the quest for solutions of dynamical strong gravity problems , such as black hole formation or mergers and gamma - ray burst production , that are astrophysically relevant and accurate enough to predict gravitational wave forms , has occupied much of the last half of the century and beyond .its history has been frustrated by the need to address several unforeseen numerical problems , including 1 ) proper initial data to begin the evolution , 2 ) the development of coordinate and physical singularities during the evolution , and 3 ) the growth of numerical instabilities in the time - dependent solutions .these problems have been dealt with by addressing each in turn with such techniques as 1 ) elevation of the initial data problem to a complete sub - field ; 2 ) development and analysis of appropriate gauge / slicing conditions that avoid coordinate singularities ; 3 ) excision of black hole centers , inside horizons , to avoid physical singularities ; and 4 ) the use of symmetric / hyperbolic equations to enhance numerical stability .one of the several remaining problems in this field is that , for some problems and some coordinate systems , the constraint - violating modes will grow exponentially .these eventually overwhelm any solution in only a short period of time ( ) , rendering a long simulation ( and the computation of any gravitational wave forms ) impossible .for very high - resolution simulations the exponential growth of errors begins early and continues until the errors diverge . on the other hand , for simulations with coarse spatial resolutionthe errors begin and remain at the truncation level until the much smaller constraint violations grow to a level that exceeds the truncation accuracy .then the solution joins the general exponential growth seen in the high - resolution simulations , blowing up in much the same manner as in the high - resolution case .the similar behavior of this error at a variety of mesh spacings indicates that it may be a numerical solution to the discrete equations that are being integrated .current attempts to solve this problem include adding the constraints as penalty functions to the evolution equations and techniques that re - converge the constraint equations every few time steps .constraint propagation is also an issue in the solution of maxwell s equations .techniques for doing so in the fields of astrophysical magnetohydrodynamics ( mhd ) and electromagnetics of antennas and waves have been in place for decades .these enforce the constraints not just stably , but _ to machine accuracy_.finite difference methods for constraint propagation in astrophysical mhd are known as the evans - hawley constrained transport ( ct ) method and now are an integral part of publicly used codes , such as zeus and zeus-3d . in engineering electromagneticsthese are known as the yee algorithm , and have many variants .they all involve building a mesh that is staggered in space , and often in time as well , and then defining appropriate vector and scalar quantities at whole or half mesh points .  while the success of ct for electromagnetics is certainly due in part to the linearity of the physical equations , its ability to maintain accuracy of the solenoidal ( ) and coulomb ( ) constraints , to a few parts in over tens or hundreds of thousands of time steps , is enticing .if such an algorithm can be found for numerical relativity , it could be as useful as excision and other such proven methods .  in this paper the properties of the electromagnetic ct method ( and of spacetime itself ) are examined , and a similar method is developed for numerical relativity .it is the thesis of this paper , and subsequent ones in this series , that ct methods work because they build spacetimes in which the temporal and spatial derivatives commute ( = o(\epsilon_r) ] are just the opposite .they are zero everywhere except at the hypercube _ body centers_. they look much like the identity tensors , but on a grid that is shifted one - half step in each dimension .furthermore , as the levi - civita tensor expresses the volume element \nonumber\end{aligned}\ ] ] its placement at the hypercube center creates a natural scheme for forming volume integrals over those hypercubes .for reasons that are developed more fully below , we will use mixed tensors to define the problem of numerical relativity . as noted above, such tensors will lend themselves easily to contraction .the classic problem of general relativity is to solve einstein s field equations ( ) for the metric coefficients .the einstein curvature tensor is derived from the ricci curvature tensor with being the ricci curvature scalar .the ricci tensor is the contraction of the riemann tensor on the first and third indices the riemann tensor is the full statement of curvature of the spacetime . in its mixed formit is given by for a coordinate basis .the doubly - raised connection coefficients are given by we assume that the metric has a unique inverse such that in this paper we will treat only the vacuum problem ( the source of stress - energy ) so that equation ( [ ein_field ] ) becomes the riemann tensor possesses several symmetries , including algebraic antisymmetry on and ( and on and ) and differential symmetries ( bianchi identities ) } & \equiv &  { { \rm r}^{\alpha \beta}}_{\gamma  \ , \delta  ; \ , \epsilon } ~ + ~  { { \rm r}^{\alpha \beta}}_{\epsilon \ , \gamma  ; \ ,\delta } ~ + ~ { { \rm r}^{\alpha \beta}}_{\delta  \ , \epsilon ; \ , \gamma } ~ = ~ 0\end{aligned}\ ] ] the reader will note that the mixed riemann tensor is missing one additional symmetry that is possessed by the covariant version : .( a raised index can not be swapped with a lower one . ) when contracted on the first and third indices , the bianchi identities become simply _i.e. _ , the divergence free condition on the einstein tensor .these four conditions are responsible for propagating the four constraints if the other equations are satisfied     how does satisfying the bianchi identities propagate the constraints numerically ? this is easy to show in riemann normal coordinates . for the vacuum problem ,the constraints are given by and we seek a scheme in which the constraints propagate to machine accuracy but satisfying the bianchi identities ( equation [ rn_bi ] ) will mean that the contracted identities are also satisfied to machine accuracy or   all that needs to be shown is that also .for the three momentum constraints ( ) this is straightforward , because are the field equations being computed , so is by definition .is accurate to only .however , if we use this truncation - accurate solution to re - compute , _ using exactly the same mathematical definition of terms that we used in the evolution equation _ , then that re - computed will be zero to machine accuracy .( it will not be so only if we use a different differencing scheme than the one used in the original evolution equation . ) as a simple example , consider a line of code that computes . then, if we later compute the function , by definition , will be zero to machine accuracy .] so the spatial gradients will be , thereby propagating to machine accuracy .  for the hamiltonian constraint ( ) we require satisfaction of the momentum constraints on the hypersurface , _i.e. _ , .therefore , as long as the momentum constraints propagate to machine accuracy on each hypersurface , which we have shown above to be the case , the hamiltonian constraint will propagate also .note , however , if the gradient of the momentum constraints has a constant bias , then the hamiltonian constraint will grow . on the other hand , if the divergence fluctuates randomly , the the hamiltonian constraint will fluctuate only randomly as well .   because a riemann normal coordinate system is local only and can not be used to cover the entire spacetime , we are forced to deal with non - zero connection coefficients .but we still will attempt to propagate the constraints in the same manner with a scheme that enforces the bianchi identities by employing temporal and spatial differences that commute .it is important to note that the bianchi identities are never _ explicitly _ calculated in the ct method .instead , we develop a scheme in which is satisfied to at least truncation accuracy for all time .( we will find that machine accuracy may be an unattainable goal .) however , in a general coordinate system , the update of will depend on an `` advection '' of curvature from surrounding cells , due to the covariant derivative connection terms .so , we equivalently seek a scheme in which to at least truncation accuracy .if we set up a grid in which } & = & o(\epsilon_{tr})\end{aligned}\ ] ] then we can proceed in much the same manner as in riemann normal coordinates , but with covariant derivatives .if the bianchi identities are satisfied to truncation accuracy , then their contracted form also will hold so , if the momentum constraints are satisfied on each hypersurface , the hamiltonian constraint will propagate , albeit along geodesics , not coordinate lines . andthe momentum constraints will propagate if the field equations are satisfied to at least truncation accuracy .   in order to demonstrate cancellation of terms in the bianchi identities , we choose the following form for the mixed riemann tensor with the singly - raised connection coefficient _re_-computed from the doubly - raised ones and the gradient of the inverse metric this form has the following properties  1 . possesses explicitly all of the algebraic symmetries discussed in section [ prob_statement ] .2 .  when the bianchi identities are formed and the symmetry in the last two indices of , the connection terms involving and will cancel , just as they did for the antisymmetric faraday tensor bianchi identities in equation ( [ maxwell_curved2 ] ) . ] and equation ( [ alt_raised_gamma ] ) is inserted , the terms will cancel _ to machine accuracy _ , as described before in section [ ct_in_rn ] .   in equation ( [ cb_full_bi ] )the terms will cancel explicitly _ algebraically_. will they cancel numerically also ?the answer is yes , if we apply the following numerical procedures  1 .use linear interpolation ( averaging ) to determine quantities at intermediate grid points 2 .when forming a product , such as , first average the factors to the grid point in question , then form the products and finally the sum .do not form the products on different grid points and then average .    while we are reasonably confident that , with these measures , terms in the bianchi identities will cancel , it is clear that the terms will _ not _ cancel to machine accuracy .these are all produced by connection of the double- terms in the riemann tensor .however , when forming the bianchi identities , the connection takes place on an already - multiplied and summed product .one can not undo the sums and products , form averages , and then re - form the triple- product . and the product of averages does _ not _ commute with the average of products .the bianchi identities , therefore , will be left with terms proportional to the truncation error and .unless some clever averaging scheme can be found to allow the triple- terms to also cancel to machine accuracy , any ct scheme developed along these lines will be subject to truncation error .the hope , then , is that cancellation of second and first order derivatives of the connection ( third and second order derivatives of the metric coefficients ) will be sufficient to improve the stability of the discrete evolution scheme .therefore , while the staggered grid ct method clearly works in riemann normal coordinates , detailed numerical experiments will be needed to see if it maintains its desirable properties when applied in a general coordinate basis .the goal of this ct scheme is to produce an interlocking , staggered four - dimensional grid of tensor values by successively adding new half and whole temporal hypersurfaces . because interpolation is in time , as well as space , the scheme necessarily will be an implicit one and , therefore , iterative .our approach then will be to first produce an initial guess for on the next level of hypersurfaces using an existing explicit scheme .this solution will not satisfy the interlocking staggered grid equations , so it will be iterated using the latter until it does .the exact iterative scheme is still under development , but one possible implementation is a multigrid technique in which the variables are the spatial and the equations are the field equations .this method appears similar to recently - suggested approaches in which the constraints are re - solved at each time step .there are , however , two key differences .first , the constraints are not solved explicitly . instead, the evolution equations are solved in such a way that they are implicitly enforced .this ensures a scheme in which the evolution and constraints are fully compatible and not tracking different numerical solutions to the differential equations .secondly , the equations being iterated are implicit hyperbolic , not elliptic .in addition to information on the new hypersurface , at each iteration these equations utilize information from the _ previous _ hypersurface information that constraint equations do not have . with thisadded information , if implemented properly , the iterative scheme should exhibit faster convergence than a regular constraint solver would have .figure [ fig5 ] shows implementation of a simple 1 + 1 space - time problem .comparison of this figure with figure [ fig4 ] will give insight into implementation of the full 4-d interlocking grid . and are quantities known from the previous time step , the first being a gauge condition and the second being the solution . and are gauge conditions on the _ new _ hypersurfaces , and is the new solution that will be determined by the iterative numerical scheme .specification of the coordinate gauge is similar to doing so in classic 3 + 1 schemes : is freely specified at cube corners on whole hypersurfaces , and is freely specified at cube edge centers on half hypersurfaces . is related to the shift vector , and both are properly located at half time steps vector points . is related to the lapse , and properly located in this scheme at scalar points .the are the six unique gravitational potential fields that we intend to solve .a typical time step then begins with components known on hypersurfaces , , and .one then specifies the vector on hypersurface and the scalar at .the explicit predictors for at then complete the new field , and the iteration on the can then begin . if the are dependent on the field ( which generally will be the case here ) , then the gauge conditions need to be updated at each iteration for a consistent solution .   the inverse of is often needed to raise the connection coefficients for use in the evolution equations . in principlethese should also be located at second - ranked tensor grid points . in practice , however , we never need the actual staggered fields . instead , we need their interpolated averages at third - ranked tensor points . furthermore , in order for the raising and lowering of connection coefficient indices to commute , the staggered field of values must be constructed in such a way that its average , and that of , are orthogonal at those third - ranked tensor points so construction of the index - raising tensor is a straightforward matter of averaging to places where the are computed , and then inverting that average _ locally _ at those grid points .the actual staggered fields of values , whose averages should give us these at third - ranked tensor points , never need to be determined .this procedure is fast , gives us raising and lowering operators that commute , and follows the afore - mentioned rule of averaging first and then multiplying and summing second .the initial data problem for a staggered grid will be a little more complicated than that for a non - staggered scheme . in the latter case , there are 12 unknowns ( and ) and four equations ( ) on the initial hypersurface at , for a net total of eight degrees of freedom . in the staggered case , is computed at , but the are computed at .solution of the momentum constraints , given at , is fairly straightforward in the staggered case .they are not functions of second - order time derivatives of the metric ( _ e.g. _ , ) , so the placement of those constraints at allows them to directly relate at to .all can be computed in a staggered manner .the momentum constraint solution at , then , would have six unknowns ( ) and three constraints , leaving three degrees of freedom , just like the non - staggered grid case .  the hamiltonian constraint , however , presents a problem .while also does not involve any second - order time derivatives , nevertheless it does involve _ first _ order time derivatives .those still are defined on half - hypersurfaces and , therefore , need to be _ interpolated _ to .that is , we need the at both and anyway , even if we are not going to compute . at first glance there does not appear to be a method of providing an accurate field to properly compute this interpolation .of course , one simply could extrapolate at forward to ( _ i.e. _ , assume ) and then solve there .the hamiltonian constraint solution then would have the six unknowns and one constraint five degrees of freedom just like the non - staggered case .but a serious problem still remains ._ there is no means of enforcing the hamiltonian constraint at ._ while the staggered grid is , in principle , capable of doing that , it can do so only through the evolution equations .but at this stage we have not yet begun to enforce the evolution .one possible method of solving this is to do nothing .just accept the fact that , at , is good only to truncation accuracy . choosing a very small ( a `` thin sandwich '' ) would keep this error small .a second approach would be to also explicitly enforce the constraint on , which would reduce the number of degrees of freedom of the hamiltonian problem from five to four .while producing a more constrained problem , this solution would result in two successive hypersurfaces on which the hamiltonian constraint is satisfied . yeta third alternative would be to locate on the hypersurface and extrapolate quantities like _ forward _ to .the problem with this approach is that , while the hamiltonian constraint is satisfied for the _ extrapolated _ field , when the evolution is begun , the hamiltonian constraint that is implicit in the evolution equations will use fields that are _ interpolated _ in time between and .there will be , therefore , an implicit constraint violation injected into the evolution at the outset .  the elegant , and proper , method of solving this problem is to _ solve all ten of the einstein field equations simultaneously on the initial hypersurfaces_. there will be 18 unknowns ( at , , and ) and 10 equations , leaving 8 degrees of freedom , just like the non - staggered case .the resulting fields will be properly staggered , and the will be properly staggered and interpolated .the hamiltonian constraint will be satisfied at using the correctly interpolated fields , and it will be satisfied at ( and even ) as well , because the evolution equations ( and therefore the bianchi identities ) are fully enforced .similarly , the momentum constraints will be satisfied at and for the same reason , regardless of whether they are actually applied at or at .the reader will , of course , recognize that this is more than solving an initial data problem ; in actuality the proposed scheme solves the initial data problem plus the first file evolutionary time step simultaneously .this is done to ensure that the initial data on the first three hypersurfaces are solutions of the discrete , staggered evolutionary field equations .no constraint violation will be introduced implicitly other than what is naturally present in the evolutionary method already .   when and are specified on the same boundary , where is the boundary normal , the boundary data problem is similar to the initial data problem . in all such cases , the boundary constraints are given by and for rectilinear grids with the boundary normal being a coordinate unit 1-form , this yields where the symbol is a label indicating the boundary direction in question , not strictly a coordinate index .the equation ( no sum ) is located at 3-cube corners and plays the role of boundary constraint in much the same manner as the hamiltonian constraint does at .similarly , the equations ( ) play the same role as the momentum constraints did earlier . by analogy ,then , the boundary problem is as follows .there are 12 unknowns ( [ , at and at , _i.e. _ , on the boundary and one ghost node _ beyond _ the boundary ) , and there are 4 equations ( [ boundary_constraints ] ) .this leaves 8 degrees of freedom again , which must be specified with additional boundary conditions .the diagonal constraint is applied at at 3-cube corners .two of the off - diagonal constraints (  $ ] ) are applied on whole hypersurfaces at , and the final constraint is applied also at but on time half - hypersurfaces .the reader will note that this latter set of equations is related to the set of momentum constraints , but the momentum constraints are defined at , , , ... , and propagated forward by the evolution equations .the constraint at must be explicitly enforced in most cases , along with the three others and the eight freely - specified at and .any number of additional boundary data problems are possible . could be specified at the upper boundary , while could be be specified at the lower boundary , for example .also , some of the eight free could be specified on opposing boundaries as well .  for periodic boundary conditions ,the six unknown at are set to those near , and those at are set to those near .no constraints are applied explicitly , only implicitly through the wrapping conditions and the evolutionary solution of the field equations .it is useful to summarize how a complete problem will proceed .the computation begins by using gauge conditions to specify at , , and and at and .the full initial data plus time - step problem , including the field equations , is then solved for on , , and , applying eight freely - specified ( or eight functions thereof ) in the process .appropriate boundary constraints also need to be applied in order to obtain a consistent solution .the field equations are generated as follows .an initial guess for the is obtained by some means , perhaps using conformal or other existing initial value methods or , for the evolution , an explicit forward integration scheme .the full field values then are differenced onto third - rank tensor grid points , and the are formed .the values are also averaged to those same third - rank tensor points , and an inverse of that average is used to raise the connection coefficients ( equations [ singly_raised_gamma ] and [ doubly_raised_gamma ] ) .the , in turn , then are differenced and averaged to second - rank tensor points for the riemann tensor calculation .( note that there will be no need for values at other fourth - rank tensor points [ hypercube body centers ] , as riemann will be immediately contracted into ricci . )equation ( [ mixed_riemann ] ) should suffice for the riemann calculation ( _ i.e. _ , the explicit version [ equation [ new_mixed_riemann ] ] should not be needed ) , because our raising and lowering operators commute , rendering the results of equations ( [ singly_raised_gamma ] ) and ( [ alt_raised_gamma ] ) the same to machine accuracy .contraction to ricci is trivial , as it sums riemann components already computed at second - rank tensor points , and the are then tested to see if they are zero . if not , the local values of are modified in an appropriate manner , and the computation of the ricci components is repeated until convergence .it is important to remember that at grid boundaries , the boundary constraints ( which are evolutionary equations in their own right ) must be applied and iterated upon as well , and the freely - specified boundary conditions must be applied .when the iterative solution is acceptable , the computation continues to the next hypersurface , first specifying the gauge conditions and then solving the field equations .the largest uncertainty in the proposed method is the effect of the truncation errors introduced by non - cancellation of the triple- terms .analytically we can be sure of exact constraint propagation only if the bianchi identities are satisfied exactly , and that is not the case in a general coordinate basis . on the other hand , the fact that the method does work in a local riemann normal system , and has some attractive properties of a finite differential calculus , are encouraging .furthermore , most of the time - dependent derivatives of do cancel in the general case , and this should enhance stability. analytical investigation of the stability of this method is difficult , so we have chosen to do so numerically .two numerical implementations of this scheme are being developed currently .the first assumes symmetries in two spatial dimensions , rendering the scheme explicit and avoiding the need for iteration .the second is a full implementation using staggered grids in four dimensions .results of these studies will help in determining the stability of the method .  at best, the method is expected to be conditionally stable .the evans - hawley and yee methods have this property , and are subject to a courant - like condition on the time step . with ,this implies , which a typical condition applied in most numerical relativity implementations .if it can be shown that such techniques provide stable constraint transport for finite difference implementations of numerical relativity , then other implementations might benefit from similar schemes that use four - dimensional grids and that have temporal and spatial derivatives that commute . in the finite element case ,this would begin by extending the elements into the time direction , rather than simply time - stepping a three - dimensional finite element grid .however , some additional features would have to be introduced , analogous to grid staggering , to ensure commutation of spatial and time derivatives across element boundaries .the pseudo - spectral may be intractable at the present time .as spatial derivatives are computed using the full spatial extent of the grid , in order to create a method in which these commuted with time derivatives , one could imagine using many , if not all , previous and future time hypersurfaces to compute the latter .this would be a truly four - dimensional grid method and involve solving the entire spacetime structure in one giant iterative procedure .present - day computers still struggle with three - dimensional explicit schemes , so a four - dimensional implicit one is clearly beyond current technology .however , in the not - too - distant future such codes might begin to be feasible .the author has benefited greatly from discussions with f. estabrook , l. lindblom , m. miller , d. murphy , and m. scheel .he is also grateful for a jpl institutional research and development grant , and for the continued hospitality of the tapir group at caltech .this research was performed at the jet propulsion laboratory , california institute of technology , under contract to the national aeronautics and space administration .clarke , d. a. 1996 , , 457 , 291320 . de raedt , h. , kole , j. s. , michielsen , k. f. l. , & figge , m. t. 2002 , arxiv : physics/0210035 .evans , c. r. & hawley , j. f. 1988 , , 332 , 659677 .stone , j. m. & norman , m. l. 1992a , , 80 , 753790 .stone , j. m. & norman , m. l. 1992b , , 80 , 791818 .scheel , m. a. , kidder , l. e. , lindblom , l. , pfeiffer , h. p. , & teukolsky , s. a. 2002 , phys .d , 66 , 124005 .yee , k. s. 1966 , ieee transactions on antennas and propagation , 14 , 302307 .

Abstract: a scheme is presented for accurately propagating the gravitational field constraints in finite difference implementations of numerical relativity . the method is based on similar techniques used in astrophysical magnetohydrodynamics and engineering electromagnetics , and has properties of a finite differential calculus on a four - dimensional manifold . it is motivated by the arguments that 1 ) an evolutionary scheme that naturally satisfies the bianchi identities will propagate the constraints , and 2 ) methods in which temporal and spatial derivatives commute will satisfy the bianchi identities implicitly . the proposed algorithm exactly propagates the constraints in a local riemann normal coordinate system ; _ i.e. _ , all terms in the bianchi identities ( which all vary as ) cancel to machine roundoff accuracy at each time step . in a general coordinate basis , these terms , and those that vary as , also can be made to cancel , but differences of connection terms , proportional to , will remain , resulting in a net truncation error . detailed and complex numerical experiments with four - dimensional staggered grids will be needed to completely examine the stability and convergence properties of this method .  if such techniques are successful for finite difference implementations of numerical relativity , other implementations , such as finite element ( and eventually pseudo - spectral ) techniques , might benefit from schemes that use four - dimensional grids and that have temporal and spatial derivatives that commute .


neyman and scott ( 1952 , 1953b ) , using statistical techniques based on counts - in - cells , showed that the galaxy distribution is statistically clustered rather than statistically uniform , and on the other , abell actually identified from palomar sky survey plates , 2712 operationally well - defined rich clusters of galaxies ( abell 1958 ) . to better describe the clustering property , the two - point correlation function ( peebles 1973 ) and continual high order correlation functions ( croton et al 2004 ) are widely used .high density regions , such as filaments ( wu et al . 2012 ) , superclusters ( connolly et al . 1996 ) and low density regions , such as voids ( danny et al . 2012 ) , are discovered and well discussed . at the same time , people begin to have interest for the comparison of high density and low density region .if the early universe is grown from quantum fluctuations , then positive and negative fluctuations will have the same probability and a sponge - like structure ( high density regions have the same topology with low density regions ) is doomed ( gott 1986 ) . with the evolution dominated by gravitational instability ,the sponge - like structure is most likely remained whatever the growth stays in the linear regime or is damped ( bond & szalay 1983 ) .gott ( 1986 ) found that both two regions have similar properties from the measured density contours , such as sponge - like shape , connectedness .however , gott ( 1986 ) mentioned a direct test more than measuring density contours is needed .here we use the hexagon cell method to investigate the equivalent of high density and low density regions .our hexagon cell method is inherited from the dodecahedron cells ( kiang 2003 , kiang et al 2004 ) ; it is actually a 2-d application of dodecahedron cells method .different cells have been applied to analyze the topology of galaxy distribution , such as truncated octahedrons ( coxeter 1937 ; gott 1967 ; gott 1986 ) , voronoi foams ( icke and weygaert 1987 ) , dodecahedron cells ( kiang 2004 ) ; 2d topological structure of the observed universe has been investigated by variety of methods , such as genus method ( davies & coles 1993 , colley 1997 , hoyle & vogeley 2001 , hoyle et al 2002 ) , percolation and ?lamentarity ( pandey & bharadwaj 2008 ) .colley ( 1997 ) found that the 2d genus of the las campanas redshift survey is very close to a gaussian random field , hoyle claimed the genus of 2 degree field galaxy redshift survey team ( 2dfgrs ) and sloan digital sky survey ( sdss ) reveals a smaller number of isolated voids than clusters ( hoyle & vogeley 2001 , hoyle et al 2002 ) .  the motivation of this paper is to investigate the topological relation between low density regions ( such as voids ) and high density regions ( such as clusters ) to get more understanding of the early universe .our article is organized as follows . in section 2we present the hexagon cells method . in section 3we study the data from the 2mass .we summarize the results in section 4 and have conclusions in section 5 .if we partition the 2-d space into cells with the shape of hexagon , we can get hexagonal lattice .hexagonal lattice is one of the five 2d lattice types and is the only lattice which can fill the 2d space inseparably with more edges than square . in the natural world, honeycomb is an interesting example of hexagonal lattice . for bees , hexagon is the most efficient shape for the bees living and storing of food . in the reality ,the hexagon structure provides the maximum strength and that it is used in designing airplane wings and satellites walls .the reason of using hexagon cells in this paper is that we want to have the direction detection as many as possible for our unit cells .hexagon cells could detect the neighbors in six directions and provide us the rich topology information , and at the same time fill in the space without overlap or missing region .for an application of hexagon method on galaxy distribution , similar with the definition of rhombic cell in kiang ( 2003 ) , if a hexagon cell contains no galaxies , then we call it an empty cell ; otherwise , it is a filled cell . automatically empty cells represent the low density region and filled cells represent the high density region . by this definition, we could discover if the filled and empty regions have the same morphology and give us a quantitative description of the topology .we know the hexagons have 6 neighbors and 6 edges shared with neighbors .if two neighbor cells are both filled or both empty , then we define their common edges as inner edge ; otherwise we call it as outer edge .we use n to define the number of inner edges .the physical meaning of n is to find the same kind of neighbors for each hexagon and investigate the aggregation pattern for filled and empty cells . in this wayn is the approximation of the two point correlation function in a specific small scale .if filled cells are not distributed in space randomly , we can expect the distribution of n will be depart from the well - known binomial form , same thing for empty cells .this in turn will tell us the morphology similarity and difference of high density and low density regions .based on the definition of , we could divide the six edges into inner edges and outer edges two groups and consider the connectedness of inner edges or outer edges .we use to represent the number of connected inner edges group and use to represent the number of connected outer edges group , that is ( see fig .1 ) . so means a single cell surrounded by 6 unlike neighbors ( same with ) and means a single cell surrounded by 6 like neighbors ( same with ) . means likely that the cell is on the boundary of the clusters or voids , so its two kind of neighbors are connected separately . represents a string - like structure and the string penetrates the cell and results in two groups for each of kind neighbors . is complicated and its special shape possibly implies a mesh structure .to better understand the and distribution difference between observed and random sample , similar with the method mentioned by kiang ( 2003 ) , we define statistic as a measure of the degree to which the observed frequency of cells exceeds its random expectation for each different structure defined by and . is for observed sample , is for random sample .   to investigate the uncertainties of the results , we use same method which kiang ( 2004 ) used in rhombic cell analysis .obviously the present application of hexagon cell analysis revealed that and results are highly sensitive to the exact placing of the grid of hexagon cells . because when the galaxies are fixed , if the cells are moved it will get same galaxies to lie in different cells , we call it as the zero offset .we start our calculations by placing the centre of our zeroth cell ( 0 , 0 , 0 ) at galactic coordinates ( 0,0,0 ) .what we found was that if we displace our entire grid of cells by an amount up to and including one unit of a0 , in any combination of the three directions , then the resulting and will generally be different , but slightly .if we define the length of cell is 1 , we simply consider 4 shifts which are ( 0 , 0.5 ) , ( -0.5,0 ) , ( 0.5,0 ) , ( 0.5,0.5 ) .the four shifts will be combined with the origin gridding to get average and value with stand deviation .generally the shift will result in the unequal number of filled and empty cells , but it is very tiny ( around 1/1,000 ) , moreover , it makes more sense for us to keep the length of the cell rather than adjusting it to get equal number of filled and empty cells , so we fix the length of cells when we shift .the 2mass ( xsc , jarrett et al . 2000 ) data we use is provided by cao et al ( 2006 ) , detailed information can be found in their paper . simply speaking , they select galaxies inside an elliptical isophote with surface brightness of 20 mag in ks band . to have 2-d plane data , they subject the 2mass data to an equal - area projection by the lambert azimuthal algorithm :    where is a relative scale factor , is the galactic latitude and is the galactic longitude .this hemisphere scheme projects the whole sky into two circular planes , a northern and southern sky .we select a square with in the central part of each circular plane .we have two fields of in the northern and southern skies .the northern square has 323,000 galaxies and the southern includes 342,000 galaxies .before we analyze the data , we could get the binomial distribution for reference .the binomial distribution gives the discrete probability distribution of obtaining exactly successes out of bernoulli trials ( where the result of each bernoulli trial is true with probability and false with probability , as given by eq.4 below : ) .  since for any neighbors of a cell , the probability of a filled or empty cell is the same ( we have the same number of filled and empty cells ) , so and . for an distribution , ( 6 walls for hexagon ) , and n=(0,1,2,4,5,6 ) . from eq . ( 4 ) the probability function of a binomial distribution is ( 1/64 , 6/64 , 15/64 , 20/64 , 15/64 , 6/64 , 1/64 ) for n= ( 0 , 1 , 2 , 3 , 4 , 5 , 6 ) . for , since this is related to specific arrangements , it is better to list all combinations and calculate the probability than using eq.(4 ) .since there are 6 walls , each wall could be inner wall or outer wall , so the total combinations give arrangements .if we use `` 1 '' for inner wall , and `` 0 '' for outer wall , we could write number 0 63 with 6-digits binary format , such as 000000 , 000001 .... then the 64 binaries are exactly all possible arrangements of .we list the results in table 1 :  . [ cols="^,^,^,^,^,^,^,^",options="header " , ]   from =6 ( or ) filled cells have a much stronger ( 21 for northern sky and 15 for southern sky ) aggregation property than empty cells .this is consistent with hoyle ( 2002 ) who found that the number of isolated voids is smaller than the number of clusters in the 2df survey . the most general case is ( 20/64 probability for standard distribution ) ; however , we can see the value at is smaller than standard distribution .this can be explained as that aggregation makes filled and empty cells have more probability on than standard distribution , hence reduce the probability at . for =1 , filled cells are closer ( 12 for northern sky and 6 for southern sky ) to standard distribution than empty cells , which implies that empty cells have less or longer fiber structure than filled cells as the end of fiber always has =1 .=1 must result in a structure , we do not see much difference between filled and empty cells for in table 1 .this suggests =1 structure is the minority in most structures , in other words , includes much more cells on the boundary of filled and empty cells than of the end . has also significant difference ( 12 for northern sky and 8 for southern sky ) between filled and empty cells .empty cells have much more frequently penetrated by filled cells than filled cells penetrated by empty cells , which implies that empty cells have less fiber structure than filled cells and is consistent with =1 results .with a direct measurement by the hexagon cell method , we found there are significant differences in topology between high density regions and low density regions in 2mass data .the accuracy is assured by the zeroshift technique .we found low density regions have less fiber structure and are less clustered than high density regions to high accuracy .the difference could be caused by the evolution , which suggests that evolution might include more things other than gravitational instability ( bond szalay 1983 ) ; or caused by the original asymmetry of quantum fluctuation in the early universe ( turner 1998 ) . overall speaking ,low density regions have slightly closer topology to random distribution than high density regions .we also found the topology of 2mass southern sky map is slightly closer to random distribution tan northern sky .our work provides a supplement for genus analysis , which shows that high density and low density regions have almost identical properties ( park et al , 2001 ; gott 1986 ) or are slightly different ( hoyle et al 2002 ) .

Abstract: we introduce the new 2-d hexagon technique to probe the topological structure of the universe . the 2-d matter space has been filled with hexagon unit cells , by detecting whether the cell includes matter or not we define the filled cells and empty cells , corresponding to high density region and low density region respectively . the number of the filled cells and empty cells are kept same by controlling the size of the cells . by analyzing the six neighbors of each hexagon we can get and compare statistical topological properties of the high density region and low density region in the universe , in order to have a more understanding of the evolution of the universe . we apply this hexagon method on 2-mass data and discover the significant topological differences between the high density and low density regions , even both of them have significant topological shift from the binomial distribution , or the random distribution .


the direct imaging and characterization of earth - like planets , especially those capable of supporting life , is one of the outstanding goals of modern astrophysics and science in general . currently , high - angular resolution astronomical imaging can be achieved interferometrically by combining the wavefronts from spatially separated telescopes  . interferometric imaging and nulling require that the light from multiple apertures be combined .although beam combining can be performed using bulk optics , integrated optic ( io ) implementations offer a number of important advantages .these include spatial filtering , enhanced stability , on - chip fringe scanning , compactness and scalability .io beam combiners for astronomical imaging were first proposed by kern , malbet , schanen - duport , and benech in 1996  .using silicate - based glass io devices , laboratory and on - sky stellar interferograms were demonstrated at astronomical h ( 1.5 m 1.8 m ) and k ( 2.0 m 2.4 m ) bands  . in many situations , especially those involving nulling and exoplanet search , operation in the infrared beyond 3 m , where silicate - based glasses are not transparent ,is required .on - sky interferometric measurements have been performed in the l band ( 3.0 m 4.0 m ) using a guided - wave device , consisting of a two - beam fluoride glass fiber coupler , but this fiber - based technology is not easily scalable to multiple apertures  .we have recently developed a prototype , single - mode , integrated optic , astronomical , beam combiner fabricated by titanium - indiffusion in an x - cut , lithium niobate ( linbo ) wafer  .the device operates in the 3.2 m 3.8 m spectral region , which lies in the l band , and has on - chip , electro - optically ( eo ) controlled , fringe scanning capabilities .our results confirm that io devices are well suited to perform the beam combining function for astronomical imaging . in the infrared wavelength region ,a host star is normally a million times brighter than the planet orbiting it , which presents major difficulties when trying to image the planet directly . nulling interferometry offers the possibility to overcome this problem by attenuating the stellar light , thus enhancing the visibility of the planetgenerally , achromatic phase shifting and broadband achromatic beam combining functions are required for deep nulling , with nulling depths of 10 or better over the infrared spectrum . although we can image the planet by nulling at a single wavelength , there are key reasons to use a broad wavelength band .first , there are several key biomarkers in the infrared spectrum from 6 m to 18 m .second , the total integration time needed to detect a planet increases as the spectral bandwidth is reduced .  one key component for achieving a broadband deep null is the broadband achromatic beam combiner . generally , there are two basic types of planar , integrated optic beam combiners : reversed - y combiners and directional couplers .the operation of a reversed - y combiner is achromatic by symmetry .a reversed - y combiner , however , when used as part of a nulling interferometer , suffers an inevitable 3 db loss of signal . on the other hand, directional coupler type beam combiners can capture the entire signal , and hence are theoretically lossless provided that both interferometric outputs are recorded and subtracted .unfortunately , the operation of the directional coupler is chromatic , and this wavelength dependence prevents the device from achieving deep broadband nulls .various approaches have been suggested to mitigate chromaticity of 2 by 2 couplers and achieve a wavelength flattened response .these approaches include ( i ) asymmetric directional couplers , where the two constituent waveguides differ in width but are uniform along the direction of propagation  , ( ii ) mach - zehnder type interferometers that are wavelength - insensitive  , and ( iii ) tapered velocity couplers , where the waveguide widths are varied along the propagation direction  .the operation of the devices described by ( i ) and ( ii ) above depend on the interference between modes , while the tapered velocity mode coupler ( iii ) is meant to operate adiabatically , where there is little coupling between the local normal modes .the later device is also referred to as a mode - evolution coupler .unfortunately these approaches do not yield an ultra - broadband achromatic response nor do they preserve the initial phase difference of the input beams in the interference term at the output .lack of broadband chromaticity in either power splitting or phase can seriously degrade the deep nulls required for exoplanets search . in this paper, we propose and present a theoretical design of an achromatic , broadband , polarization - insensitive , mode - evolution , integrated optic , beam combiner suited for space - based nulling interferometry .the design is based on a system of three coupled waveguides along the lines of  and  .the proposed device can be realized using germanium - on - silicon or germanium air - bridge waveguide structures  .  in section [ lnm ] ,the theory of normal modes of a three coupled waveguide system is presented . a design of an achromatic , polarization - insensitive, mode evolution beam combiner is proposed in section [ proposeddesign ] . in section [ cmtoflnm ] ,the derivation of the coupled mode equations that describe the mode coupling between the local normal modes is given , and a condition for adiabatic operation is presented . in section[ abcdesign ] , a numerical design example of the proposed achromatic beam combiner , based on candidate technologies for waveguide fabrication in mid - infrared wavelength region , is presented and the device performance is numerically evaluated .the results of the paper are summarized in section [ abcdis ] .before we go into the details of the design of the proposed fully - achromatic , broadband , mode - evolution beam combiner , we first develop representations for the normal modes of a symmetric three coupled waveguide system shown in the fig .[ fig : threewg ]  .the waveguides shown in fig .[ fig : threewg ] are assumed to be uniform , i.e. , do not change along the direction of propagation .we also assume that ( this assumption is valid when the waveguides are weakly coupled ) the total scalar ( te or tm ) field, , of the combined three waveguide structure can be approximately written as the weighted sum of the normalized waveguide modes , , , , associated with each individual waveguide expressed in the single , , coordinate system shown in fig .[ fig : threewg ] .thus , where is a real - valued quantity which describes the spatial profile of the mode and is the associated propagation constant . the coupled - mode equations describing this system are  where and are mode propagation constants for outer waveguides and is that of the center waveguide , is the coupling coefficient between waveguides and , and , , and are the complex - valued mode amplitudes associated with the individual waveguide modes in waveguides 1 , 2 and 3 , respectively .we further assume that the outer waveguides are identical and equidistant from the center waveguide , and thus = and = .considering a longitudinally varying waveguide structure , the amplitudes of the local normal modes are not constant as a function of position along the waveguide .these longitudinally - varying local normal modes will be designated by , and and expressed as where and , are the local normal modes of the individual waveguides .thus we define as and write by substituting eq .  into eq . , we obtain the following local normal modes of the three coupled waveguide structure and the waveguide parameter is defined as , where . in general , the mode profile of a single local normal mode will evolve as it propagates along a longitudinally - varying waveguide structure .if we consider the three local normal modes of the combined longitudinally - varying three - waveguide structure , it is easily verified that , associated with the antisymmetric mode , remains constant along the whole structure .the other two symmetric local normal modes are identified by their mode order , i.e. , by the relative magnitudes of their propagation constants , and , respectively . the lowest order mode will be identified by the subscript , and the third mode is identified by the subscript ( where ) .generally , power transfer occurs between the local normal modes of a longitudinally - varying structure .the degree of power transfer between the local normal modes is a function of the overlap parameter between their spatial mode profiles , and where .this overlap parameter is given by  when , as occurs for a structure that does not change along the propagation direction , there is no power exchange .similarly if the structure varies slowly enough in z , then will be small and will be approximately zero .thus there will be little power exchange between the local normal modes .we propose a design of waveguide structures that are neither invariant along z nor vary slowly along z but for which and . when there is very little power transfer between the local normal modes , the mode propagation is said to be _based on the functional forms of the local normal modes of a three coupled waveguide system as shown in eq . , we propose to use the symmetric three coupled waveguide structure shown in fig .[ fig : abclayout](a ) to achieve broadband achromatic beam combining .the waveguide parameters are given in table [ tab : wgparameters ] .if monochromatic input beams with complex amplitudes and , respectively , are launched into the outer waveguides 1 and 3 , then the excited input field at = 0 can be expressed as the linear combination of the local normal modes and of the coupled structure at = 0 as indicated below \nonumber \\   & = |a_1|\exp(j\phi_1)[\frac{1}{\sqrt{2}}a_0(0)+\frac{1}{\sqrt{2}}a_+(0 ) ]   \label{eq95}\end{aligned}\ ] ] \nonumber \\   & = |a_3|\exp(j\phi_3)[\frac{-1}{\sqrt{2}}a_0(0)+\frac{1}{\sqrt{2}}a_+(0 ) ]   \label{eq95a}\end{aligned}\ ] ] (0 ) \nonumber \\   & + [ \frac{1}{\sqrt{2}}|a_1|\exp(j\phi_1)+\frac{1}{\sqrt{2}}|a_3|\exp(j\phi_3)]a_+(0 )   \label{eq95b}\end{aligned}\ ] ] at the output of the device we wish to recover the input phase difference and the full power , i.e. , , of the incoming beams .the width of the center waveguide is kept fixed while the widths of the outer waveguides are equal to one another at all points along but are varied along to achieve the desired dephasing .note that the separation between waveguide 1 and 3 is assumed to be sufficiently large so that and thus can be neglected .the coupling term , , and the dephasing term , , of the waveguide structure are chosen as indicated in table [ tab : wgparameters ] .it will be shown in sec .[ abcdesign ] that the waveguide transition can be made nearly adiabatic ( i.e. , , ) .thus the power in each local normal mode will remain constant as the mode propagates along the length of the structure , although the shape of each local normal mode will gradually evolve due to the adiabatic change of the waveguide parameters .the antisymmetric mode and the lowest order symmetric mode at will evolve into the following forms , respectively , at the output .thus by superposition the total output at becomes therefore , the intensities at outputs of waveguides 1 , 2 , and 3 at are given from these expressions we see that the total power in all three waveguides combined is , and thus there is no loss in power . furthermore , the input phase difference , , is preserved in the output interference term , , which is wavelength and polarization independent , guaranteeing fully achromatic and polarization - insensitive operation .the definition of an adiabatic waveguide transition is a transition between two waveguide structures that takes place in such a manner that negligible power transfer occurs between the normal modes as they propagate from one structure to the other .lack of mode conversion facilitates device design and can lead to wavelength insensitive operation as seen in sec .[ proposeddesign ] .conventionally , the suppression of mode conversion has been achieved by slowly varying the waveguide structure along the direction of propagation .it was recently suggested , however , that a slow longitudinal variation , though sufficient , is not necessary to achieve adiabatic operation .more generally , the suppression of mode conversion is rooted in the small magnitude of the nonadiabatic term itself  .thus conventional notions of adiabaticity based on slowly varying changes can be abandoned , and a new and different type of adiabaticity based on a controlled interaction can be exploited . in this section , the coupling equations governing the power transfer between local normal modes and the expression for the nonadiabatic term will be derived .   in order to calculate the spatial variation of electric field and magnetic field of the optical waveguide structure shown in fig .[ fig : abclayout ] , we will use the vector wave equation derived from time - harmonic form of maxwell s equations the time dependence of the field is of the form .the dielectric constant of a waveguide is related to its refractive index by , where is the free space electric permittivity , and the magnetic permeability is assumed to be its free space value ( ) everywhere . is the free space wavenumber , is the vacuum speed of light and is the wavelength of light in free space .although it is not necessary to do so , we will adopt a scalar analysis , and thus the last term in eq .  will be neglected .considering quasi - te mode ( i.e. , and ) or quasi - tm mode ( i.e. , and ) , the scalar ( either y - component te or x - component tm ) electric field of the waveguide can be written as where is the slowly varying field profile along the propagation direction , is the nominal propagation constant and is the nominal effective index of the optical mode .if the waveguides have no abrupt changes , then the transverse field will vary very little on the scale of a wavelength , and hence the paraxial approximation , is valid . if we substitute eq .  into eq . ,apply the paraxial approximation , and replace the vector operator by the scalar laplacian , we obtain next if we substitute or back into eq .  or, we obtain the following equation where for the quasi - te mode and for the quasi - tm mode . the propagation operator is given by it can be easily shown that the operator is hermitian . for the sake of simplicity in exposition, we will restrict the remaining discussion to the te - mode and will designate the operator simply as .the analysis for the tm - mode is similar .the three local normal modes , and of the waveguide structure shown in fig .[ fig : abclayout ] are eigenfunctions of eq ., i.e. ,    in the following analysis , the derivation of the coupling equation of the local normal modes will be carried out .consider a system of three coupled waveguides with corresponding local normal modes , and , which are assumed to be real - valued ( without loss of generality ) since the operator appearing in eq .is hermitian .the local normal modes will be normalized such that and the inner product written in bra - ket notation is defined as it follows from eq .  and the fact that , and are real - valued that since the operator is hermitian it also follows that the local normal modes are orthogonal , that is and therefore radiation modes will be neglected , and thus the total electric field , , in the coupled waveguides can be written as the following linear combination of the local normal modes substituting the above equation into the paraxial wave equation  yields where , and are the local propagation constants associated with the local normal modes , and .thus multiplying both sides of eq .by , integrating over and using eqs . ,  and , yields similarly by multiplying both sides of eq .  by or and integrating over yields  equations ,  and  can be written in matrix form as follows where when there is no coupling ( i.e. , power exchange ) between the local normal modes , and , the modes are said to evolve adiabatically . from equation , it is clear that mode evolution will occur adiabatically when for all .it follows from eq.  that mode evolution will be adiabatic when the waveguide structure is uniform or changes slowly along the propagation direction since for these cases equation , however , need not be satisfied for adiabatic operation , and the more general condition is given by .in future discussions , will be referred to as the nonadiabatic terms . according to the results in sec .[ lnm ] , the three local normal modes of the three waveguide structure can be written as where are the local normal modes of each of the three individual waveguides . without loss of generality, the will be assumed to be real - valued and normalized such that it follows from eq .that since the two outer waveguides are identical and symmetrically placed relative to the central waveguide , it also follows that combining eqs .- , it may be easily verified that consequently , thus for adiabatic operation we need only consider the coupling between the symmetric local normal modes and .   because the derivation of the nonadiabatic term , , is similar for both quasi - te and -tm modes, we now will only consider the quasi - te mode and denote the corresponding operator as , , and for the coupled three - waveguide system , the coupled waveguide system consisting of the individual waveguides 1 and 3 alone , and the individual waveguide 2 alone , respectively , with the following distribution of dielectric constants : where and are the refractive indices of the waveguide cores and cladding regions , respectively .the waveguides and refractive index profiles are shown in fig .[ fig : wgprofile ] for a buried waveguide structure . the local normal modes of the coupled system and the constituent waveguide modesare defined as the eigenmodes of the corresponding operator . where the index is used to distinguish the lowest ( ) and the other ( ) symmetric local normal modes of the coupled three waveguides system ( ) . is the lowest order local normal mode associated with the coupled waveguide system consisting of waveguides 1 and 3 alone . is the fundamental mode associated with waveguide 2 alone . since the operators are hermitian , the eigenfunctions of at each fixed z ( i.e. the local normal modes ) are orthogonal and the corresponding eigenvalues are real - valued . without loss of generality , the local normal modes are normalized as follows : it is easily verified that and are given by  \label{eq80a } \\\psi_2(x , y;z)&=\phi_2(x , y;z )  \label{eq80b}\end{aligned}\ ] ] where , and are the fundamental waveguide modes of the individual three waveguides . forthis to be true , we need to assume weak coupling between waveguides 1 and 3 . is the overlap integral between and , i.e. , , and thus when the fundamental modes of the individual waveguides are known and the coupling between the individual waveguides is not very strong , the local normal modes of the coupled system can be approximated as a linear combination of the individual waveguide modes and thus substituting eq .  into the definition of the local normal modes of the coupled system , i.e. , eq . , andusing eq . , we obtain the following eigenvalue equation for the coefficients : where the matrix representation of the operator of the coupled system and the coupling coefficients between constituent waveguides are , respectively , defined as  where  if we treat the coupled waveguide system consisting of the individual waveguides 1 and 3 alone ( i.e. , ) as waveguide _ a _ and the individual waveguide 2 ( i.e. , ) as waveguide _ b _, then the coupling coefficient , , defined by eq . , is equivalent to the coupling coefficient between waveguides _ a _ and _ b _ provided that there is no coupling between individual waveguides 1 and 3 of the three coupled waveguide system . by taking the inner product of eq .  with and making use of the normalization conditioneq .  and eq . , the following additional constraint on is obtained the solution of the simultaneous equations  and  can be obtained using the method developed in  and further expanded upon by ishikawa  .weak coupling will be assumed , so that .it will also be assumed that and . if we introduce new parameters and as where the solution of eqs .  andcan be expressed as  where using eqs .  and, the nonadiabatic term , , can be expressed as follows : the nonadiabatic term is now expressed as a derivative of local waveguide parameters , i.e. , coupling coefficients and propagation constants .our goal is to minimize the nonadiabatic term , and thus design a nearly adiabatic device . if the center waveguide is chosen to be uniform along its length , then and thus ( since ) the last term in eq .in practice , a fully adiabatic transition can not be achieved .therefore , there will be some power exchange between the local normal modes and as they propagate down the structure .this power coupling , however , can be minimized by careful waveguide design .the design process will be illustrated in this section by considering a germanium / silicon raised strip three coupled waveguide system .equation  will be used to estimate the magnitude of the nonadiabatic term at each for the design example .the power that couples from local normal mode to , as the local normal mode propagates from the beginning ( i.e. , ) to the end ( i.e. , ) of the device , will be determined by solving eq .  numerically .both chalcogenide elements  and group iv elements , especially silicon and germanium  , have been studied as candidate materials for mid - ir waveguides .there are several promising substrate materials and fabrication methods for realizing mid - infrared waveguide circuits .the most promising candidates include :  1 .chalcogenide rib waveguides on chalcogenide substrates 2 .chalcogenide rib waveguides on silicon substrates 3 .silicon or germanium rib waveguide membranes with an air bridge underneath the rib to serve as a lower cladding layer 4 .ge / si heterostructure rib waveguides with the ge rib on top of a si substrate 5 .hollow air core arrow waveguides with a sige / si antiresonant cladding  the widespread use of silicon - based electronics , especially si - cmos technology , has become a major driving force beyond the development of silicon photonics  .the successful developments of silicon photonic devices have aroused interests in deploying silicon for mid - infrared applications  .silicon has low loss in the spectral bands 1.2 m 6 m and 24 m 100 m , while multiphonon absorption prevents its use between 6 m and 24 m .eo modulation using the free carrier plasma effect becomes more efficient at wavelength beyond near ir , thus enhancing the functionality of silicon optical circuits in the mid - infrared regions .traditional substrate cladding materials , such as sio , which are suitable at near - ir are too lossy to be used for device operation beyond approximately 2 m . suspended rib waveguides , as shown in fig .[ fig : mirwg](a ) , consisting of silicon membrane clad below and above by air have been proposed as a potential solution  .another promising approach is to use a ge / si heterostructure consisting of a ge raised strip waveguide of strain - relaxed crystal ge epitaxially grown upon a silicon substrate as shown in fig .[ fig : mirwg](b )  .crystalline ge has low loss from 1.9 m out to approximately 12.5 m . under certain conditions , the high refractive index difference between ge and si may permit si to be used as the lower cladding level even it is lossy in the 6 m 12 m band , since the penetration of the optical field into the cladding is limited .we will choose the ge / si heterostructure raised strip waveguide as the waveguide fabrication technique to illustrate our design method .the heteroepitaxial growth of multiple microns of high quality germanium film on silicon by multiple steps of growth and hydrogen annealing has been successfully demonstrated  . if the silicon cladding induces too much loss , a ge rib waveguide with an air bridge underneath ( see fig .[ fig : mirwg](a ) ) would be a logical choice instead .   the schematic plot of the achromatic mode - evolution beam combiner is shown in fig .[ fig : gesi2d ] along with cross - section of the waveguide geometry based on ge / si raised strip waveguide .the refractive indices of the germanium strip and silicon substrate in the n band are taken from reference  and are summarized in table [ tab : indexofgesi ] .due to relatively large refractive index difference between germanium and silicon , the optical mode of ge / si raised strip waveguide is well confined inside germanium strip . thus , the results obtained for the buried waveguide as described in previous sections can be similarly applied to the waveguide geometry based on ge / si raised strip waveguide . using the 3d structure together with a commercial , semi - vectorial 3d mode solver ( rsoft beamprop ) , the waveguide parameters that correspond to single - mode operation in the n band at a nominal wavelength of = 10 m for both te and tmmodes are determined .the ge strip height is chosen to be 3.5 m and the strip width for the center waveguide is fixed to be 3.5 m as well , while the widths of the outer waveguides shown in fig .[ fig : gesi2d](a ) are varied from 3.58 m at = 0 to 3.42 m at = . since the central waveguide is uniform with width = 3.5 m , the last term of eq .vanishes , and thus the nonadiabatic term is reduced to under weak coupling condition , , and thus according to eq . .therefore is approximately given by thus , nearly adiabatic operation of the device can be achieved by minimizing , where ( see eq . ) and ( see eqs .  and ) if the outer waveguides are identical to the center waveguide and uniform along the direction of propagation , it can be easily verified that according to eqs .  - .thus if the variation of the width of the outer waveguides is much smaller than their nominal widths and the gap between the outer and the center waveguides , then the following relations approximately hold therefore , eqs . ,  and can be approximately written as a three waveguide beam combiner is designed that achieves nearly adiabatic , achromatic operation for the te mode at a center wavelength of = 10 m .the waveguide parameters ( i.e. , and ) from to are varied according to table [ tab : wgparameters ] such that is small along the length of the device to achieve desired operation .the device s operation at 8 m and 12 m is also numerically evaluated .  the fundamental te and tm modes profiles for = = 3.5 m , and = 10 mare shown in fig .[ fig : gesitetmwl10 ] .the effective refractive index , , of te and tm modes are 3.6017 and 3.6454 , respectively .the mode profiles are used to compute the coupling coefficient , , at different center - to - center gap spacings between outer and center waveguides for both te and tm polarizations .the dephasing term , , is also computed as a function of the width of the outer waveguides .although not shown here , the coupling coefficients and the dephasing terms at 8 m and 12 m wavelengths have also been computed for the purpose of device verification at other wavelengths in the n band .the coupling coefficient between the outer waveguides and the center waveguide is restricted to positive values , while the dephasing term can be either positive , 0 or negative . in order to estimate the power transferred from the to the local normal mode, we use the equivalent two - waveguide system described by eq . . where  , , and are the modal amplitude of each individual waveguide , and and are the equivalent modal amplitudes of the straight and curved waveguides as shown in fig .[ fig : gesi2d](a ) , respectively .the coupling coefficient between the two waveguides in the two - waveguide system is related to the coupling coefficient between the outer and center waveguides of the three - waveguide system by . as indicated earlier the design approach which will be takenis to vary and along the length of the device in a manner consistent with table [ tab : wgparameters ] and in such a way that , where .such a design can be realized by choosing where is the maximum coupling coefficient ( which occurs at the device center ) , is the maximum dephasing term ( which occurs at the beginning and the end of the device , i.e. , at and ) , and is a monotonically ( or nearly monotonically ) increasing function of , with and , that controls the rate of change of and .the waveguide parameter is equal to . the nonadiabatic term according to eq .can be written as where .therefore , the nonadiabatic term can be expressed in terms of the variation of the waveguide parameter , , as follows the coupling between the local normal modes and for the te mode at = 10 m is computed with the nonadiabatic term described in eq .  for the different functions listed below  : maximum coupling coefficient , , and the maximum dephasing term , , are chosen to be 0.0068 and 0.012 , respectively .these values are selected such that the single mode condition for each individual waveguide and the weak coupling condition between outer and center waveguides are maintained at wavelengths from 8 m to 12 m , as the waveguide parameters from to are varied according to table [ tab : wgparameters ] .the fraction of power transferred ( i.e. , ) from the local normal mode to the local normal mode is numerically computed using eq .  with and .the result is shown for between 0 and 10000 m for the various functions and plotted in fig .[ fig : residuepkvsl ] . among different functions , for 3000 m , the raised cosine and the blackman functions outperform ( i.e. , smaller fractional power transferred from to ) the others and the blackman function is slightly better than the raised cosine function . based on fig .[ fig : residuepkvsl ] , we choose = 6000 m for our device .better performance can always be achieved by using a longer value of . for the blackman function and = 6000 m , the coupling term , , and the dephasing term , , along with the corresponding nonadiabatic term , ,are shown in fig .[ fig : kappanonadterm ] , and the power transfer characteristics between the local normal modes along the propagation direction is shown in fig . [fig : powertransferte10 ] . using computed coupling coefficients for both polarizations , together with eqs . , , and with = 0.0068 , = 0.012 , = 6000 m , and = 10 m , the device layout is obtained .the width variation of the outer waveguides and the separation between the outer and center waveguides , as a function of the position along the device , are plotted in fig .[ fig : widthdesign](a ) and [ fig : widthdesign](b ) , respectively .the performance of the broadband operation of the device design is also numerically evaluated using eq .  at wavelengths of 8 m and 12 m and the results are summarized in table [ tab : residuecheck ] . based on the data, it is seen that there is little coupling between the and the local normal modes , and hence the device operates adiabatically .consequently , according to sec .[ proposeddesign ] , the performance of our mode - evolution beam combiner is relatively insensitive to the operating wavelength and the polarization , even though the coupling coefficient , , and the dephasing term , , are highly wavelength - dependent .  the sensitivity of the performance of the device due to fabrication errors has also been numerically investigated .cases where small perturbations to the coupling coefficient , , or the propagation constant , , introduce asymmetries , i.e. , and , are considered .these perturbations are modeled here by the relationships where the deviation is a constant independent of z. after introducing these perturbations , the coupled mode equations  are solved for the output ( i.e. , , , and ) assuming that the symmetric mode ( i.e. , , , and ) was launched into the three waveguide structure . if the device operates adiabatically then the output should all remain in the mode , i.e. , , , and . the fraction of power coupled to the other modes , i.e. , , is given in table [ tab : residuecheckdev ] as a function of the deviation . according to table [ tab :residuecheckdev ] , the device operation is more sensitive to perturbations in the propagation constant than to perturbations in the coupling coefficient .the predictions in table [ tab : residuecheckdev ] .however , may be a bit pessimistic in practice since they assume that the errors are constant along the entire length of the device .we have presented the numerical design of a broadband , polarization insensitive , achromatic beam combiner for the operation in the astronomical n band based on ge / si heterostructure raised strip waveguides .the beam combining is intrinsically achromatic because of the symmetric arrangement of the three coupled waveguides .as opposed to a reversed - 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[ cols="^,^,^,^,^,^,^",options="header " , ]   [ tab : residuecheckdev ]1 . three coupled channel waveguides .waveguides 1 and 3 are identical and equidistant from waveguide 2 .( left ) the variation of the coupling coefficient , , and the dephasing term , , along propagation direction ( right ) the nonadiabatic term , , for the blackman function for the te mode at = 10 m .  fig .( top ) total fraction of power remaining when only the local normal mode is excited at , i.e. , and .( bottom ) fraction of launched power in mode ( and ) transferred to the local normal mode .

Abstract: integrated optic beam combiners offer many advantages over conventional bulk optic implementations for astronomical imaging . to date , integrated optic beam combiners have only been demonstrated at operating wavelengths below 4 m . operation in mid - infrared wavelength region , however , is highly desirable . in this paper , a theoretical design technique based on three coupled waveguides is developed to achieve fully achromatic , broadband , polarization - insensitive , lossless beam combining . this design may make it possible to achieve the very deep broadband nulls needed for exoplanet searching .