τ ω,L E f ω,L = τ ω,L γ ( E inω ) τ ω, (1 − ρ ω, ρ ω,L e − ik ω L − α ω L ) × iL sinc (cid:18) (∆ k − iα ω / iα ω ) L (cid:19) e (∆ k − iα ω/ iαω ) L × − ρ ω, ρ ω,L e − i k ω L − α ω L × (cid:16) ρ ω, ρ ω,L e − ik ω L e − i k ω L e − α ω L/ − α ω L (cid:17) e − α ω L/ − ik ω L . (9)This equation is split into four terms in order to highlight the factors thatcontribute to the observed interference fringes, as noted by Berger [10]. Thefirst term represents the Fabry-Perot interference of the fundamental field;the second term is the spectrum of the second harmonic signal generated ina single pass; the third term is the Fabry-Perot interference of the secondharmonic field and the final term represents the phase mismatch betweenthe nonlinear polarization and the second harmonic field over half of a cavityround trip, or equivalently, the phase between the forward and backwardspropagating second harmonic waves.The second harmonic power exiting the system when pumped at wave-length λ is then given by squaring the field (9), I ω ( λ ) = | E out ω ( λ ) | .The profile of I ω ( λ ) depends on the complex facet reflectivities ρ = | ρ | e iφ at z = 0 and z = L , the fundamental and SH losses α ω/ ω and on the cavitylength L . Qualitatively, one can observe that these parameters affect theshape of I ω ( λ ) in different ways: the length L of the sample affects thewidth of the spectrum and the free spectral range (FSR) of the primaryfrequency component of the fringing, the contrast of the fringes dependson the magnitude of both the fundamental and second harmonic losses and5he complicated internal structure of the fringing is dependent on the facetreflectivities and the crystal length. In the following section it is shownthat it is possible to find an optimized fit to these free variables, therebyproviding an estimate of the value of α ω . The fit of the theory to the measured data is undertaken in steps in orderto constrain the range of some of the parameters to physically acceptablevalues. First, both the model I ω ( λ ) and the measured data I meas ( λ ) arenormalized to have unitary maximum intensity. Next, the loss of the funda-mental field α ω is fixed to the value measured using the standard low-finesseloss technique [5]. This measurement is performed scanning the fundamen-tal field over wavelengths slightly shifted away from phasematching so thatthe second harmonic process does not influence the measurement. Next, thelength L of the sample is retrieved from the free spectral range of the fun-damental field. In particular, by Fourier transforming I meas ( λ ), the length L is estimated from the FSR or the primary frequency components, corre-sponding to the interference of the fundamental field (1). Subsequently, thecentral phasematching wavelength λ pm is estimated from the data using aweighted average of the recorded wavelengths, where the second harmonicspectral intensity is used as weights. From λ pm , the poling period Λ thatbest centres the phasematching is chosen.After determining the center values of these parameters, the theoreticalphasematching spectrum I ω ( λ ) is then fitted to the measured data I meas ( λ ).As there are a total of 9 free parameters to be optimised (four reflectivityamplitudes | ρ | ω/ ω, /L , four reflectivity phases φ ω/ ω, /L and α ω ), the fit ofthese quantities is performed in two steps. The phases φ are first optimisedassuming α ω = α ω and | ρ | ω/ ω, /L as given by the corresponding Fresnelequations. The optimal φ are used as initial parameters in the second step ofthe fit, where the model I ω ( λ ) is fitted again to the measured data. At thisstage, the length L , the poling period Λ , the second harmonic losses α ω ,the modules and phases of the facet reflectivities ρ ω/ ω, /L are considered asfitting parameters. The length L is constrained to a 500 µ m range around L ,the poling period Λ is constrained to be within 1% of Λ , while the phasesretrieved in the first step of the optimisation are used as initial parametersfor the fitting algorithm.The fitting routine solves a nonlinear least square minimisation prob-lem using the Trust Region Reflective algorithm that minimizes the meansquared error (MSE) between the model I ω ( λ ) and the data I meas ( λ ). Dueto the complexity of the model, the initial values for the reflectivities of thefacets and α ω are initialised with random weights and the minimisation isrepeated 10 times to find the best set of parameters. To obtain physically6eaningful results, we bound the parameters of the fit during the minimi-sation. In particular, the phases of ρ are constrained between [0,2 π ] and thereflectivities are permitted to vary by a few percent from the calculated val-ues obtained by the Fresnel equation. Moreover, as some measured spectrashowed asymmetries attributable to waveguide inhomogeneities [12], onlythe central lobe of the second harmonic spectrum was used during the fit.Note that the length and the poling period are allowed to vary slightlyin this fit in order allow some flexibility, required due to phasematchingdistortions in the measured data. Furthermore, the mirror reflectivities aretreated as complex numbers in order to account for an unknown phase shifton reflection at the end facets of the sample. This phase shift can be ex-tended in order to include the unknown phase shifts present in a quasi-phasematched sample. In such samples the length of the first and final domainsare generally unknown and will impart an unknown phase shift on the twofields, which can be absorbed by the phase term in the complex reflectivities.As a final note, the model presented in (9) requires the refractive indicesof both the fundamental and second harmonic fields as the fundamentalpump field is varied - the Sellmeier equation. For the titanium-indiffusedlithium niobate waveguides investigated here these dispersion relations havebeen calculated using a finite element solver written in Python implementingthe model described in [13]. This model provides a very accurate descrip-tion of the dispersion, with a predicted poling period within 0.25 µ m of thenominal one (1% error). In contrast, the bulk model for lithium niobatecrystals [14] predicts poling periods 1.3 µ m away from the nominal ones (8%error). We apply the described measurement technique in order to retrieve thelosses of a 31.2mm long 7 µ m-wide titanium indiffused waveguide quasi-phasematched (with a 16.8 µ m poling period) for type 0 second harmonic gen-eration in the TM00 spatial mode when pumped with a fundamental fieldat 1525nm. This system also supports second harmonic generation in theTM01 mode at around 1480nm. The losses α ω of each of these phase-matching processes at the fundamental wavelength are first found slightlyoff phasematching. At around 1525nm the fundamental field losses werefound to be 0.21 ± ± unable laser Sample Si-PDLock in amplifierPC ChopperF.I. Pol.
Figure 2: Setup for the measurement of the second harmonic. The lightfrom an IR laser tunable in the range 1460nm-1640nm (EXFO TUNICS)passes through a chopper used in conjuction with a lock-in amplifier toenhance the second harmonic readout. The IR field then passes through aFaraday isolator (F.I.) that suppresses any backreflection from the sample.A polariser is used in front of the sample to set the input polarisation of thefundamental field. Anti-reflection (AR) coated 8mm focal length asphericlenses are used for the in- and out-coupling. Finally, the second harmoniclight is measured via a silicon photodiode connected to a lock-in amplifier.3a. An excellent qualitative fit between the measured profile and the theoryis observed. The frequency of the fringing and the envelope of this centralregion overlap well. In particular, the insets show zoomed-in regions of thefits that highlight the fact that even the highly complex structure of theinterferences is reproduced by the theory. It can be seen, however, that thepresence of waveguide imperfections affects the fit of the “side lobes” of theprofile. The minimisation routine results in losses of 1.2 ± ± α ω . In particular, the MSE must exhibit a globalminimum for the fit to converge to a reasonable value for the SH losses, asis the case for the waveguide analysed in Figures 3 and 4.8 a) TM00(b) TM01 Figure 3: Measured (blue line) and theoretical fit (orange dotted) for theTM00 and TM01 second harmonic mode phasematching spectra.9igure 4: Mean squared error of the sum squared residuals between themodel and the data for the TM00 and TM01 second harmonic modes.However, a global minimum for the MSE was not always observed. Thiswas seen when investigating a 10mm long waveguide from a second 7 µ m-widetitanium indiffused waveguide. The process under investigation was againa quasi-phase matched, type 0 second harmonic generation in the TM00spatial mode with Λ =16.8 µ m. This waveguide was found to have losses α ω =0.12 ± − dB/cm) second harmonic losses.However, as displayed in Figure 5, a visual inspection of the fit for differ-ent values of α ω reveals that, qualitatively, the measured spectrum can bereproduced very well with losses up to α ω (cid:46) α ω can still be used to determine an upper bound on thelosses by choosing a threshold value in relation to the asymptote. For ex-ample, setting a 1% threshold for the variation of the MSE with respect toits minimum value provides an upper bound of α ω ≤ WG1 TM00 WG1 TM01 WG2 TM00
Fund. Loss 0.21 ± ± ± ± ± ≤ . In this paper we have introduced a new method for characterizing the loss ofspatially multi-mode waveguides. A model is introduced that describes theexpected phasematching spectrum of the generated second harmonic power,including interferences due to the Fabry-Perot effect from the uncoated endfacets. Experimental data is obtained by scanning the wavelength of thefundamental pump field over the phasematching spectrum corresponding toa chosen, single spatial-mode of the second harmonic field. In this way itis possible to determine the losses of a chosen spatial-mode of the secondharmonic. The presented technique is then applied to two waveguides. Inone case a reasonable estimate of the losses is found, and in the other anupper bound on the second harmonic losses is obtained. The presentedapproach is very general and can be extended to other nonlinear processesin virtually any high quality waveguide system.
Funding
The work was supported by the European Union via the EU quantum flag-ship project UNIQORN (Grant No. 820474) and by the DFG (DeutscheForschungsgemeinschaft).
References [1] M. Stefszky, R. Ricken, C. Eigner, V. Quiring, H. Herrmann, and C. Sil-berhorn. Waveguide cavity resonator as a source of optical squeezing.
Phys. Rev. Applied , 7:044026, 2017.11
526 15280.00.20.40.60.81.0 I n t e n s i t y [ a . u . ] SH : 0.001 dB/cm SH : 0.1 dB/cm SH : 10 dB/cm FF [nm]0.00.20.40.60.81.0 I n t e n s i t y [ a . u . ] FF [nm]0.00.20.40.60.81.0 1527.0 1527.2 1527.4 FF [nm]0.00.20.40.60.81.0 Figure 5: Central portion of the measured (blue) and fitted (orange) phase-matching profiles. It can be qualitatively seen that the fit works very wellfor low losses but that both the structure and envelope of the fit for highersecond harmonic losses is degraded. The bottom row shows a zoom-in onthe region around 1527.2nm, highlighting the ability of the model to fit thefine structure of the measured spectrum.12 SH [dB/cm]10 M S E Figure 6: Mean squared error between the measured and fitted central por-tion of the phasematching profiles as the second harmonic losses are in-creased. It can be clearly seen that the mean squared error increases rapidlywith losses greater than around 0.1 dB/cm. Of note is that the mean squarederror does not increase with vanishingly small second harmonic losses.[2] M. Stefszky, V. Ulvila, C. Silberhorn, and M. Vainio. Towards opticalfrequency comb generation in continuous-wave pumped titanium indif-fused lithium niobate waveguide resonators.
ArXiv e-prints , December2017.[3] F. Kaiser, B. Fedrici, A. Zavatta, V. D’Auria, and S. Tanzilli. A fullyguided-wave squeezing experiment for fiber quantum networks.
Optica ,3(4):362–365, Apr 2016.[4] David Barral, Nadia Belabas, Lorenzo M. Procopio, Virginia D’Auria,S´ebastien Tanzilli, Kamel Bencheikh, and Juan Ariel Levenson.Continuous-variable entanglement of two bright coherent states thatnever interacted.
Phys. Rev. A , 96:053822, Nov 2017.[5] R. Regener and W. Sohler. Loss in low-finesse Ti:LiNbO optical waveg-uide resonators. Appl. Phys. B , 36:143, 1985.[6] K. H. Park, M. W. Kim, Y. T. Byun, D. Woo, S. H. Kim, S. S. Choi,Y. Chung, W. R. Cho, S. H. Park, and U. Kim. Nondestructive prop-agation loss and facet reflectance measurments of GaAs/AlGaAs strip-loaded waveguides.
J. Appl. Phys. , 78:6318, 1995.137] D.F. Clark and M. S. Iqbal. Simple extension to the fabry-perot tech-nique for accurate measurement of losses in semiconductor waveguides.
Opt. Lett. , 15:1291, 1990.[8] Alfredo De Rossi, Valentin Ortiz, Michel Calligaro, Loc Lanco, SaraDucci, Vincent Berger, and Isabelle Sagnes. Measuring propagation lossin a multimode semiconductor waveguide.
Journal of Applied Physics ,97(7):073105, 2005.[9] H. R¨utz, K-H. Luo, H. Suche, and C. Silberhorn. Towards a quantuminterface between telecommunication and UV wavelengths: Design andclassical performance.
Appl. Phys. B , 122:13, 2016.[10] V. Berger. Second-harmonic generation in monolithic cavities.
J. Opt.Soc. Am. B , 14(6):1351–1360, Jun 1997.[11] M. Fujimura, T. Suhara, and H. Nishihara. Theoretical analysis of res-onant waveguide optical second harmonic generation devices.
Journalof Lightwave Technology , 14(8):1899–1906, Aug 1996.[12] M Santandrea, M Stefszky, V Ansari, and C Silberhorn. Fabricationlimits of waveguides in nonlinear crystals and their impact on quantumoptics applications.
New Journal of Physics , 2019.[13] E. Strake, G. P. Bava, and I. Montrosset. Guided Modes of Ti:LiNbO3Channel Waveguides: A Novel Quasi-Analytical Technique in Com-parison with the Scalar Finite-Element Method.
Journal of LightwaveTechnology , 6(6):1126–1135, 1988.[14] D.H. Jundt. Temperature-dependent Sellmeier equation for the in-dex of refraction, n(e), in congruent lithium niobate.