Observation of nonlinear bands in near-field scanning optical microscopy of a photonic-crystal waveguide
Amandev Singh, Georgios Ctistis, Simon R. Huisman, Jeroen P. Korterik, Allard P. Mosk, Jennifer L. Herek, Pepijn W. H. Pinkse
OObservation of nonlinear bands in near-field scanning optical microscopy of aphotonic-crystal waveguide
A. Singh,
1, 2
G. Ctistis, a) S.R. Huisman,
1, 2
J.P. Korterik, A.P. Mosk, J.L. Herek, and P.W.H. Pinkse b) Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, PO Box 217,7500 AE Enschede, The Netherlands Optical Sciences (OS), MESA+ Institute for Nanotechnology, University of Twente, PO Box 217,7500 AE Enschede, The Netherlands
We have measured the photonic bandstructure of GaAs photonic-crystal waveguides with high energy andmomentum resolution using near-field scanning optical microscopy. Intriguingly, we observe additional bandsthat are not predicted by eigenmode solvers, as was recently demonstrated by Huisman et al. [Phys. Rev. B86, 155154 (2012)]. We study the presence of these additional bands by performing measurements of thesebands while varying the incident light power, revealing a non-linear power dependence. Here, we demonstrateexperimentally and theoretically that the observed additional bands are caused by a waveguide-specific near-field tip effect not previously reported, which can significantly phase-modulate the detected field.
I. INTRODUCTION
For the investigation of the propagation of lightin nanosized stuctures, near-field scanning optical mi-croscopy (NSOM) is a powerful and unique technique asit allows for measurements with a high spatial, energy,and momentum resolution. Its ability to tap light outof light-confining structures such as integrated opticalwaveguides and cavities, makes NSOM an invalu-able and popular tool in nanophotonics.
One very im-portant class of such nanophotonic structures consists ofphotonic-crystal waveguides, which are two-dimensionalphotonic-crystal slabs with a line defect wherein the lightis guided. Their importance is found in their uniqueproperties such as their dispersion relation, slow-lightpropagation, strong confinement of light and thus en-hanced light-matter interaction.
With NSOM, onehas the tool to determine the dispersion relation in thesewaveguides and thus the bandstructure, spatially mapoptical pulses, observe slow-light propagation and phe-nomena such as disorder-induced formation of Andersonlocalized modes near the band edge, which other-wise would not be accessible.Here, we present our results on measuring the band-structure of a photonic-crystal waveguide. We show thatour measurements reveal additional bands which are notaccounted for in eigenmode solvers. We analyze thesenew modes by controlling the incident light intensity inpower-scaling measurements. Their nonlinear scaling be-havior is different from the linear response of the knownbands. We explain in detail the origin of these nonlin-ear bands as a consequence of mode coupling caused bythermal perturbation of the standing waves formed in afinite-size GaAs photonic-crystal waveguide. We demon-strate that these new bands are in fact not modes belong-ing to the photonic-crystal waveguide but a measurementartefact caused by the presence of the NSOM tip. a) Electronic mail: [email protected] b) Electronic mail: [email protected] T unab l e La s e r ( C W ) Δω SMF BS
SMF: Single mode fiber BS: Beam splitter: AOM’s : Si-PD Detector Δω (a)(b) (c) xy z BS D : Quarter wave plate : Polarizer: Half wave plate BS D SMF
Figure 1. (Color online) (a) Schematic of the heterodynenear-field scanning optical microscope setup. (b) Scanningelectron micrograph of the NSOM tip with an aperture of ≈
160 nm. (c) Scanning electron micrography of the GaAsphotonic-crystal waveguide membrane structure used in theexperiments. The W1 waveguide is formed by the missingrow of holes in the center of the photonic-crystal slab.
II. EXPERIMENTAL DETAILS
Figure 1(a) shows the experimental setup. It is an ex-tended version of the setup described in Ref. [ ]. There-fore, we will concentrate on the implemented improve-ments. A tunable continuous-wave laser (Toptica DLpro940) with a wavelength range between 907 and 990 nmand a linewidth of 100 kHz is used. In order to extractboth amplitude as well as phase out of the NSOM, aheterodyne interferometric detection technique is used. The signal path is equipped with a motorized combina-tion of λ/
4- and λ/ , see Fig. 1(a)). The output of D behind the beam-splitter BS , i.e. , the induced photocurrent, measuredas a voltage drop behind a fixed resistor, is measuredagainst a calibrated power meter on the signal input,resulting in a one-to-one mapping of voltage to inputpower. Using this arrangement, we have the ability to a r X i v : . [ phy s i c s . op ti c s ] O c t tune the input power very accurately over a large range, i.e. , 50 µ W to 15 mW, throughout the entire wavelengthrange of the laser. After this calibration, the signal in-put light is coupled to the cleaved end facet of a GaAsphotonic-crystal waveguide by means of a microscope ob-jective (NA = 0 .
55) and propagates there along the pos-itive ˆ x -direction. The polarization of the incident lightis kept thereby at approximately 45 ◦ with respect to thenormal of the waveguide allowing us to excite both TE-and TM-like modes simultaneously. The field pattern iscollected ≈ µ m away from the coupling facet – toavoid direct light scattering into the collection part – us-ing an Al-coated fiber tip with an aperture of ≈
160 nm(Fig. 1(b) shows a scanning electron micrograph (SEM)of such a tip). The tip is thereby kept at a fixed distanceof ≈
20 nm above the sample surface using shear-forcefeedback control. The used height is a trade off betweenpicking up enough evanescent light and disturbing thelight field by the tip’s presence. The picked-up light isheterodyned with the local oscillator and detected on aSi photodiode (D ) allowing to accurately measure am-plitude and phase with subwavelength spatial resolution.The parameters of the GaAs photonic-crystal slabwaveguide, which are also used in the calculations, areextracted from SEM images, such as the one shownin Fig. 1(c) and are: pitch size of the triangular lat-tice a = 240 ±
10 nm, normalized hole radius r/a =0 . ± .
002 nm, and slab thickness h = 160 ±
10 nm.The length of the waveguide l of approximately 1 mm isderived from optical microscope images. III. RESULTS
With our NSOM we map the amplitude of the lightfield inside the photonic-crystal waveguide with high spa-tial resolution. Figure 2(a) shows a detailed part ofsuch a scan, which extends normally 52 µ m × . µ m inthe ˆ x - and ˆ y -direction, respectively, showing a standing-wave pattern of the light inside the waveguide. Thewavelength of the light coupled into the waveguide is λ = 964 . ± .
01 nm. Analyzing such a spatially re-solved measurement further allows the extraction of allthe (Bloch-)periodic optical signals in the waveguide bymeans of a spatial Fourier transform (SFT).
Fig-ure 2(b) shows the result of the SFT performed on ourmeasurement in Fig. 2(a), revealing four bands and thelight line (LL).We can reconstruct the photonic bandstructure of oursystem very accurately by collecting more near-field im-ages across the whole frequency range of the laser source.Figure 3 displays the so reconstructed bandstructure forour photonic-crystal waveguide. Furthermore, we in-cluded the eigenmodes of the system as calculated us-ing the MIT photonic bands eigenmode solver (MPB). Comparing the results, we can match the measured bandsto those in the calculations. Strikingly, our measure-ments demonstrate additional bands that are not pre- (a)(b) ya LL P FB k x ( ) P FC P FC+1 N FD SFT Amplitude (mV)
Measured Amplitude (mV) y x Figure 2. (Color online) (a) Measured amplitude profile ofthe photonic-crystal waveguide at an incident wavelength of λ = 964 . ± .
01 nm. (b) Amplitude coefficients of the spatialFourier transform (SFT) of the measured near-field pattern.Modes marked with P B F , P CF , N DF and P C+1F correspond tobands responsible for the standing-wave pattern present inFig. 2(a). -1 -2 SFT Amplitude along x-axis (mV) D N F r equen cy ( a ) λ W a v e l eng t h ( n m ) k x ( ) a LL FE N F E+1 N FF+1 N BF N BA P B A P F C P F B P F B+1 P F C+1 P F A+1 P F A+1 P B Figure 3. (Color online) Measured bandstructure of aphotonic-crystal waveguide. The measured bands are labeledfrom A to F (in the superscript), LL denotes the light line.Furthermore, the results of MPB simulation are inserted asblack dots. Modes present in experiment and calculation arelabeled P (predicted). Bands measured but not appearing inthe calculations are shown in magenta and labeled N (not pre-dicted). The subscript denotes backward (B) or forward (F)propagation. The green dashed line denotes the frequency ofmore detailed analysis. dicted in the eigenmode solvers. To be certain that thebands calculated by the MPB eigenmode solver are pre-dicted correctly, we varied several parameters, such as thesystem size, the unit cell as well as the resolution in thecalculations. The measured new bands will subsequentlybe analyzed in more detail.For clarity, we introduce here the following nomencla-ture: Measured bands which show also up in the MPBcalculations are denoted P (predicted) while the bands in = 6.5 mWP in = 1.0 mW (a) (b) k x ( ) -17 -15 -13 -11 N FD P FC+1
LL P FC P FB F ou r i e r S i gna l I n t en s i t y ( V ) F ou r i e r S i gna l I n t en s i t y ( V ) -17 -15 -13 -11 -9 Incident Power (mW) -1.5 -1 -0.5 (Scaling behavior)(Scaling behavior) P FC N FD Figure 4. (Color online) Power scaling experiment to measurethe intensity-dependent behavior of waveguide bands. (a) To-tal intensity in the Fourier signal (as shown in Fig. 2(b))as function of the spatial frequency k x . The curves repre-sent two different incident power levels: P in = 6 . in = 1 . B F , P CF , N DF andP C+1F ) are clearly visible as peaks. (b) Fourier signal inten-sity of P CF and N DF measured as a function of incident power.The dotted lines represent linear (blue), quadratic (red), andcubic (green) fits to the data, whereby N DF is fitted only forincident powers exceeding 1 mW ( cf. text). measured but not appearing in the calculations are de-noted N (not predicted). Moreover, from the slope inthe bandstructure, as derived from Fig. 3, one can derivethe group index and energy velocity, thus knowing if itis a forward (F) or backward (B) propagation, which ismarked in subscript. The band name appears alphabeti-cally in superscript. For example, a forward propagatingpredicted mode A is denoted as P AF , a backward prop-agating structure F as N FB , and a forward propagatingpredicted mode C in the second Brillouin zone as P C+1F .Finally the light line appears in its usual abbreviation asLL.In Fig. 3 we identify three new bands, labeled D, E,and F. We can construct these new bands from predictedbands following a general rule: k (new band) x = 2 k (predicted band) x ∓ k (second predicted band) x (1)In case of D, E, and F this leads to: k x (N DF ) = 2 k x (P CF ) + k x (P CF ) = 3 k x (P CF ) k x (N EF ) = 2 k x (P AF ) − k x (P CF ) k x (N FB ) = 2 k x (P AB ) − k x (P CF ) , (2)where they are constructed from the predicted modes Aand C.Due to the complexity of the bandstructure, we restrictour further analysis without loss of generality to one spe-cific wavelength below the band edge, i.e. , λ = 964 . DF . We performed power-dependent measurementsto study the origin of these new bands. In that respectwe took NSOM images as shown in Fig. 2(a) in a series ofpower levels ranging from 0 . − . y -direction. Figure 4(a) showsthe resulting Fourier signal intensity vs. k x for two in-put powers of P in = 6 . in = 1 . CF and N DF in Fig. 4(b). Surprisingly, the two bandsshow a different scaling behavior with the incident power.For the predicted mode P CF a linear dependence of theintensity is observed as expected. The new band N DF onthe other hand shows a highly nonlinear behavior. Here,the power law appears to be cubic at powers P > IV. THEORETICAL MODEL
To understand the experimental findings and whythe eigenmode calculations cannot predict the measuredbands, we first look at the difference between calcula-tions and experiment. In contrast to the MPB calcula-tions, which assume an infinite sample length and linearmaterial response, our experiment consist of a finite pho-tonic crystal waveguide. Moreover, GaAs is known forits highly nonlinear material response. Therefore, theeigenmodes of the waveguide as calculated by MPB canonly be a starting point for the explanation of the real sit-uation, since it cannot account for any coupling betweenmodes. I in I refl Tipa L F ou r i e r S i gna l I n t en s i t y ( V ) -17 -15 -13 -11 (c) k x ( ) P BC P FC N FD (a) Position I n t en s i t y (b) Figure 5. (Color online) (a) Schematic representationof a standing-wave intensity pattern formed by counter-propagating waves in a finite photonic-crystal waveguide. Thesignal is then picked-up with the NSOM tip. (b) Schematicrepresentation of the position-dependent total intensity insidethe photonic-crystal waveguide. The intensity drop duringforward and backward propagation is caused by light scat-tering and absorption of light due to the presence of the tip.(c) Intensity of the measured Fourier signal ( cf.
Fig. 4(b)),as a function of spatial frequency k x at an incident power ofP in = 6 . CB and P CF correspond to twocounter-propagating waves and N DF is the new band appearingin this experiment. In the following we will show that this measurement-induced coupling mechanism leads to virtual mode cou-pling and is essential for the observation of new bandsin our experiment. We restrict our analysis, without lossof generality, thereby to the effects caused by the bright-est eigenmode C. The same effects will also be present inother (weaker) eigenmodes. Furthermore, we can omitthe Bloch periodicity in the analysis of the electric fieldbecause it does not affect the appearance of the observednew bands.In the finite structure, reflections from the end facets ofthe waveguide will lead to a formation of a standing-wavepattern due to forward (+ k x ) and backward ( − k x ) prop-agating modes, as is schematically depicted in Fig. 5(a).The complete electric field associated with the forwardpropagating bright eigenmode C at position r and fre-quency ω is given by E P CF ( r, ω ) = A P CF ( r, ω ) e − ik c x/a ,where A P CF ( r, ω ) is the field amplitude. The back-ward propagating mode is then given by E P CB ( r, ω ) = A P CB ( r, ω ) e ik c x/a . The total intensity I ( r, ω ) inside thewaveguide then becomes I ( r, ω ) = I BG + 2 | A P CB ( r, ω ) A P CF ( r, ω ) | cos (2 k c xa ) , (3)where I BG corresponds to the non-sinusoidal backgroundintensity. Assuming an intensity dependence for the re-fractive index and inserting Eq. 3 as expression for the in-tensity, the refractive index n ( r, ω ) = n l ( ω )+ n ( ω ) I ( r, ω )can then be written as: n ( r, ω ) = n l ( ω ) + n ( ω ) I BG + n · | A P CB ( r, ω ) A P CF ( r, ω ) | cos (2 k c xa ) , (4)where n l ( ω ) is the linear refractive index of GaAs and n ( ω ) ≈ n is the corresponding nonlinear refractive in-dex at the incident optical frequency ω and intensity I ( r, ω ).We need one more ingredient to explain the appear-ance of the new bands, namely the perturbative nature ofNSOM measurements. While scanning across the surfaceof the waveguide, the tip causes losses predominantly atthe maxima of the standing-wave pattern ( cf. Fig. 5(a)).This, however, is a loss mechanism for the total power in-side the waveguide leading to a variation of the refractiveindex in the waveguide as a function of the position of thetip. Using I ( r, ω ) (Eq. 3) and leaving out the explicit po-sition and frequency dependence of the amplitudes, theperturbation of the refractive index of a large fraction ofthe waveguide caused by the moving near-field tip can bedescribed through n ( r, ω ) = n [1 + n I ( r, ω )]= n [1 + n I BG + n · | A P CB A P CF | cos (2 k c xa )]= n + ∆ n. (5)Here we assume that the field in the waveguide is domi-nated by band C as justified by Fig. 3.As a consequence of this refractive-index change anadditional phase φ is introduced to the propagating light.The change in phase for any forward propagating modebetween the front facet and the NSOM tip is thus∆ φ ( r, ω ) = k c r ∆ n = k c rn n [ I BG + 2 | A P CB A P CF | cos (2 k c xa )]= φ BG + k c rn n | A P CB A P CF | cos (2 k c xa ) , (6)where φ BG corresponds to the phase term caused by theaverage temperature rise and hence does not contributeto the new bands. Backward propagating modes experi-ence a similar phase shift. The measured field can there-fore be described as the unperturbed signal with an ad-ditional phase. Expanding the phase by a Taylor seriesresults in the general expression for the field detected bythe NSOM measurements: E det ( r, ω ) = Ae − ik xa e −(cid:52) φ ( r,ω ) = Ae − ik c xa e − φ BG · [1 + ik c rn n · A P CB A P CF cos (2 k c xa )+ ( ik c rn n · A P CB A P CF cos (2 k c xa ))
2! + ... ]= ∞ (cid:88) N =0 Ae − ( ik c xa + φ BG ) · ( e − i k c xa + e i k c xa ) N N N ! · [ ik c rn n · A P CB A P CF ] N , (7)where A is the amplitude of any real mode in the waveg-uide with wave vector k . In particular, A could be A P F C .Equation 7 gives the general expression of the field asso-ciated with mode C of the intensity-perturbed photonic-crystal waveguide. Its Fourier transform will reveal allspatial frequencies present in the measured field data.The bands will thereby follow the general expression: k det = 2 N k c + k , where N = 0 , , , ... . Pure modeC corresponds to the term N = 0 in Eq. 7 and the fieldassociated with this mode is A P CF exp[ − ik c xa + φ BG ]. Theintensity is thus given by | A P CF | , scaling linearly with theincident intensity as expected for a pure mode.A new band as observed in the experiment correspondsto N = 1 in Eq. 7 and its field is given by E N =1 ( r, ω ) = ik c rn n · A P CB A P CF · A P CF e − ( ik c xa + φ BG ) e − i k c xa + e i k c xa A P CB A P CF A P CF e φ BG ik c rn n · [ e ( − i k c xa ) + e (+ ik c xa ) ] . (8)It contains bands corresponding to 3 k c and k c and theassociated field amplitude for both these bands is pro-portional to A P CB · A P CF · A P CF . In the case of N DF will scaleas | A P CB | · | A P CF | · | A P CF | , therefore the intensity of thenew band should scale cubic with the incident intensity.In the same way the new bands E and F can be explainednot with mode C but with mode A as perturbed signalfield. V. THERMAL ORIGIN OF NONLINEARITY
In principle, several nonlinear effects can be respon-sible for the refractive index modification as a conse-quence of an intensity change. The Kerr effect and aheat-induced index change are the most obvious ones.Our measurements alone do not allow us to dissect theeffects, but a thermal origin of the new bands seems themost likely as we will argue below. With a Kerr coeffi-cient of the order of 10 − cm / W, − . On a length of a mm this causes a 10 − rad phase shift.To understand the effect of heating, let us recall theprocesses leading to a temperature increase inside thewaveguide. The photon energy coupled into the waveg-uide is between 1 .
25 and 1 .
36 eV and therefore no directone-photon absorption of the light is possible (band gapof GaAs: E g , GaAs = 1 .
42 eV). However, due to the con-finement of the light inside the waveguide, the intensitycan increase such that the probability of two-photon ab-sorption processes becomes non-negligible. Accompany-ing the photon absorption is a heating up of the sample.In addition, contaminations or defects at the slab’s sur-face can lead to additional heating. In fact, we haveobserved similar waveguides to be destroyed by a run-away thermal degradation at slightly higher excitationpowers. Taking the thermal diffusivity and conductivityof GaAs into account, the time it takes the materialto thermalize on the length scale of the wavelength λ isof the order of 10 − s, which means that the inducedlocal heating pattern (due to the standing wave) is com-pletely washed out and therefore homogeneous. As ex-plained before, however, the total intensity depends onthe position of the NSOM tip relative to the intensitymaxima in the standing light wave. The alteration ofthe band intensity correlates to a temperature changeas ∂ x T ( r, ω ) = α · ∂ x I ( r, ω ), with T ( r, ω ) the absolutetemperature of the photonic-crystal waveguide as a func-tion of spatial position r of the near-field tip. The re-sulting refractive index due to the heating is given by n th = n (1 + n T ), where n and n for GaAs are 3.255and n = 4 . × − K − , respectively. A rough esti-mate of the temperature rise given the intrinsic absorp-tion coefficient of GaAs and the heat conductivity of theGaAs slab yields a temperature rise of about 1 K per mWpump power. Given n , this is three orders of magnitudemore than the index change caused by the Kerr effect. VI. CONCLUSIONS AND DISCUSSION
Our model based on the perturbation caused by a near-field tip correctly explains the appearance of the observednew bands. Hence, these bands are not optical modes ofthe waveguide but a waveguide-specific NSOM tip effectnot previously observed, where the NSOM tip can causea large phase shift. There are several circumstances en-abling us to observe the new bands in our experiment:1) Use of a direct-band gap material with a large ther-mal coefficient, 2) A high thermal isolation provided bythe free-standing perforated waveguide samples, and 3)Phase-sensitive measurement allowing to map and sep-arate individual spatial signals in Fourier space. Theobserved scaling factor ( cf.
Fig. 4(b)) matches the onepredicted by our model for higher powers. Competingextrinsic as well as intrinsic refractive index perturbingprocesses cannot be ruled out, though. Therefore, eventhough we successfully explained the observed modal ap-pearance as near-field tip-induced thermal perturbation,we cannot exclude the influence of intensity-dependentnonlinear optical effects, since there has already been ex-perimental evidence of a very large nonlinear optical re-sponse measured in InGaAs photonic-crystal waveguidestructures, yet without clarity on the origin of the ef-fect. VII. SUMMARY
In conclusion, we have observed and explained the ex-istence of new bands in NSOM measurements of photoniccrystal waveguides not predicted by standard eigenmodecalculations. Our experimental results demonstrate anintriguing effect caused by coupling of modes in a finite-sized photonic-crystal waveguide and could be explainedas a result of position-dependent tip losses and accompa-nying temperature changes.
VIII. ACKNOWLEDGMENTS
We thank D.J. Dikken, L. Kuipers, A. Lagendijk, P.Lodahl, H.L. Offerhaus, S. Stobbe, and W.L. Vos forstimulating discussions and F.B. Segerink and C.A.M.Harteveld for technical support. This work was sup-ported by NWO-nano and FOM, a subsidiary of NWO.
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