ACCURATE prediction of the behavior of distributed

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 50, NO. 6, JUNE Characterization of Distributed Bragg Reflectors Andrew Grieco and Yeshaiahu Fainman, Fellow, IEEE Abstract We propose and demonstrate a novel method to characterize the coupling coefficient and loss coefficient of a distributed Bragg reflector (. The method is based on a coupled-mode analysis of periodic structures, and involves the linewidth comparison of a pair of Fabry Pérot resonators. The method is shown to be independent of coupling efficiency and wavelength dependence of the parameters. Index Terms Bragg gratings, integrated optics, metrology. I. INTRODUCTION ACCURATE prediction of the behavior of distributed Bragg reflectors (s is critical to the design of a number of photonic applications, including optical digital signal processing [1], switching [], [3], wave mixing [], and has implications for the stability analysis of such devices [5]. The reflectance of a distributed Bragg reflector created by the periodic perturbation of a dielectric waveguide may be described within the paradigm of coupled-mode theory [6]. The effect of the perturbation is to transfer energy from one mode to another when the difference between the propagation constants of the modes is matched to the period of the perturbation. In this context the spectral response of a is fully determined by a coupling coefficient κ that determines the strength of the interaction between the modes, and a loss coefficient α that describes the net attenuation (or amplification of the optical field, in addition to physical parameters such as refractive index of the guiding materials and device length. Previous characterization methods involve curve fitting either to the stopband of a single [7], or a Fabry-Pérot resonator formed by a pair of s [8]. For low loss devices, the latter is more effective because the resonator response is more sensitive to the parameters. To be truly general, a fitting process must account separately for the wavelength dependence of the parameters of the and resonant cavity, as well as the possibility of coupling loss between the mode of the and that of the cavity. The sheer number of independent parameters has a detrimental effect on the robustness of the fit. An improvement on the fitting Manuscript received November, 013; revised April, 01; accepted April 11, 01. Date of publication April 16, 01; date of current version April 30, 01. This work was supported in part by the National Science Foundation (NSF, in part by the NSF Engineering Research Center for Integrated Access Networks, in part by the Defense Advanced Research Projects Agency, in part by the Office of Naval Research, and in part by the Cymer Corporation. The authors are with the Department of Electrical and Computer Engineering, University of California at San Diego, San Diego, CA USA ( agrieco@ucsd.edu; fainman@eng.ucsd.edu. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JQE method for characterizing loss has been demonstrated, based on the spectral comparison of a series of s [9]. This method is independent of coupling efficiency and wavelength dependence of the parameters, however it assumes that the loss coefficients of the s and waveguide are the same. In this manuscript we describe a novel method to characterize s based on the comparison of the resonant linewidth of two Fabry-Pérot resonators. The method is shown to be independent of coupling efficiency, and of the wavelength dependence of the parameters. In Section II we start from the theoretical background underpinning the characterization method, followed by Section III describing in detail the process of sample fabrication and experimental characterization. Section IV contains a discussion of the experimental results, and methods of avoiding potential sources of error and conclusions of the presented study. II. THEORY A. Spectral Response of a Fabry-Pérot Resonator To begin, consider the spectral response of a Fabry-Pérot resonator formed by a pair of symmetric mirrors [10]: C I C O CM T FP = T M exp( α C L C { [1 C M R M exp( α C L C ] } + CM R M exp( α C L C sin ( ϕ Total (1 ϕ Total = π n ef f L C + ϕ λ Mirror, ( where the resonator and individual mirror transmittances are T FP and T M respectively, R M is the mirror reflectance, the cavity length is indicated by L C,andα C is the cavity loss (or equivalently net gain coefficient. Equations (1 and ( explicitly include terms that account for coupling losses associated with the device: C I is the fraction of power coupled from the light source to the resonator, and C O is the fraction of power coupled from the output of the resonator to the detector. Finally, we also include a term to account for the loss due to mode mismatch between the mirror and the cavity, which is a possibility in guided wave devices. This term, C M, is the fraction of power coupled from the mirror into the cavity (and vice versa. The total phase shift, ϕ Total, includes the linear phase shift from traversing the cavity, as well as the phase shift incurred upon reflection, ϕ Mirror.Heren ef f is the effective refractive index of the cavity, and λ is the free space wavelength of light. In general, all of the quantities besides cavity length may vary with the wavelength of light. The transmittance of such a resonator is maximal when the total phase shift is an integer multiple of π. The full width at half maximum (FWHM linewidth of such a resonance may be IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 5 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 50, NO. 6, JUNE 01 derived from (1 by dividing by the resonance transmittance and extracting the phase shift ϕ FWHM : [ ] 1 C ϕ FWHM = sin 1 M R M exp( α C L C C M R M exp( α. C L C (3 When applied to a resonator formed from s this expression may be simplified further. Specifically, consider a resonator in which the cavity is approximately one period in length. For such a device the loss from the cavity will be negligible in comparison to the loss from the elements, which are many periods in length. Additionally, it is also generally accurate that the phase shift upon reflection from the is much greater than the cavity phase shift. Finally, it is generally possible to manufacture a device in which the loss due to mode mismatch between the and cavity is negligible. This may be accomplished either by relying on a small perturbation such that the mode mismatch between the cavity and is negligible, or by tapering the transition adiabatically [11], provided the taper is small compared to the total length. In these limits the expression in (3 reduces to: ( 1 ϕ sin 1 RM R. ( M Here the quantity ϕ is the FWHM phase shift due to reflection from the elements. The expression in ( is significant because it indicates that the characteristics of the resonance lines are entirely determined by the reflectance characteristics. Consequently, given an expression for the reflectance and phase shift in terms of the parameters κ and α it is possible to infer them by comparing the FWHM linewidth of two resonators composed of different length s (as these are equivalent to a set of two simultaneous equations in two variables. This observation is the main result of this manuscript. B. Reflectance and Phase Characteristics The analytical expression for the coefficient of reflection of a lossy has been described previously [6]: = ( s = i β+α i κ + s L tanh ( s L κ κ + i β α + α (5 ( β (6 β = β F ( β B m π. (7 Here the coefficient of reflection is and the length is L. The propagation constant of the forward and backward modes coupled by grating order m are respectively β F and β B (where m is an integer. It is pertinent to note that the center of the stopband occurs when β = 0. The reflectance follows directly from the coefficient of reflection, and all that remains is to derive an expression for the phase shift upon reflection. To begin the derivation, we explicitly express the dependence of the argument of on β: d arg ( = dim [ ln ( ] = d Im [ ln ( i κ ] [ Im ln ( i β+α + s L tanh ( s L ]. Next note that κ is a purely imaginary number and simplify the expression accordingly: d arg ( { [ ( d i β +α = Im ln L + s L ]} tanh ( s L ( i L + ds d s L tanh ( ds s L = Im i β+α + s L tanh (. (9 s L For the Fabry-Pérot resonator considered in ( with a cavity length of one period, the resonance line will occur at the center of the stopband. At the band center β = 0and consequently s, tanh(s, and their derivatives are real numbers. In this case the expression becomes: d arg ( β=0 = α + κ κ + α ( tanh κ κ + α 1 (8. (10 The combination of (, (5, and (10 is the necessary prerequisite for the determination of parameters using the method described above. The expression for phase shift upon reflection is slightly simplified when β F = β B = β in (7. This corresponds to the situation in which the forward and backward propagating fields of a given mode are coupled by the grating, and is the most common configuration (in a single mode waveguide it is the only possible configuration. In this case the quantity ϕ in ( becomes: ϕ = d arg ( where dβ FWHM = d arg ( dβ is the resonance linewidth. dβ FWHM dβ FWHM, (11 III. EXPERIMENTAL DEMONSTRATION A. Fabrication and Characterization The characterization method proposed in this paper requires the fabrication of two symmetrical Fabty-Pérot resonators, each with different length elements. To demonstrate

3 GRIECO AND FAINMAN: CHARACTERIZATION OF s 55 Fig.. Diagram of the experimental characterization setup. Fig. 1. Scanning electron micrograph of an uncladded section of. Note that the scale bar label is 500 nm. this approach experimentally, these devices were fabricated in silicon-on-insulator (SOI strip waveguides. Specifically, the devices are created from a SOI substrate with a 50 nm silicon top layer with [100] orientation, and a 3 μm buried oxide layer composed of thermally grown silicon dioxide. The substrate is then coated with a hydrogen silsesquioxane (HSQ mask, and the waveguides are patterned by electron beam lithography. The resist is then developed in a tetramethylammonium hydroxide (TMAH solution. The waveguides proper are formed by subjecting the sample to an inductively coupled plasma reactive-ion etch. The sample is completed by cladding the waveguides with a layer of plasma-enhanced chemical vapor deposition silicon dioxide. It is not necessary to remove the remaining resist after the etching step, as it is converted to silicon dioxide during the development process. Both devices were fabricated together on the same sample to minimize the random fabrication variations between them. The dimensions of the waveguides used in this experiment are 50 nm height and 53 nm width. The elements are produced by periodically modulating the waveguide width by ±50 nm (Fig. 1. The grating period is μm, and the s are 80 periods long in the first resonator and 110 periods long in the second. Transitions from the waveguide to the are made by a single period ±5 nm taper to eliminate loss due to mode mismatch. Moreover, we also use parabolic inverse tapered couplers [1], narrowing to 110 nm at both the input and output terminals of the device. Measurement uncertainty in the device dimensions is ±1%. Characterization of the device was performed by using a lensed tapered fiber to couple light into the waveguides and a 0. numerical aperture microscope objective to collect the output signal (lens 1 in Fig.. The light is then imaged on the detector by two sequential F systems (lenses 1 and 3 in Fig.. A tunable laser (Agilent model 81980A was used as a light source. The laser output was polarization scrambled and polarized to excite only the TE mode of the waveguide. Following the microscope objective at the output Fig. 3. Experimental resonance lines of the 80 period Fabry-Pérot resonator. The transmittance is normalized to the resonance maximum. an iris in the focal plan is used to eliminate stray light, and a polarizer in the Fourier plane is used to reject any TM mode component that may arise from imperfect alignment of the lensed fiber or through scattering. Measurements are automated by a computer which coordinates the laser source with the Newport model 918D-IG-OD3 detector and Newport model 931-C power meter. The uncertainty in each power measurement is ±1%. B. Results The transmission spectra of the two structures with corresponding s lengths of 80 and 110 periods about the resonance wavelengths are presented in Fig. 3 and Fig.. The spectra are normalized to the resonance transmittance. In the fabricated devices the absolute positions of the resonances do not exactly coincide. This minor variation ( 0.08 nm is the result of drift in the laboratory temperature between measurements, local heating of the cavity from absorption of the optical field, and fabrication error. The offset has the potential to introduce error into the calculation when the reflectance is considered in (. This error is negligible because the stopband is very broad ( 0 nm between null points and in accordance with (5 the reflectance varies little (<10 5 over the interval. Simultaneous solution of the governing equations is accomplished using the native algorithm of the software Mathcad. The FWHM of the experimental resonators are ± nm for the cavity with 110 period s,

4 56 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 50, NO. 6, JUNE 01 Fig.. Experimental resonance lines of the 110 period Fabry-Pérot resonator. The transmittance is normalized to the resonance maximum. and ± nm for the cavity with 80 period s. The wavelength dependence of the effective index of the unperturbed waveguide may be obtained numerically using the software COMSOL to be dn ef f /dλ = ± /μm (for this calculation the index of refraction of silicon is 3.8, the index of refraction of silicon dioxide is 1.6, and the uncertainty in waveguide width is ±5 nm. With this information the parameters are calculated to be κ = 960 ± 0 /cm and α = 1.±0.5 /cm (equivalently 6.0 ±.3 db/cm. For the sake of completeness, the cavity phase was not neglected during the calculation (its contribution is less than 3% of the reflector phase. The uncertainty of the loss coefficient α is most sensitive to uncertainty in the FWHM linewidth measurements. In the current experiment this is limited by the wavelength accuracy of the tunable laser, and therefore may be improved with higher precision equipment. The uncertainty of the coupling coefficient κ is most sensitive to the measurement precision of the period, and is unlikely to see dramatic improvement. The loss measurement is comparable to, if not slightly larger than, the loss measured in a straight waveguide of similar dimensions, which is 1.0 ± 0.1 /cm (equivalently.5 ± 0.6 db/cm [6]. This is not surprising, as the loss is dominated by scattering from roughness in the waveguide sidewall, and the periodic perturbation increases the total sidewall area. Strictly, the field distribution within the periodic structure is also somewhat different, which may increase its susceptibility to scattering loss. It is possible to infer a lower bound on κ of 900 /cm from [6] by considering the case in which losses are neglected (an upper bound is not attainable because the to waveguide mode mismatch loss is unknown. This is in good agreement with the results of the current experiment. IV. DISCUSSION AND CONCLUSION A. Discussion The novel method of characterization proposed here has a number of advantages. Since the only measurements involve relative transmittance, the method is independent of absolute coupling efficiency. This is significant for nanoscale applications as the absolute coupling efficiency is difficult to characterize because it is particularly subject to fabrication uncertainty and prone to nonuniformity across multiple devices. Since the method involves resonators, it is possible to accumulate a larger total amount of loss over a smaller physical device length. Additionally, the method only involves narrow spectral features which eliminate any uncertainty that might be associated with the wavelength dependence of the parameters. The limitations of the method should be explicitly considered, however. It is necessary that the coupling efficiency of the device not drift over time, otherwise error will be introduced into the FWHM measurement. Likewise, the coupling efficiency must not vary significantly with wavelength across the resonance. Since the measurements occur over narrow resonance lines, these limitations are trivial to satisfy. The measurement time will be short (minimizing sensitivity to drift, and the bandwidth will be narrow (minimizing sensitivity to coupling variation. It is also necessary that the parameters κ and α not vary significantly across the resonator linewidth. Except near material absorption lines this condition will be trivial to meet for the same reasons. For this measurement attention should be paid to the power in the resonators. While it is desirable to use long elements to increase loss sensitivity, the high reflectance can cause enough power to be concentrated in the resonant cavity that nonlinear effects manifest [], [3]. On the other hand, if the total transmitted power is very low the resonance may be contaminated by the noise floor of the measurement system. These problems can usually be avoided by proper consideration during the resonator design. In the experiment reported here, the maximum signal rejection of the devices is 5 db, coupling loss (mostly on the fiber side is 1 db, and 6 dbm laser input power is used (prior to the polarization scrambler, producing greater than 3 dbm power at the detector, well above the noise floor of 60 dbm. Coupling at the input end was controlled by defocusing the lensed fiber to ensure the resonant cavities did not display evidence of nonlinearity. A potential source of experimental error is impedance mismatch at the coupling points along the facets of the sample. In a strict sense, these points act as additional mirrors and the entire device is as a series of coupled Fabry-Pérot resonators. If the impedance mismatch is large, it would produce complex transmission spectra, and may spoil the desired linewidth measurements. The effect would be clearly visible, however, particularly when compared with the transmittance spectra of a simple resonator. In our experiment, inverse tapers were sufficient to match the impedances and suppress facet reflections. If such features appear, reflections can be further reduced by cleaving the facet at an angle to the waveguide. This would come at the cost of more complicated fabrication and otherwise might risk reducing coupling efficiency below the noise floor. Increasing the distance between the couplers would also mitigate this effect to an extent by damping the unwanted outer resonator (at the cost of increased device footprint. In the same vein, the transition sections between waveguide and may have some effect. The simplest method of

5 GRIECO AND FAINMAN: CHARACTERIZATION OF s 57 bounding this effect is to consider the mutually extreme situations in which they act as full extensions of the, and the opposite case in which they act as plain sections of waveguide. The maximum uncertainty in these cases is comparable to the measurement uncertainty in the experimental results. As these represent the most physically unrealistic limits possible, the true uncertainty introduced by transitions will be much less than the measurement uncertainty and need not be accounted for separately. Finally, variation in the fabrication process can lead to unintended differences between the resonator structures. The gratings in this experiment were fabricated using electron beam lithography which results in a high degree of uniformity. Our measurements of similarly prepared resonators suggest that two nominally identical devices separated by 1 mm distance on the wafer may differ by up to % in full width at half maximum [6]. In this experiment such a variation would result in a 0% error in the loss coefficient α measurement, and 0.% error in the coupling coefficient κ measurement. This variation results from numerous causes, including drift of the lithography electron beam path and focusing, and height variation of the SOI wafer and lithography resist. As such, it will increase with larger device separation and longer write times. Appropriate consideration during the design stage can generally eliminate these effects. In our experiment, this error is negligible because the devices were laterally separated by only 0 μm. B. Conclusion In conclusion, we have proposed and demonstrated a method to characterize the coupling coefficient and loss coefficient of a distributed Bragg reflector based on the linewidth comparison of a pair of Fabry-Pérot resonators. The method is independent of coupling efficiency and wavelength dependence of the parameters. It is conditional upon a small number of limitations that are easily met in practice, specifically that the coupling must be time and wavelength independent at the measurement points. The devices required for the measurement have a small footprint which minimizes the uncertainty associated with nanofabrication. ACKNOWLEDGMENT The authors are indebted to Dr. Boris Slutsky for his valuable input. They would also like to thank the Nano3 staff at UCSD for support during sample fabrication, and C. Hennessey for logistical support. REFERENCES [1] Y. Wang, A. Grieco, B. Slutsky, B. Rao, Y. Fainman, and T. Nguyen, Design and analysis of a narrowband filter for optical platform, in Proc. IEEE ICASSP, Prague, Czech Republic, May 011, pp [] N. D. Sankey, D. F. Prelewitz, and T. G. Brown, All-optical switching in a nonlinear periodic-waveguide structure, Appl. Phys. Lett., vol. 60, pp , Mar [3] K. Ikeda and Y. Fainman, Nonlinear Fabry Pérot resonator with a silicon photonic crystal waveguide, Opt. Lett., vol. 31, pp , Dec [] A. S. Helmy, B. Bijlani, and P. Abolghasem, Phase matching in monolithic Bragg reflection waveguides, Opt. Lett., vol. 3, pp , Aug [5] G. P. Agrawal, Nonlinear Fiber Optics, th ed. San Diego, CA, USA: Academic, 007, pp [6] A. Grieco, B. Slutsky, and Y. Fainman, Characterization of waveguide loss using distributed Bragg reflectors, Appl. Phys. B, vol. 11, no., pp. 67 7, 01. [7] C. Kruse, H. Dartsch, T. Aschenbrenner, S. Figge, and D. Hommel, Growth and characterization of nitride-based distributed Bragg reflectors, Phys. Status Solidi B, vol. 8, pp , Jun [8] Y. Painchaud, M. Poulin, C. Latrasse, and M.-J. Picard, Bragg grating based Fabry Pérot filters for characterizing silicon-on-insulator waveguides, in Proc. IEEE 9th Int. Conf. GFP, San Diego, CA, USA, Aug. 01, pp [9] H.L.Rogers,S.Ambran,C.Holmes,P.G.R.Smith,andJ.C.Gates, In situ loss measurement of direct UV-written waveguides using integrated Bragg gratings, Opt. Lett., vol. 35, pp , Sep [10] P. Yeh, Optical Waves in Layered Media. New York, NY, USA: Wiley, 005, pp [11] A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, Robust design of slow-light tapers in periodic waveguides, Eng. Optim., vol. 1, pp , Apr [1] V. R. Almeida, R. R. Panepucci, and M. Lipson, Nanotaper for compact mode conversion, Opt. Lett., vol. 8, pp , Aug Andrew Grieco received the B.S. degrees in physics and earth science (with geology option from the New Mexico Institute of Mining and Technology in 007, and the M.S. degree in electrical engineering (with photonics option from the University of California, San Diego, in 010. From 007 to 008, he was a Post-Baccalaureate Student Researcher with the Shock and Detonation Physics Group, Los Alamos National Laboratory. Since 008, he has been a Research Assistant with the Electrical Engineering Department, University of California, San Diego. His research interests include nonlinear optics and integrated photonics. Yeshaiahu Fainman is a Cymer Professor of Advanced Optical Technologies and a Distinguished Professor of Electrical and Computer Engineering (ECE with the University of California, San Diego (UCSD, where he is currently the Chair of the ECE Department. He is directing research of the Ultrafast and Nanoscale Optics Group at UCSD, and made significant contributions to near-field optical phenomena, inhomogeneous and metamaterials, nanophotonics and plasmonics, and nonconventional imaging. His research applications target information technologies and biomedical sensing. His current research interests are in near-field optical science and technology. He contributed over 0 manuscripts in peer-reviewed journals and over 350 conference presentations and conference proceedings. He is a fellow of the Optical Society of America and the Society of Photo-Optical Instrumentation Engineers, and was a recipient of the Miriam and Aharon Gutvirt Prize, Lady Davis Fellowship, Brown Award, and Gabor Award.