modern olympics are inspired by the ancient version of the games , but based on a wider idea of globality .while ancient games were opened only to greek speaking athletes  , modern olympics were , since their beginning , considered a world event involving people from every part of the globe  .the same symbol of the olympics , composed of five interlocking rings standing for the five continents , was designed by the _ baron pierre de coubertin _ , the founder of the modern olympic games , with the aim of reinforcing the idea that the games are an international event and welcome all countries of the world  . since athens , editions of the event has been organized in different locations around the world , and , from the participants representing nations of the first edition , the games have grown to about competitors from countries at the latest edition of the summer games of beijing .the olympics are one the most important events worldwide not only for sports , but also for politics and society. many important facts of the last century history , such as the nazism  , the israeli - palestinian conflict  , and the cold war  , have influenced the regular organization of the games .also , the olympics generally play a fundamental and positive role for the economic and urban development of the city that hosts the event  .+ performance data of athletes at the olympics are available for each modern edition of the games organized so far , and represent an optimal proxy for the study of human limits in sport performances for three main reasons : ( i ) data cover more than a century of sport performances since the first edition of the olympics dates back to ; ( ii ) olympic data provide a detailed record of sports performances at regular -year intervals ; ( iii ) the performances of olympic medalists truly reflect the best achievements that could be obtained in a given historic moment because , in the vast majority of sport disciplines , the games have always represented the most important event during the career of an athlete , and consequently all the greatest athletes have always taken part in the olympics .+ latest years have witnessed the appearance of a large number of statistical studies of data coming from professional sports .examples include basketball  , baseball  , soccer  , tennis  , etc .also olympic performance data have been the subject of many analyses  .some of them focused on models aimed at the description of performance progression along time , including linear models  that can even lead to unrealistic results  , s - shaped curves  and logistic functions  .others studied statistical properties of performance patterns , such as the power - law relation between time ( or speed ) and length of running events  .in addition , performance data of athletes at the olympics have been used to tune the parameters of complicated models aimed at the determination of physiological limits in sport performances  .for example , according to a mathematical model for human running performance that accounts for various energetic factors , such as capacity of anaerobic metabolism , maximal aerobic power and reduction in peak aerobic power , perronet and thibault predicted the limiting times that athletes can reach in various running events in athletics  .+ in spite of the numerous efforts however , we still miss a general description for the performances of athletes . we still miss a universal way to predict limiting performance values and calculate the probability of future achievements in sport . in this paper , we address all these issues by generating a simple and coherent picture for the description of the performances obtained by olympic medal winners in all specialties of athletics and swimming .we analyze historic performance data and provide empirical evidence about the discovery of a novel statistical law governing performances of medal winners at the olympic games . with a self - consistent approach we simultaneously ( i ) show that performance improvements obey a universal law , ( ii ) estimate limiting performance values , ( iii ) predict future achievements at the olympics .while former statistical studies have mainly analyzed the progression of absolute performance values along the various editions of the games , here we change point of view and focus our attention on relative improvements in performances between two consecutive editions of the olympics .let us indicate with the value of the performance obtained by the gold medalist in a specific specialty at the edition of year of the olympic games .depending on the specialty , may indicate time ( running and swimming ) , length ( long and triple jumps ) , height ( high jump and pole vault ) , or distance ( discus and hammer throws , shot put ) .we define the relative improvement of the gold - medal performance in the games of year with respect to the gold - medal performance in the previous edition of the olympics as where represents the gap between the performance value of the gold medalist in year and the asymptotic performance value .the asymptotic or limiting performance value is a unknown parameter representing the physiological limit that can be achieved in the specialty by an athlete .[ eq:1 ] defines the relative improvement towards the asymptotic performance value of the gold medalist in year with respect to the performance of the gold medalist in year .note that the same definition can be used for the measurement of the relative improvements of silver and bronze medalists , and in principle for athletes who have reached any arbitrary rank position .+ for reasonable values of , we find that the distribution of the relative performance improvements is statistically consistent with a normal distribution .we determine the best estimate of the asymptotic performance value as the value of for which the statistical significance ( -value ) of the normal fit is maximized ( see materials and methods section ) .the procedure is generally accurate and allows us to identify reasonable values of in all specialties considered in this study . in fig . 1 for example , we report the results obtained by analyzing performance data of male athletes in meters sprint .the best estimate of the asymptotic time is seconds . for this value of , we find that relative improvements obey a normal distribution with average value and standard deviation .statistical significance , however , can be used not only for the determination of the best estimate of the asymptotic performance value , but also , in a broader sense , to define confidence intervals for . in the case of meters sprint of male athletes for example ,we find that , at significance level , is in the range to seconds . at significance level ,the interval is restricted and is in the range to seconds , while , at significance level , is expected to be between and seconds .the results shown in fig . 1 are obtained by analyzing the relative performance improvements of gold - medal winners .similar results are , however , obtained when considering the performances of silver and bronze medal medalists ( fig .interestingly , the finiteness of the data does not affect the reliability of the best estimate of the limiting performance value since compatible values of can be detected by removing results of the latest editions of the games from the analysis ( fig .+   lower than the actual olympic record , we evaluate the goodness of the fit of performance improvements with a normal distribution . is determined as the value of the asymptotic time that maximizes the statistical significance ( -value ) . for men meters sprint ,our best estimate is seconds , where we find that relative performance improvements are normally distributed with a confidence of . for this value of ,the best empirical estimates of the average value and standard deviation are respectively and . *b. * the cumulative distribution function of the -scores obtained for ( red curve ) is compared with the standard normal cumulative distribution ( black curve ) . * c. * normal sample quantile are plotted against normal theoretical quantiles  .the dashed line corresponds to the theoretically expected behavior in case of a perfect agreement between sample and theoretical distributions . *d. * -scores of relative performance improvements between consecutive editions of the games.,scaledwidth=45.0% ]   of the asymptotic performance value , while in the insets we provide a graphical comparison between the sample cumulative distributions ( red line ) and the standard normal cumulative distribution ( black line ) . * a * and * b. * we report the results obtained by the analysis of the performances of male athletes in marathon ( seconds , -value ) and female athletes in long jump ( meters , -value ) .* c * and * d. * we show the outcome of our method for performances of men and women in meters sprint ( respectively , seconds and -value , seconds and -value ) ., scaledwidth=45.0% ]  the normality of the relative improvements towards the asymptotic performance value is a simple and strong result . at each new edition of the games ,gold - medal performances get , on average , closer to the limiting performance value .the average positive improvement observed in historic performance data can be motivated by several factors : as time goes on , athletes are becoming more professionals , better trained , and during the season have more events to participate in ; the pool for the selection of athletes grows with time , and , consequently there is a higher level of competition ; the evolution of technical materials favors better performances . on the other hand , there is also a non null probability that winning performances become worse than those obtained in the previous edition of the games ( i.e. , relative improvement values are negative ) .all these possibilities are described by a gaussian distribution that accounts for various , in principle hardly quantifiable , factors that may influence athlete performances : meteorological and geographical conditions , athletic skills and physical condition of the participants , etc .the accuracy of the normal fit is not only testified by its high statistical significance , but also by graphical comparisons between the sample distribution and the theoretical normal distribution ( see figs .1b and c ) .it is also important to note that the values of the relative improvements do not depend on the particular edition of the games , and thus their distribution is stationary ( fig .1d ) . the strength of our results , however , is not only in the significance of the fits , but especially in its generality .we repeated the same type of analysis for a total of different specialties , and found that performance improvements are governed by a universal law .first of all , the law holds for all running events in athletics .this is valid for an heterogeneous set of running distances ranging from to meters ( marathon , fig . 2 and supporting information s1 ) .second , our analysis suggests that relative improvements are normally distributed not only when considering time performances , but also performances regarding length or height ( jumps ) and distance ( throws ) . in fig.2b for example , we report the outcome of our method when applied to performance data of female gold medalists in long jump .other examples can be found in supporting information s2 .finally , the law is valid for performance improvements of athletes in swimming specialties ( supporting information s3 ) .+ given the attention received in the recent past  , we reserve a special consideration to the comparison in performances between female and male athletes in meters sprint . in fig . 2c and 2d, we report the results obtained through the analysis of olympic performances in this specialty . according to our analysis ,the best estimate of the limiting time for males is seconds , while for females we identify the best estimate for the asymptotic time at seconds .our statistical analysis predicts that women will be always slower than men and that the gap will saturate at about , consistent with the estimation by sparling _ but in disagreement with what predicted by the unrealistic model of atkinson _ et al _it should be noted that for women the statistical significance is less predictive than the one measured for men . while for menwe observe that statistical significance is clearly peaked around and goes rapidly to zero as decreases , the same does not happen in the case of women .we believe that the statistics are less accurate because the analysis is based on editions instead of since women started to run the meters sprint only in amsterdam , while men already in athens .in particular , the lack of sufficient data provides high statistical significance also for the unrealistic seconds .we expect , however , that the future addition of more data point will suppress this effect . despite these problems ,our analysis still produces meaningful estimates of the upper bound of the asymptotic time : at significance level , the asymptotic value is expected to be lower than seconds , while at significance level , should be lower than seconds . also ,our best estimates of the limiting performance values are probably not as accurate for this specialty ( or other short distances ) because there is not enough reliable performance data regarding the first editions of the games ( automatic time was introduced in mexico city ) .the removal of data points for male meters sprint before amsterdam ( and in general of a few data points from the entire time serie ) leads also to the impossibility to determine the best estimate of the asymptotic time as a global maximum of statistical significance ( see fig .s3 ) . for meters sprint , we have performed therefore an additional analysis in which we aggregated together the results of gold , silver and bronze medalists and obtained slightly different estimates for the limiting performance values [ seconds for men ( fig .s4 ) and seconds for women ( fig .s5-s6 ) ] . +   and the length of the race for men running events in athletics ( red circles ) .we excluded from the analysis relay and hurdles events .we find that , and the best estimate of the power - law exponent is ( black line ) .* b. * probability density functions of the winning time for the men meters sprint in future editions of the games .the dashed line represents the winning time in the latest edition of the olympics in beijing .this value is used as initial condition for the prediction of future performances . * c. * the probability density of the winning time in men meters predicted by our model is compared to past performance data ( black circles ) .the density plot is obtained by convoluting the various prediction curves derived from real data . *d. * probability that athletes will breach challenging walls in various specialties of athletics as a function of time.,scaledwidth=45.0% ]  [ cols="<,<,>,>,>,>,>,>,>,^",options="header " , ]   in general , our approach produces good results for specialties with a sufficiently long tradition in the games .this is basically the case of all male specialties in athletics. data about female performances typically provide less accurate results , but still , in the majority of the cases , the predictions of the asymptotic performance values are reasonable .we summarize in table 1 the results obtained for some specialties , while we refer to the supplementary information for a systematic analysis of all of them .it should be noted that there are also a few cases in which things do not work perfectly . in women800 meters , for example , statistical significance does not exhibit any peak value ( supporting information s1 ) .there are also a few specialties in which the best estimate of the limiting performance value does not correspond to the global maximum of statistical significance ( supporting information s1 ) . in these cases ,statistical significance is a non monotonic function of the and more maxima are present .still the peak value that appears more plausible can be used as an estimate of . finally , there are three specialties in athletics in which a clear peak in statistical significance is visible only by excluding performance data of sidney , but this exclusion is fully justified by the fact that the top athletes of the moment did not take part in the competition ( supporting information s1 ) .for example , about the men 200 meters sprint of sidney , the web site sports-reference.com reports : this race was expected to be between the americans _maurice greene _ and _ michael johnson_. greene was the best in the world at meters and johnson at meters , and their race in the middle distance was highly anticipated .but neither qualified for the team at the olympic trials , succumbing to minor injuries , although they both made the team in their better events . "+ the good accuracy of our best estimates of the limiting performance values is supported also by the power - law relation between these quantities and the length of the running events in athletics ( see fig .3a ) . as already observed by katz and katz , world record times ( ) and running distances ( )are related by the power - law relation and katz studied the relation between world record performances and running distances in various epochs , and found that the power - law exponent value is always slightly larger than but decreases for more recent epochs .for example , they measured in , and in .on the basis of our measurements , we claim that the asymptotic value of the exponent will be exactly , when limiting performance values , and thus definitive world records , will be reached in all specialties of athletics . +a final application of our findings is the prediction of future performances at the olympics .the performance value of the gold medalist in london 2012 , for example , can be estimated as , where is a random variate extracted from the normal distribution with mean value and standard deviation .similar equations can be written also to predict performance values of the other editions after london .for each future edition of the games , we can draw a distribution of performance values ( see fig .the distribution is normal for the edition of , but diverges from normality as time grows .in particular , while the expected performance value decreases exponentially towards the asymptotic performance value as time increases , the standard deviation initially grows as we move further in future until predictions become again more accurate because of the boundary effect of ( see fig .+ by simply looking at the performances expected at the next edition of the games in london , we can ask what is the probability that the winner of the gold - medal will beat the actual world record of her / his specialty . in table 1 , we list these probabilities for some specialties together with the most likely performance values that gold - medal winners will obtain . in athletics , there are not negligible chances ( about ) that the actual world records of meters , meters hurdles and marathon will be lowered by men . in swimming specialties ,the expectations are more promising : there is a good probability ( higher than ) that the world record of 1,500 meters freestyle will be beaten by male athletes .+ relevant limits are unlikely to be broken at the next olympics ( fig .we will have to wait until in order to have a chance that a man will run the meters in less than seconds .for other specialties , expectations ( probability higher than ) are even less promising : men will run the meters in less than seconds and the marathon in less than two hours ( seconds ) only after , women will run the meters sprint in less than seconds only after , and finally the wall of minutes ( seconds ) in meters will likely be breached by male athletes only after year .in conclusion , our paper shows that the performance of olympic medal winners in athletics and swimming obey , independently of the type of specialty , a simple universal law .if performance improvements are calculated with respect to an asymptotic performance value , then the relative difference between improvements obtained in two different editions of the games is a random variate following a normal distribution .this is the common property of a broad class of natural phenomena that be described by the theory of biased random walks  , such as the locomotory movements of organisms responding to an external stimulus  , the activity of spiking neurons  , the trends of daily temperatures  , stock prices  , capital markets  , etc .+ the normality of the relative improvements can not be explained in trivial terms , especially in this case where the statistics is performed on extremal properties of the system .remember in fact that the performance values analyzed here are those obtained by the best athletes of a given edition of the olympics ( i.e. , potentially the best performers on the earth ) , and thus it is natural to expect that absolute performance values obey statistical laws of extremes  .more importantly , since the distribution is normal , it makes sense to refer to average trajectories of top performance values along editions of the games .our findings in fact allow to say that , on average , the absolute performance value of top athletes at the olympics gets closer to the limiting performance value in an exponential fashion , with a rate of about in athletics and in swimming .more in detail , the average trajectory of the performance value can be described by the equation where is an arbitrary initial edition year of the olympics and is the performance value measured in year .eq.2 can be derived directly from eq.1 and the fact that relative improvements are normally distributed but only under the assumptions that the edition year of the olympics is considered as a continuous variable and that .note that this observation is important for stressing the difference between our fitting procedure and a more straightforward analysis based on the exponential fit of absolute performance values , as the one used to find that the progression of world record performances follows a piecewise exponential decaying pattern  .note also that the analysis of the only olympic performances differs from the one of world record performances for the following reasons : ( i ) the relative change between two world records , if defined in a similar manner as eq.1 , can be only a positive quantity ; ( ii ) the time difference between two world record performances is not a constant , but a random variate by itself . because the number of events in which new world records can be established is higher today than it was one century ago ( and they had been growing in the course of the years ) , in any analysis of the progression of world record performances time should be rescaled to account for that  .+ the asymptotic performance value is an _ a priori _ unknown variable whose value can be self - consistently determined by maximizing the statistical significance of the normality fit .it is particularly important to stress that our simple methodology provides good estimates of performance limits that are in general consistent with those obtained through complicated physiological models  .for example , perronet and thibault predicted that the limiting time for men in marathon is hour , minutes and seconds  . with our minimalistic model, we are able to predict that this limiting time is between hour , minutes and seconds and hour , minutes and seconds ( for men marathon the peak of statistical significance is wide , see fig .at the same time , it is also important to stress that our minimalistic analysis can also lead to little inconsistencies .for example , the best estimates of obtained here state that , asymptotically , the average pace in marathon would be higher than the one in meters .this means that according to our estimates , the first meters in marathon would be run in less than minutes , while the entire race of meters would be run asymptotically in more than minutes .this inconsistency can be partially explained by the fact that the statistics for meters is less reliable because based only on events , while the one for marathon on the results of editions of the games . in general, it is very important to remark that , at the moment , we are able to provide only good estimates of the asymptotic performance values because such estimates are based on a relatively small set of empirical data ( at best editions of the olympics ) , and therefore must be taken with a grain of salt .we expect in fact that , while the normal law governing performance improvements will likely continue to hold , the accuracy in the estimation of the asymptotic performance values will improve with the addition of more data points in the future , starting already from the next edition of the games in london .medal lists and results of all editions of the olympic games have been collected from the web sites www.sports-reference.com and www.databaseolympics.com . whenever possible , we considered automatic measures of time instead of manual ones .we included in our study all results obtained in the editions of the modern olympic games since athens 1896 , but we excluded from the analysis data about the so - called `` intercalated '' edition of the games held in athens in .we focused on sports classified as `` track & field '' and `` swimming '' , and particularly on specialties of these sports that have been performed at least in the latest ten editions of the olympic games .we compared only performances between subsequent editions of the games held at four years of difference .we excluded therefore comparisons between either the consecutive editions of stockholm and antwerp ( separated by world war i ) , and those of berlin and london ( separated by world war ii ) . + for consistency , we considered only specialties whose rules or techniques have not changed during time .for example , we excluded javelin throw because of the javelin redesign in .we also excluded performances in high jump before mexico city when athletes started for the first time to adopt the modern jump style called `` fosbury flop '' .+ data are made available for download at filrad.homelinux.org/resources .the results reported in the paper are based on the normality test introduced by anderson and darling  .given a value of , we compute the best estimates of the mean and the standard deviation as and , respectively .the relative improvement is defined in eq.1 . indicates the number of results between consecutive editions of the olympic games that are included in the analysis .we then compute the -scores as and rearrange them in ascending order such that .the anderson - darling distance is computed with the formula $ ] , where is the standard normal cumulative distribution function .we further use the modified statistics , suitable in the case in which both the mean and standard deviation are estimated from the data as suggested by stephens  .+ we evaluate the goodness of the fit by generating random number sequences of length extracted from the standard normal distribution .the statistical significance of the normality test ( -value ) is calculated as the number of artificial sequences whose is larger than the one measured for real data divided by the total number of generated sequences . note that there is a trivial monotonic relation between the -value and the anderson - darling distance , and therefore the maximum of the -value corresponds to the minimum of .+ we used the normality test by anderson and darling because this test is considered one of the best empirical distribution function statistics for detecting most departures from normality , and can be used for testing the normality of very small sample sizes  .we verified , however , the robustness of our results by using other standard normality tests , including those based on the criteria of kolmogorov - smirnov , cramr - von mises and shapiro - wilk  .we also verified the consistency of our results with normality tests based on the moments of the distributions ( see fig .+ furthermore , we tested the accuracy of our fitting method by implementing a bootstrap procedure  , and found that our fitting method is able to well recover the correct parameter values in artificial sequences generated according to our model ( see fig .we thank c. castellano , p.s .dodds , e. ferrara and a. hockenberry for comments and suggestions on the manuscript .

Abstract: inspired by the games held in ancient greece , modern olympics represent the world s largest pageant of athletic skill and competitive spirit . performances of athletes at the olympic games mirror , since , human potentialities in sports , and thus provide an optimal source of information for studying the evolution of sport achievements and predicting the limits that athletes can reach . unfortunately , the models introduced so far for the description of athlete performances at the olympics are either sophisticated or unrealistic , and more importantly , do not provide a unified theory for sport performances . here , we address this issue by showing that relative performance improvements of medal winners at the olympics are normally distributed , implying that the evolution of performance values can be described in good approximation as an exponential approach to an _ a priori _ unknown limiting performance value . this law holds for all specialties in athletics including running , jumping and throwing and swimming . we present a self - consistent method , based on normality hypothesis testing , able to predict limiting performance values in all specialties . we further quantify the most likely years in which athletes will breach challenging performance walls in running , jumping , throwing and swimming events , as well as the probability that new world records will be established at the next edition of the olympic games .


we show how large - scale computational optimization can be used to design superior and non - intuitive structured illumination patterns that achieve 20-fold enhancements ( for fixed incident - field intensity ) of the optical torque on sub - micron particles , demonstrating the utility of an optimal design approach for the many nanoscience applications that rely on optical actuation of nanoparticles .recent advances in nanoparticle engineering and holographic beam - generation via spatial light modulators ( slms ) and other phase - manipulation techniques have created many new degrees of freedom for engineering light particle interactions beyond traditional optical tweezers . enhanced and unusual optical forces and torques can be engineered by designing material objects and/or structured illumination , with the latter including `` tractor beams '' and beams carrying optical angular momentum .these increased degrees of freedom pose an interesting design challenge : for a given target object , what is the optimal illumination pattern to produce the strongest optical force or torque ? while a small number of gaussian beam parameters can be manually calibrated for optimal performance using manual trial - and - error , an arbitrary 3d vector field requires a more targeted approach .moreover , optimization of a 3d vector field is highly nonconvex by nature , and posesses many local optima due to wave interference and resonance .when exploring so many parameters , a large number of scattering problems must be solved efficiently , which requires careful design of the optimization framework .research in computational optimization of optical actuation has focused on the design of new material geometries and on the improvement of multiplexed optical traps for microscale dielectric particles ( holographic optical tweezers ) .however , no computational method has been available to design structured illumination for unconventional target objects that are nonspherical , lossy , or nanometer - scale , thereby requiring a costly full - wave numerical simulation for computing optical force and torque .we present a compact and rapid computational framework to optimize structured illumination for the mechanical actuation of an arbitrary target object .we combine ( i ) a compact bessel - basis representation ( sec.[subsec : analytical - representation - of ] ) ; ( ii ) a numerical solver based on boundary element method ( bem ) that discretizes the surfaces of a 3d scattering problem to form a bem matrix , and solves hundreds of thousands of incident - field configurations using the same matrix ( sec.[subsec : numerical - solver ] ) ; ( iii ) an appropriate optimization algorithm that exploits the smoothness of the nonconvex and nonlinear optimization problem ( sec.[subsec : optimization - algorithm ] ) ; and ( iv ) a suitable figure of merit ( fom ) and optimization constraints ( sec.[subsec : figure - of - merit ] ) . as a result, we rapidly attain many - fold improvements in optical forces and torques over random field or plane wave illuminations ( sec.[subsec : optimization - results ] ) , and discuss the tradeoff between enhancement and robustness of optimization ( sec.[subsec : robustness ] ) .furthermore , a given material object may have scattering resonances at various frequencies , and the choice of frequency for the incident field has several important implications . when comparing interactions with two different resonances , we were able to distinguish the impact of the change in the resonant field pattern from the change in the resonance lifetime .controlling for the change in lifetime , we found that torque seems to favor higher - order ( _ e.g. _ quadrupole ) resonances with greater angular momentum , while force seems to favor lower - order ( _ e.g. _ dipole ) resonance with greater field intensity within the particle ( sec.[subsec : force ] ) .a structured - illumination optimization aims to find the best 3d vector field , i.e. , the one that maximizes a desired fom for a given scattering problem ( see fig.[fig : schematic ] ) .our choice of the bessel basis expansion is described in sec.[subsec : analytical - representation - of ] .sec.[subsec : numerical - solver ] and sec.[subsec : optimization - algorithm ] discuss the bem solver and the optimization algorithm .lastly , sec . [subsec : figure - of - merit ] explains force and torque foms and the corresponding optimization constraints .the entire numerical framework is implemented with ` c++ ` ., height=264 ]   the computational design of structured illumination requires a compact analytical representation of an arbitrary 3d vector field .the vector field will contain spatial variations in both intensity and phase , and must satisfy the vector wave equation .the electric field can be represented using a basis expansion , where the complex scalar coefficient determines the relative intensity and phase of each mode .the choice of coordinates and the basis functions * * depends on the problem geometry .while the spherical coordinate system is a common choice in mie scattering , it requires a very large number of modes to represent light propagating along a linear axis and potentially interacting with flat substrates or slms .the cartesian coordinate system is also ill - suited because it requires large to describe laser beams with a finite radius .we find that the cylindrical coordinate system is well suited , requiring a small number of modes to describe structured illumination with varying distributions of linear and angular momentum .  among the wide menu of cylindrical basis functions ( e.g. , bessel , laguerre - gaussian , hermite - gaussian , and so forth ) , we choose the bessel basis for its compact analytical expression , derived from the scalar generating function where is the mth - order bessel function , is the transverse wavevector in , and is the longitudinal wavevector in , satisfying . the ratio between and specificies the numerical aperture of the basis , .higher _ na _ represents greater transverse momentum that can increase optical torque .but the permissible range of _ na _ is often dictated by experimental considerations , and the range of _ na _ in the optimization can be set accordingly .  taking spatial derivaties of eq.([eq : besspsi ] )gives where is the unit vector in , and and are the bases for azimuthal and radial polarizations , respectively ( right inset of fig.[fig : schematic ] ) .the incident electric field can be expressed as : where and are the complex scalar coefficients .the bessel basis produces the most compact expressions for and because the magnitude of does not vary with , reducing terms in eqs.([eq : mi],[eq : ni ] ) .note that we intentionally decouple our optimization framework from the idiosyncratic differences in the spatial resolution of the slms .a wide variety of experimental methods ( e.g. , superposed pitch - fork holograms ) can be used to generate beams expressed as eq.([eq : eexpand ] ) , for a finite and _ na_. in this paper , we consider numerical apertures with opening angles and 12 basis functions ( ) .the optimization process itself has no restrictions on the choice of the numerical solver , so the biggest consideration is the computational cost : the smaller the the better .we choose the boundary element method ( bem ) for several reasons . in comparison to other scattering methodologies such as the finite - difference or finite - element methods ,bem is particularly well - suited to the type of large - scale optimization problem requiring a rapid update of the incident field for a given geometry and a given wavelength . in eq.([eq : bem ] ) , the bem matrix remains fixed for a given geometry and frequency , while the column representing the incident field is rapidly updated at each step of the optimization process .this allows hundreds of thousands of scattering configurations to be computed on the order of a few hours .in addition , bem projects the 3d scattering problem onto a 2d surface mesh , thereby reducing the computation volume by a factor of 1/2000 in the nanoparticle scattering problem we consider .lastly , recent improvements have significantly increased the speed with which optical froce and torque can be computed in bem .structured - illumination optimization is nonlinear and nonconvex , such that searching for a global optimum is prohibitively expensive .therefore we choose a local algorithm with random starting points .we choose one of the simplest solutions available : constrained optimization by linear approximation ( coblya ) , a derivative - free algorithm that exploits the smoothness of the problem .an open - source implementation of cobyla is available through nlopt .we consider two types of optical actuation with respect to the object coordinate ( left inset of fig.[fig : schematic ] ) ; the force and torque . in order to avoid the optimizer from increasing the brightness of the beam indefinitely , we choose to divide the force and torque by the average incident - field intensity on the particle surface ( , where is the impedance of free space ) , which is easily computed in bem .we choose the incident - field rather than the total - field intensity to avoid penalizing high extinction efficiency . is measured on the particle surface because we want to account for the portion of the beam that interacts with the target particle , rather than the entire beam .we choose nondimensionalized figures of merit : where the constants in parentheses reflect ideal single - channel scattering .the largest scattering cross - section into a single ( spherical harmonic ) channel is , which when multiplied by single - photon changes in linear ( ) and angular ( ) momentum per photon , divided by the photon energy ( ) , yields the constants in eqs.([eq : forcefom],[eq : torquefom ] ) .optimization constraints can be added to suppress actuation in undesired directions .we suppress actuations in directions other than using smooth constraints : and , where the limiting value 0.01 is set to ensure that and exceed of the force and torque magnitudes and , respectively .we demonstrate our illumination - field optimization framework on the gold nanotriangle illustrated in fig.[fig : schematic ] . our previous work analyzes the optical force and torque on such a particle for circularly - polarized ( cp ) planewave illumination .cp planewave is a common incident - field choice for torque generation due to its intrinsic spin angular momentum , but we find in our computational optimization that highly optimized field patterns can show 20x improvement of .the wavelength of illumination in each optimization , , is chosen to correspond to the plasmonic resonance wavelengths of the model particle .sec.[subsec : optimization - results ] presents the distribution of 2000 local - optimization results and discusses the optimized field - patterns .sec.[subsec : force ] analyzes the wavelength - dependence of optical force and torque for optimized illuminations , based on the choice of , and compares the results with the reference force and torque from cp planewave .lastly , the tradeoff between robustness and enhancement is discussed in sec.[subsec : robustness ] .   -axis indicate the initial ( triangle ) andfinal ( circle ) fom when the optimization starts from a circularly polarized plane wave .representative incident fields are plotted in the right .[ fig : distribution - dip],height=226 ]  , height=264 ]  the illumination - field design space is nonconvex and littered with local optima , due primarily to wave - optical interference effects .we survey this broader design space by restarting our local - optimization algorithm 2000 times with randomly selected initial configurations that are constructed using eq.([eq : eexpand ] ) , where the complex coefficients and are uniform random numbers bounded by .the results are summarized in figs.[fig : distribution - dip]-[fig : distribution - quad ] at ( dipole resonance ) and 625 nm ( quadrupole resonance ) , respectively . at both wavelengths, we find that more than 50% of local optimizations from random starting points can achieve over 5x enhancement of compared to cp planewave reference , and that the optimized field patterns contain various combinations of bessel - basis modes without a systematic convergence to one over the others .the distributions are plotted in log - scale to increase the visibility of small bins .  in fig.[fig :distribution - dip ] , the median at 0.913 is very close to the best at 1.01 and the distribution is predominantly concentrated to the right : the 4 rightmost bars represent 69% of all samples , which all acheive over 5x enhancement of compared to cp planewave reference at 0.169 ( marked with a red triangle ) .the insets show the optimization results from two different starting points random field ( top ) and cp planewave ( bottom ) that reach similar optimized field patterns .we also observe that a variety of other patterns , dominated by different combinations of bessel - basis modes , can produce a nearly identical or superior fom .  in fig.[fig :distribution - quad ] , the final distribution is more dispersed between the median at 3.934 and the best at 10.94 , which respectively achieve over 5x and 14x enhancement compared to cp planewave reference at 0.779 . as in fig.[fig : distribution - dip ] , the results concentrate heavily around the median ; however , in fig.[fig : distribution - quad ] a small number of samples achieve a remarkable improvement above 14-fold .the top inset shows four different field patterns that produce a nearly identical above the median , and the bottom inset shows the field - pattern with the highest .a comparison of all optimized field patterns at 1028 nm and 625 nm shows that the latter contains more higher - order bessel - basis contributions .    and ( b ) .the spectrums for the best ( blue line ) and the median ( red dashed line ) optimized field configurations are each labeled with the factor of enhancement , with respect to cp planewave reference ( black dashed line ) .[ fig : force - spectrum ] , height=226 ]  , ( b ) , and ( c ) .the spectrums for the best ( blue line ) and the median ( red dashed line ) optimized field configurations are each labeled with the factor of enhancement , with respect to cp planewave referrence ( black dashed line ) .( d ) the total - field distributions of the initial random field ( left ) and the final optimized field ( right ) for the best optimized field configurations at the three target wavelengths .scalebar is .[ fig : torque - spectrum],height=302 ]  we further investigate the influence of by plotting optical force and torque per incident - field intensity as a function of illumination wavelength .in figs.[fig : force - spectrum]a-[fig : force - spectrum]b , the reference planewave force spectrum ( black dashed line ) is identical in both plots , clearly dominated by a broad dipole resonance with smaller peaks at higher - order resonances . through illumination - fieldoptimization , we can enhance the force at the dipole mode while suppressing higher - order modes ( fig.[fig : force - spectrum]a ) and also enhance the force at the quadrupole mode while suppressing the dipole mode ( fig.[fig : force - spectrum]b ) .  in figs.[fig: torque - spectrum]a-[fig : torque - spectrum]c , the reference planewave torque spectrum ( black dashed line ) is identical in all three plots and peaks at both dipole and quadrupole resonances with nearly equal heights ( explained in detail in ) .illumination - field optimization at dipole resonance and off - resonance achieve a similar 6x - boost at the chosen value without suppressing the quadrupole resonance .the optimized total fields in fig.[fig : torque - spectrum]d at 1028 nm and 805 nm both exhibit a phase change around the circumference of the particle , resembling a quadrupole resonance .  in [ fig :torque - spectrum]c , on the other hand , the best optimization at quadrupole resonance achieves a remarkable 20x improvement while suppressing much of the dipole resonance ; the median optimization also achieves 12x improvement while suppressing the dipole resonance to a lesser extent . the optimized total field in fig.[fig : torque - spectrum]d at 625 nm shows a distinct , highly resonant distribution .when comparing interactions with two different resonances , we were able to distinguish the impact of the change in resonant field pattern from the change in the resonance lifetime .controlling for the change in lifetime , we found that torque seems to favor higher - order ( _ e.g. _ quadrupole ) resonances with greater angular momentum , while force ( ) , which closely correlates with extinction power , seems to favor lower - order ( _ e.g. _ dipole ) resonance with greater field intensity within the particle . with the use of structured illumination ,higher - order resonances can be excited more effectively , which contributes to higher optical torque after optimization .our numerical optimization framework allows a systematic search of the illumination - field design space to maximize force and torque on lossy , non - spherical particles with multipolar scattering channels .in addition , we think a rigorous analytical study of the fundamental upper bounds on opto - mechanical responses , similar to the analysis performed on light extinction , would be useful in the future.    with respect to fractional random error added to the best ( blue ) and the median ( red ) optimized fields , at ( right ) and ( left ) .the error bar represents the standard deviation for 100 samples.[fig : robustness],height=226 ]  experimental generation of the designed illumination via slms may suffer various types of manufacturing errors .fig.[fig : robustness ] shows the tradeoff between enhancement and robustness to experimental errors .the fractional error is added to the beam using where is a complex - valued random error bounded by the magnitude of the largest coefficient multiplied by the fractional weight .fig.[fig : robustness ] shows the decrease in as a function of . at ,increasing from to decreased the best fom from to , and the median fom followed a similar trend . on the other hand, of the best optimized field at dropped from to , and the median optimized field changed from to .fig.[fig : robustness ] demonstrates a clear tradeoff between field enhancement and error tolerance , as one might expect due to the need to couple strongly to the underlying particle resonances . for the tolerance requirements of a given experimental setup, the approach we outline here could easily be adapted to a robust - optimization framework in which the expected variability is included and optimized against .we present a numerical framework for computer optimization of structured illumination that maximizes optical force and torque on arbitrary scatterers , and show a 20-fold enhancement in optical torque per intensity on an example plasmonic nanoparticle , compared to a circularly polarized planewave .previously , the major bottleneck has been the cumbersome computation .we overcome this bottleneck with a compact cylindrical bessel basis and a fast boundary element method .we are optimistic that such computational framework for 3d vector fields can be generalized and applied to other design problems in opto - mechanics , nanophotonics , and 3d imaging .the authors thank george barbastathis for helpful discussions .10        k. l. kelly , e. coronado , l. l. zhao , and g. c. schatz , `` the optical properties of metal nanoparticles : the influence of size , shape , and dielectric environment , '' the journal of physical chemistry b * 107 * , 668677 ( 2003 ) .e. karimi , b. piccirillo , e. nagali , l. marrucci , and e. santamato , `` efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q - 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Abstract: this paper aims to maximize optical force and torque on arbitrary micro- and nano - scale objects using numerically optimized structured illumination . by developing a numerical framework for computer - automated design of 3d vector - field illumination , we demonstrate a 20-fold enhancement in optical torque per intensity over circularly polarized plane wave on a model plasmonic particle . the nonconvex optimization is efficiently performed by combining a compact cylindrical bessel basis representation with a fast boundary element method and a standard derivative - free , local optimization algorithm . we analyze the optimization results for 2000 random initial configurations , discuss the tradeoff between robustness and enhancement , and compare the different effects of multipolar plasmon resonances on enhancing force and torque . all results are obtained using open - source computational software available online .   ^1^department of mechanical engineering , massachusetts institute of technology , cambridge , massachusetts 02139 , usa + ^2^department of mathematics , massachusetts institute of technology , cambridge , massachusetts 02139 , usa + ^2^currently with the department of applied physics , yale university , new haven , connecticut 06520 , usa  ^*^nicfang.edu  oscis code ( 350.4855 ) optical tweezers or optical manipulation;(140.7010 ) ; ( 090.1760 ) computer holography ; diffraction and gratings : optical vortices ; ( 250.5403 ) plasmonics ; ( 290.2200 ) extinction


photonic crystals ( phcs ) remain to attract a considerable attention of the scientific community due to their unique properties and potential technological applications . to a large extent ,phcs applications are based on the photonic bandgap effect and involve sophisticated defect structures , cavities and wave guides in the periodic crystal host .the full dynamics of the electromagnetic field in such structures may be studied in principle rigorously by direct numerical solution of maxwell s equations using the finite difference time domain ( fdtd ) method . for the calculation of stationary modes , however , the numerical effort may be substantially reduced by using the galerkin method , i.e. , by expanding the electromagnetic field in terms of an appropriate orthonormal set of basis functions which renders the stationary electromagnetic wave equation as a discrete matrix eigenvalue problem . in this casethe proper choice of the set of basis functions is crucial in order to obtain an accurate description while keeping the dimension of the eigenvalue problem minimal .  while for extended wave problems an expansion in terms of bloch functions , the eigenmodes of the unperturbed phc , is natural , for the description of defect structuresthe use of wannier functions as a basis set is in principle superior , because wannier functions may be constructed as being localized in space and are still an exact representation of the point symmetry group of the host phc .effective solutions developed for the electronic case , namely , maximally localized generalized wannier functions , have been applied only recently to the electromagnetic case . since then , the theory of photonic wannier functions has been applied to the analysis of 2d phc cavities , waveguides , waveguide crossings and phc heterostructures .the generalization of the approach to the case of 2d slab phcs and 3d phcs has been also reported in .however , the construction of well - localized wannier functions involves a multidimensional optimization problem , with the arbitrary phase of each bloch mode ( defined below ) as optimizations parameters .therefore , the practical importance of this approach has been fairly limited up to now . in the present workwe systematically analyze this optimization problem employing local as well as global optimization procedures .we propose a novel locality measure for the wannier functions which allows for a highly efficient optimization of the locality of the wannier function .wannier functions are defined as the fourier transform of the bloch modes with respect to the wave vector , the bloch mode is a solution of the corresponding wave equation in a periodic medium , where is a lattice - periodic envelope function , and we have explicitly denoted the arbitrary phase of the bloch function , the bloch phase . is the band index . for simplicity ,in the present work we do not consider mixing of different bands in the construction of wannier functions from bloch modes .from the definitions ( [ eq : bloch_def ] ) , ( [ eq : wan_def ] ) it is seen that the wannier functions would be -localized in space only if the envelopes were constant . by contrast , for arbitrary bloch phases the wannier functions have actually a rather large spatial extension due to the oscillatory character of the , a feature which becomes more and more pronounced for higher band indices .however , the gauge freedom in the bloch phases may be employed to adjust , for each , the value of so as to construct maximally localized wannier functions with respect to some locality criterion .this constitutes a complex , multidimensional optimization problem .the choice of the locality measure is not unique .the common choice used in the literature is to minimize the second moment ( sm ) of the modulus square of the wannier function .unfortunately , it turned out that the sm , as a functional of the set of bloch phases , has multiple local minima , so that local optimization procedures , like the conjugate gradient method , tend not to find the global minimum and the locality of the wannier function optimized in this way depends sensitively on the initial set of bloch phases .global optimization procedures are , on the other hand , slow and computationally exceedingly expensive .the purpose of this paper is a systematic analysis of different locality measures using both local and global optimization methods . in section [ sec : sm ]we construct sm optimized maximally localized wannier functions , comparing conjugate gradient ( local ) and genetic - algorithm - based ( global ) methods .this analysis leads us in section [ sec : im ] to propose a new locality measure , namely the integrated modulus square ( i m ) measure , resulting in an optimization problem with a single extremum only .this allows one to use fast and efficient local optimization techniques to construct maximally localized wannier functions . in section [ sec : bc ] we show that , if the bloch modes conform certain conditions , it is possible to find an optimal set of bloch phases with respect to the i m locality measure analytically . although the required constrains will not be fulfilled in the general case , the wannier functions calculated using such an analytical set of phases show strong tendency towards localizations and can be used as an efficient starting point for the numerical optimization . in section [ sec : applications ] using several examples of phc defect structures , the quality of the constructed wannier functions is demonstrated .section [ sec : conclusion ] concludes the paper .in what follows , we limit ourselves to the two - dimensional ( 2d ) case , which is characterized by the periodic dielectric functions , , with denoting a 2d vector in the --plane and being a lattice vector of some 2d lattice . in this case , the wave equation for time harmonic tm ( transverse magnetic ) , , and te ( transverse electric ) , , polarization reads , respectively , the wave operators and are hermitian with respect to the corresponding inner products :  is the 2d volume of the crystal .it is important to mention , that the completeness and orthogonality of the bloch modes and translates into the completeness and orthogonality of wannier functions with respect to the corresponding inner product : moreover , the translation property of the wannier functions , follows from the periodicity of the envelope functions .  in this work four different two - dimensional photonic crystals are analyzed for both fundamental polarizations .square ( sq ) and triangular ( tr ) lattices of dielectric rods in air ( d ) and air rods in dielectric ( a ) are considered . in what followswe will refer to these systems as sq - d , tr - d , sq - a and tr - a , respectively .the radius of rods and the dielectric constant of dielectric material and air are chosen to be , and , respectively .we adopt the following definition of the second moment of the wannier function : with being the wannier center .two positions of the wannier center will be analyzed further , ( i ) in the center of the scatterer ( `` on - site '' ) and ( ii ) in the geometrical center between four ( three ) scatterers in the case of square ( triangular ) lattices ( `` between '' ) . for example , in a square lattice with lattice constant , the wannier center can be set to ( on - site ) or ( between ) , where .( inverse locality ) of the wannier functions in the band , minimized by using the conjugate gradient method .four different , randomly chosen initial sets of bloch phases , a , b , c , d , were used for the cg optimization .top : sq - d crystal , tm polarization .bottom : tr - d crystal , te polarization . ]the commonly adopted way of finding maximally localized wannier functions is to minimize the corresponding second moment with respect to the set of bloch phases in the band .note that , by definition of the sm , the regions far away from the wannier center ( where the wannier functions usually have an increasingly complex structure ) contribute to the sm with quadratically increasing weight . for this optimizationwe first apply a standard conjugate gradient ( cg ) method .this method tends to get trapped in the local extrema and , hence , requires a careful choice of the initial set of bloch phases . in figure[ fig : cg_results_sm ] representative examples of the locality of the sm - optimized wannier functions are shown for sq - d ( tm polarization ) and tr - d ( te polarization ) structures . here and in the following, the bloch modes have been calculated using the plane wave expansion method .the minimized sm ( i.e. , the inverse locality ) of the wannier functions in the first eight bands are displayed for four different , random distributions of the initial bloch phases .as expected , the optimal locality obtained using the cg method depends crucially on the choice of the initial phases .this is an indication that the sm of the wannier functions possesses several local minima .  ) of the sm - optimized wannier functions as a function of ga generation for the sq - d structure , tm polarization , 3rd band ( top ) and for the sq - a structure , te polarization , 5th band ( bottom ) .the dashed , green line shows the locality of the best - localized wannier function in each generation .every 100 generations , these wannier functions served as a starting point for the subsequent cg optimization step ( red crosses ) .on the right hand side , the modulus square of the sm - optimized wannier function is shown for an early ( top : 40000th , bottom : 38300th ) and a later ( top : 60000th , bottom : 100000th ) generation , respectively.,title="fig : " ] + ) of the sm - optimized wannier functions as a function of ga generation for the sq - d structure , tm polarization , 3rd band ( top ) and for the sq - a structure , te polarization , 5th band ( bottom ) .the dashed , green line shows the locality of the best - localized wannier function in each generation .every 100 generations , these wannier functions served as a starting point for the subsequent cg optimization step ( red crosses ) . on the right hand side ,the modulus square of the sm - optimized wannier function is shown for an early ( top : 40000th , bottom : 38300th ) and a later ( top : 60000th , bottom : 100000th ) generation , respectively.,title="fig : " ]   the proper choice of the initial set of bloch phases is in general not trivial . even if such a choice can be justified , one can never be sure that the resulting sm - optimized wannier functions indeed correspond to the global minimum of the sm . to solve the global optimization problem and to examine the validity of the sm optimization in more detail , we have applied a global , stochastic - based optimization technique , namely a genetic algorithm ( ga ) method .taking the biologic evolution in nature as a model , the ga method works with a population of individuals which pass through a selection procedure and can reproduce themselves .each wannier function represents an individual .the set of bloch phases , which determines the wannier function , is represented as a large , binary string .the ga method starts with a population of random wannier functions and passes them through a selection procedure where only that one half of the wannier functions are retained which are most strongly localized with respect to the given locality criterion ( most fit individuals ) . in a second step ,these survived individuals are allowed to reproduce themselves by randomly mixing their strings of phases , thus passing their attributes to the offsprings . together with their off - springs ,the survived individuals , which correspond to better localized wannier functions , comprise the new generation . by iterating this procedure over several thousands of generationsthe algorithm will converge slowly but definitely towards the global extremum .once the ga procedure has reached the valley of the global extremum , the cg method should be applied subsequently to the ga algorithm in order to accelerate the convergence and improve the accuracy of the solution .  in figure [ fig : ga_sm ]the evolution of the ga results is depicted for two representative systems and polarizations .every 100 generations a wannier function with highest locality in the current population was taken as a starting point for the subsequent cg optimization . over the first several thousand generationsthe locality of the resulting wannier functions is varying strongly , indicating hopping of the solution among different local minima due to the stochastic nature of the algorithm . at the top panel of figure [ fig : ga_sm ]one can observe how the algorithm is stuck in a local minimum over several thousands of generations , before it escapes and reaches the global minimum valley at around the 50000th generation .an improvement of the locality for later generations is clearly seen from the modulus square of the optimized wannier functions ( figure [ fig : ga_sm ] , right panels ) .the discontinuous nature of the ga method ensures with stochastic certainty that the global minimum of the sm is found , providing the best localization of the wannier functions with respect to a given locality measure . at the same time, however , the numerical load of the ga method exceeds the one of the cg method by far , making it inappropriate for routine application for an efficient construction of maximally localized wannier functions .) of the wannier functions optimized with respect to the i m locality measure using the cg method .four different randomly chosen initial sets of bloch phases were used for the cg optimization .top : sq - d crystal , tm polarization .bottom : tr - d crystal , te polarization . ]the complicated structure of the wannier functions at large distances , which is expressed by several local minima of their sms and the associated difficulties in the construction of maximally localized wannier functions , motivates the search for a simpler criterion for the locality of wannier functions . here, we introduce a new criterion , the integrated modulus square ( i m ) , defined as where the integration region is the first unit cell ( uc ) around the wannier center ( uc ) .we choose the function in such a way , that the i m is equal to unity for a wannier function which is completely confined within such a unit cell .i.e. , a well localized wannier function corresponds to a large i m , and one needs to maximize the i m in order to obtain the maximally localized wannier functions .the i m is a very sharp criterion , since it does not depend on the structure of the wannier functions outside of the integration region at all .we examined the i m locality measure as a localization criterion for the same four different physical systems and both polarizations as it was done in the sm case .to that end we construct maximally localized wannier functions using the same four different , randomly chosen initial sets of bloch phases as in section [ sec : sm ] .representative examples of cg optimization are shown in figure [ fig : cg_results_im ] .in contrast to the sm case , the localities of the resulting wannier functions coincide for all four sets of phases .this is the case for all considered systems and polarizations , strongly indicating that the i m locality measure does not possess any local extrema .  )of the im - optimized wannier functions as a function of ga generation for the sq - d structure , tm polarization , 3rd band ( top ) and for the sq - a structure , te polarization , 5th band ( bottom ) .the dashed green line shows the locality of the best - localized wannier function in each generation .every 100 generations these wannier functions served as a starting point for the subsequent cg optimization step ( red crosses ) . on the right - hand side ,the modulus square of the sm - optimized wannier functions is shown for an early ( 20000th ) and a later ( 60000 ) generation , respectively.,title="fig : " ] + ) of the im - optimized wannier functions as a function of ga generation for the sq - d structure , tm polarization , 3rd band ( top ) and for the sq - a structure , te polarization , 5th band ( bottom ) .the dashed green line shows the locality of the best - localized wannier function in each generation .every 100 generations these wannier functions served as a starting point for the subsequent cg optimization step ( red crosses ) . on the right - hand side ,the modulus square of the sm - optimized wannier functions is shown for an early ( 20000th ) and a later ( 60000 ) generation , respectively.,title="fig : " ]  to support this hypothesis , we applied the ga method to solve the optimization problem .for all considered structures no significant variation of the locality has been observed for different ga generations .representative examples are shown in figure [ fig : ga_im ] .one can clearly see , that over many thousands of generations the locality of the wannier functions optimized with respect to i m criterion stays constant .this proves numerically that the considered optimization problem possesses a single extremum , making the procedure independent on the choice of the initial set of bloch phases . as a consequence ,the use of the i m as a locality criterion together with the cg as an optimization method represents a fast as well as reliable method for the construction of maximally localized wannier functions. figures [ fig : mlwf_dielrods_tm ] and [ fig : mlwf_airrods_te ] show the maximally localized wannier functions for the first several bands of the sq - d ( tm polarization ) and sq - a ( te polarization ) structures , respectively . in both casesthe i m criterion was used along with the cg method .the optimized wannier functions demonstrate good locality which degrades slowly with increasing band indices , since the envelope functions become more and more oscillatory , reflecting the not - plane - wave like nature of bloch modes .it is worth noting , that the wannier functions optimized with respect to the i m and sm criteria are in general not equal , even if the global minimum of the sm or the global maximum of the i m has been reached .for example for the system of dielectric rods in air ( tm polarization ) the sm- and im - optimized wannier functions coincide for the 1st and the 2nd band ( not shown ) , but not for the third band ( top right wannier function in figures [ fig : ga_sm ] and [ fig : ga_im ] ) .sometimes , it is beneficial to use the time - reversal symmetry of wave operators and to construct real wannier functions .this is possible , since the envelope functions of the bloch modes transform as under time reversal ( inversion of the reciprocal space ) , and the bloch phases can be constrained by the condition by restricting the parameter space of the optimization problem in such a way , we found that both sm and i m optimization criteria possess multiple extrema , making the use of local optimization method not efficient without special choice of the initial bloch phases .however , using real - valued wannier functions is of advantage for the considerations of the next section .here we propose an analytical expression for a generic set of bloch phases to be used as a starting point for the optimization procedure .it is based on the following theorem .  _theorem : _ suppose that ( i ) the wannier functions are real - valued , i.e. , , and the bloch functions conform to the following conditions : has the same sign ( i ) for all points in the unit cell and ( ii ) for all wave vectors in the first brillouin zone ( for each component of the vector bloch function ) . then maximizing the i m of the wannier functions , with respect to the bloch phases is equivalent to maximizing the i m of the real part of the bloch functions in the first unit cell around the wannier center , for each wave vector in the first brillouin zone separately .this theorem holds in any spatial dimension , even though in this paper we consider two - dimensional systems only .note that in the above expressions we have set the weight factor appearing in eq .( [ eq : def_im ] ) equal to unity for simplicity .the proof is straightforward . due to the translation property of the wannier function , , it is sufficient to prove the equivalence for only . in this case   ) optimized wannier functions for a square lattice of dielectric rods in air ( tm polarization ) .for n=1,2,4 the bloch criterion is equivalent to the im - criterion for real wannier functions .for n=3,5,6 the wannier functions show at least a tendency to localize around the wannier center . ]it is easy to recognize that the equality holds , if the conditions ( ii ) and ( iii ) are fulfilled .since the wannier functions are chosen to be real - valued ( i ) , the functional on the left - hand side is maximized by the same set of as , which proves the theorem .the maximization problem can be solved analytically leading to the following set of bloch phases where ^{1/2}$ ] is the ( complex ) amplitude of the vector bloch function .relation defines the bloch optimization criterion .note that the phases which are transformed by , or equivalently , also fulfill .furthermore , the phases which fulfill also obey and , therefore , the wannier functions optimized with respect to the bloch criterion are real - valued , as initially assumed .  due to the strongly oscillatory nature of bloch functions in coordinate space ( especially for the higher order bands ) , condition ( ii ) will hardly be fulfilled exactly in a realistic system .but the conditions ( i ) and ( iii ) can always be fulfilled simply by correct choice of the phase factor sign . in figure[ fig : blochoptwf ] an example of the bloch criterion optimization is shown for the sq - d structure ( tm polarization ) . for the 1st , 2nd and 4th bandsbloch functions fulfill bloch criterion conditions at least approximately , and an analytical set of bloch phases leads to well localized wannier functions . even in the case when the bloch functions do not maximize the bloch criterion eq .( [ eq : blochcrit ] ) exactly , the wannier functions optimized with respect to exhibit a tendency to localize around its wannier center . the same tendency has been obtained for all four test structures in both fundamental polarizations .this suggests to use the analytical set of bloch phases as an initial set of phases for the numerical optimization , using either second moment or integrated modulus methods .this should help avoiding the local minima trapping problem and can reduce the computation time considerably .in a a square lattice of dielectric rods in air ( tm polarization ) .the dots indicate the results of the wannier function approach by taking the first eight bands into account .they are in complete agreement with plane wave calculations ( red line ) . at the bottom the real part of two defect modes with and is shown . ]  in this section the quality of i m optimized wannier functions is demonstrated by using them as a basis set for photonic crystal defect structure analysis . in figure[ fig : reprbs ] the reconstructed tight - binding band structure of a square lattice of dielectric rods in air ( tm polarization ) is shown .the number of lattice sites taken into account in the nearest neighbor approximations were increased successively .the deviation of the reconstructed band structure from the original band structure decreases by increasing the number of next neighbors taken into account . by restricting to lattice sites separated by up to four lattice constants the band structureis reproduced well except for small deviation at higher bands and symmetry points .the slight mismatch for higher bands is due to the fact that the higher band wannier functions are not as well localized as the lower ones .we further calculated the modes and frequencies of a point defect structures which consist of a single rod with deviating permittivity at in the sq - d structure .the light propagation inside the band gap between the first and the second band is forbidden and the formation of a localized defect mode is possible .we split up the total permittivity into a periodic term which corresponds to the unperturbed system , and a defect term .an expansion of the -component of the electric field in terms of maximally localized wannier functions leads to a generalized eigenvalue problem with sparse matrices .the defect mode frequencies are obtained from the solution of this sparse system ( fig .[ fig : pointdefect ] ) .the first eight wannier functions have been taken into account .there are monopole like defect modes for a defect rod permittivity of and doubly degenerated , dipole like modes for higher permittivities in the defect rod .the results are in complete agreement with plane wave calculations . to analyze contribution of the individual wannier functions to the defect mode, the band index contribution where the normalization factor is given by is shown in figure [ fig : bandcontri ] for the defect modes and .one can see , that rapidly decrease for higher band indices .since the defect mode frequencies are in between the 1st and the 2nd band , only the lower band wannier functions contribute to the defect modes .therefore it is justified to cut of the band index which reduces the numerical load .the procedure to construct maximally localized wannier functions by bloch phase optimization was analyzed for several two dimensional photonic crystals for both fundamental polarizations , using two different locality measures .although the stochastic , genetic algorithm is numerically too costly for routine application , it has , as a global optimization method , provided us an important benchmark to judge under which conditions the faster and less memory intensive , but local conjugate gradient method finds the global optimum of a given locality measure .we found that the commonly used second moment locality measure has generically multiple extrema , which makes it difficult to construct maximally localized wannier functions by local optimization techniques .one may conjecture that this multiplicity results from the complex oscillatory behavior of the ( not yet optimized ) wannier functions at large distances from the wannier center , which makes the dominant contribution to the second moment .this led us to propose a new locality measure which is controlled by the behavior close the wannier center , the integrated modulus square measure .we showed numerically by comparison of conjugate gradient and genetic algorithm optimization that this measure does not feature multiple extrema and is , therefore , suitable for fast and efficient local optimization techniques , like the standard conjugate gradient method .because this result presumably originates from the local nature of the integrated modulus measure , it should hold generally , not only for two - dimensional systems , but also in three dimensions for photonic as well as electronic lattices .we also presented and tested an analytical formula for the set of bloch phases to be used as a starting point of the optimization process .this initial set of bloch phases is suggested because , albeit it does not solve the optimization problem in general , it does generate maximally localized wannier functions in special cases where the optimization problem can be solved analytically .we expect that these two main results may significantly increase the efficiency of the wannier function approach for the description of defect structures in photonic lattices .y. jiao , s. mingaleev , m. schillinger , d. miller , s. fan , and k. busch , `` wannier basis design and optimization of a photonic crystal waveguide crossing , '' ieee photonics technol . lett . * 17 * , 18751877 ( 2005 ) .

Abstract: wannier function expansions are well suited for the description of photonic - crystal - based defect structures , but constructing maximally localized wannier functions by optimizing the phase degree of freedom of the bloch modes is crucial for the efficiency of the approach . we systematically analyze different locality criteria for maximally localized wannier functions in two - dimensional square and triangular lattice photonic crystals , employing ( local ) conjugate - gradient as well as ( global ) genetic - algorithm - based , stochastic methods . besides the commonly used second moment ( sm ) locality measure , we introduce a new locality measure , namely the integrated modulus ( i m ) of the wannier function . we show numerically that , in contrast to the sm criterion , the i m criterion leads to an optimization problem with a single extremum , thus allowing for fast and efficient construction of maximally localized wannier functions using local optimization techniques . we also present an analytical formula for the initial choice of bloch phases , which under certain conditions represents the global maximum of the i m criterion and , thus , further increases the optimization efficiency in the general case .