explosive phenomena in stars are often associated with the rapid release of nuclear energy and a lavish production of nuclear species . in any calculation dealing with novae or supernovae explosions it is highly desirable to implement the nuclear part of hydrodynamic codes as better as possible in order to get a suitable representation of these explosions and nucleosynthetic yields .a complication arises because of the huge quantity of nuclei involved , which makes it difficult to simultaneously handle hydrodynamics and nuclear combustion .recently , however , there have been performed hydrodynamic simulations in spherical symmetry incorporating several thousand nuclei ( rauscher et al .also , calculations in two or three dimensions of both kinds of supernovae are quite common . in these multidimensional calculationsthe mission of the nuclear network is to provide a reasonable nuclear energy rate at a minimum computational cost .this is usually accomplished by using small networks of 7 or 13 isotopes ( timmes , hoffman & woosley 2000 ) .detailed nucleosynthesis is calculatedted _ a posteriori _ by using a postprocessing technique with a larger network .the most common procedure used for integrating the nuclear networks handles the discretized chemical equations in their implicit form .this enhances the stability of the nuclear system , which is very rigid ( arnett 1996 ) , allowing it to take longer time - steps .it is worthy of note that despite the great dependence of the rates on the temperature , the effect of the latter is rarely incorporated implicitly into numerical codes .this enforces to make a reliable extrapolation of the temperature and to take small time - steps to guarantee stability , especially when the nuclei approach the nuclear statistical equilibrium ( nse ) regime .several methods have been suggested for handling nuclear combustion around the nse regime .the most simple and fastest way is to use a conventional integration method when the nuclear system is not too close to equilibrium and to rely on statistical techniques to treat the phase in which direct and reverse reactions are almost or completely balanced .nevertheless , this leads to the problem of how to make a smooth transition between the two regimes .another possibility is to always use a standard integration method , which takes care of adequately restricting the time - step near equilibrium . in this casethe computational effort increases .there are also mixed schemes ( hix & thielemann 1996 ) in which reduced networks are used to describe the evolution of light and a few intermediate - mass elements ( generally below  or ) and a statistical approach is taken for handling the remaining nuclei . while this method is a very promising tool for explosive nucleosynthesis studies , due to its ability to handle hundreds of nuclei with a low computational burden , the decoupling between the thermal evolution and nuclear kinetics again imposes a constraint on time - steps near nse .the importance of solving the nuclear network by coupling molar fractions and temperature in the same matrix was demonstrated by mller , who used a small network of 13 nuclei , from helium to nickel .he showed how the inclusion of the temperature effects was decisive in avoiding the numerical oscillations which otherwise appear during and after relaxation to the nse regime .however , in that paper there were presented only very few test calculations , limited to isochoric combustion .moreover , the use of an entropy equation instead of the most frequently used energy equation was preferred to calculate the temperature evolution .in this paper we extend mller s previous work to consider a larger nuclear network and to analyze in greater detail the transition between time - dependent combustion to complete nuclear equilibrium and viceversa .we also propose a method for integrating based on the energy equation that is robust and straightforward to implement , giving very good results in all the realized tests .in addition we have exhaustively checked our numerical scheme against a large variety of thermodynamic conditions of astrophysical interest .these include not only isochoric combustion , as considered in mller seminal work , bul also adiabatic , isothermal and isobaric processes , as well as the particular case of combustion within a strongly coupled coulombic plasma .nowadays the effort to write the proposed numerical scheme for larger networks ( even of ` kilo - nuclide ' size ) is considerably less than a decade ago , as we have the great advantage of a recently published library of theoretical reactions rates ( i.e. rauscher & thielemann 2000 or the nacre compilation , angulo et al . 1999 ) .an efficient algorithm of this kind could be suitable for handling a variety of vigorous nuclear combustion modes such as detonation , flames or silicon photodisintegration .it is especially useful whenever the operator splitting technique has to be applied to save computational time .currently , this often happens in hydrodynamic calculations of both kinds of supernovae ; in these situations the implicit thermal coupling could help to enlarge the time - steps without degrading neither the accuracy nor the stability of the system .there is also the additional advantage to this scheme that it is more compatible with the philosophy behind the post - processing technique .a necessary condition for obtaining a detailed nucleosynthesis from the thermal history of an explosive event , calculated beforehand using a reduced network , is that the small and large networks give a similar energy release . integrating the chemical equations along with the energy conservation helps to fulfill this requirement .  in 2 we describe the method of calculation and its implementation in two nuclear networks with 14 nuclei and 86 nuclei respectively .the ability of the scheme to monitor the quasi ( qse ) and complete statistical equilibria is discussed . in 3 the results of three tests are provided : adiabatic expansion from the nse stage , hydrostatic and explosive silicon burning and nuclear flame propagation , all of them of evident interest to current studies on supernovae . the role of coulomb corrections on the chemical composition of burned matter at high densitiesis also discussed in 3 . finally ,  4 is devoted to summarizing the conclusions .to integrate the set of differential equations associated with the nuclear species we have chosen a scheme based on the simple and well checked implicit method by arnett & truran .the main motivation behind such election is that arnett&truran method has been traditionally used in nucleosynthetic calculations , always giving satisfactory results with low computational effort .in addition it is straightforward to modify in order to incorporate the thermal feedback and other proposed changes into the basic scheme .there are some differences between our integration method and that used in mller worth to mention . in mller swork the evolution of the system was followed by coupling the nuclear evolution with the entropy changes at constant density .moreover , it was assumed that entropy changes were only due to the release of nuclear energy , thus no other sources of entropy were considered . instead of the entropy equation our formulation uses energy conservation which is more often found in current calculations of explosive combustion . moreover , the proposed algorithm is quite general , not restricted to isochoric processes , and potential entropy variations coming from heat exchange were also included .minor effects , such as the incidence of coulombic corrections to the ionic component of the plasma and screening enhancement factors have been also incorporated to our scheme .a simplified description of our method can be found in garca - senz&cabezn ( 2003 ) .starting from a typical differential equation governing the evolution of the molar fractions of the i - specie , :    where  stands for particle reactions and  is for photodisintegrations and decays . as usual , reaction rates between identical particles have to be divided by the corresponding factor ( two or six for binary and ternary reactions respectively ) .the triple - alpha reaction was also incorporated to equations ( 1 ) .other possible three body reactions can be included in the same way .this set of equations is then linearized by taking    where  is the molar fraction of the i - nuclei , n the integration step and  a parameter which allows the quality of the integration to be controlled .choosing  leads to a totally implicit first order integration ;  is centered ( second order ) implicit and  is totally explicit .then , neglecting the terms in , the linearized version of the chemical equations is written . with the same level of accuracy we propose to take  and approximate the nuclear reaction rates by taking {t^n}~   \theta\delta t\ ] ]if we also neglect second order terms in temperature , such as , we can approximate a typical element on the right part of chemical equations ( 1 ) by :  \theta\delta y_j+ \left[y_j^n~r_{ij}(t^n)\right]\theta\delta y_i+\\ \left[y_i^n~y_j^n \frac{dr_{ij}}{dt}\right]\theta\delta t+y_i^n y_j^n r_{ij}(t^n)\end{aligned}\ ] ]  then , equation ( 1 ) reads :    note that the derivative of nuclear rates with respect temperature ,  and , do not pose a practical problem since the recently published fitting formulae of rauscher & thielemann can be used to calculate them in an efficient way . using that compilation through all stages ofthe combustion also leads to a complete internal coherence between direct and inverse rates when the equilibrium is approached , which is an essential condition for the proper handling of nse .  for a network of species we have linear equations with  unknowns .an additional equation , energy equation , is needed in order to close the system . for a process not involving particle difussionit writes :    where p is pressure , u is the energy per gram of material ,  represents the nuclear binding energy of a mol of the i - species and  can also include sources or sinks of energy other than nuclear .it is useful to write that equation by using the thermodynamical identity :    thus , using , the discretized form of the energy equation is easily written .for example , for an adiabatic process equation ( 6 ) reads :    for an ideal gas of ions , which is usually much lower than the q - value of the reactions .coulomb corrections to the ideal gas must be also included when the ionic coupling constant parameter , , becomes non negligible .this often happens in high temperature combustion of degenerate matter because of the strong dependence of  on the mean charge of the system .such dependence also favours  of the products of charged particle reactions against that of the reactant particles .thus , on the whole , both contributions to  ( ideal gas and coulomb corrections ) work in the same direction , slightly increasing the equilibrium temperature for a given density .therefore , regardless the density regime we have _ always kept _  both contributions to  in equationinterestingly the implicit treatment of temperature also allows for the proper handling of quasi - hydrostatic and isobaric processes .let s suppose that the value of the actual pressure of the system , p , remains always close to the external pressure , p , so that . then , writing  and taken out  the energy equation ( eq.[6 ] ) reads :    once this equation is discretized ( in a similar way that we did for eq.[6 ] ) and solved jointly with equations ( 5 ) the solution ,  and , can be used along with p  to find .equation ( 9 ) has been used in 3.3 to solve the structure of a nuclear flame .  from here on we will focus on the case of  which is expected to give the most stable system behaviour of the nuclear system .a recalculation of the first test presented in the next section with  did not lead to relevant quantitative changes in the results .the standard criterion to assign the integration time - step was to demand that the fractional change in temperature and molar fractions of the more abundant species would not exceed 2% per model .nevertheless , one can safely relax that criterion even to 10 - 15% variation in molar fractions without seriously modify the quantitative results .this set of linear equations can be efficiently solved by using standard sparse matrix algorithms .we in particular used the one found in prantzos , arnould & arcoragi .the resulting algorithm led to a satisfactory mass conservation , the error in the sum of mass fractions was always lower than , close machine precision , for the calculations described in 3 .there is not a large loss of computational efficiency when the linearized thermal coupling is turned on .the main source of degradation of that type is the computation for the derivatives of the rates rather than the increase in the size of the matrix from  to  . from the tests carried out we estimate that our numerical treatment leads to a time overload under 15% and 2% for the 14 and 86-nucleus systems respectively .several tests were run which demonstrated the ability of the scheme to monitor rapid isochoric combustion followed by adiabatic expansion in the hydrodynamic time scale .adding either diffusive terms or those related to the artificial viscosity formulation to the right side of equations ( 6 ) and ( 9 ) allows us to include heat transport and shock wave heating in the scheme . an extreme case of heat transport , the propagation of a nuclear flame , is also presented and checked in  3 .even though all the test presented below were calculated solving the linearized equations ( 5 ) and ( 8) an improvement of the algorithm based on the iterative refinement of the solution is also given and discussed in 3.1.2 .three test cases were calculated in order to check the method .all of them are of an indisputable interest in explosive combustion : a ) isochoric combustion followed by adiabatic expansion once complete nse has been achieved , b ) endoenergetic disintegration of silicon and c ) isobaric nuclear flame propagation .the nuclear network consists of 86 nuclei spanning from  to  including , p and n particles , as depicted in figure 1 .even though this is not a large network , but rather a moderate one , it is certainly more complex than the -chain considered in mller . on another note , to avoid an excessive computational burden , current multidimensional simulations of supernovae can only support very reduced networks . for this reason, we also carried out several runs of the test using an -network of 14 nuclei , a subset of the previous sample of 86 nuclei .this reduced set was similar to the sets of 7 and 13 nuclei considered in the comparative study about the performance of small nuclear networks carried out by timmes et al .nuclear rates on light elements , up to , were taken from caughlan & fowler , and from rauscher & thielemann above neon .a common feature of all the tests shown here is that nuclear burning proceeds through the so - called e - process , thus it manages to achieve complete nuclear equilibrium at high temperature . in nature , that equilibrium does not last very long because it is cut by the reduction in density as the expansion progresses . at some pointthe temperature becomes too low to allow the nuclear reactions to proceed and the chemical abundances freeze out .the precise value of the resulting abundances is a function of the initial entropy of the material in the nse phase , the degree of neutronization of the material and the rate of the expansion , as many studies have demonstrated ( see for instance meyer , krishnan & clayton  1998 for a recent work and references therein ) .the equation of state ( eos ) used in all the tests was quite complete , consisting of a mix of electrons treated as a partially degenerate relativistic gas , an ideal gas of ions with coulomb and other minor corrections , and radiation . although important at high densities , electron capture on protons and nuclei have not been included in these calculations .nevertheless , they can be incorporated into the scheme with the same formulation expressed in equations ( 2 ) and ( 3 ) .the treatment of electrostatic corrections deserves further discussion . at high densitiesthe energy of the plasma deviates from that of a pure ideal gas of nuclei and electrons .electrostatic interactions must be taken into account in this regime , even in nse , owing to the strong dependence of such interactions with the average charge of the system .the electrostatic energy of the k - specie associated to the coulomb interactions ,  ( erg .g ) , can be approximated by the following expressions ( ogata & ichimaru 1987 , yakolev & shalybkov 1987 ) :    where  is the ionic coupling constant parameter of the k - nuclei , a=-0.8980 , b=0.96786 , c=0.2207 , d=-0.86097 ,  and .the chemical potentials , , can be easily derived from expression ( 10 ) .nevertheless , the inclusion of coulombic interactions affects not only the thermodynamics , they also slightly alters the rates of electron captures and the rate of nuclear reactions through the corresponding screening factors . as we have not included weak interactions in the network we will not address the first point .the second point , the incorporation of screening corrections to the nuclear rates in the scheme should be straightforward . however , there is also necessary to compute the derivatives of such corrections with respect to temperature , as we did with the nuclear reaction rates .  in the following , we are mainly interested in the behaviour of the system when it approaches the nse . in this caseit is enough to make a simplified treatment of the screening . in particular , we take the enhancement factors given by mochkovitch & nomoto who , among other things , studied the equilibrium of direct and reverse reactions when strong screening corrections are taken into account .they found that the dominant term in the enhancement factor , ef , to direct capture reactions is :  \ ] ]  where  is the chemical potential of the reactants minus that of the products .the above expression is especially adequate for non - resonant reactions at high temperature as well as for resonant reactions with large enough resonance energy .note that , in this approximation the use of the enhancement factors given by equation ( 11 ) does not affect to the photodisintegration reactions calculated through the detailed balance , because the q - value of the reaction becomes  which exactly compensates the screening correction . in the limit of complete nuclear equilibrium this approach leads to the same results as the statistical description given by the saha equation with the chemical potentials corrected for electrostactic interactions ( bravo & garca - senz 1999 ) .isochoric combustion is a very interesting test because the original fuel ( usually helium or carbon and oxygen in many explosive events ) goes through several modes of burning : initial fast burning followed by quasi - statistical and complete nuclear equilibrium .further expansion will take the combustion out of nse , which leads to a rapid freezing of the abundances .in hydrodynamic calculations , that regime might be associated to a detonation wave passing through degenerate material .  to begin , the combustion of a system composed of equal parts  and  at  g. was followed _ without _ including the implicit coupling between abundances and temperature .the result is shown in figure 2 , where we can see that mass fractions begin to _ oscillate _ as soon as the combustion approaches the equilibrium regime .this behavior is characteristic of any stiff system of differential equations solved with explicit rather than implicit schemes , demanding very short time - steps to control the system (  s ) . the evolution _ with _thermal coupling is depicted in figure 3 ; here the oscillations _ did not show up _ and the qse and nse regimes were handled without problems , despite the large time - steps (  s , even larger than the hydrodynamic timescale , see below ) taken once equilibrium was achieved . from tables 1 and 2 we can see that the most abundant nuclei correspond to fe - peak elements , especially  ( 27% , in mass ) and also to helium ( almost 20% ) .such abundances and the high temperature achieved during the isochoric combustion are in agreement with the current picture of nse ( arnett 1996 ).  the integration scheme also allowed us to follow a rapid adiabatic expansion in the hydrodynamic timescale  s immediately after the nse regime was reached :  \ ] ]  during the expansion there was no need to impose any particular freeze - out temperature for the nucleonic system . in table 1we provide the mass fractions at five different fiducial temperatures achieved along the cooling path . from these numberswe can see that there is a large spread in the freezing temperatures .many species are still changing even below  k , although their abundances are generally too low to be significative .the most abundant element , , freezed around  k which is not a high enough temperature to keep the nse going . thus , whenever the saha equation is used to describe nuclear statistical equilibrium above  k one has to be cautious and switch to a time - dependent network calculation below that critical temperature because the material is not completely burned yet .in table 2 we provide the freeze - out temperature and final composition for each of the five most abundant nuclei as a function of the initial thermodynamical state and chemical composition . as one can see , while the critical temperature for achieving complete nse was the same for all nuclei this was certainly not true for the freeze - out temperature .when the initial fuel was composed of carbon and oxygen with no neutron excess ,  ( ) , the resulting abundances were largely dominated by . under the conditions of our numerical experiment,  froze at , while other fe - peak isotopes were still reacting with residual protons and neutrons .therefore their freezing temperatures were lower .a similar calculation ( same initial density and temperature ) is shown in the second row of table 2 , but the initial composition was  ,   and  , resulting in a moderate neutron excess of .this time the dominant nucleus after the adiabatic cooling was not  but instead were  and  in accordance with well - known results ( i.e. , clayton 1968 ) .there is , however , a clear difference in the freezing temperatures of these dominant isotopes .  the third row of table 2 and figure 4 summarizes the evolution of a system initially composed of pure helium at  g. .in this case  and the combined value of temperature and density in nse resulted in a larger entropy than in the two earlier cases .now , the period of relaxation to nse was larger than in the two precedent tests ; however there was no sign of instability during the integration .once the nse was achieved , at  k , the composition was again dominated by fe - peak elements with a lower concentration of light particles  and p , ( 0.66% and 0.74% by mass respectively ) than in the two precedent cases . for a given initial densitythe outcome of the adiabatic cooling depends on the adopted expansion rate . for very fast expansions the alpha particles have some difficulty to react , owing the low - density environment , and the final distribution is rich in  particles and intermediate - mass elements ( woosley , arnett & clayton 1973 ) . in our calculationthe expansion rate was not rapid enough to allow the synthesis of intermediate - mass elements in appreciable amounts . according to table 2the final composition was completely dominated by  ( 98% ) and fe - peak elements plus a little amount of unburnt helium .as mentioned above , the enhancement factors to the nuclear reaction rates given by equation ( 11 ) were taken into account in all the realized tests . generally speaking , their inclusion in explosive processes ,although important , is not very crucial ; its main effect being to shift the temporal evolution and the maximum value achieved by temperature .therefore , there is expected to be some changes in the mass fractions of the species in nse when screening factors are incorporated into the scheme , especially at high density . in figure 5 thereis represented the evolution of temperature during the isochoric combustion of a 50%-50% mix of carbon and oxygen at constant density of  g. . as we can see the profiles with and without enhancement factors are not very different , although the induction time is shorter and the equilibrium temperature is no longer the same :  ( no screening included ) and  ( screening included ) .again , there is a stable numerical behaviour of the system when the equilibrium is approached .  to analyze the nucleosynthesis during the nse stageit is more convenient to compare at the same temperature . in doing thatnot only the comparison among abundances becomes more reliable , but a direct check with the results obtained by using the nuclear saha equation with the mass excesses corrected from the coulombic part of the chemical potentials deduced from equation ( 10 ) is also possible .the results of such comparison for the 14-nuclei network are depicted in figure 6 . from that figureit becomes clear that the main effect of screening corrections is to shift the distribution of the species towards higher atomic masses and that , the larger the density the bigger the change .in fact , the change in the nse distribution of the nuclei depicted in figure 6 looks rather spectacular .this is in part due to the limited size of the 14-nuclei chain ; when the same calculation was repeated with the 86-nuclei network such differences were _ spread out and smoothed_. for example , at   the mean molecular weight calculated from the -network was  g.mol ( no screening factors ) and  g.mol ( with screening factors ) whereas the corresponding values for the 86-nuclei network were  g.mol  and  g.mol  respectively .the comparison between the resulting mass fractions calculated through the network and the nuclear saha equation also confirms that the adequate way to handle the coulombic interactions in the nse regime is by substracting the electrostatic chemical potentials from the nuclear mass excesses ( bravo&garca - senz 1999 ) .   the integration scheme developed in 2 can be considered as the first iteration step of a more general newton - raphson based method for solving the full non - linear equations governing the evolution of the nuclear system .when using these methods there is often an enhancement of the computational efficiency despite that the time invested per model is larger .in addition they also have the advantage that a check on accuracy is possible , allowing for a more consistent choice of the time - steps ( timmes 1999 ) .fortunately it is straightforward to modify our numerical scheme , with minimum changes , in order to allow the iterative refinement of the solution , as described below .the resulting algorithm is equivalent to the standard implicit multidimensional newton - raphson calculation . for a given time - stepthe iterative process leads to a sequence of ` solutions '  and  for the i - specie and temperature which , after  iterations , should converge to the true values for  and .we define a set of functions , which represents the discretized form of the chemical equations ( 1 ) :  = f_i[({\bf y}^{n+1})^k,(t^{n+1})^k]~\delta t- [ ( y_i^{n+1})^k - y_i^n]\qquad  k=1 ... m\end{aligned}\ ] ]  where } and  are the terms on the right of equation ( 1 ) .a similar function , , representing the energy equation ( 8) is also defined :  = \sum_i\left(\mathrm{be}_i-\frac{\partial u}{\partial y_i}\right )   [ ( y_i^{n+1})^k - y_i^n)]\qquad\qquad\qquad\\   \qquad\qquad-\left\{\left(\frac{\partial u}{\partial t}\right)- \frac{\delta\rho}{\rho^2 }  \left(\frac{\partial p}{\partial t}\right)\right\ }  [ ( t^{n+1})^k - t^n]+   t^n\frac{\delta\rho}{\rho^2 }  \left(\frac{\partial p}{\partial t}\right)\end{aligned}\ ] ]  the newton - raphson iterative method demands that , which allows to solve for the corrections  to the approximate values  and  found in the latest iteration . in this casethe jacobian matrix associated to  is _ exactly the same _ as the matrix of the linear system associated to equations ( 5 ) and ( 8) .for the first iteration we take  and  so that when m=1 the extended arnett&truran method given by equations ( 5 ) and ( 8) is recovered . for  an iterative sequence results which usually ends when the -corrections  to  and  to  becomes negligible or , better , when the  functions defined by equations ( 13 ) and ( 14 ) , conveniently normalized , go to zero ( for example we took  and a similar expression for ) . usually convergence is achieved in 3 - 4 iterations . the criterion to choose the time - step was not very restrictive , we allowed for a  relative variation in temperature and  in molar fractions . in figure 7 thereis shown the evolution of the errors in mass fractions of several species during the isochoric carbon combustion , estimated by comparing the size of the corrections given by the first iteration with the algebraic sum of the lowest - order corrections given by the remaining iterations .as we can see in that figure the first iteration always led to a much larger correction , usually of two orders of magnitude or more , than that of the sum of the remaining iterations .thus , the evolution of the mass fractions and temperature can reasonably be described taking only one iteration and neglecting second and higher - order corrections . for this particular test , a further relaxation of the criterion that restrict the time step led to several negative abundances .nevertheless , the iterative newton - raphson scheme would also be in trouble in this case , due to the lack of convergence .therefore , taking only one iteration along as a moderate time - step would generally be enough , especially if we are mainly interested in the thermal evolution of the system and in the gross features of the nucleosynthesis . for larger time - steps or for detailed nucleosynthetic studiesit is better to rely on the iterative scheme which has the additional advantage that a check of the accuracy is feasible ( timmes 1999 ) .hydrostatic combustion of silicon is a quasi - equilibrium process from the beginning due to the important role played by photodisintegration reactions .one goal of this calculation is to check the ability of our integration scheme to describe silicon photodissociation at constant temperature and density .we also studied explosive silicon burning within an expanding medium , until the reactions were totally quenched . in both caseswe compared our results with those of timmes et al . , which were calculated under the same environmental conditions .thus , a system composed of 100%  at t= and  g. was considered . in the hydrostatic case ,the temperature was kept constant , therefore the feedback between nuclear kinetics and thermal evolution no longer took place .nevertheless , our scheme was also able to keep track of this limiting case without introducing any modification . instead to rewrite our code to not include  terms in the matrixwe have artificially raised the value of the specific heat during the silicon dissociation to keep the temperature constant . in the adiabatic expansion testthe specific heat was again restored to its real value .the results of these calculations for the small network are shown in figures 8,9 and 10 . a comparison between the evolution of the nuclear energy generation rate , provided in figure 8 , with respect to that shown in figure 1a of timmes et al . is satisfactory .the main difference is the tiny bump shown around t , which was probably due to the different libraries used to compute the nuclear rates .a comparison of their abundances ( their fig .1c ) with ours ( fig .9 ) also shows a good agreement .  as a second check a piece of pure siliconwas enforced to follow an adiabatic expansion in the hydrodynamic timescale as shown in equation ( 12 ) . in this case , the temperature evolution was calculated self - consistently from the network .the results of this calculation for our are summarized in figures 8 and 10 .these figures can be compared with figures 2a,2c in timmes et al . , in which the temperature is evolved with an approximate analytical formula .the evolution of the nuclear energy rate follows a close path in both calculations .the comparison between abundances is also quite satisfactory , although there are divergences above t=0.1 s , which is about the hydrodynamical time .these differences probably arise from the slightly different path followed by temperature in both calculations .we note that timmes et al .( 2000 ) only included the contribution of the radiation to the specific heat thus , for a given density , we found a higher value of temperature .in particular , our final silicon group abundance ( up to titanium ) was about five times higher than theirs . on the wholeits composition is characteristic of an freeze - out with a large abundance of  ( 92% in mass ) followed by  ( 7% ) .thermonuclear flame propagation is a subsonic mode of energy propagation which takes place under nearly isobaric conditions .flames could appear due to the combination of sharp thermal gradients , high conductivity and potentially explosive fuels such as helium or carbon and oxygen .these requirements may be fulfilled in the interior of those massive white dwarfs that are reactivated by the accretion from a companion star , giving rise to a type ia supernova .this calculation is more complex than those carried out in the previous subsections because both , the temporal evolution and the spatial structure of the flame have to be solved . therefore our nuclear subroutine was coupled to the diffusion equation within an initially uniform grid of shells representing a physical system with planar geometry and constant density . given the adequate initial conditions ( see below ) , the advance of the flame was calculated directly from the nuclear network of 86 isotopes along with the isobaricity assumption ( timmes & woosley 1992 ) .a diffusive term was added to the energy equation so that the heat is transported from the burned to the unburned zone , mainly by electronic conduction .thus , we take ,    in equation ( 9 ) and set  in the same equation in order to keep the pressure constant during the flame propagation .the conductivity , , was that of khokhlov , oran and wheeler which includes electronic and radiative contributions to the opacity .as the dependence of conductivity on temperature is much lower than it is in nuclear reactions this new term , equation ( 15 ) , can be added to the independent terms in the corresponding discretized version of equation ( 9 ) . in order to build a nuclear flamethe left side of a tube containing equal parts of carbon and oxygen was incinerated at constant pressure until nse was achieved . after a brief transitory period the heat exchange between the burned the and cold material gave rise to a steady planar flame that moved at a constant velocity through the fuel with constant density  g. . behind the flame front the original fuelwas rapidly transformed into intermediate - mass elements , especially silicon ; the temperature rose to  k ; and the density dropped to  g. to keep the pressure constant . under these conditionsthe ashes were in qse and began to slowly relax to complete equilibrium , as can be seen in figure 11 . during the total elapsed time( about  s ) the nuclear system did not show any sign of instability .the stationary velocity of the flame front was  km.s , which is about 50% higher than that obtained by timmes & woosley for the same density .such discrepancy is well within the factor two error usually attributed to the input physics ( conductivities , nuclear libraries and eos ) .current standard integration methods for chemical equations encounter some difficulty when handling high temperature combustion near equilibrium .the reason for this is the extreme sensitivity of nuclear rates to the temperature , which makes it difficult to reach a time - independent situation ( equilibrium ) when using a time - dependent solver for equations that describe the evolution of molar fractions .mller pointed out that the solution is to implicitly couple the equations that give abundance changes with those that describe energy or entropy evolution .a practical drawback of such a procedure is that the derivatives of the nuclear reaction rates have to be worked out at each iteration .nowadays , however , very complete libraries are available , which detail nuclear reaction rates for many species with simple formulae ( i.e. rauscher & thielemann 2000 ) . from these ,then , derivatives can be evaluated without causing excessive computational overload .the main purpose of this study was to build and check an integration method , inspired by mller idea , that would be flexible enough to accommodate a variety of combustion modes involving quasi / total nuclear equilibrium at some point along the path of its evolution .screening enhancement factors were also incorporated to our numerical scheme .good astrophysical examples of the potential applicability of this algorithm would be for nuclear flames and detonation waves in type ia supernovae and for silicon burning in type ii supernovae respectively .  in spite of the simplicity of the proposed numerical algorithm , described in 2, we found that it was able to pass many tests related to high - temperature combustion , 3 . in each of them stabilitywas always preserved , even in the nse stage , without restricting the time - step .the results also matched well those obtained in similar conditions by other authors using different integration schemes .a comparison of the numerical efficiency of the small chain ( of 14 nuclei ) and the moderate - sized one ( of 86 nuclei ) showed that the time overload introduced by thermal coupling was under 15% , and 2% for the small and large networks respectively .these factors arise mainly from the calculation of the derivatives of the nuclear rates rather than from the inversion of the matrix .recently , the 14-isotope -chain with thermal coupling was incorporated to a multidimensional sph code designed to simulate type ia supernova explosions .in that situation , the average time degradation is far below the aforementioned 15% as the time expended per model is largely controlled by hydrodynamics , especially when one is searching for neighbours or calculating gravity .crrrrr|rrrrrr n & 5.7(-5)&&3.0(-16)&8.6(-25)&4.0(-85)&&2.5(-7 ) & 3.1(-14 ) & 2.2(-12 ) & 1.1(-8)&1.3(-8 ) + p & 2.1(-2 ) & 1.7(-3 ) & 5.2(-4 ) & 2.6(-4 ) & 1.6(-6 ) & & 2.3(-5 ) & 7.1(-12 ) & 2.2(-12 ) & 2.6(-11 ) & 2.6(-11 ) + & 1.95(-1 ) & 3.4(-3 ) & 4.5(-3 ) & 4.4(-3 ) & 4.4(-3 ) & & 5.6(-3 ) & 6.6(-8 ) & 1.3(-7 ) & 4.6(-7 ) & 4.6(-7 ) + & 2.5(-5 ) & 1(-8 ) & 1.1(-7 ) & 4.1(-7 ) & 5.1(-7 ) & & 1.3(-3 ) & 1.7(-11 ) & 1.9(-12 ) & 1.5(-18 ) & 1.3(-21 ) + & 5.3(-5 ) & 1.7(-9 ) & 3.2(-9 ) & 1.1(-9)&3.7(-10 ) & & 7.1(-4 ) & 6.4(-13 ) & 7.6(-15 ) & 1.2(-20 ) & 6.8(-27 ) + & 3(-6 ) & 3.3(-11 ) & 1.3(-10 ) & 1.6(-11)&4.7(-12 ) & & 1.5(-5 ) & 4(-12 ) & 4.8(-12 ) & 4.2(-11 ) & 1.8(-10 ) + & 4.3(-8 ) & 6.2(-17 ) & 2.3(-20 ) & 8.8(-30 ) & 2(-44)& & 5.5(-5 ) & 7.9(-13 ) & 6.6(-13 ) & 3.5(-16 ) & 3(-19 ) + & 5(-9 ) & 1.4(-19 ) & 2.5(-22)&7.5(-32 ) & 1.3(-65 ) & & 6.2(-4 ) & 4.6(-12 ) & 1.2(-12 ) & 6.9(-15 ) & 1.9(-17 ) + & 4(-8 ) & 7.7(-14)&6.2(-13 ) & 1.9(-12)&5.1(-12 ) & & 4.8(-8 ) & 1.1(-14 ) & 8.8(-14 ) & 3.3(-10 ) &8(-10 ) + & 7.2(-8 ) & 9.4(-15)&3.1(-16 ) & 4.6(-22 ) & 2(-40 ) & & 1.8(-6 ) & 2.6(-13)&4.8(-12 ) & 8.3(-12 ) & 8.3(-12 ) + & 4.1(-7 ) & 3.1(-14 ) & 5.2(-14 ) & 1.1(-16 ) & 4.3(-25 ) & &1.2(-3 ) & 1.3(-8 ) & 6(-8 ) & 7.9(-8 ) & 5.5(-8 ) + & 1.3(-9 ) & 4.1(-14 ) & 1.4(-11 ) & 4.9(-8 ) & 5.9(-8 ) & &1.7(-3 ) & 2.1(-10 ) & 1.4(-12 ) & 5.3(-20 ) & 3.9(-24 ) + & 1.4(-7 ) & 7.9(-12 ) & 7.8(-11 ) & 7.9(-11)&7.9(-11 ) & & 4.6(-3 ) & 5.4(-10 ) & 3.9(-13 ) & 2.4(-19 ) & 1.9(-23 ) + & 1.3(-4 ) & 7.5(-11 ) & 4.2(-10 ) & 1.1(-11 ) & 9.1(-12 ) & & 5.8(-6 ) & 7(-12 ) & 3.3(-11 ) & 3(-10 ) & 1.5(-9 ) + & 3.9(-6 ) & 7.3(-16 ) & 3.7(-19 ) & 2.6(-29 ) & 1.3(-40 ) & & 1.3(-4 ) & 7.1(-11 ) & 1(-11 ) & 2.4(-15 ) & 2.7(-18 ) + & 9.1(-7 ) & 1.2(-17 ) & 1.1(-19 ) & 8.4(-29 ) & 2.2(-51 ) & & 4.8((-3 ) & 7.8(-9 ) & 1.7(-10 ) & 6.5(-13 ) & 1.1(-13 ) + & 1.1(-6 ) & 1.1(-13 ) & 1.5(-12 ) & 6.7(-13 ) & 1.2(-11 ) & & 2.9(-8 ) & 2(-13)&2.2(-11 ) & 1.5(-7 ) & 1.8(-7 ) + & 4.2(-6 ) &4.7(-14)&1.7(-15)&2.9(-21 ) & 1.8(-32 ) & & 7.9(-6 ) & 5.7(-11 ) & 2.5(-10 ) & 2.6(-10 ) & 2.6(-10 ) + & 4.5(-6 ) & 6.9(-13 ) & 3.6(-12 ) & 7.8(-15 ) & 2.8(-17 ) & & 4(-3 ) & 4.3(-6 ) & 2.2(-7 ) & 2(-7 ) & 5.7(-8 ) + & 3.1(-8 ) & 5.3(-14 ) & 2.3(-11 ) & 3.4(-8 ) & 3.5(-8 ) & & 1.2(-2 ) & 6.9(-7 ) & 2.1(-9 ) & 2.8(-16 ) & 2.6(-19 ) + & 5.3(-6 ) & 2(-11)&2(-10 ) & 2(-10 ) & 2(-10 ) & & 5.2(-2 ) & 2(-6 ) & 1.6(-12 ) & 5.7(-20 ) & 7.3(-24 ) + & 8.9(-3 ) & 5(-10 ) & 3.3(-9 ) & 3.6(-11 ) & 1(-12 ) & & 2.2(-5 ) & 3.9(-9 ) & 3.4(-10 ) & 5.4(-9 ) & 3.5(-9 ) + & 6(-4 ) & 4.5(-14 ) & 9.7(-19 ) & 1.4(-25 ) & 1.7(-34 ) & & 7.5(-4 ) & 1.3(-7 ) & 5.7(-9 ) & 1.2(-12 ) & 7.1(-15 ) + & 1.7(-4 ) & 5.3(-16 ) & 1.1(-17 ) & 1.3(-26 ) & 7.1(-42 ) & & 3.1(-2 ) & 2.1(-5 ) & 6.2(-10)&1.8(-13 ) & 1.3(-14 ) + & 6.5(-5 ) & 9.8(-13 ) & 1.8(-11 ) & 1.3(-11 ) & 2.5(-12 ) & & 1.3(-7 ) & 2.5(-11 ) & 2.4(-11 ) & 3.3(-7 ) &4.8(-7 ) + & 2.2(-4 ) & 3(-13 ) & 2.7(-14)&6(-19 ) & 7.1(-27 ) & & 3.6(-5 ) & 7.3(-8 ) & 8.6(-8 ) & 9.4(-8 ) & 9.4(-8 ) + & 9.2(-4 ) & 6.2(-13 ) & 2.8(-12 ) & 1.4(-15 ) & 3.1(-19 ) & & 1.3(-2 ) & 2.1(-3 ) & 8.9(-6 ) & 7.6(-6 ) & 7.5(-6 ) + & 9.2(-7 ) & 5.6(-14)&1.5(-11 ) & 1.6(-8 ) & 1.7(-8 ) & & 5.1(-2 ) & 4.1(-4 ) & 1.6(-5 ) & 7.7(-12 ) & 6.8(-14 ) + & 4.6(-5 ) & 3.3(-12 ) & 2.4(-11 ) & 2.4(-11 ) & 2.4(-11 ) & & 2.7(-1 ) & 3(-3 ) & 5.5(-7 ) & 3.6(-16 ) & 8.4(-20 ) + & 8.8(-3 ) & 1.2(-9 ) & 1.1(-8 ) & 2.7(-10 ) & 2.6(-12 ) & & 2.5(-5 ) & 4.1(-7 ) & 2(-9 ) & 1.3(-8 ) & 1.3(-8 ) + & 1.5(-3 ) & 3.6(-13 ) & 1.8(-15 ) & 1(-23 ) & 1.4(-30 ) & & 2.1(-3 ) & 4.8(-5 ) & 2.3(-5 ) & 1(-8 ) & 3.4(-10 ) + & 7.6(-4 ) & 2.1(-14 ) & 1.5(-15 ) & 2.5(-23 ) & 1(-34 ) & & 9.6(-2 ) & 1.9(-2 ) & 9.9(-5 ) & 3.7(-10 ) & 8.4(-12 ) + & 6.5(-5 ) & 1.2(-12 ) & 2(-11 ) & 1.3(-11 ) & 1.8(-12 ) & & 7.8(-8 ) & 1.3(-9 ) & 5.2(-11 ) & 1.2(-7 ) & 2.3(-7 ) + & 3.2(-4 ) & 6.1(-13 ) & 7.7(-14 ) & 2.1(-18 ) & 1.1(-25 ) & & 5.3(-5 ) & 9.1(-6 ) & 8.9(-5 ) & 1.3(-4 ) & 1.3(-4 ) + & 1.9(-3 ) & 3.1(-14 ) & 2.1(-11 ) & 2.5(-14 ) & 2.8(-17 ) & & 2.3(-2 ) & 9.18(-1 ) & 9.65(-1 ) & 9.63(-1 ) & 9.63(-1 ) + & 8(-7 ) & 7.4(-14 ) & 2.3(-11 ) & 3.6(-8 ) & 3.6(-8 ) & & 5.5(-2 ) & 3.4(-2 ) & 2.7(-2 ) & 1.8(-2 ) & 1(-2 ) + & 4.5(-5 ) & 3.4(-12 ) & 3(-11 ) & 3.1(-11 ) & 3.1(-11 ) & & 1.14(-1 ) & 1.8(-2 ) & 1.4(-3)&1.6(-8 ) & 7.2(-10 ) + & 6.3(-3 ) & 2.2(-9 ) & 8.2(-9 ) & 2.94(-9 ) & 7.9(-10 ) & & 2.2(-5 ) & 9.5(-6 ) & 3.9(-6 ) & 4.7(-6 ) & 5.3(-6 ) + & 1.3(-3 ) & 1(-12 ) & 2.6(-15 ) & 1.9(-23 ) & 2.74(-29 ) & & 5(-4 ) & 6.2(-5 ) & 1.4(-4 ) & 6.2(-3 ) & 6.8(-3 ) + & 1.1(-3 ) & 1.9(-13 ) & 5.4(-15 ) & 8(-22 ) & 6.5(-30 ) & & 5.4(-3 ) & 6(-4 ) & 2.4(-4 ) & 5.8(-7 ) & 1.7(-7 ) + & 2.8(-5 ) & 6.8(-13 ) & 3(-12 ) & 1.2(-11 ) & 7.1(-11 ) & & 1.2(-8 ) & 6(-10 ) & 4.5(-10 ) & 1.4(-8 ) &2.7(-9 ) + & 2.3(-4 ) & 2.3(-12 ) & 1.6(-13 ) & 7(-18 ) & 1.1(-22 ) & & 1.8(-6 ) & 8.7(-8 ) & 6.7(-7 ) & 2.8(-3 ) & 1.02(-2 ) + & 2.3(-4 ) & 3.1(-11 ) & 7.9(-11 ) & 8.5(-13 ) & 1.1(-13 ) & & 1.3(-4 ) & 1.5(-4 ) & 1.7(-3 ) &4.9(-3 ) & 4.9(-3 ) +  crrrrrr|rrrrc & ash & &&&& & &&&& + co&&&& & & & && & & + & t&8.03&8.03&8.03&8.03&8.03&3.20&1.31&1.37&1.35&2.10 + & ash & &&&& & &&&& + cone&&&& & & & && & & + & t&8.51&8.51&8.51&8.51&8.51&5.52&4.77&4.75&4.45&4.63 + & ash&&&&& & &&&& + he&&&& & & & && & & + & t&4.30&4.32&4.32&4.32&4.32&2.95&1.14&1.11&1.23&2.43 +

Abstract: a method for integrating the chemical equations associated with nuclear combustion at high temperature is presented and extensively checked . following the idea of mller , the feedback between nuclear rates and temperature was taken into account by simultaneously computing molar fraction changes and temperature response in the same matrix . the resulting algorithm is very stable and efficient at calculating nuclear combustion in explosive scenarios , especially in those situations where the reacting material manages to climb to the nuclear statistical equilibrium regime . the numerical scheme may be useful not only for those who carry out hydrodynamical simulations of explosive events , but also as a tool to investigate the properties of a nuclear system approaching equilibrium through a variety of thermodynamical trajectories .


improvement of solar cell technology as well as cost reduction is an increasingly challenging topic in the quest of renewable energy sources .one of the strategy currently followed to reduce the fabrication cost is the use of ultrathin active layers .this approach could help reducing the cost of photovoltaic technologies since a smaller quantity of material is used .however , reducing the thickness brings new problems .the fact that the film thickness could be much lower than the absorption length leads to a significant reduction of the absorption .ultrathin technologies need solutions to keep solar light absorption high .one already well - known solution is the use of front - side or / and back - side surface texturing to help coupling incident light into the active layer via light trapping techniques .the optimization of light - trapping structures ( see fig . [ fig1](a ) ) in solar cell is still of high interest . in this approach ,numerical computations are needed to investigate how electromagnetic field propagates in those devices .usually , numerical methods used for calculating the absorption inside a solar cell ( finite - difference time - domain method ( fdtd ) , rigorous coupled wave analysis ( rcwa ) , ... ) basically consider the cell response under coherent incident light .unfortunately , it is known that the absorption of an optical device depends on the coherent or incoherent nature of the light .coherent light leads to oscillations ( such as fabry - perot ) in the absorption spectrum .incoherent light instead , leads to the disappearance of these oscillations due to destructive - interference effects : a fact that is well known from anyone performing measurements in solar cells .therefore , the calculated absorption spectrum of the active layer is strongly affected by the choice of the incident source ( coherent or not ) .since the absorption spectrum is used to determine the photocurrent and the efficiency of solar cells , it should be recommended to use only incoherent absorption spectrum .solar light is indeed a strongly incoherent light source : incoming waves from the sun have a finite coherence time ( finite spectral width ) .the photocurrent supplied by the solar cell is given by : where is the absorption spectrum and the global power spectral density of the sun .in most studies , the absorption spectrum is computed from numerical codes which propagate the electromagnetic field .nevertheless , such computed is calculated from coherent fields and then we write .this quantity does not correspond to the required effective incoherent absorption experienced by the solar cell . in order to theoretically predict the performance of a solar cell, it is therefore very important to improve numerical methods and to take incoherent incident light into account , i.e. to use in eq .( [ current0 ] ) .it must be noted that , as far as the propagation of the electric field of incident optical radiation is concerned , a solar cell , whatever the complexity of its structure is , behaves as a linear system ( thanks to maxwell equations ) , which is fully characterized by its scattering matrix ( see appendix [ formalism ] ) .however , as soon as energy fluxes need to be calculated , linearity does not stand anymore since the intensity ( poynting vector flux ) is proportional to the electric field squared , i.e. . the way the square modulus of the fields enters into the calculation of the absorption ( reflectance , transmittance ) of the system is far from being trivial ( see , for example , the case of rcwa method , in appendix [ rcwa ] ) . in any case, there exists no transfer function relating _ linearly _ the solar cell absorption to the incident poynting vector flux . the fact that the relevant quantities ( absorbed flux , photocurrent , ... ) do not obey the superposition principle ( since ) prevents us from applying common principles of the linear response theory .in particular , the incoherent response can not be treated in the same way as the coherent one , a fact that is known for long time in optics , at least for simple cases , such as thin films .  in order to overcome this limitation ,numerical solutions have been proposed for modeling incoherent processes .basically , light propagation is computed many times for various incident waves , and the final result relies on a global numerical statistical analysis .many recipes and algorithms have been proposed to achieve this task . nevertheless , in the present article , we show that it is not necessary to upgrade numerical codes or to perform time consuming statistical analysis in order to deal with incoherent light .indeed , the efficiency of solar cells under incoherent light illumination can be analytically calculated from the coherent response computed with usual numerical codes , as explained hereafter .results obtained with our method are compared with those from previous studies .) with many diffraction orders. domains and are respectively the incident and emergence media .( b ) model of field amplitude for the incoherent monochromatic incident light . is the corresponding modulation which characterizes the random phase switching at different times . is the coherence time , equal to the average value .,title="fig:",width=12 ]  when the active domain of a solar cell is illuminated by a coherent monochromatic light , a steady state is reached in which the intensity of the electromagnetic field exhibits a typical stationary pattern .the absorption coefficient is then given by : where is the imaginary part of the permittivity of the active medium of the solar cell and is the power of the incident light . is the volume of the active domain .by contrast , if the light is incoherent , the incident electromagnetic field exhibits a time - dependent fluctuating behaviour with a characteristic time , defined as the coherence time of the incident light .therefore , light inside the active volume can not exhibit the same constructive interference pattern as in the fully coherent case : incoherence affects the way the light propagates .in addition , a specific medium mainly interacts with light through electronic and ionic motions of its components . as a consequence ,the response of a medium to an incident electromagnetic wave is not instantaneous and occurs with a given response time related to dielectric relaxation time and/or to charge carrier generation and recombination . if , then the medium undergoes many different field intensity patterns during the characteristic time . as a result ,the absorption and the generated photocurrent both fluctuate in time .the effective incoherent absorption ( and the photocurrent ) is the time averaged value during a typical time about ( see appendix [ formalism ] ) .it is important to point out that the incoherent absorption is physically different from the coherent one indeed , can be considered as an intrinsic property of the solar cell which only depends on its geometry and on its constitutive materials . by contrast , reflects the effective measured response of the solar cell while it interacts with its environment , i.e. for instance when the photocurrent is produced and flows in a closed electrical circuit .  in a numerical computation approach , the absorption coefficient can be obtained from at least two ways . in fdtd for instance ,( [ pow ] ) can be used as such , since maps of the electromagnetic field are directly computed . in rcwa ,reflection and transmission are calculated and is deduced from . in the following ,we consider a rcwa formalism of the light propagation ( see appendix [ rcwa ] ) , and we use it to determine the effective incoherent absorption ( see appendix [ formalism ] ) .the main results of appendix [ formalism ] are summarized hereafter .let us consider an incoherent radiation spectrum , the solar spectrum for instance .such a spectrum can be considered as constituted by an infinite set of incoherent quasi - monochromatic spectral lines , each line being expressed by the temporal signal : the bracket notation denotes the fact that is a supervector which contains the time - dependent complex fourier components related to each diffraction orders ( see appendix [ rcwa ] ) . describes the incident wave amplitudes , assumed to be non zero only for the zeroth diffraction order ( see fig .[ fig1](a ) and appendix [ rcwa ] ) . is the angular frequency .the function is a modulation which ensures the incoherent behaviour of the spectral line . is a stochastic function which has an average time fluctuation equal to the coherence time . as an illustration , a simple but non - restrictive modulationis shown in fig .[ fig1](b ) .the autocorrelation function of a random process has a spectral representation given by the power spectrum of that process ( wiener kinchine s theorem ) .the function can therefore be characterized by its spectral density : with .we then introduce the normalized incoherence function , which characterizes the incoherence of incident light . assuming that describes a random process which follows a gaussian distribution , the incoherent source illuminating the device is characterized by a gaussian spectral density . as a result, the incoherence function is simply written as : with a full width at half maximum .though the coherence time is estimated from the solar spectrum , it must be noted that is not a model of the solar spectrum . simply describes the stochastic behaviour of each spectral line composing the whole solar spectrum itself .it can then be shown ( see appendix [ formalism ] ) that the incoherent absorption results from the convolution product , noted , between the coherent absorption and the incoherence function :    this extremely simple formula is easy to use in practice .but its demonstration is not obvious and is therefore detailed in appendix [ formalism ] .( [ f1 ] ) can also be used to compute incoherent reflection and transmission spectra from their coherent counterparts ( see appendix [ formalism ] ) .nm - thick c - si slabs .the coherent spectra were obtained using rcwa and the incoherent ones using our convolution formula , eq .( [ f1 ] ) .( a ) absorption spectra of the planar slab according to various coherence times .( b ) absorption spectra of the corrugated slab according to various coherence times .inset : corrugated structure ; nm , nm , nm , nm , nm . the structure follows a super - gaussian profile with ( see ref .).,title="fig:",width=12 ]  in order to illustrate the usefulness of the convolution formula , i.e. eq . ( [ f1 ] ) , we use it for the calculation of the incoherent absorption of a nm - thick crystalline silicon ( c - si ) slab either planar ( see fig .2(a ) ) or corrugated ( see fig .2(b ) ) .the first step is the numerical calculation of the coherent absorption using the rcwa method ( see fig .2 , red lines ) . in the rcwa simulations , we use unpolarized light at normal incidence , where the plane is the plane of incidence ( polar and azimuthal angles equal to ) ( see fig .the permittivities of materials are taken from the literature .the second step is the convolution of the coherent absorption spectrum with the incoherence function .this step leads to the incoherent absorption spectra .nm ) deposited on a nm - thick au film on a glass substrate .( b ) reflectance spectrum of the whole grating structure.,title="fig:",width=12 ]  the first step is the longest one .for example , using orders of diffraction ( plane waves along the direction and plane waves along the direction ) , it takes a few hours .the second step is very fast .it takes only a few minutes on a personal computer .it is an important improvement in terms of computational time , in comparison with other computational methods . in these methods ,the first step is performed several times for various incident waves , and the final result relies on a global numerical statistical analysis .five different coherence times are considered here ( fs , fs , fs , fs and fs ) .these values are chosen according to the ones used by w. lee and coworkers in a previous work .  in the case of a planar slab , computed spectraare compared with the approximate incoherent absorption spectrum obtained from a standard analytical expression of the absorption using fresnel equations , but in which propagation phases are roughly set to zero to mimic incoherence ( orange curve in fig .2(a ) ) . we note that this kind of analytical expression must be considered carefully in the present case .indeed , such an expression considers the effect of incoherence on light propagation only and does not consider the incoherent response of the permittivities of the materials , contrary to our present approach using convolution ( see appendix [ formalism ] ) . in spite of that ,both spectra ( orange and black curves in fig .2(a ) ) are quite similar , which validates our method ( in our method , the incoherence limit is approached by taking fs ) .the weak discrepancies are due to the absence of incoherent permittivity response in the fresnel approach .we also reproduced some results reported in the literature .two structures were studied under tm polarization .the first one is a nm - thick film of crystalline silicon ( c - si ) , deposited on a nm - thick layer of gold ( au ) on a glass substrate .the second one is a nm - thick c - si film deposited on a gold grating on a glass substrate .the gold grating has a height of nm , a period of nm and a fill factor of .the grating is built on a nm - thick gold layer .a sketch of both structures can be found in .the results obtained with the convolution formula are presented in fig .3 and can be compared with those shown in figures 2 and 4 in .some small discrepancies are observed , which are due to the fact that we did not use the same numerical method ( rcwa vs fdtd ) .however , our results give the same trends as the results obtained by w. lee _et al_. it is therefore possible to account for the light incoherence with a simple method , which does not require long computational times contrary to other previously reported methods .coming back to fig .2 , the absorption spectra of a three - dimensional corrugated nm - thick c - si slab were calculated using the same method as for the planar slab .the corrugation follows a super - gaussian profile such as the one defined in ( ) .the period of the corrugation is nm and its height is nm . the absorption spectra according to various coherence times are shown in fig .we notice that the value of the coherence time affects the absorption spectrum . the highest coherence time ( here fs ) leads to the strongest oscillations in the spectrum .physically , these oscillations are due to fabry - perot resonances ( in the thin - slab ) and to guided mode resonances ( enabled by the corrugation ) . with the decrease of , these oscillations are expected to be smoothened which is actually observed when fs .this smoothing effect is indeed observed when measuring absorption spectra with an incoherent source .we finally calculated the photocurent of the corrugated slab , according to eq .( [ current0 ] ) ( see table [ photocurrent ] ) .the aim was to quantify the effect of the coherence time on the value of the photocurrent .the photocurrent fluctuates as the coherence time varies .as expected , the incoherent light behaviour affects both the absorption spectrum and the photocurrent ..computed photocurrents related to the corrugated device of fig .2(b ) for various coherence times .photocurrent was integrated over the spectral range : 300 nm - 1200 nm . [ cols="<,<",options="header " , ]we demonstrated that reflection , transmission , and absorption spectra of a photonic structure illuminated with incoherent light can be easily calculated .the only input is the knowledge of their coherent counterparts and of the coherence time of the source ( used to determine the incoherence function ) .the incoherent response is simply the convolution between a function accounting for the incoherent source and the coherent response .this result is theoretically demonstrated from first principle and is confirmed by numerical simulations . in analogy with signal processing theory , it can be interpreted as the consequence , in the frequency domain , of the temporal filtering associated with the intrinsic incoherent modulation .the proposed method allows to significantly simplify the description of incoherent phenomena which are regarded as key problem in solar cells optimization .the inherent simplicity of our method allows to minimize the computational time cost and the algorithm complexity .reflection , transmission , and absorption spectra , but also the photocurrent , were shown to depend on the coherence time .   the following demonstration is based on the formalism of the rigorous coupling wave analysis ( rcwa ) .this analysis takes into account the periodicity of the device and describes the permittivity ( ) through fourier series . the electromagnetic fieldis then described by bloch waves also expanded in fourier series . in this formalism , the maxwell s equations take the form of a matricial first - order differential equation in the variable .the axis is perpendicular to the plane ( , ) where the permittivity is periodic ( fig .1(a ) ) . the essence of the method is to solve this equation but is not the topic of the present article .  reflected and transmitted field amplitudesare linked to the incident field amplitudes by the use of the scattering matrix ( ) which is calculated by solving maxwell equations using fourier series ( see appendix [ rcwa ] ) .let us define as the scattered field and as the incident field , such that the associated supervectors are : , \left| f_{in}\right\rangle = \left [  \begin{array}{c } \overline{n}_i^{+ } \\\overline{x}_i^{+ } \\  \overline{n}_{iii}^{- } \\\overline{x}_{iii}^{- } \end{array } \right ] .\label{incscat}\ ] ]  the subscripts and denote the incidence and emergence media , respectively .the superscripts and denote the positive and negative directions along the axis associated with the field propagation .all the fourier components related to the reciprocal vectors are contained in the vector or . for each vector of the reciprocal lattice , and the and polarization amplitudes of the reflected field , respectively . and represent the and polarization amplitudes of the transmitted field .accordingly , and define the and polarization amplitudes of the incident field , respectively . is connected to _ via _ the scattering matrix by : the flux of the poynting vector through a unit cell area for the incident plane wave in homogeneous medium is given by : for the refracted ( ) or transmitted ( ) wave , it is given by where we define the connection matrices , c_r(\omega ) = \left [  \begin{array}{cccc } 0 & 0 & 0 & 0 \\  0 & 0 & 0 & 0 \\  0 & 0 & q_i(\omega ) & 0 \\  0 & 0 & 0 & q_i(\omega ) \end{array } \right ] ,  \label{matrix}\ ] ] with .\label{matrix2}\ ] ] in ( [ matrix2 ] ) , ^{1/2}$ ] with and being the wave vector component parallel to the surface and the pulsation of the incident plane wave , respectively . represents either the permittivity of the incident medium ( ) , or of the emergence medium ( ) .we now introduce a convenient notation : where with      let us first consider a coherent monochromatic incident wave : with an ( optical ) angular frequency .the response of the device is given by  where denotes the convolution product .the response can then be written explicitely as : where is the fourier transform of scattering matrix defined in ( [ nota2 ] ) .we note that since must be real .  from ( [ resp2b ] ) ,we set with we remind here that stands either for the reflected ( ) or the transmitted ( ) wave . reflection and transmission coefficients associated with a coherent process can be obtained thanks to : and the absorption is then simply given by the energy conservation law : i.e. .the incident ponyting flux , i.e. turns out to be time independent ( see eq . ( [ a13 ] ) in appendix [ rcwa ] ) .therefore , the time - averaged incident flux is identical to  this result is only valid as far as the incident wave is coherent .  let us now consider an incoherent quasi - monochromatic incident wave : by quasi monochromatic , we mean that the spectral line has a finite though narrow spectral width . as explained in the present article ,the function is a modulation which ensures the incoherent behaviour . on average, has a coherence time equal to ( see fig .thanks to the wiener - khinchine theorem , the autocorrelation function of a random process has a spectral decomposition given by the power spectrum of that process .the function is then equally characterized by its spectral density : where is the fourier transform of .the device response is then calculated by the same convolution product as in ( [ resp1 ] ) : where again stands for the reflected ( ) or transmitted ( ) wave but now    with      upon incoherent illumination , the device with its structure and combination of various materials undergoes a set of many incident wave trains randomly dephased with respect to each other .these wave trains have the same pulsation and an average duration equal to the coherence time .the coherence time plays a key role while light propagates since it prevents constructive interferences from taking place .but it also changes the way matter responds to light .indeed , a specific medium mainly interacts with light thanks to electronic and ionic motions of its components . as a consequence ,the response of a medium to an incident electromagnetic wave is not instantaneous and occurs according to the dielectric relaxation time .if the incoherence time  is short enough in comparison with the relaxation time , the medium response is not coherent . in this case , the medium response is not simply given by the value of the permittivity at .  in this way, we must consider that the full response of the medium is a time averaged value of the response recorded during a typical time about such that .the time is the sampling time interval which reflects the non - instantaneity of any measurement process .for instance , a spectrophotometric measurement is characterized by such a sampling time .likewise , the photocurrent of a solar cell is a measure of the response of the solar cell to the incident light . in this context, the sampling time corresponds to the recombination / generation time of charged carriers ( carrier lifetime ) .for instance , in silicon , carrier lifetime ranges from ns to ms according to the doping density .these delays are very large compared to the coherence time of sunlight , which is about fs .therefore , the condition is well verified in solar cells .the flux of the poynting vector for an incoherent incident light is given by : the integral can be easily expressed according to the fourier transform of , noted : we note that where is the dirac distribution . since , should have a frequency distribution spreading over around zero frequency , quite comparable to , the spectral density of the random process .we can therefore assume that the angular frequencies for which is significantly different from zero are such that , i.e. roughly speaking .we can then fairly consider the following substitution : ( . as a result, we get :      since , using ( [ eff ] ) , we have ( sign denotes the adjoint matrix , i.e. the conjugate transpose of the matrix ) we can then deduce from eqs .( [ ope ] ) , ( [ jincfi ] ) and ( [ part ] ) : where from ( [ ope ] ) , we deduce that and , where sign denotes the transpose of the matrix .then , ( [ ix ] ) becomes : from which , by writing explicitly the convolution product , one deduces since and , one obtains : then , using eqs .( [ sd ] ) and ( [ fluxinc ] ) , we can deduce      let us estimate the flux of the incident incoherent wave . for a non dispersive incident medium ,we get : following the same argument as in section [ response ] , at a given angular frequency , we assume that the effective flux impinging on the device is given by the time average : where is the incident flux for the coherent wave , see eq .( [ incoav ] ) . using the same calculation as in section [ response ] for estimating the time average , one deduces : as a consequence , one gets :      reflection and transmission coefficientscan always be written as the ratio of a scattered flux to the incident flux , i.e. .then , from ( [ jfin ] ) and ( [ fincof ] ) , we can deduce since does not depend on . for the coherent case , we showed previously ( [ rco]-[tco ] ) that therefore , the incoherent response can be expressed as a function of the coherent one : where is the normalized spectral density of .as the quantity stands for and ( reflection and transmission coefficients ) and since the absorption is defined by , we can deduce the main result of our first principle calculation , i.e. eq .( [ f1 ] ) , that is the absorption resulting from an incoherent scattering process can be related to the same quantity resulting from a coherent scattering process , i.e. . the two quantities are related through a convolution product which involves the normalized spectral density of , i.e. the incoherence function .  the reader will find more details about the present approach in references .we consider as an example a planar dielectric layer with a bidimensional periodic array described by the dielectric function : in the layer , the dielectric function does not depend on the normal coordinate , which is used to define the layer thickness , i.e. the layer extends from to ( fig .one sets , according to the unit cell basis ( , .in such a system , bloch s theorem leads to the following electromagnetic field expression in the layer :   = \sum_{\mathbf{g}}\left [  \begin{array}{c } \mathbf{e}_{\mathbf{g}}(z ) \\\mathbf{h}_{\mathbf{g}}(z ) \end{array } \right ] e^{i(\mathbf{k+g})\cdot \mathbf{\rho } } e^{-i\omega t}.  \label{a2}\ ] ]  it can then be easily shown that maxwell equations can be recasted in the form of a first - order differential equation system , = \left [  \begin{array}{cc } 0 & a \\  \widetilde{a } & 0 \end{array } \right ] \left [  \begin{array}{c } \overline{e}_{//}(z ) \\\overline{h}_{//}(z ) \end{array } \right ] ,  \label{a3}\ ] ] where and are the electric and magnetic field components parallel to the layer surface . in the layer ,one can then write : = \exp \left\ { \left [  \begin{array}{cc } 0 & a \\\widetilde{a } & 0 \end{array } \right ] ( z_i - z_{iii})\right\ } \left [  \begin{array}{c } \overline{e}_{//}(z_{iii } ) \\  \overline{h}_{//}(z_{iii } ) \end{array } \right ] .\label{a4}\ ] ]  let us define the following unit vectors :  one can then expand the electric and magnetic fields ( parallel components ) in fourier series : ^{i(\mathbf{k+g})\cdot \mathbf{\rho } }  \nonumber  \label{a8b}\end{aligned}\ ] ] and ^{i(\mathbf{k+g})\cdot \mathbf{\rho } }.  \nonumber\end{aligned}\ ] ] the subscripts and stand for the incident medium and emergence medium respectively , and the superscripts and denote the positive and negative direction along the axis for backward ( ) and forward ( ) field propagation . for each vector of the reciprocal lattice , and the and polarization amplitudes of the reflected field , respectively , and and , those of the transmitted field . similarly , and define the and polarization amplitudes of the incident field , respectively .we can then define a transfer matrix  : = \left [  \begin{array}{cc } t^{++ } & t^{+- } \\t^{-+ } & t^{-- } \end{array } \right ] \left [  \begin{array}{c } \overline{n}_{iii}^{+ } \\  \overline{x}_{iii}^{+ } \\  \overline{n}_{iii}^{- } \\  \overline{x}_{iii}^{- } \end{array } \right ] .\label{a10}\ ] ] alternatively , we can express the scattered field against the incident field and we define a scattering matrix  : = \left [  \begin{array}{cc } s^{++ } & s^{+- } \\  s^{-+ } & s^{-- } \end{array } \right ] \left [  \begin{array}{c } \overline{n}_i^{+ } \\  \overline{x}_i^{+ } \\  \overline{n}_{iii}^{- } \\  \overline{x}_{iii}^{- } \end{array } \right ] .  \label{a11}\ ] ] the flux of the poynting vector through the unit cell is : we get then : \theta ( \varepsilon _ i(\omega ) \frac{\omega ^2}{c^2}-\left| \mathbf{k}+\mathbf{g}\right| ^2 )  \label{a13}\ ] ] for the incident flux in medium , \theta ( \varepsilon _ { iii}(\omega ) \frac{\omega ^2}{c^2}-\left| \mathbf{k}+\mathbf{g}\right| ^2 )  \label{a14}\ ] ] for the transmitted flux in medium , and \theta ( \varepsilon _ i(\omega ) \frac{\omega ^2}{c^2}-\left| \mathbf{k}+\mathbf{g}\right| ^2 )  \label{a15}\ ] ] for the reflected flux in medium . in ( [ a13]-[a15 ] ) , is the heaviside function : i.e. if and otherwise .m.s . is supported by the cleanoptic project ( development of super - 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Abstract: because of the temporal incoherence of sunlight , solar cells efficiency should depend on the degree of coherence of the incident light . however , numerical computation methods , which are used to optimize these devices , fundamentally consider fully coherent light . hereafter , we show that the incoherent efficiency of solar cells can be easily analytically calculated . the incoherent efficiency is simply derived from the coherent one thanks to a convolution product with a function characterizing the incoherent light . our approach is neither heuristic nor empiric but is deduced from first - principle , i.e. maxwell s equations . usually , in order to reproduce the incoherent behavior , statistical methods requiring a high number of numerical simulations are used . with our method , such approaches are not required . our results are compared with those from previous works and good agreement is found . + authors have contributed equally to this work . _ _