The three-dimensional finite-difference time-domain method is used to obtain loss per unit length in a two-dimensional photonic crystal membrane waveguide by simulating three different length guides. Results are shown for propagation both above and below the light line. The results are compared with a Fourier expansion method and good agreement is obtained above and below the light lin

Craddock, IJ

Cryan, MJ

Railton, CJ

Rorison, JM

Wong, DCL

Yu, S

English

The three-dimensional finite-difference time-domain method is used to obtain loss per unit length in a two-dimensional photonic crystal membrane waveguide by simulating three different length guides. Results are shown for propagation both above and below the light line. The results are compared with a Fourier expansion method and good agreement is obtained above and below the light lineThe three-dimensional finite-difference time-domain method is used to obtain loss per unit length in a two-dimensional photonic crystal membrane waveguide by simulating three different length guides. Results are shown for propagation both above and below the light line. The results are compared with a Fourier expansion method and good agreement is obtained above and below the light lin

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Calculation of losses in 2-D photonic Crystal membrane waveguides using the 3-D FDTD method

                          Cryan, M. J., Wong, D. C. L., Craddock, I. J., Yu, S., Rorison, J. M., &Railton, C. J. (2005). Calculation of losses in 2-D photonic crystal membranewaveguides using the 3-D FDTD method. IEEE Photonics TechnologyLetters, 17(1), 58 - 60. 10.1109/LPT.2004.837914Link to published version (if available):10.1109/LPT.2004.837914Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.58 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 1, JANUARY 2005Calculation of Losses in 2-D Photonic CrystalMembrane Waveguides Using the 3-D FDTD MethodM. J. Cryan, D. C. L. Wong, I. J. Craddock, S. Yu, J. Rorison, and C. J. RailtonAbstract—The three-dimensional finite-difference time-domainmethod is used to obtain loss per unit length in a two-dimen-sional photonic crystal membrane waveguide by simulating threedifferent length guides. Results are shown for propagation bothabove and below the light line. The results are compared with aFourier expansion method and good agreement is obtained aboveand below the light line.Index Terms—Finite-difference time-domain (FDTD), photoniccrystals (PCs), two-dimensional photonic crystal waveguides (2-DPC-WGs).I. INTRODUCTIONTHERE IS much interest in two-dimensional (2-D) photoniccrystal waveguides (PC-WGs) which show the potentialfor low-loss guiding [1] and enhanced functionality over simpleuniform high-contrast waveguides due their strong confinementand dispersion properties [2]. 2-D PC-WGs can be broadlydivided into two categories, those with weak and those withstrong vertical confinement. The latter are known as membraneguides and can be either suspended in air or used in a silicon-on-insulator configuration [2]. This high vertical contrast leadsto wide wavelength ranges where the propagating modes arebelow the light line which implies zero coupling to radiationmodes and, hence, zero loss (neglecting effects such as surfaceroughness and material loss). In the former case of weakvertical confinement, propagation below the light line is muchmore difficult to achieve and losses in these structures tendto be higher. Although when used in actual circuits includingbends and Y-junctions improved relative performance has beensuggested [2], [3]. Here, we will concentrate on the strongvertically confined structures since these have most publishedloss results and the below light-line region gives a useful checkon the accuracy of the results.II. MODELLINGWhile measured loss values in PC-WGs are reducing quiterapidly, there is still a strong requirement to fully understandthe lower limits that can be achieved in the various guidingtechnologies. Below the light line, these are reasonably clearsince surface roughness and material loss are well understoodfor standard high-contrast waveguide [2]. However, in the morecomplex propagation regimes possible in PC-WGs, furtherwork is still required. Above the light line is a more interestingManuscript received December 17, 2003; revised June 10, 2004.The authors are with the Department of Electrical Engineering, University ofBristol, Bristol BS8 1UB, U.K. (e-mail: m.cryan@bristol.ac.uk).Digital Object Identifier 10.1109/LPT.2004.837914case where intuitively one would expect high losses, however,low loss guiding [4], [5] has been predicted.In order to understand these lower limits, three-dimen-sional (3-D) modeling is required. There are in general twoapproaches: First one can calculate the complex propagationconstant for the waveguide modes, the imaginary part thengives the loss directly [4], [6], [7]. An alternative approach is touse time-domain numerical methods such as finite-differencetime-domain (FDTD) [8], the transmission line method [9], orthe finite element method in time domain. In [9], the wholewaveguide structure is discretised in 3-D space and the totalpower, derived from the Poynting vector, is calculated at twoclosely spaced planes in the guide and from this a loss per unitlength can be calculated. In [8], the approach of using twodifferent length guides including input and output waveguideswas introduced. This has the advantage of producing largerlosses for the long individual guides which should improve theaccuracy of the final loss per unit length figure and gives veryrealistic excitation. While the approach outlined here and in [8]is computationally intensive, it has a number of advantages.First, the structures need not be periodic in any sense, theycan include surface roughness, wavelength-dependent mate-rial losses, and even nonlinear loss [10] such as two-photonabsorption. Second, and possibly more importantly, where thewaveguide is multimoded, the guide can be excited in a veryrealistic way, for example from a deep etched ridge guide. Thisallows the mode from the input guide to couple to the correctcombination of guided modes, resulting in accurate propagationmodeling.This work uses the FDTD method which has been widelyused in microwave engineering for almost 40 years since Yee’sseminal paper [11]. In recent years with the advent of low-costrandom access memory (RAM) and high-performance desktopmachines, FDTD has been applied in the optics regime [10].Here, an in-house code which has been developed over a numberof years is used.III. RESULTSThe structure to be studied is shown in Fig. 1. This isthe middle length guide( ), two other simulations withm and m are also performed.The structure has uniform meshing in the plane, withnm and nm. This results in 20 cellsper lattice constant, in the direction and 37 cells perin the direction. This produces uniform meshing across theholes and keeps staircasing errors constant across the structure.In the vertical direction, nonuniform meshing is used with anm, resulting in 16 cells in the membrane.1041-1135/$20.00 © 2005 IEEECRYAN et al.: CALCULATION OF LOSSES IN 2-D PC MEMBRANE WGs USING THE 3-D FDTD METHOD 59Fig. 1. PC membrane waveguide with r=a = 0:3016, a = 430:55 nm, h =0:6a = 258:3 nm, L = 6:028 m; 10:76 m; 19:37 m, L =11a;22a;42a, d = 870 nm, W1 guide in   K direction, n = 3:4.This results in million cells for the shortguide, million cells for the middle guide,and million cells for the long guide.The PC is hexagonal with a lattice constant of 430.55 nm, witha one missing hole (W1) waveguide in the direction.It should be pointed out here that only six rows of holes havebeen used in the “walls” of the waveguide in these simulations,whereas a minimum of seven are used in [4]. The structure is ex-cited with the fundamental transverse-electric (TE) [ domi-nant] mode of the input waveguide. This mode is obtained froma separate 2-D FDTD simulation which finds the eigenmodesof the cross section of the guide. This is crucially importantin exciting the structure realistically. Furthermore, it enablesa specialized transmission calculation to be performed whichwe have called the Modal S-parameter calculation [12]. Here,an overlap integral between the input mode and the fields at aplane (known as a snapshot) specified in the output waveguideare stored at each time step. This enables a modal S-parameter ortransmission coefficient to be defined. That is the ratio of outputpower in the fundamental mode of the output guide to that at theinput. In fact a separate 3-D simulation is performed with justa straight waveguide in order to obtain the incident power. Thismeans that higher order modes coupled into the output wave-guide will be ignored—this is quite realistic since in 100 mof output guide that would in reality be present, the high lossesof these modes generally means that they would not be mea-sured. Since the waveguide has a symmetry plane at its centerand the structure is being excited by the fundamental TE mode,a perfect electric conducting (PEC) boundary can be introducedthere without altering the results. This allows the simulationsize to be halved. The computational box must be terminatedwith absorbing boundaries (apart from the center where a PECFig. 2. Comparison of losses in PC-WG of Fig. 1 between FDTD with Murand PML boundaries and Fourier expansion method [2], [3].boundary is used). Two different boundaries have been used.Firsy, Mur first-order [10] and second perfectly matched layers(PML) [10]. The longest waveguide with PML boundaries re-sults in a simulation requiring 800 MB of RAM and taking140 h on a Pentium 4 2.4-GHz machine. It must be remem-bered that for this one time-domain simulation, we obtain thou-sands of usable frequency points, since the time-domain datais Fourier transformed. This results in run times per frequencypoint of around 6 min.The essence of this approach is to calculate the transmissioncoefficient for two different length guides. The difference be-tween these two results, with a knowledge of the difference inlengths provides a loss per unit length figure. This is closely re-lated to the cut-back method used for measured data [1]. Theuse of different length waveguides provides some correction foreffects such as coupling loss into the PC guide. More advancedcorrection techniques using full two-port S-parameters are cur-rently being investigated.Fig. 2 shows a comparison between 3-D FDTD with Murand PML boundaries with the results from Lalanne [4]. FDTDresults for and are also shown. The loss resultshave been scaled in the same way as those shown in [13], toaccount for different lattice constants. This scaling is valid sinceMaxwell’s equations scale with length, however, we are usinga fixed unit length for comparison of losses. A lattice constantof 430.55 nm was chosen here to place 1550 nm below the lightline.The first point to note is that the light line for this waveguidemode is around 1.52 m and the reduction in loss is clearlyshown in both methods. Below the light line, the Fourier methodgives very low losses as might be expected for a method whichcalculates the Bloch modes of the waveguide. The FDTDresults, however, are much less idealized calculations and, asshown in Fig. 2, have a strong ripple especially below the lightline of around 10 dB/mm. The average of this ripple isclose to zero which suggests that, fundamentally, the FDTDresults are low loss but they have an error superimposed uponthem. The origin of this ripple will be discussed later. Themain point to observe in Fig. 2 is that good agreement is beingmaintained over a wide wavelength range. It is seen that theand give very similar results suggesting that60 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 1, JANUARY 2005Fig. 3. Comparison of output power density with and without Hammingwindowing for L1 data.Fig. 4. Comparison between loss results for Fourier method and FDTD withHamming windowing.the shorter length guides could be sufficient for loss calculation.This results in much shorter simulation times. The results alsohighlight the importance of using PML boundaries.The most obvious explanation for the observed ripple wouldbe multiple reflections from the input and output of the wave-guide, but this was found not to be the case. In fact, the causewas found to be in the early truncation of the time-domain data.Fig. 3 shows the standard FDTD output power density and it isseen that at around 2.25 ps, the simulation stops. This resultsin an abrupt discontinuity in the time-domain data which trans-lates to ripple in the frequency domain. To counteract this, thesimulation would need to be run for a longer time. An alterna-tive approach is to use a windowing function. Fig. 3 also showsthe effect of applying a Hamming windowing function to thetime-domain data. As can be seen, the discontinuity is now muchreduced.A Hamming window has been applied to the data sets forthe three different length guides and the losses recalculated.The results are shown in Fig. 4. As can be seen, the ripple is nowmuch reduced and the data is close to the average of the previousripple. It is seen that much improved agreement below the lightline has been obtained. Around 1.4 m, there is a small discrep-ancy between FDTD and the Fourier method; this is thought tobe due to the different discretization schemes being used in thetwo methods [14] and is currently under investigation.IV. CONCLUSIONThis letter has presented an approach based on the 3-D FDTDmethod to determine the loss per unit length in PC membranewaveguides. Good agreement with a Fourier expansion methodhas been shown. The problem of early truncation of time-do-main data has been overcome with the use of a Hammingwindow. The use of the FDTD method enables effects such assurface roughness, disorder, and nonlinear losses to be studied.The use of modal excitation and modal S-parameters allowsvery realistic simulations to be performed.ACKNOWLEDGMENTThe authors would like to thank P. Lalanne and C. Sauvan foruseful discussion of results.REFERENCES[1] S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic inte-grated circuit with membrane-type photonic crystal waveguides,” Opt.Express, vol. 11, no. 22, pp. 2927–2939, Nov. 2003.[2] T. F. Krauss, “Planar photonic crystal waveguide devices for integratedoptics,” Phys. Stat. Solidi, vol. 197, no. 3, pp. 688–702, 2003.[3] S. Olivier et al., “Coupled-mode theory and propagation losses in pho-tonic crystal waveguides,” Opt. Express, vol. 11, no. 13, pp. 1490–1496,Jun. 2003.[4] P. Lalanne, “Electromagnetic analysis of photonic crystal waveguidesoperating above the light cone,” IEEE J. Quantum Electron., vol. 38, no.7, pp. 800–804, Jul. 2002.[5] T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittanceof a hexagonal photonic crystal slab,” Phys. Rev. B, vol. 63, no. 125107,pp. 1–7, 2001.[6] G. R. Hadley, “Out-of-plane losses of line-defect photonic crystalwaveguides,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 642–644,May 2002.[7] L. C. Andreani and M. Agio, “Intrinsic diffraction losses in photoniccrystal waveguides with line defects,” Appl. Phys. Lett., vol. 82, no. 13,pp. 2011–2013, Mar. 2003.[8] M. Cryan et al., “Analysis of losses in 2D photonic crystal waveguidesusing the 3D finite difference time domain (FDTD) method,” in IEEELEOS, Glasgow, U.K., Nov. 2002, pp. 516–517.[9] Y. Desieres et al., “Propagation losses of the fundamental mode in asingle line-defect photonic crystal waveguide on an InP membrane,” J.Appl. Phys., vol. 92, no. 5, pp. 2227–2234, Sep. 2002.[10] A. Taflove, Computational Electrodynamics—The FDTDMethod. Norwood, MA: Artech House, 2000.[11] K. S. Yee, “Numerical solution of initial boundary value problem in-volving Maxwell’s equations in isotropic media,” IEEE Trans. AntennasPropag., vol. AP-14, pp. 302–306, Mar. 1966.[12] I. J. Craddock et al., “Applications of single mode extraction from finitedifference time domain data,” in Proc. Inst. Elect. Eng. Microwave An-tennas Propag., vol. 146, Apr. 1999, pp. 160–162.[13] C. Sauvan et al., “Accurate modeling of line-defect photonic crystalwaveguides,” IEEE Photon. Technol. Lett., vol. 15, no. 9, pp. 1243–1245,Sep. 2003.[14] P. Lalanne and C. Sauvan, private communication, Jun. 2004.

Accurate modeling of line-defect photonic crystal waveguides,”IEEEPhoton.Technol.Lett.,vol.15,no.9,pp.1243–1245,

Analysis of losses in 2D photonic crystal waveguides using the 3D ﬁnite difference time domain (FDTD) method,”

Applications of single mode extraction from ﬁnite difference time domain data,” in

Computational Electrodynamics—The FDTD Method.

Coupled-mode theory and propagation losses in photonic crystal waveguides,”

Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,”

Electromagnetic analysis of photonic crystal waveguides operating above the light cone,”

Intrinsic diffraction losses in photonic crystal waveguides with line defects,”

Numerical solution of initial boundary value problem involving Maxwell’s equations in isotropicmedia,” IEEETrans.

Out-of-plane losses of line-defect photonic crystal waveguides,”

Planar photonic crystal waveguide devices for integrated optics,”

private communication,

Propagation losses of the fundamental mode in a single line-defect photonic crystal waveguide on an

Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,”

Calculation of losses in 2-D photonic crystal membrane waveguides using the 3-D FDTD method

The three-dimensional finite-difference time-domain method is used to obtain loss per unit length in a two-dimensional photonic crystal membrane waveguide by simulating three different length guides. Results are shown for propagation both above and below the light line. The results are compared with a Fourier expansion method and good agreement is obtained above and below the light line.The three-dimensional finite-difference time-domain method is used to obtain loss per unit length in a two-dimensional photonic crystal membrane waveguide by simulating three different length guides. Results are shown for propagation both above and below the light line. The results are compared with a Fourier expansion method and good agreement is obtained above and below the light line

Calculation of losses in 2D photonic crystal membrane waveguides using the 3D FDTD method

Cryan, MJ., Wong, DCL., Craddock, IJ., Yu, S., Rorison, JM., &Railton, CJ. (2005). Calculation of losses in 2-D photonic crystalmembrane waveguides using the 3-D FDTD method. IEEE PhotonicsTechnology Letters, 17(1), 58 - 60.https://doi.org/10.1109/LPT.2004.837914Peer reviewed versionLink to published version (if available):10.1109/LPT.2004.837914Link to publication record on the Bristol Research PortalPDF-documentUniversity of Bristol – Bristol Research PortalGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/brp-terms/58 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 1, JANUARY 2005Calculation of Losses in 2-D Photonic CrystalMembrane Waveguides Using the 3-D FDTD MethodM. J. Cryan, D. C. L. Wong, I. J. Craddock, S. Yu, J. Rorison, and C. J. RailtonAbstract—The three-dimensional finite-difference time-domainmethod is used to obtain loss per unit length in a two-dimen-sional photonic crystal membrane waveguide by simulating threedifferent length guides. Results are shown for propagation bothabove and below the light line. The results are compared with aFourier expansion method and good agreement is obtained aboveand below the light line.Index Terms—Finite-difference time-domain (FDTD), photoniccrystals (PCs), two-dimensional photonic crystal waveguides (2-DPC-WGs).I. INTRODUCTIONTHERE IS much interest in two-dimensional (2-D) photoniccrystal waveguides (PC-WGs) which show the potentialfor low-loss guiding [1] and enhanced functionality over simpleuniform high-contrast waveguides due their strong confinementand dispersion properties [2]. 2-D PC-WGs can be broadlydivided into two categories, those with weak and those withstrong vertical confinement. The latter are known as membraneguides and can be either suspended in air or used in a silicon-on-insulator configuration [2]. This high vertical contrast leadsto wide wavelength ranges where the propagating modes arebelow the light line which implies zero coupling to radiationmodes and, hence, zero loss (neglecting effects such as surfaceroughness and material loss). In the former case of weakvertical confinement, propagation below the light line is muchmore difficult to achieve and losses in these structures tendto be higher. Although when used in actual circuits includingbends and Y-junctions improved relative performance has beensuggested [2], [3]. Here, we will concentrate on the strongvertically confined structures since these have most publishedloss results and the below light-line region gives a useful checkon the accuracy of the results.II. MODELLINGWhile measured loss values in PC-WGs are reducing quiterapidly, there is still a strong requirement to fully understandthe lower limits that can be achieved in the various guidingtechnologies. Below the light line, these are reasonably clearsince surface roughness and material loss are well understoodfor standard high-contrast waveguide [2]. However, in the morecomplex propagation regimes possible in PC-WGs, furtherwork is still required. Above the light line is a more interestingManuscript received December 17, 2003; revised June 10, 2004.The authors are with the Department of Electrical Engineering, University ofBristol, Bristol BS8 1UB, U.K. (e-mail: m.cryan@bristol.ac.uk).Digital Object Identifier 10.1109/LPT.2004.837914case where intuitively one would expect high losses, however,low loss guiding [4], [5] has been predicted.In order to understand these lower limits, three-dimen-sional (3-D) modeling is required. There are in general twoapproaches: First one can calculate the complex propagationconstant for the waveguide modes, the imaginary part thengives the loss directly [4], [6], [7]. An alternative approach is touse time-domain numerical methods such as finite-differencetime-domain (FDTD) [8], the transmission line method [9], orthe finite element method in time domain. In [9], the wholewaveguide structure is discretised in 3-D space and the totalpower, derived from the Poynting vector, is calculated at twoclosely spaced planes in the guide and from this a loss per unitlength can be calculated. In [8], the approach of using twodifferent length guides including input and output waveguideswas introduced. This has the advantage of producing largerlosses for the long individual guides which should improve theaccuracy of the final loss per unit length figure and gives veryrealistic excitation. While the approach outlined here and in [8]is computationally intensive, it has a number of advantages.First, the structures need not be periodic in any sense, theycan include surface roughness, wavelength-dependent mate-rial losses, and even nonlinear loss [10] such as two-photonabsorption. Second, and possibly more importantly, where thewaveguide is multimoded, the guide can be excited in a veryrealistic way, for example from a deep etched ridge guide. Thisallows the mode from the input guide to couple to the correctcombination of guided modes, resulting in accurate propagationmodeling.This work uses the FDTD method which has been widelyused in microwave engineering for almost 40 years since Yee’sseminal paper [11]. In recent years with the advent of low-costrandom access memory (RAM) and high-performance desktopmachines, FDTD has been applied in the optics regime [10].Here, an in-house code which has been developed over a numberof years is used.III. RESULTSThe structure to be studied is shown in Fig. 1. This isthe middle length guide( ), two other simulations withm and m are also performed.The structure has uniform meshing in the plane, withnm and nm. This results in 20 cellsper lattice constant, in the direction and 37 cells perin the direction. This produces uniform meshing across theholes and keeps staircasing errors constant across the structure.In the vertical direction, nonuniform meshing is used with anm, resulting in 16 cells in the membrane.1041-1135/$20.00 © 2005 IEEECRYAN et al.: CALCULATION OF LOSSES IN 2-D PC MEMBRANE WGs USING THE 3-D FDTD METHOD 59Fig. 1. PC membrane waveguide with r=a = 0:3016, a = 430:55 nm, h =0:6a = 258:3 nm, L = 6:028 m; 10:76 m; 19:37 m, L =11a;22a;42a, d = 870 nm, W1 guide in   K direction, n = 3:4.This results in million cells for the shortguide, million cells for the middle guide,and million cells for the long guide.The PC is hexagonal with a lattice constant of 430.55 nm, witha one missing hole (W1) waveguide in the direction.It should be pointed out here that only six rows of holes havebeen used in the “walls” of the waveguide in these simulations,whereas a minimum of seven are used in [4]. The structure is ex-cited with the fundamental transverse-electric (TE) [ domi-nant] mode of the input waveguide. This mode is obtained froma separate 2-D FDTD simulation which finds the eigenmodesof the cross section of the guide. This is crucially importantin exciting the structure realistically. Furthermore, it enablesa specialized transmission calculation to be performed whichwe have called the Modal S-parameter calculation [12]. Here,an overlap integral between the input mode and the fields at aplane (known as a snapshot) specified in the output waveguideare stored at each time step. This enables a modal S-parameter ortransmission coefficient to be defined. That is the ratio of outputpower in the fundamental mode of the output guide to that at theinput. In fact a separate 3-D simulation is performed with justa straight waveguide in order to obtain the incident power. Thismeans that higher order modes coupled into the output wave-guide will be ignored—this is quite realistic since in 100 mof output guide that would in reality be present, the high lossesof these modes generally means that they would not be mea-sured. Since the waveguide has a symmetry plane at its centerand the structure is being excited by the fundamental TE mode,a perfect electric conducting (PEC) boundary can be introducedthere without altering the results. This allows the simulationsize to be halved. The computational box must be terminatedwith absorbing boundaries (apart from the center where a PECFig. 2. Comparison of losses in PC-WG of Fig. 1 between FDTD with Murand PML boundaries and Fourier expansion method [2], [3].boundary is used). Two different boundaries have been used.Firsy, Mur first-order [10] and second perfectly matched layers(PML) [10]. The longest waveguide with PML boundaries re-sults in a simulation requiring 800 MB of RAM and taking140 h on a Pentium 4 2.4-GHz machine. It must be remem-bered that for this one time-domain simulation, we obtain thou-sands of usable frequency points, since the time-domain datais Fourier transformed. This results in run times per frequencypoint of around 6 min.The essence of this approach is to calculate the transmissioncoefficient for two different length guides. The difference be-tween these two results, with a knowledge of the difference inlengths provides a loss per unit length figure. This is closely re-lated to the cut-back method used for measured data [1]. Theuse of different length waveguides provides some correction foreffects such as coupling loss into the PC guide. More advancedcorrection techniques using full two-port S-parameters are cur-rently being investigated.Fig. 2 shows a comparison between 3-D FDTD with Murand PML boundaries with the results from Lalanne [4]. FDTDresults for and are also shown. The loss resultshave been scaled in the same way as those shown in [13], toaccount for different lattice constants. This scaling is valid sinceMaxwell’s equations scale with length, however, we are usinga fixed unit length for comparison of losses. A lattice constantof 430.55 nm was chosen here to place 1550 nm below the lightline.The first point to note is that the light line for this waveguidemode is around 1.52 m and the reduction in loss is clearlyshown in both methods. Below the light line, the Fourier methodgives very low losses as might be expected for a method whichcalculates the Bloch modes of the waveguide. The FDTDresults, however, are much less idealized calculations and, asshown in Fig. 2, have a strong ripple especially below the lightline of around 10 dB/mm. The average of this ripple isclose to zero which suggests that, fundamentally, the FDTDresults are low loss but they have an error superimposed uponthem. The origin of this ripple will be discussed later. Themain point to observe in Fig. 2 is that good agreement is beingmaintained over a wide wavelength range. It is seen that theand give very similar results suggesting that60 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 1, JANUARY 2005Fig. 3. Comparison of output power density with and without Hammingwindowing for L1 data.Fig. 4. Comparison between loss results for Fourier method and FDTD withHamming windowing.the shorter length guides could be sufficient for loss calculation.This results in much shorter simulation times. The results alsohighlight the importance of using PML boundaries.The most obvious explanation for the observed ripple wouldbe multiple reflections from the input and output of the wave-guide, but this was found not to be the case. In fact, the causewas found to be in the early truncation of the time-domain data.Fig. 3 shows the standard FDTD output power density and it isseen that at around 2.25 ps, the simulation stops. This resultsin an abrupt discontinuity in the time-domain data which trans-lates to ripple in the frequency domain. To counteract this, thesimulation would need to be run for a longer time. An alterna-tive approach is to use a windowing function. Fig. 3 also showsthe effect of applying a Hamming windowing function to thetime-domain data. As can be seen, the discontinuity is now muchreduced.A Hamming window has been applied to the data sets forthe three different length guides and the losses recalculated.The results are shown in Fig. 4. As can be seen, the ripple is nowmuch reduced and the data is close to the average of the previousripple. It is seen that much improved agreement below the lightline has been obtained. Around 1.4 m, there is a small discrep-ancy between FDTD and the Fourier method; this is thought tobe due to the different discretization schemes being used in thetwo methods [14] and is currently under investigation.IV. CONCLUSIONThis letter has presented an approach based on the 3-D FDTDmethod to determine the loss per unit length in PC membranewaveguides. Good agreement with a Fourier expansion methodhas been shown. The problem of early truncation of time-do-main data has been overcome with the use of a Hammingwindow. The use of the FDTD method enables effects such assurface roughness, disorder, and nonlinear losses to be studied.The use of modal excitation and modal S-parameters allowsvery realistic simulations to be performed.ACKNOWLEDGMENTThe authors would like to thank P. Lalanne and C. Sauvan foruseful discussion of results.REFERENCES[1] S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic inte-grated circuit with membrane-type photonic crystal waveguides,” Opt.Express, vol. 11, no. 22, pp. 2927–2939, Nov. 2003.[2] T. F. Krauss, “Planar photonic crystal waveguide devices for integratedoptics,” Phys. Stat. Solidi, vol. 197, no. 3, pp. 688–702, 2003.[3] S. Olivier et al., “Coupled-mode theory and propagation losses in pho-tonic crystal waveguides,” Opt. Express, vol. 11, no. 13, pp. 1490–1496,Jun. 2003.[4] P. Lalanne, “Electromagnetic analysis of photonic crystal waveguidesoperating above the light cone,” IEEE J. Quantum Electron., vol. 38, no.7, pp. 800–804, Jul. 2002.[5] T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittanceof a hexagonal photonic crystal slab,” Phys. Rev. B, vol. 63, no. 125107,pp. 1–7, 2001.[6] G. R. Hadley, “Out-of-plane losses of line-defect photonic crystalwaveguides,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 642–644,May 2002.[7] L. C. Andreani and M. Agio, “Intrinsic diffraction losses in photoniccrystal waveguides with line defects,” Appl. Phys. Lett., vol. 82, no. 13,pp. 2011–2013, Mar. 2003.[8] M. Cryan et al., “Analysis of losses in 2D photonic crystal waveguidesusing the 3D finite difference time domain (FDTD) method,” in IEEELEOS, Glasgow, U.K., Nov. 2002, pp. 516–517.[9] Y. Desieres et al., “Propagation losses of the fundamental mode in asingle line-defect photonic crystal waveguide on an InP membrane,” J.Appl. Phys., vol. 92, no. 5, pp. 2227–2234, Sep. 2002.[10] A. Taflove, Computational Electrodynamics—The FDTDMethod. Norwood, MA: Artech House, 2000.[11] K. S. Yee, “Numerical solution of initial boundary value problem in-volving Maxwell’s equations in isotropic media,” IEEE Trans. AntennasPropag., vol. AP-14, pp. 302–306, Mar. 1966.[12] I. J. Craddock et al., “Applications of single mode extraction from finitedifference time domain data,” in Proc. Inst. Elect. Eng. Microwave An-tennas Propag., vol. 146, Apr. 1999, pp. 160–162.[13] C. Sauvan et al., “Accurate modeling of line-defect photonic crystalwaveguides,” IEEE Photon. Technol. Lett., vol. 15, no. 9, pp. 1243–1245,Sep. 2003.[14] P. Lalanne and C. Sauvan, private communication, Jun. 2004.

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Calculation of losses in 2-D photonic crystal membrane waveguides using the 3-D FDTD method

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