We present a method to analyse the coupling of lateral displacements in
nanoscale structures, in particular waveguide grating mirrors (WGM), into the
phase of a reflected Gaussian beam using a finite-difference time-domain
simulation. Such phase noise is of interest for using WGMs in high-precision
interferometry. We show that WGMs do not couple lateral displacements into the
phase of a reflected beam to a precision of 10^{-7} rad and that WGMs are
therefore not subject to the same stringent alignment requirements as
previously proposed layouts using diffraction gratings.Comment: 3 Pages, 4 figure

Brown, Daniel

Brückner, Frank

Carbone, Ludovico

Freise, Andreas

Friedrich, Daniel

Schnabel, Roman

English

arXiv

We present a method to analyze the coupling of lateral displacements in nanoscale structures, in particular waveguide grating mirrors (WGMs), into the phase of a reflected Gaussian beam using a finite-difference time-domain simulation. Such phase noise is of interest for using WGMs in high-precision interferometry. We show that, to the precision of our simulations (10-7 rad), waveguide mirrors do not couple lateral displacement into phase noise of a reflected beam and that WGMs are therefore not subject to the same stringent alignment requirements as previously proposed layouts using diffraction gratings

Brückner, F.

Brown, D.

Carbone, L.

Freise, A.

Friedrich, D.

Schnabel, R.

University of Birmingham Research Portal

Invariance of waveguide grating mirrors to lateral displacement phase shifts

We present a method to analyse the coupling of lateral displacements in nanoscale structures, in particular waveguide grating mirrors (WGM), into the phase of a reflected Gaussian beam using a finite-difference time-domain simulation. Such phase noise is of interest for using WGMs in high-precision interferometry. We show that WGMs do not couple lateral displacements into the phase of a reflected beam to a precision of 10^{-7} rad and that WGMs are therefore not subject to the same stringent alignment requirements as previously proposed layouts using diffraction gratings

MPG.PuRe

arXiv:1303.0544v2  [physics.optics]  30 Apr 2013Invariance of waveguide grating mirrorsto lateral displacement phase shiftsDaniel Brown1,∗, Daniel Friedrich2,∗∗, Frank Bru¨ckner1,Ludovico Carbone1, Roman Schnabel2 and Andreas Freise11University of Birmingham, School of Physics and Astronomy, Edgbaston, Birmingham, B15 2TT, UK2Institut fu¨r Gravitationsphysik, Leibniz Universita¨t Hannover and Max-Planck-Institut fu¨r Gravitationsphysik(Albert-Einstein-Institut), Callinstrasse 38, 30167 Hannover, Germany∗Corresponding author: ddb@star.sr.bham.ac.uk∗∗Currently working at the Institute for Cosmic Ray Research, University of Tokyo, JapanCompiled May 2, 2013We present a method to analyse the coupling of lateral displacements in nanoscale structures, in particularwaveguide grating mirrors (WGM), into the phase of a reflected Gaussian beam using a finite-differencetime-domain simulation. Such phase noise is of interest for using WGMs in high-precision interferometry. Weshow that, to the precision of our simulations (10−7rad), waveguide mirrors do not couple lateral displacementinto phase noise of a reflected beam and that WGMs are therefore not subject to the same stringent alignmentrequirements as previously proposed layouts using diffraction gratings. c© 2013 Optical Society of AmericaOCIS codes: 050.2770, 050.5080, 230.4040, 050.1755, 050.6624.The sensitivity of high precision interferometry experi-ments such as the preparation of entangled test masses[1], frequency stabilisation with rigid cavities [2] andgravitational wave detectors [3] are eventually limited bythe quantum noise of the interrogating light or the ther-mal noise of the optical components. Quantum shot noiseof the detected light can typically be reduced by increas-ing the laser power. However, this induces larger ther-mal distortions in the high-reflectivity (HR) coatings andsubstrates. A number of new techniques have been sug-gested to reduce the distortions from high power beamsas well as thermal noise: the use of non-fundamentalbeam shapes [4], all-reflective interferometer layouts us-ing dielectric gratings to reduce absorption of the laserin optics [5] and the use of waveguide grating mirrors(WGM) [6]. WGMs would replace HR mirror coatings,reducing their thickness by a factor of 10 to 100; whichpromises to reduce the Brownian coating thermal noise[3,7]. However, gratings couple lateral displacements intothe phase of diffraction orders |m| > 0 [8,9]. This placesstringent requirements on the alignment and stability ofgratings and the incident laser beams for high-precisioninterferometry [10]. WGMs rely on diffraction into thefirst order and could potentially be subject to the samedisplacement phase noise effects. However, so far no the-oretical or experimental evidence has been presented todemonstrate whether WGMs suffer from this.In this Letter we apply a rigorous Finite-DifferenceTime-Domain (FDTD) based simulation with Gaussianbeams to show that WGMs do not couple lateral dis-placements into the phase of a reflected laser beam. Wefurther provide a simplified ray picture to illustrate thisresult.Ray pictures have already been used to describe sev-eral WGM features [11] as depicted in Fig.1. WGMs intheir most simplistic form consist of 2 layers: (i) a waveg-uide layer applied to some substrate material; and (ii) agrating layer which couples the incident laser light intothe waveguide layer (typically etched into the waveguidelayer). In this case both the waveguide and grating layersare made from a high refractive index nh material withthe substrate material’s lower refractive index denotedby nl. The geometrical grating parameters must be care-fully chosen to reach a theoretical maximum reflectiv-ity [6,7]. For given materials and laser light wavelength,the grating period d is chosen such that the normallyincident beam is diffracted into the 1st order within thewaveguide layer at an angle that allows total-internal-reflection (TIR) at the waveguide-substrate boundary tooccur. TIR at the substrate boundary along with thegrating create a waveguide in which the±1st orders prop-agate. These undergo diffraction at the grating multipletimes, coupling out into the vacuum where it interfereswith the reflected specular laser light. The remaininggrating parameters, namely the thickness of the waveg-uide layer s, fill-factor f and groove depth g can all betuned to provide destructive interference in the substrateand constructive in the vacuum, ideally providing 100%reflectivity.A lateral displacement δx of some grating structureversus the incident beam induces a phase shift of∆Φm = 2pimδx/d (1)relative to a non-displaced beam for diffraction orderm [10]. For WGMs we require that any rays couplingout into the vacuum do not have any phase terms de-pendent on δx. From Fig.1 each time a ray is diffractedand picks up a ∆Φm term, an ∗ is added as a su-perscript. The ray −1T ∗∗ diffracted into the vacuumhas collected two ∆Φm terms, its total phase is thenΦ−1T∗∗ = Φo(s, d, g, f, nl, nh)+∆Φ+1+∆Φ−1, where Φois a collection of all phase terms depending on the WGM1vacuumFig. 1. The incident beam (black) is coupled into the waveg-uide layer by the grating intom = ±1. Orders±1T propagatealong the waveguide coupling back into the vacuum (blue) tointerfere with the initially reflected light, 0R, picking up ∆Φmphase terms with each interaction with the grating. Furthercoupling into the vacuum is also possible (magneta) whichinvolved further ∆Φm terms. The ∗ superscript refers to thenumber of diffractions beam has undergone. R =Reflectionfrom grating, T =Transmission from grating, first number isorder of diffraction m.parameters, but not δx. Φo is tuned with simulations byadjusting each parameter to produce 100% reflectivity.From Eq.(1) we see the ∆Φ−1 and ∆Φ+1 terms cancel,with a similar argument being valid for all other raysthat couple out into the vacuum such as +1T ∗∗∗. Thus,following this strongly simplified picture any of the phasenoise effects outlined in Ref. [10] for gratings should notapply to WGMs under normal incidence.In order to provide a rigorous and physicallycorrect computer model of a finite beam reflectedfrom a WGM we have implemented a numericalfinite-difference time-domain (FDTD) based algo-rithm, which provides the ability to model a varietyof grating structures as well as arbitrary and fi-nite incident electromagnetic field distributions.The simulation tool is coded in Java, open sourced(http://kvasir.sr.bham.ac.uk/redmine/projects/fdtd)and was based on Ref. [12]. A 2D FDTD simulationsufficed for our needs as only a displacement of theWGM in one direction orthogonal to a normally inci-dent Hermite-Gaussian (HG) beam was required; thusspeeding up computation time significantly. Two extrafeatures were also required for the simulation [12]:Total-Field Scattered-Field (TFSF) for separating theincident and reflected beam from the WGM and complexperfectly matched layers (CPML) to reduce reflectionsfrom the simulation boundaries. The simulation packagewas validated by reproducing known dependencies(found in Ref. [6]) of the reflectivity as a function of thegrating parameters and by investigating the phase noiseof standard diffraction gratings [9].The aim of the simulation was to measure the wave-front of a HG beam reflected from a WGM whilst dis-placing it from δx = 0 → d. Along the wavefront thephase can then be deduced and plotted against δx toview any apparent phase shifts. The simulation setupis depicted in Fig.2, where a HG TEM00 is injected inthe xˆ direction along the TFSF boundary and the elec-Fig. 2. Schematic layout of 2D FDTD simulation for testingWGM shift invariance. Gaussian beam injected along TFSFboundary onto WGM (Red). The reflected beam (Green)then propagates to the measurement line where the phaseis measured. The CPML absorbs outgoing waves to reducereflections from boundaries.tric field of the reflected beam is measured along themeasurement line 15µm away to avoid near-field vari-ations. The Courrant stability factor [12] for the simu-lation was chosen as S = c∆t/∆x = 1/√2; where ∆tis the simulation timestep and ∆x = ∆y = 25nm arethe size of the 2D discretisation of the simulation spacewith dimensions Ly = 250∆y and Lx = 4000∆x. Theinjected beam had a wavelength λ = 1064nm and waspositioned such that the waist was at the WGM withsize w0 = 800∆x = 20µm. The WGM parameters cho-sen were d = 28∆x = 700nm, g = 14∆x = 350nm,f = 0.5 and s = 5∆x = 125nm which provided a reflec-tivity of 99.8% for the incident beam (in agreement withRef. [6]). The indices of refraction used were fused silicafor nl = 1.45 and Ta2O5 for nh = 2.084 which are thetypical materials used for 1064nm optics.Eq.1 states the phase shift for m = ±1 is periodicwith displacements of the grating period d = 28∆x.Thus for this effect to be visible the simulation was run28 times for offsets δx = p∆x with p = 0, 1, 2, ..., 28.Approximately 3000 timesteps were required for the re-flected beam to reach an approximate steady-state ineach simulation. At this point 1024 time samples of theelectric field at each point along the measurement linewere taken, Ep(x, t). The generalised Goertzel algorithm(based on Fast Fourier Transforms (FFT)) [13] was usedto extract the amplitude Ap(x) and phase φp(x) of thereflected beam along the measurement line for the inci-dent laser frequency f0 = c/1064nm. φp(x) was obtainedfor each offset δx = p∆x of the WGM with the changein phase with displacement defined as ∆φp(x) = φp(x)−φ0(x). Our model showed that displacement phase shiftfor WGMs are at least 105 smaller than for an equiva-lent grating setup, see Fig.3: The central plot shows thephase change as a function of the displacement alongthe beam profile; the satellite plots provide the scale forthe central plot. The top plot shows ∆φ14 increasingslightly towards the edge of the beam, this is expectedto occur when Ap(x) → 0, which degrades any accu-rate calculation of the phase as the signal–to–numericalnoise ratio decreases. At the beam peak, shown in the2right plot, the phase change is ∆φp(x = 0) ≈ 20µradand shows no correlation with δx. This result is 5 or-ders of magnitude smaller than what Eq.(1) states fordisplaced grating structures. To determine whether theoscillations seen in ∆φp(x) were near field effects or nu-merical artefacts the value max{∆φp(x)} was computedat increasing distances from the WGM for a displace-ment over one grating period p = 0 → 28 at the centreof the beam (x = −d/2 → d/2). As seen in Fig.4, thenear field phase shifts from the initial imprint of thegrating can be seen at y < 3µm which decays rapidlywith distance, after which a flat noise is present. Fig.4shows 3 different FFT windowing functions agreeing aty < 3µm but possessing different noise floors, the lowestbeing max{∆φp(x)} ≈ 10−7rad using a Blackman FFTwindow. Numerical errors present from the FDTD arenot thought to be limiting, increased spatial and tempo-ral resolution (∆x→ ∆x/2) does not offer any improve-ment in the noise levels as seen in Fig.4. This suggestsspectral leakage from the FFT is limiting the accuracyof phase measurements and the oscillations present in∆φp(x) measured at 15µm are purely numerical arte-facts, similar results were seen at varying distances fromthe WGM.This work presents the successful implementation ofan FDTD simulation to analyse displacement inducedphase shifts in a reflected Gaussian beam from a WGM.No such phase shifts were found within the precisionlimit of ≈ 10−7rad set by numerical errors. This lowerlimit is seven orders of magnitude lower than the phasenoise estimated for previously proposed layouts withdiffraction gratings, which raised concerns regardingthe stability and alignment [10] of such configurations.Therefore, the absence of this phase shift for WGMsstrengthens the argument for their usage in future high-precision interferometry experiments.We acknowledge the Science and Technology FacilitiesCouncil (SFTC) for financial support in the UK. D.F.was supported by the Deutsche Forschungsgemeinschaft(DFG) within the Collaborative Research Centre TR7.This document has been assigned the LIGO LaboratoryDocument number ligo-p1300019.References1. H. Mu¨ller-Ebhardt, H. Rehbein, R. Schnabel, K. Danz-mann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008)2. K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett.93, 250602 (2004)3. G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kit-telberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fe-jer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nak-agawa, CQG 19, 897 (2002)4. S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79,122002 (2009)5. K.-X. Sun and R. L. Byer, Opt. Lett. 23, 567 (1998)6. A. Bunkowski, O. Burmeister, D. Friedrich, K. Danz-mann, and R. Schnabel, CQG 23, 7297 (2006)7. F. Bru¨ckner, T. Clausnitzer, O. Burmeis-δ x [nm]∆φp(x) against WGM displacement, δ x−60 −40 −20 0 20 40 600350700−60 −40 −20 0 20 40 60−202φ 14(x) [ mrad] Change in phase ∆φ14(x) δ x = 350nm035070002040∆φp(x=0) [µrad]−60 −40 −20 0 20 40 6000.51x [µm]|A 0(x)|2Fig. 3. Central plot shows phase change in the reflectedbeams wavefront ∆φp(x) against WGM displacement, δx.Top plot shows cross section of phase at δx = 350nm wherethe phase change is maximum. The right plot shows the vari-ation in the phase of at x = 0 against δx. Bottom plot showsthe intensity of the reflected beam along x.0 5 10 15 20 25 30 35 40 451010105100Distance from WGM surface [µm]max{p} [rad]Rectangular Hanning Blackman Rectangular   x/2Fig. 4. Maximum change in phase at beam peak intensityas the WGM is displaced from δx = 0 → d with increasingdistance from the WGM surface and different FFT window-ing functions. Large near field phase shifts are seen close tothe WGM and then a flat noise. Increasing the spatial andtemporal resolution of FDTD ∆x→ ∆x/2 did not appear tooffer reductions in numerical noise caused by FDTD.ter, D. Friedrich, E.-B. Kley, K. Danzmann,A. Tu¨nnermann, and R. Schnabel, Opt. Lett. 33,264 (2008)8. S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller,D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen,A. Tu¨nnermann, E. Kley, and T. Clausnitzer, Phys. Rev.Lett. 95, 013901 (2005)9. D. Lodhia, D. Brown, F. Bru¨ckner, and A. Freise, inpreparation (2013)10. A. Freise, A. Bunkowski, and R. Schnabel, New Journalof Physics 9, 433 (2007)11. A. Sharon, D. Rosenblatt, and A. A. Friesem, J. Opt.Soc. Am. A 14, 2985 (1997)12. A. Taflove and S. C. Hagness, Computational Electro-dynamics: The Finite-Difference Time-Domain Method3rd ed. (Artech House Publishers, 2005)13. P. Sysel and P. Rajmic, EURASIP Journal on Advancesin Signal Processing 2012, 1 (2012). 10.1186/1687-6180-2012-563

Invariance of waveguide grating mirrors to lateral displacement phase shifts

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