sound source localization with sensor arrays involves the estimation of the direction - of - arrival ( doa ) of ( usually a few ) sources from a limited number of observations .compressive sensing ( cs ) is a method for solving such underdetermined problems with a convex optimization procedure which promotes sparse solutions .  solving the doa estimation as a sparse signal reconstruction problem with cs , results in robust , high - resolution acoustic imaging , outperforming traditional methods for doa estimation .furthermore , in ocean acoustics , cs is shown to improve the performance of matched field processing , which is a generalized beamforming method for localizing sources in complex environments ( e.g. , shallow water ) , and of coherent passive fathometry in inferring the number and depth of sediment layer interfaces .one of the limitations of cs in doa estimation is basis mismatch which occurs when the sources do not coincide with the look directions due to inadequate discretization of the angular spectrum . under basis mismatch , spectral leakage leads to inaccurate reconstruction , i.e. , estimated doas deviating from the actual ones .employing finer grids alleviates basis mismatch at the expense of increased computational complexity , especially in large two - dimensional or three - dimensional problems as encountered in seismic imaging for example .  to overcome basis mismatch, we formulate the doa estimation problem in a continuous angular spectrum and introduce a sparsity promoting measure for general signals , the atomic norm .the atomic norm minimization problem , which has infinitely many unknows , is solved efficiently over few optimization variables in the dual domain with semidefinite programming . utilizing the dual optimal variables, we show that the doas are accurately reconstructed through polynomial rooting .it is demonstrated that grid - free cs gives robust , high - resolution reconstruction also with non - uniform arrays and noisy measurements , exhibiting great flexibility in practical applications .polynomial rooting is employed in several doa estimation methods to improve the resolution .however , these methods involve the estimation of the cross - spectral matrix hence they require many snapshots and stationary incoherent sources and are suitable only for uniform linear arrays ( ula) .grid - free cs is demonstrated not to have these limitations .finally , we process acoustic data from measurements in the north - east ( ne ) pacific with grid - free cs and demonstrate that the method provides high - resolution acoustic imaging even with single - snapshot data .  in this paper , vectors are represented by bold lowercase letters and matrices by bold uppercase letters .the symbols , denote the transpose and the hermitian ( i.e. , conjugate transpose ) operator respectively on vectors and matrices . the symbol denotes simple conjugation .the generalized inequality denotes that the matrix is positive semidefinite .the -norm of a vector is defined as . by extension, the -norm is defined as .the paper makes heavy use of convex optimization theory ; for a summary see app .[ sec : appoptimizationproblems ] .the doa estimation problem involves the localization of usually a few sources from measurements on an array of sensors . for simplicity , we assume that the sources are in the far - field of the array , such that the wavefield impinging on the array consists of a superposition of plane waves , that the processing is narrowband and the sound speed is known . moreover , we consider the one - dimensional problem with a uniform linear array of sensors and the sources residing in the plane of the array .the location of a source is characterized by the direction of arrival of the associated plane wave ,  ] , into angular directions the doa estimation problem is expressed in a matrix - vector formulation , where is the vector of the wavefield measurements at the sensors , is the unknown vector of the complex source amplitudes at all directions on the angular grid of interest and is the sensing matrix which maps the signal to the observations , .\label{eq : sensingmatrix}\ ] ] in the presence of additive noise , the measurement vector is described by , the noise is generated as independent and identically distributed ( iid ) complex gaussian. the array signal - to - noise ratio ( snr ) for a single - snapshot is used in the simulations , defined as snr= , which determines the noise -norm , .practically , we are interested in a fine resolution on the angular grid such that and the problem  is underdetermined . a way to solve this ill - posed problem is to constrain the possible solutions with prior information .traditional methods solve the underdetermined problem  by seeking the solution with the minimum -norm which fits the data as described by the minimization problem , the minimization problem  is convex with analytic solution , .however , it aims to minimize the energy of the signal rather than its sparsity , hence the resulting solution is non - sparse .  conventional beamforming ( cbf )is the simplest source localization method and it is based on the -norm method with the simplifying condition .cbf combines the sensor outputs coherently to enhance the signal at a specific look direction from the ubiquitous noise yielding the solution , cbf is robust to noise but suffers from low resolution and the presence of sidelobes .  a sparse solution is preferred by minimizing the -norm leading to the minimization problem , however , the minimization problem  is a non - convex combinatorial problem which becomes computationally intractable even for moderate dimensions .the breakthrough of compressive sensing ( cs ) came with the proof that for sufficiently sparse signals , , , and sensing matrices with sufficiently incoherent columns the minimization problem  is equivalent to the minimization problem , where the -norm is replaced with the -norm .the problem  is the closest convex optimization problem to the problem  and can be solved efficiently by convex optimization even for large dimensions .  for noisy measurements ,the constraint in  becomes , where is the noise floor , i.e. , .then , the solution is , which has the minimum -norm while it fits the data up to the noise level .herein , we use the cvx toolbox for disciplined convex optimization which is available in the matlab environment .it uses interior point solvers to obtain the global solution of a well - defined optimization problem .interior point methods solve an optimization problem with linear equality and inequality constraints by transforming it to a sequence of simpler linear equality constrained problems which are solved iteratively with the newton s method ( iterative gradient descent method ) increasing the accuracy of approximation at each step .cs offers improved resolution due to the sparsity constraint and it can be solved efficiently with convex optimization . however , cs performance in doa estimation is limited by the coherence of the sensing matrix ( see ref . ) , described by the restricted isometry property , and by basis mismatch due to inadequate discretization of the angular grid .herein , we demonstrate a way to overcome the limitation of basis mismatch by solving the -minimization problem on a grid - free , continuous spatial domain .the fundamental assumption in cs is the sparsity of the underlying signal in the basis of representation , i.e. , the sensing matrix .however , when the sources do not match with the selected angular grid , the signal might not appear sparse in the selected dft basis  .figure [ fig : basismismatch ] shows the degradation of cs performance under basis mismatch due to inadequate discretization of the doa domain in fft beamforming .= sensors , and snr =  db .cbf and cs ( * ) reconstruction of two sources ( o ) ( a ) at and on a grid [ :: , ( b ) at and on a grid [ :: and ( c ) at and on a grid [ ::.,width=325 ]  to increase the precision of the cs reconstruction , malioutov _ _ et al.__ and duarte and baraniuk propose an adaptive grid refinement .the adaptive grid refinement aims at improving the resolution of cs reconstruction without significant increase in the computational complexity by first detecting the regions where sources are present on a coarse grid and then refining the grid locally only at these regions .grid refinement is an intuitive way of circumventing basis mismatch .however , the problem of basis mismatch is avoided only if the problem is solved in a continuous setting , particularly for moving sources .in the continuous approach , the -sparse signal , , is expressed as , where is the complex amplitude of the source , is its support , i.e , the corresponding doa , on the continuous sine spectrum  ] , where is the angle between and .then , \ ! & = \ !\lvert x_{i}\rvert \!\left [ 1 \ ! - \ ! \lvert\left(\mathcal{f}_{m}^{h}\mathbf{c}\right)_{i}\rvert \cos\phi_{i}\right]\\ & \geq \lvert x_{i}\rvert \!\left [ 1 \ ! - \ !\lvert\left(\mathcal{f}_{m}^{h}\mathbf{c}\right)_{i}\rvert\right ] . \end{aligned } \label{eq : infimumdualfunction}\ ] ] the lower bound in  is nonnegative if is less than one , , and the infimum is zero .otherwise , < 0 ] at the infimum , which for every yields , i.e. , and , as both and .thus , for , is a unit vector in the direction of , maximizing the dual function  constitutes the dual problem , \ ; \text{subject to } \ ;  \lvert\mathcal{f}^{h}_{m}\mathbf{c}\rvert_{\infty}\leq 1 .\label{eq : dualproblem}\ ] ] since the primal problem  is convex with linear equality constraints , strong duality holds assuring that the maximum of the dual problem  is equal to the minimum of the primal problem .the dual problem  selects a vector which is maximally aligned with the measurement vector while its beamformed amplitude is bounded by unity across the whole angular spectrum . at the angular direction corresponding to the doa of an existing source ,the beamformed dual vector  is equal to the normalized source amplitude .the dual problem is a semi - infinite programming problem with a finite number of optimization variables , , and infinitely many inequality constraints , which is still intractable .define the dual polynomial , note that is a trigonometric polynomial , of the variable , , with the dual variables ^{t} ] .( a ) the dual polynomial .( b ) the nonnegative polynomial .( c ) the support is estimated by the angle of the roots , , of for which .,width=325 ]   once the support is recovered by locating the roots of the polynomial in eq .that lie on the unit circle , the source amplitudes ( the complex weights in eq . )are recovered from , where denotes the pseudoinverse of with columns ^{t}t_{i}} ] with amplitudes  ] .the minimum separation condition  is a consequence of the coherence of the sensing process which is related to the beampattern ; see sec .iv.d . in ref . .to guarantee a well - posed sparse signal reconstruction , it is required that the columns of the inverse fourier operator , the steering vectors , are sufficiently uncorrelated .the continuous formulation  implies that adjacent steering vectors are in arbitrarily close directions , hence fully coherent .however , the requirement  inhibits closely spaced ( i.e. , highly correlated ) steering vectors , hence prevents the sparse reconstruction problem from being too ill - posed due to coherence .figure [ fig : solution7sources21ula ] shows the reconstruction for the maximum number of sources possible . for positive source amplitudes , , the bound  suffices to ensure a unique solution .grid - free cs achieves super - resolution even for doas in general position ; see figs .[ fig : solution7sources21ula](a)(b ) . inserting an additional source at 71.81 , thus exceeding the maximum number of resolvable sources , results in a non - informative dual polynomial , , for all , fig .[ fig : solution7sources21ula](c ) , and inaccurate reconstruction where only 7 out of the 11 sources are resolved , fig .[ fig : solution7sources21ula](d ) .for complex source amplitudes , , an additional constraint  on the minimum separation of doas is required along with the bound on the number of sources  to ensure a unique solution , figs .[ fig : solution7sources21ula](e)(f ) . violating the minimum separation condition ,the cs doa estimation becomes extremely ill - posed due to the coherence of the underlying steering vectors resulting in inaccurate reconstruction characterized by the presence of spurious sources , figs .[ fig : solution7sources21ula](g)(h ) .   sensors and to localize the possible maximum number of sources ( o ) , .( a ) the dual polynomial and ( b ) reconstruction with grid - free cs ( * ) and cbf for sources with positive amplitudes ,  ] .( e ) the dual polynomial and ( f ) reconstruction for sources with complex amplitudes ,  ] with amplitudes  ] with amplitudes  ] , which resides in the noise subspace ( compare with the dual vector  which resides in the signal subspace ) such that , where is the number of sources .all the noise subspace eigenvectors , i.e. , the columns of have the property in eq . .however , if the vector is chosen as a linear combination of the noise subspace eigenvectors the algorithm tends to be more robust .the minimum - norm method selects a vector , , in the noise subspace with minimum -norm and unit first element , .the vector can be constructed from the noise eigenvectors as , where the vector is the first row of .equivalently , the vector can be constructed from the signal eigenvectors as , where the vector is the first row of .the minimum - norm spectrum is ,    the root version of the doa estimation methods is based on the fact that for ulas the null spectrum has the form of the trigonometric polynomial in eq .  with ( since  ] ) .thus , evaluating the spectrum is equivalent to evaluating the roots of the polynomial on the unit circle .more analytically , let be the null spectrum , such that the spectrum is . for mvdr , ( ref ., p.1147 ) , for music , ( ref . , p.1159 ) and for the minimum - norm method , ( ref ., p.1163 ) . then , where is the sum of the elements of the hermitian matrix along the diagonal and .the set of doas , , is estimated from the roots of the polynomial , or equivalently the polynomial , which lie on the unit circle , as , after the support is recovered , the amplitudes can be estimated through an overdetermined problem as in eq . .even though the root forms of doa estimation methods have , often , more robust performance than the corresponding spectral forms , they require a regular array geometry to form a trigonometric polynomial and detect its roots behavior . to achieve a robust estimate of the cross - spectral matrix many snapshotsare required , , i.e. , stationary sources .furthermore , eigendecomposition based methods fail to discern coherent arrivals .forward / backward smoothing techniques can be employed to mitigate this problem and make eigendecomposition based methods suitable for identification of coherent sources as well , but they still require a regular array geometry and an increased number of sensors .the high - resolution capabilities of sparse signal reconstruction methods , i.e. , cs for doa estimation , and the robustness of grid - free sparse reconstruction even under noisy conditions and with random array configurations are demonstrated on ocean acoustic measurements .the interest is on single - snapshot reconstruction for source tracking and the results are compared with cbf .the data set is from the long range acoustic communications ( lrac ) experiment recorded from 10:00 - 10:30 utc on 2010 in the ne pacific and is the same as in ref . to allow comparison of the results .the data are from a horizontal uniform linear array towed at knots at  m depth .the array has sensors , with intersensor spacing  m. the data were acquired with a sampling frequency of  hz and the record is divided in  s non - overlapping snapshots .each snapshot is fourier transformed with samples .the data are post - processed with cbf and cs on a discrete doa grid [ :: as well as grid - free cs at frequency  hz ( ) . to facilitate the comparison of the results , the grid - free cs reconstruction is also presented on the grid [ :: by rounding the estimated doas to the closest integer angle and using the maximum power within each bin .the results are depicted in fig .[ fig : duallracdata ] both with all sensors active , figs .[ fig : duallracdata](a)(d ) and by retaining only sensors active in a non - uniform configuration , figs .[ fig : duallracdata](e)(h ) . both array configurations , figs .[ fig : duallracdata](a ) and [ fig : duallracdata](e ) , have the same aperture thus the same resolution .  the cbf map  in fig .[ fig : duallracdata](b ) indicates the presence of three stationary sources at around , and .the two arrivals at and are attributed to distant transiting ships , even though a record of ships in the area was not kept .the broad arrival at is from the towship r / v melville .the cbf map suffers from low resolution and artifacts due to sidelobes and noise .the cs reconstruction  ( =3.5 , fig .[ fig : duallracdata](c ) ) results in improved resolution in the localization of the three sources by promoting sparsity and significant reduction of artifacts in the map . the grid - free cs solution , fig .[ fig : duallracdata](d ) , provides high resolution and further artifact reduction due to polynomial rooting .retaining only of the sensors on the array in a non - uniform configuration degrades the resolution of cbf due to increased sidelobe levels , fig .[ fig : duallracdata](f ) . however , both cs on a discrete doa grid , fig.[fig : duallracdata](g ) , and grid - free cs , fig.[fig : duallracdata](h ) , provide high - resolution doa estimation without a significant reconstruction degradation .   sensors and the corresponding ( b ) cbf , ( c ) cs on a discrete grid , [ :: , and ( d ) grid - free cs reconstruction .( e ) non - uniform array with sensors and the corresponding ( f ) cbf , ( g ) cs on a discrete grid and ( h ) grid - free cs reconstruction.,width=325 ]  the single - snapshot processing , fig .[ fig : duallracdata ] , indicates that the sources are adequately stationary .therefore , the 200 snapshots can be combined to estimate the cross - spectral matrix and employ cross - spectral methods for doa estimation .figure [ fig : duallracrootmethods](a ) compares the power spectra of mvdr , music and the minimum - norm method and fig .[ fig : duallracrootmethods](b ) the corresponding root versions .the root versions of cross - spectral methods , especially the root music and the root minimum - norm method , provide improved resolution compared to the corresponding spectral forms .however , the root cross - spectral methods require both many snapshots ( i.e. , stationary sources ) for a robust estimate of the cross - spectral matrix and uniform arrays .grid - free cs does not have these limitations .sensors and is used.,width=325 ]doa estimation with sensor arrays is a sparse signal reconstruction problem which can be solved with compressive sensing ( cs ) .discretization of the problem involves a compromise between the quality of reconstruction and the computational complexity , especially for high - dimensional problems .grid - free cs assures that the sparsity promoting optimization problem in cs can be solved in the dual domain with semidefinite programming even when the unknowns are infinitely many .grid - free cs achieves high - resolution doa estimation through the polynomial rooting method .in contrast to established doa estimation methods , cs provides high - resolution acoustic imaging even with non - uniform array configurations and robust performance under noisy measurements and single - snapshot data .finally , the grid - free cs has the same performance both with coherent and incoherent , stationary or moving sources while other doa estimation methods based on polynomial rooting fail to discern coherent arrivals and have degraded resolution for moving sources as they require many snapshots .this work was supported by the office of naval research , under grant no .n00014 - 11 - 1 - 0320 .this section summarizes the basic notions and formulations encountered in convex optimization problems , as presented analytically in ref . .a generic optimization problem has the form , where is the optimization variable , the function is the objective ( or cost ) function , the functions are the inequality constraint functions and the functions are the equality constraint functions .the optimization problem  is convex when are convex functions and are affine ( linear ) functions .the set of points for which the objective and all constraint functions in eq .are defined is called the domain of the optimization problem , a point is called feasible if it satisfies the constraints in eq . .the optimal value of the optimization problem , achieved at the optimal variable , is , for all and .the lagrangian , , of an optimization problem is obtained by augmenting the objective function with a weighted sum of the constraint functions .the lagrangian of the generic optimization problem  is , \ !, \label{eq : lagrangian}\ ] ] where is the lagrange multiplier associated with the inequality constraint , , and is the lagrange multiplier associated with the equality constraint , . the vectors and are the dual variables of the problem .the dual function of the problem  is the minimum value of the lagrangian  over for and , since the dual function is the pointwise infinum of a family of affine functions of , it is concave , even when the problem  is not convex .the dual function  yields lower bounds on the optimal value  for any ( where represents componentwise inequality ) and any , since for every feasible point .the dual function  gives a lower bound on the optimal value of the optimization problem , which depends on the dual variables with ; see eq . .the best lower bound , i.e. the lower bound with the greatest value , is obtained through the optimization problem , which is the dual problem to the optimization problem .the dual problem  is a convex optimization problem , since the objective function to be maximized is concave and the constraints are convex , irrespectively whether the primal problem  is convex or not .the optimal value of the dual problem , achieved at the dual optimal variables is , the dual maximum is the best lower bound on the minimum of the primal problem , that can be obtained from the lagrange dual function .the inequality , holds even if the primal problem  is non - convex and is called weak duality .  the non - negative difference is called the duality gap for the optimization problem , since it gives the gap between the minimum of the primal problem and the maximum of the dual problem .when the duality gap , , is zero , strong duality holds characterized by the equality , strong duality holds when the optimization problem  is convex and there exists a strictly feasible point , i.e. , the inequality constraints hold with strict inequalities . the constraint qualification which implies strong duality for convex problemsis called slater s condition , when the primal problem is convex and slater s condition holds there exist a dual feasible such that , i.e. , the optimal value of the primal problem can be obtained by solving the dual problem .the slater s condition holds also with a weaker constraint qualification , when some of the inequality constraint functions , , are affine ( instead of convex ) , the weaker constraint qualifications  imply that strong duality reduces to feasibility when both the inequality and the equality constraints are linear .let be a square hermitian matrix partitioned as , where is also square hermitian . if then the matrix , is called the schur complement of in .a useful property related to the schur complement is that if then if and only if .this section presents useful results for bounded trigonometric polynomials and their roots as presented in refs . .let ^{t} ] .a ( causal ) trigonometric polynomial can be written in terms of the basis vector as , where ^{t } \in \mathbb{c}^{l} ] . defining and as in , yields . from lemma 4.23 in , implies , where and are the autocorrelation matrices of the coefficient vectors ^{t} ] of the polynomials and respectively . through a schur complement ( see sec .[ sec : schurcomplement ] ) , is equivalent to semidefinite matrix , let the polynomial have amplitude uniformly bounded for all  ] , }{\max } \lvert h(\omega ) \rvert \leq \gamma,\\ & \lvert h(\omega ) \rvert \leq \gamma , \ ; \forall \omega\in[-\pi , \pi ] . \end{aligned }\label{eq : boundedlinfty}\ ] ]   for a bounded trigonometric polynomial , we can construct a polynomial , which is by definition real - valued and nonnegative , thus it can not have single roots on the unit circle .the degree of the polynomial is .therefore , the polynomial has at most distinct roots on the unit circle . at a root , , we have and subsequently .the algorithm in table [ table : matlabcode ] for the implementation of the method described in sec .[ sec : noiselessreconstruction ] is an adaptation of the code by fernandez - granda in ref . .code for sec .[ sec : noiselessreconstruction ] [ cols= " < " , ]   [ table : matlabcode ]in the case that the measurements  are contaminated with additive noise such that , the primal problem of atomic norm minimization  is reformulated to the problem  or equivalently , the lagrangian for  is formulated by augmenting the objective function with a weighted sum of the constraints , + \xi\left(\mathbf{n}^{h}\mathbf{n}-\epsilon^{2}\right ) , \end{aligned } \label{eq : lagrangiannoise}\ ] ] where are the dual variables related to the equality constraints , , and is a lagrange multiplier related to the inequality constraint , .the dual function is the infimum of the lagrangian , , over the optimization variable x , + \xi\left(\mathbf{n}^{h}\mathbf{n}-\epsilon^{2}\right)+\\ & + \underset{x}{\inf } \ ; \left ( \lvert x\rvert_{\mathcal{a } } - \mathrm{re}\left[\mathbf{c}^{h}\mathcal{f}_{m}x\right ] \right ) . \end{aligned } \label{eq : dualfunctionnoise}\ ] ] minimizing over the unknown noise , yields the optimal noise vector , .the dual function evaluated at is , -\frac{\mathbf{c}^{h}\mathbf{c}}{2\xi } + \xi\left(\frac{\mathbf{c}^{h}\mathbf{c}}{4\xi^{2}}-\epsilon^{2}\right)+\\ & + \underset{x}{\inf } \ ; \left ( \lvert x\rvert_{\mathcal{a } } - \mathrm{re}\left[\mathbf{c}^{h}\mathcal{f}_{m}x\right ] \right ) . \end{aligned } \label{eq : dualfunctionnoiseno}\ ] ] further , maximizing over the dual variable , we obtain the optimal value for the dual variable .finally , the dual function evaluated at the optimal values and becomes , -\epsilon\lvert\mathbf{c}\rvert_{2}+\\ & + \underset{x}{\inf } \ ; \left ( \lvert x\rvert_{\mathcal{a } } - \mathrm{re}\left[\mathbf{c}^{h}\mathcal{f}_{m}x\right ] \right ) , \end{aligned } \label{eq : dualfunctionnoisenoxo}\ ] ] and the dual problem is formulated by maximizing the dual function , , over the dual variables similarly to the process detailed in sec .[ sec : dualproblem ] , -\epsilon\lvert\mathbf{c}\rvert_{2 }  \ ; \text{subject to } \ ;  \lvert\mathcal{f}^{h}_{m}\mathbf{c}\rvert_{\infty}\leq 1 .\end{aligned } \label{eq : dualproblemnoiseapp}\ ] ]                            j. j. fuchs .sparsity and uniqueness for some specific under - determined linear systems . in _ieee international conference on acoustics , speech , and signal processing , icassp05 _ , volume * 5 * , pages 729732 .ieee , 2005 .a. barabell .improving the resolution performance of eigenstructure - based direction - finding algorithms . in _ieee international conference on acoustics , speech , and signal processing , icassp83 _ , volume * 8 * , pages 336339 .ieee , 1983 .

Abstract: the direction - of - arrival ( doa ) estimation problem involves the localization of a few sources from a limited number of observations on an array of sensors , thus it can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing ( cs ) to achieve high - resolution imaging . on a discrete angular grid , the cs reconstruction degrades due to basis mismatch when the doas do not coincide with the angular directions on the grid . to overcome this limitation , a continuous formulation of the doa problem is employed and an optimization procedure is introduced , which promotes sparsity on a continuous optimization variable . the doa estimation problem with infinitely many unknowns , i.e. , source locations and amplitudes , is solved over a few optimization variables with semidefinite programming . the grid - free cs reconstruction provides high - resolution imaging even with non - uniform arrays , single - snapshot data and under noisy conditions as demonstrated on experimental towed array data .


the construction of a new terminal at the schenzen airport , southeast china , has been used to question the current strategies of infrastructure growth in developing countries  .schenzen is a large city , but its airport is directly connected by an eight kilometer ferry to the hong kong airport , which can handle twice as many passengers .is it reasonable to invest more than one billion dollars increasing the capacity of such a large infrastructure with another one nearby ?opponents to this investment classify it as a white elephant and as one example of the misbegotten infrastructure growth in developing countries .the ones in favor , argue that not only the costs of investing in schenzen are lower than in hong kong but also global connectivity can profit from cooperation between stakeholders of air - transportation systems in the region . herewe address , from a network science perspective , how such proximity and cooperation between local actors is key to the robustness of spatial networks .the simplified representation of complex systems as a network of nodes and links has provided important insights into the design of a variety of systems , such as power grids  , maritime commerce  , and communication networks such as the internet  . in many cases ,nodes are spatially embedded according to the spatial coordinates of the elements  .this simplification allows us to focus on the topological aspects of the system and to easily extend our results to many applications .we characterize the robustness of a network as its capacity to maintain global connectivity under a sequence of node removals and describe a strategy based on local cooperation to improve robustness under possibly realistic constraints .we discover a continuum transition when changing the distance for which nodes are allowed to swap links , a cooperation range .we calculate the critical exponents of this transition and show that the key factor controlling the value of the critical exponents is the exponent of the algebraic decay of the connection probability with the node distance .  as an ubiquitous infrastructure system ,we explain our method in the context of the worldwide air - transportation network ( wan ) , though our results impact the whole class of spatially embedded networks .it is paramount that the wan works in an extremely reliable and efficient fashion , as any temporary airspace closure , such as the one caused by the eruption of the volcano eyjafjallajkull in 2010 , may cause huge losses worldwide  .complex networks have been used to study airflight networks .simple abstractions of flights and airports have been used to characterize its robustness  , study its structural properties  and evolution  . hereinstead we go one step further and propose modifications to the robustness of the wan and identify new properties associated with it .we summarize data provided by openflights in the year 2011 as a single static network with airports ( nodes , modeled as points distributed across the surface of a sphere with distance calculated according to the haversine formula ) and around eighteen thousand links  .links are undirectional , assuming that each flight should return to its origin , and weighted according to the number of possible flights between two airports .airports are weighted by the total number of passengers transported that year .information regarding the number of passengers transported between airports is not available .  as a self - organized system , in which preferential attachment is expected to play a pivotal role ,the wan is quite fragile to targeted attacks , i.e. , intentional removal of the most connected nodes causing the collapse of the giant connected component  .the aim of our optimization strategy is to create a robust yet economically feasible wan .since robustness can be defined in different ways , we consider that a robust wan should be capable of transporting passengers even in face of a targeted attack , in contrast with previous works where only the size of the largest connected component is considered  .we simulate a sequence of airport closures ( node removal ) and quantify robustness as : where is the total number of nodes , is the number of nodes removed from the network , and is the number of passengers in the largest component after a fraction of nodes were removed , i.e. , the sum of nodes weight on the largest component .closures are executed from the most to the least connected node .the location of airports are mostly determined by economical forces , such as to cater to local demands . in many cases ,airports are located within a short distance from each other , sometimes only a few kilometers away as , e.g. , airports in the schenzen - hong kong area , or a few hundred kilometers but still easily reachable , such as the airports in the northeast of the united states .we assume that a flight rerouted to an airport within a cooperation range of the original destination has a similar attractiveness .if need be , a passenger landing at a different airport could easily take another means of transportation , such as the local train network or a shuttle bus , to go to the desired destination . for transportation networks ,the cooperation range is defined as the geographical distance between nodes , but other spatial networks might require other metrics , such as travel time or cost .we increase network robustness through link swaps , with the probability to swap a route inversely proportional to the weight of a link between airports and , so that important connections are affected with less priority .this tends to keep the transportation capacity of the system stable , and the introduction of complicated interventions on airports , such as building new runaways , might be avoided .moreover , a connection is only rerouted to an airport within the cooperation range of the original destination .as an example , flights could be distributed between hong kong and schenzen or among the airports surrounding london .  given a cooperation range and a metric , which calculates the distance between nodes and ,the following swap strategy is performed :  1 .  select a node randomly having at least one neighbor ; 2 .  select a neighbor of node with probability inverselyproportional to the weight of ; 3 .  select a pair of connected nodes and so that and with probability inversely proportional to the weight of ; 4 .  remove links and and create links and .this strategy is illustrated in fig .[ fig::diagram ] .swaps change the network robustness but we only perform swaps that increase . from this point on ,we call such a swap , a _geo swap_. a fixed number of geo swaps of the order of the number of links is executed .to compare networks of different sizes and populations , we normalize the robustness as , in which is the value of for km , the minimum value for which a geo swap will be considered in the wan , and the maximum robustness obtained for km , which is approximately half of the planet perimeter .this strategy builds on top of previous work which focused on different acceptance mechanisms  or topological characteristics  , but differs significantly by focusing on geographic limitations and low - weight links ., a node at distance is selected ( panel a ). if nodes , neighbor of , and , neighbor of , also have then links and , in blue , are swapped ( panel b ) . ]the cooperation range limits the area of possible swaps to guarantee that a geographically acceptable change is performed .a too small value of does not provide sufficient room for robustness improvement . while a too large value of leads to reroutes of connections to impractically far away airports . by tuning the values of , we observe a critical value of the cooperation range km at which a significant improvement in the wan is first registered .this range in fact yields the highest variance of robustness increase among all possibilities , as shown in fig .[ fig::rwan ] .km , in which the standard deviation reachs the maximum . ]  by considering continental airflight networks - constructed from the division of the wan into continents ( details in * s1 * table ) - together with other spatially embedded networks ( the european power grid and the european rail network ) , we observe that is positively correlated with ( spearman s correlation coefficient ) , where is the total area in which the nodes are embedded ( fig .[ fig::vcrit_x_density ] ) .being the combination of local and intercontinental flights , the wan lies slightly off the trend , but in general we can conclude that the typical radius served by an airport is correlated to the minimum distance at which swaps become effective . artificially generated random networks also confirm this relationship ( * s1 * figure ) .other topological characteristics of the optimized networks are detailed in * s4 * figure .is positively correlated with , where is the total area in which nodes are embedded .the three different symbols represent the types of infrastructure networks in which the geo swap is applied . ]  close to the critical cooperation range , the evolution of scales with for the continental networks . applying the finite - size scaling , ,\ ] ] where and are critical exponents and  ] , in which is the maximum link weight on the wan .we observe that changes in the value of affect consistently the slope in the data - collapse ( fig .[ fig::gravity]*b * ) .we find a value of similar to the one of the continents , without australia , for $ ] . for , the finite - size scaling in fig .[ fig::gravity]c allows us to estimate : and ( fig .[ fig::gravity]*a * and fig .[ fig::gravity]*c * ) .further analysis also show that when the ratio between the average length of routes and the average distance between two airports is similar to that found for the continents ( * s2 * figure ) .interestingly , the empirical probability distribution of link lengths in the wan is a power law of exponent ( * s5 * figure ) .this suggests that correlations as the ones developed in the distance - decay model are consistent with the ones found for the wan and characteristic for the universality class .after successive applications of the geo swap on random networks generated through the distance - decay model with . * a * , value of at with scaling with the number of nodes ( ) as , with . *b * , impact on for the distance - decay model with different values of . * c * , curves for different system sizes with , scaled using and . ]in order to provide applicable insights , any network modification strategy should take into account realistic constraints naturally imposed by the problem .the geo swap contains a simple set of rules specifically designed to improve the robustness of spatial networks .it is important to note however that a probabilistic approach is more a guidance than a closed optimization recipe .we expect that future procedures built on top of our strategy should be carefully tailored to the underlying system .  in an infrastructure perspective ,the geo swap , which makes flights land in different airports , has two crucial implications .firstly , a second transportation system should be used to connect nearby airports , in line with recent works dealing with the coupling of infrastructure networks  . by taking into account the critical cooperation range these couplingscould be designed or improved for distances close to .secondly , a local level of cooperation is necessary between airports .flights in schenzen and hong kong could be rerouted to attend different yet nearby airports worldwide , further increasing the reliability of the local service and the overall wan robustness .one could also organically extended our analysis to incorporate airlines into the scenario .the already existing alliances within the airline industry do cooperate on a regular basis .this direction could be pursued by future works if data about alliances among airlines is available .our rewiring strategy is also able to show that the continents , with one exception , follow the same universality class regarding robustness improvements , as the probability that two airports are connected decays quadratically with their distance .being a continent with its own geographical idiosyncrasies , australia does not fit our analysis , for which further studies are necessary . in summary ,our results show that , for any spatial network , the universality class of the robustness improvement strongly depends on the spatial correlation of connected nodes .figure [ fig::rwan ] results from the application of tentative geo swaps for each value of in the wan .each blue circle is the average over samples while red circles stand for the variance over samples .  in fig .[ fig::vcrit_x_density ] , data for the european power grid ( red ) is retrieved from ref . and the rail transportation network was manually assembled using public data .the power grid network has nodes and links , and the rail network has nodes and links . for continents , the power grid , and the rail network , a total of tentative geo swaps are executed for several cooperation range values .the value of is selected as the highest variance point over samples , with error bars representing the values where variance is equal to .the same data for the continents is used to construct fig .[ fig::scaling_cont ] , in which symbols are larger than the standard deviation .  for all panels in fig .[ fig::gravity ] , a maximum of tentative geo swaps are executed for several cooperation ranges .each point represents the average over samples , with symbols being larger than standard deviation in panel * c*. the critical cooperation range is defined as km .* acknowledgments .* authors would like to thank the cnpq , conselho nacional de desenvolvimento cientfico e tecnolgico - brasil , the european research council advanced grant fp7 - 319968-flowcss , and the eth zurich risk center for financial support .na acknowledges financial support from the portuguese foundation for science and technology ( fct ) under contract no .we would also like to thank i. nikolakopoulos and k. j. schrenk for providing the caffeine for the discussion in which the idea of this paper was conceived .arajo a , moreira aa , makse ha , stanley he , andrade jr , js . traveling length and minimal traveling time for flow through percolation networks with long - range spatial correlations .phys rev e. 2002;66 : 046304 .cardillo a , zanin m , gmez - gardees j , romance m , garca aj , et al . modeling the multi - layer nature of the european air transport network : resilience and passengers re - scheduling under random failures .eur phys j - spec top .2013;215 : 2333 .louzada vhp , daolio f , herrmann hj , tomassini m. generating robust and efficient networks under targeted attacks . in : krol d , fay d , gabrys b , editors , propagation phenomena in real world networks , berlin : springer ; 2014

Abstract: a robust worldwide air - transportation network ( wan ) is one that minimizes the number of stranded passengers under a sequence of airport closures . building on top of this realistic example , here we address how spatial network robustness can profit from cooperation between local actors . we swap a series of links within a certain distance , a cooperation range , while following typical constraints of spatially embedded networks . we find that the network robustness is only improved above a critical cooperation range . such improvement can be described in the framework of a continuum transition , where the critical exponents depend on the spatial correlation of connected nodes . for the wan we show that , except for australia , all continental networks fall into the same universality class . practical implications of this result are also discussed .


avalanche photodiode ( apd ) is widely used in broad fields , such as single molecule detection , autocorrelated fluorescent decays and quantum information , to detect extremely weak light signals or even single photons .the latter is usually named single - photon avalanche detector ( spad ) , which acts as an essential device and may greatly influence the performance of the system in many applications , especially in quantum information fields .spads usually works in two different modes : free - running mode and gated mode . for free - running mode ,the spad is biased above its breakdown voltage to allow an avalanche happens . for gated mode , the spad is commonly biased below its breakdown voltage and avalanche process can only happen in gating time windows when additional gate voltage signals are superimposed .there are many parameters can be used to quantify the performance of spads , such as clocking rate , dark count rate , detection efficiency , timing jitter and so on . in the past decade , the performance of spad has been greatly improved and can working at gigahertz clocking frequency with up to one hundred megahertz counting rates .  however , afterpulsing effect of spad severely limits the count rate and applications in precise measurements .afterpulsing , which is correlated with the ignition avalanche , comes from detrapping of carriers that were trapped by deep energy level in the junction depletion layer .the trapped carriers then release from the traps over time by thermal fluctuation .usually , the release lifetime is dependent on the type of traps and can vary from 10 to several microseconds , which is usually much longer than that needed for quenching an avalanche .a releasing carrier may initiate another pseudo detection signal , which is usually called afterpulsing or after pulse , if the spad is biased above the breakdown voltage .the pseudo signal leads to negative effects on spads applications .for example , afterpulsing will increase quantum bit error rate in quantum key distribution and destruct the randomness of quantum random number generator . in order to reduce afterpulsing ,another hold - off time depending on is necessary .the afterpulsing effect is negligible if the hold - off time is much longer than carriers lifetime .but it is not practical for high - count - rate spads .many approaches have been proposed to reduce afterpulsing , but it remains a critical limitation for high - count - rate spads . precisely modeling the behavior of afterpulsing is the foundation to handle this problem .the trap lifetime of apd can be described by the arrhenius equation where is the temperature , is the boltzmann constant and is determined by the relevant effective states density , cross section of the trap and temperature  .the carriers releasing probability is then exponential via time . as the types of traps in apd are dependent on practical manufacturing processes , there are thereby variety of traps with different lifetime .thus , the afterpulsing probability ( ap ) during a detection window contains different exponential components  where are amplitudes of different exponential components . and means the ap of the -th detection window after the ignition avalanche .recently , different fitting methods have been studied , which employed a broad distribution of different exponential components instead of summating a few dominate discrete ones .so far , most studies on afterpulsing effect just focus on its time - dependent probability distribution ignited by a single light pulse , and do not discriminate the situation with single ignition avalanche from that with multiple ones .however , it is not the real situation when the spad works continuously for photon detection . in this case, several avalanches may happen successively within the lifetimes of trapped carriers .previous works have demonstrated that traps filled with carriers during an avalanche event are much smaller than the total number of traps .thus , avalanches happen successively will increase the number of filled traps and hence will enhance the ap .that is , the ap distribution may be correlated to the avalanching history , while prior studies have not delve down into this physical procedure .  in this paper, we studied the afterpulsing influence of multi - ignition avalanches on photon detection process of ingaas / inp spad under different gate frequencies and photon intensities and found out that afterpulsing possesses memory effect and thus correlates to the avalanching history .we call this property as non - markov property .we gave out analytical expressions of afterpulsing influences on photon detection in theory .the afterpulsing influence on photon detection can then be quantified .we also demonstrated experiments to verify the theory on afterpulsing effect .the experimental results fitted well with the theoretical model .the model is instructive to the manufacturing processes and afterpulsing evaluation of high - count - rate spads .it also offers a new perspective to deal with practical security of quantum communication with imperfect devices .   considering that trap level distribution is a technological parameter , which is determinedonce the product is fabricated and the details have been studied well , we do not deal with specific time - dependent distribution of ap in the next analysis process .the primary data of can be obtained by experimental measurements .in addition , it is reasonable to assume that the filling and release processes of different traps are independent as the filling is rare .we model the procedure according to markovian and non - markov procedure individually .firstly , we assume that ap distribution is only determined by the latest ignition avalanche as assumed in , namely , the ap distribution has no memory of prior avalanching history and is markovian .the probability of an avalanche triggered by laser pulse and dark count can be derived from the poisson distribution and avalanche probabilities of different gates ( detection windows ) are independent identical distribution ( iid ) .let be the avalanche probability per gate triggered by photon pulse , be the ap of gate- with single - ignition photon pulse and be the total avalanche probability of gate- .there is no afterpulsing for the first gate .as we assume that afterpulsing effect comes from the latest ignition avalanche only , the system should have no memory of avalanching history before that .that is , if gate- avalanches , .if gate- does not avalanche , gate- is considered . if gate- avalanches , .if gate- does not avalanche , gate- is considered , and so on .hence , the average avalanche probability of gate- is \\ & + ( 1-p_{n-1})\{p_{n-2}[p+(1-p)p_a(2)]+(1-p_{n-2})\\ & \quad\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"}. \end{aligned } \label{equ3}\ ] ] the first term of the equation means if the pre - gate avalanches , the ones before the pre - one are not considered . if the pre - gate does not avalanche , the gate before the pre - gate is under considered , and so on . if ap is small , and only the first order terms are reserved , namely , , then the approximate equation becomes .\label{equ4}\ ] ] see appendix for detailed proof of equation [ equ4 ] .then , we assume that the system has memory of prior avalanching history and ap distribution is determined by the avalanching history .thus , the derivation of avalanche probability of gate- is much different from that of markov - like model and the full - expression depending on the avalanching history is very complex .for example , +(1-p)p = p[1+(1-p)p_a(1)] ] . herewe give a formal expression of , and then give the approximate equation of . can be classified into two sets : gate-0 avalanches in the first set and does not avalanche in the second set .the second set can be expressed as .then can be expressed as   by adopting the same approximation utilized in the markov model , where , then we obtain the approximate equation of equation [ equ5 ] , .\label{equ6}\ ] ] see appendix for detailed proof of equation [ equ6 ] .in fact , the formal expression is enough for the proof .comparing equation [ equ6 ] with equation [ equ4 ] , we find that , for both cases , depends on two parameters .the first one is , the photon - ignition avalanche probability without afterpulsing .the second one is the ap distribution .the afterpulsing influence increases via , but the weight of decays exponentially .the decay rate depends on the base .in fact , the afterpulsing influence becomes saturation when is large enough .it is reasonable to truncate to an appropriate value according to the actual value of .the equation will degenerate into if no afterpulsing exists . however , the influence of afterpulsing of non - markov model is more significant .the weight of in non - markov case is , which is linear .but the corresponding weight of in markov - like case is , which decays exponentially .thus , the markovian approximation holds only for .the difference between these two model will become more and more significant as or ( or both of them ) increases .the experiments demonstrated below prove that afterpulsing effect is non - markov .the experimental setup is shown in figure [ fig : setup ] .the laser diode ( ld ) is operated in pulsed mode and is triggered by a modulating signal .figure [ fig : setup](b ) shows the modulating signal , which includes trigger pulses for every interval .the trigger frequency and interval time are adjustable .the spad ( the model of apd is : princeton lightwave pga-300 ) is operated in gated mode without dead time and the gate width is 2.5 .the detection efficiency of the spad is about 10.5% .the detection gate is triggered by external pulsed signal , as shown in figure [ fig : setup](c ) .all trigger signals are synchronized .laser pulses from the ld are attenuated by a adjustable attenuator and then sent into the spad .the spad outputs a detection signal if an avalanche happens .the detection signals are input into a time - to - digital converter ( tdc , agilent u1051a acqiris tc890 ) in real time. the tdc can records the arriving time of every detection signal with a resolution of 50 .the parameter can be modulated by adjusting the attenuator .the parameter is modulated by changing the trigger frequency .the verification work consists of three experiments . * in the first experiment * , the ap distributions under different trigger frequencies ( and mhz ) are measured with single ( ) and multi - ignition laser pulses ( ) .the average photon number per pulse is set as 0.07 .the ap is the avalanche probability of gate- after subtracting the dark count . * in the second experiment * , the average photon number is modulated ( and ) so that the avalanche probability with different can be measured , while the ap distribution varies little .* in the third experiment * , the afterpulsing distribution is modulated .the trigger frequency is modulated ( and mhz ) so that the ap distribution becomes different , while is set as constant ( ) .for all experiments , the dark count rate per gate is under the order of .the first experiment demonstrates that ap increases as increases ( see figure [ fig : afpprobdist ] ) , where .as shown in figure [ fig : afpprobdist](a ) , when mhz , while when mhz , where is the ap of gate- and is the total ap with single - ignition pulse .it also demonstrates that aps with multi - ignition pulses ( figure [ fig : afpprobdist](b ) ) are much larger than that with single ignition pulse ( figure [ fig : afpprobdist](a ) ) .for example , when mhz , , while and , where is the ap of gate- with -ignition pulses .thus , .it means that ap accumulates as increases .namely , ap depends on the avalanching history and is non - markovian .the ap becomes larger as increases .as the full width at half maximum ( fwhm ) of the detection signal output from the spad is about 40 , the saturated processing rate of the discrimination circuit is less than 20 mhz by taking the processing time into account .thus , drops and is even smaller than expected when mhz . however , the processing rate limitation is not that significant for the experiments because the average photon number , which is small , when mhz .the second experiment ( see figure [ fig : different pa ] ) shows that the difference between non - markov model and markov model is negligible when ( ) .the difference is significant when is larger , e. g. , figure [ fig : different pa](d ) shows that .the markov model can no longer fit with the experimental data .the third experiment gives the differences between non - markov model and markov model with different ( see figure [ fig : different gate ] ) .there is hardly any difference between these two model when mhz ( ) , as shown in figure [ fig : different gate](a ) .but the difference becomes for large when mhz ( ) .the experimental results above are all consistent well with the non - markov model . according to the equation [ equ6 ] , the afterpulsing influence on photon detection depends on the product of , the pulse - ignition avalanche probability per detection window , and , the total ap . in figures [ fig : different pa](a ) , . and in figure [ fig : different gate](a ) , . in both figures, the markov model also fits well with the experimental data .but in figures [ fig : different pa](b),(c),(d ) and [ fig : different gate](b),(c),(d ) , markov model can no longer fit with the data , where is in figures [ fig : different pa](d ) and is in figures [ fig : different gate](d ) .the non - markov effect will be more significant for higher .it is really the case for high - count - rate spads , of which the detection period is much smaller than the lifetime of carriers .it should be note that detection efficiency of the spad varies when trigger frequency varies . in order to make comparing of ap influence with different ,the gate voltage loaded on spad is fine adjusted so that detection efficiency varies less ( ) .otherwise , afterpulsing would be a catastrophe and the spad under mhz could no longer work properly if the voltage remains the same with that under mhz .in conclusion , we have proved that afterpulsing effect of spad has memory on avalanching history and is non - markov .we give an approximate expression ( equation [ equ6 ] ) that describes the non - markov effect well .the theoretical analysis fits well with the experimental results .the non - markov effect will be more significant for higher .it is really the case for high - count - rate spads , of which the detection period is much smaller than the lifetime of carriers .as it is the embodiment of rarely filling of traps , the non - 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order terms of only , we obtain the approximate equation . \label{equ12}\ ] ]   thus , in general , can be expressed as , where is a power series of and .for the first term of right side , we have &=[p+g_{n-1}(p_a(j))]\cdot[p+(1-p)p_a(1)]\\ & = p^2+p(1-p)p_a(1)+pg_{n-1}(p_a(j))+g_{n-1}(p_a(j))\cdot ( 1-p)p_a(1)\\  & \cong p^2+p(1-p)p_a(1)+pg_{n-1}(p_a(j ) ) . \end{aligned}\ ] ] the remainder terms become +(1-p_{n-2})\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"}\\ = & [ 1-p - g_n(p_a(j))]\{p_{n-2}[p+(1-p)p_a(2)]+(1-p_{n-2})\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"}\\ = & ( 1-p)\{p_{n-2}[p+(1-p)p_a(2)]+(1-p_{n-2})\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"}\\ & -g_n(p_a(j))\{p_{n-2}[p+(1-p)p_a(2)]+(1-p_{n-2})\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"}. \end{aligned}\ ] ] in order to simplify term +(1-p_{n-2})\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"} ] .the expressions of outer braces can be simplified by analogy , and +(1-p_{n-2})\{\cdots\{p_0[p+(1-p)p_a(n)]+(1-p_0)p\underbrace{\}\cdots\}\}}_{(n-1)"\}"}\cong g_n(p_a(j))p ] , where we only keep the first - order terms of .thus , we assume that  ] , \cong p[1+(1-p)(p_a(1)+p_a(2))] ] .thus , we assume that $ ] .if holds , then equation [ equ14 ] holds .the right side of equation [ equ13 ] includes two sets : gate-0 avalanches in the first set and does not avalanche in the second set .the second set can be expressed as .the main difference of the first set from the second set is the afterpulsing effect ignited by gate-0 .if we separate out all terms due to the afterpulsing effect of gate-0 ( that is , terms with factors of gate-1 , of gate-2 , , of gate- ) , the remaining terms of the first set can be expressed as . terms with factor of gate-1 are , where and are the abbreviations for the remaining factors . according to the approximate convention , all terms with in and abandoned .then and approximate to the same expression ^{n-1}=1,\ ] ] where is the binomial coefficient .hence , .analogously , terms with factor of gate-2 , of gate-3 , , of gate- all approximate to 0 . as there is no term for gate- ,there remains one nonzero term , where by the approximation convention .thus , . \end{aligned}\ ] ] this completes the proof .

Abstract: single - photon avalanche photodiode(spad ) has been widely used in researching of quantum optics . afterpulsing effect , which is an intrinsic character of spad , affects the system performance in most of the experiments and needs to be carefully handled . for a long time , afterpulsing has been presumed to be determined by the pre - ignition avalanche . we studied the afterpulsing effect of a commercial ingaas / inp spad ( apd : princeton lightwave pga-300 ) and demonstrated that its afterpulsing is non - markov , which has memory effect of the avalanching history . theoretical analysis and the experimental results clearly indicate that the embodiment of this memory effect is the afterpulsing probability , which increases as the number of ignition - avalanche pulses increase . the conclusion makes the principle of afterpulsing effect clearer and is instructive to the manufacturing processes and afterpulsing evaluation of high - count - rate spads . it can also be regarded as an fundamental premise to handle the afterpulsing signals in many applications , such as quantum communication and quantum random number generator .


the calar alto legacy integral field area ( califa ) survey is an ongoing large project of the centro astronmico hispano - alemn at the calar alto observatory to obtain spatially resolved spectra for 600 local ( 0.005.03 ) galaxies by means of integral field spectroscopy ( ifs ) .califa observations started in june 2010 with the potsdam multi aperture spectrograph ( pmas ) , mounted to the 3.5 m telescope , utilizing the large ( 74 ) hexagonal field - of - view ( fov ) offered by the ppak fiber bundle .ppak was created for the disk mass survey ( bershady et al .each galaxy is observed using two different setups , an intermediate spectral resolution one ( v1200 , ) , that cover the blue range of the optical wavelength range ( 3700 - 4700 ) , and a low - resolution one ( v500 , , that covers the first octave of the optical wavelength range ( 3750 - 7500 ) .a diameter - selected sample of 939 galaxies were drawn from the 7th data release of the sloan digital sky survey ( sdss ) which is described in . from this mother samplethe 600 target galaxies are randomly selected , of which we have currently observed 517 objects ( december 2014 ) , being near to its conclusion .combining the techniques of imaging and spectroscopy through optical ifs provides a more comprehensive view of individual galaxy properties than any traditional survey .califa - like observations were collected during the feasibility studies ( mrmol - queralt et al .2011 ; viironen et al .2012 ) and the ppak ifs nearby galaxy survey ( pings , rosales - ortega et al .2010 ) , a predecessor of this survey . first results based on those datasets already explored their information content ( e.g. rosales - ortega et al .2010 ; rosales - ortega et al .2012 ) .  compared with other ifs surveys, califa offers an unique combination of ( i ) a sample covering a wide range of morphological types in a wide range of masses , sampling the color - magnitude diagram for m 18 mag ; ( ii ) a large fov , that guarantees to cover the entire optical extension of the galaxies up to 2.5 for an 80% of the sample ; and ( iii ) an accurate spatial sampling , with a typical spatial resolution of kpc for the entire sample , which allows to optical spatial resolved spectroscopic properties of most relevant structures in galaxies ( spiral arms , bars , buges , hii regions ... ) .the penalty for a better spatial sampling of the galaxies is the somehow limited number of galaxies in the survey , e.g. , manga and sami . in terms of the spectral resolution , while in the red both survey have better spectral resolution than califa , in the the blue wavelength range both three have similar resolutions .  as a legacy survey ,one of the main goals of the califa collaboration is to grant public access of the fully reduced datacubes . in november 2012we deliver our 1st data release ( husemann et al . 2013 ) , comprising 200 datacubes corresponding to 100 objects . after almost two years , and a major improvement in the data reduction , we present our 2nd data release ( garcia benito et al . , 2014 ) , comprising 400 datacubes corresponding to 200 objects , the 1st of october 2014 .the data products that can be derived from the ifu datasets obtained by the califa survey comprise information on the stellar populations , ionized gas , mass distribution and stellar and gas kinematics .similar data products are derived for any of the indicated projects : atlas3d , manga or sami . in summary , they conform a panoramic view of the spatial resolved spectroscopic prorperties of these galaxies  different science goals have been already addressed using this information : ( i ) new techniques has been developed to understand the spatially resolved star formation histories ( sfh ) of galaxies ( cid fernandes et al . , 2013 , 2014 ) .we found the solid evidence that mass - assembly in the typical galaxies happens from inside - out .the sfh and chemical enrichment of bulges and early - type galaxies are fundamentally related to the total stellar mass , while for disk galaxies it is more related to the local stellar mass density ; negative age gradients indicate that the quenching is progressing outward in massive galaxies , and age and metallicity gradients suggest that galaxy bars have not alter signicantly the sfh of spirals ; ( ii ) we explore the origin of the low intensity , liner - like , ionized gas in galaxies .these regions are clearly not related to star - formation activity , or to agn activity .they are most probably relatd to post - agb ionization in many cases ; ( ii ) we explore the aperture and resolution effects on the data .califa provides a unique tool to understand the aperture and resolution effects in larger single - fiber ( like sdss ) and ifs surveys ( like manga , sami ) .we explored the effects of the dilution of the signal in different gas and stellar population properties ( mast et al . , 2014 ) , and proposed an new empirical aperture correction for the sdss data ; ( iv ) califa is the first ifu survey that allows gas and stellar kinematic studies for all morphologies with enough spectroscopic resolution to study ( a ) the kinematics of the ionized gas , ( b ) the effects of bars in the kinematics of galaxies ; ( c ) the effects of the intraction stage on the kinematic signatures ( barrera - ballesteros et al . ,submitted ) , ( d ) measure the bar pattern speeds in late - type galaxies ( aguerri et al . ,submitted ) , ( iv ) extend the measurements of the angular momentum of galaxies to previously unexplored ranges of morphology and ellipticity ( falcn - barroso et al . , in prep .) ; and ( v ) finally we explore in detail the effects of galaxy interaction in the enhancement of star - formation rate and the ignition of galactic outflows .the results based focused on the analysis of the regions will be discussed in the next section .the program to derive the properties of the regions was initiated based on the data from the pings survey .this survey acquired ifs mosaic data for a dozen of medium size nearby galaxies . in and we studied in detail the ionized gas and regions of the largest galaxy in the sample ( ngc628 ) .the main results of this studies are included in the contribution by rosales - ortega in the current edition .we then continued the acquisition of ifs data for a larger sample of visually classified face - on spiral galaxies , as part of the feasibility studies for the califa survey .the spatially resolved properties of a typical galaxy in this sample , ugc9837 , were presented by .  in presented a new method to detect , segregate and extract the main spectroscopic properties of regions from ifs data ( hiiexplorer ) . a preliminar catalog of regions and aggregations extracted from 38 face - on spiral galaxies compiled from the pings and califa feasibility studies was presented .we found a new local scaling relation between the stellar mass density and oxygen abundance , the so - called -z relation .the same catalog allows us explore the galactocentric radial gradient of the oxygen abundance .we confirmed that up to disk effective radius there is a negative gradient of the oxygen abundance in all the analyzed spiral galaxies .the gradient presents a very similar slope for all the galaxies ( dex/ ) , when the radial distances are measured in units of the disk effective radii . beyond diskeffective radii our data show evidence of a flattening in the abundance , consistent with several other spectroscopic explorations , based mostly on a few objects ( e.g. * ? ? ?_ left panel : _ distribution of the oxygen abundances at the effective radii as a function of the integrated stellar masses for the califa galaxies ( 236 , circles ) , together with those from the califa feasibility studies ( 31 , squares ) . _right panel : _ distribution of the differential oxygen abundances with respect to the solid - line shown in the left - panel ( i.e. , the dependence on the stellar mass ) , as a function of the integrated sfr for the califa galaxies.,title="fig:",width=287 ] ._ left panel : _ distribution of the oxygen abundances at the effective radii as a function of the integrated stellar masses for the califa galaxies ( 236 , circles ) , together with those from the califa feasibility studies ( 31 , squares ) . _right panel : _ distribution of the differential oxygen abundances with respect to the solid - line shown in the left - panel ( i.e. , the dependence on the stellar mass ) , as a function of the integrated sfr for the califa galaxies.,title="fig:",width=287 ]  in we presented the first results based on the catalog of regions extracted from a enlarged sample of galaxies ( ) .we studied the dependence of the -z relation with the star formation rate .we found that no secondary relation different than the one induced by the well known relation between the star formation and the mass , contrary to what was claimed other authors , based on single aperture spectroscopic data ( sdss ) .although the reason for the discrepancy is still not clear , we postulate that simple aperture bias , like the one present in previous datasets , may induce the reported secondary relation .figure [ fig : mz ] presents an updated version of these results , including the last list of analyzed galaxies , until july 2014 ( 236 galaxies from the califa sample together with 31 galaxies from the califa - pilot studies ) .the left panel shows the -z relation found for these galaxies , with color code indicating the integrated sfr for each galaxy .it is appreciated that the stronger gradient in sfr is along the stellar mass , as expected for star - forming galaxies .once subtracted the best fitted function to the -z relation , the residual of the abundance do not present any evident secondary relation with the sfr ( as it is seen in the right panel ) .thus , the results presented in are confirmed with a sample of galaxies enlarged by almost a factor two .we also confirmed the local -z relation unveiled by , with a larger statistical sample of regions ( ) .this nebular gas -z relation is flatter than the one derived for the average stellar populations , but both of them agree for the younger stars , as expected if the most recent stars are born from the chmical enriched ism . in , we confirmed that the abundance gradients present a common slope up to effective radii , with a distribution compatible with being produced by random fluctuations , for all galaxies when normalized to the disk effective radius of .1 dex/ .finally , in , we found evidence that regions keep a memory of their past , by analysing the correspondance between the properties of these ionized regiones with that of their underlying stellar populations .in summary the results from the califa survey present a coherent picture of the mass - growth and chemical enrichment of galaxies .all the results indicate that the bulk of the galaxy population presents an inside - out growth ( at the mass range covered by the survey ) , with a chemical enrichment dominated by local processes , and limited effects by processes like outflows or radial mixing .

Abstract: we present here a brief summary of the status of the on - going califa survey with an emphasis on the results that have been recently published . in particular , we make a summary of the most relevant results found regarding the properties of regions discovered using this survey , and the evidence uncovered for an inside - out growth of galaxies .