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in recent years the manufacturing of freeform surfaces has become increasingly feasible .
these freeform surfaces offer an elegant way of simultaneous irradiance and phase control .
therefore , the development of numerical algorithms for the calculation of continuous freeform surfaces for control of the irradiance and/or the phase of a beam is of great interest .
+ in this work the problem of designing continuous double freeform surfaces in a geometrical optics approximation is considered , in which two collimated beams of arbitrary irradiance are mapped onto each other .
several methods for phase and irradiance control with double freeform surfaces have been proposed in literature .
+ one of the first design methods for the mapping of two wavefronts by coupled freeform surfaces is the simultaneous multiple surface ( sms ) method , which was developed by benitez and miano @xcite .
the surfaces are thereby constructed from generalized cartesian ovals and by applying constant optical path length ( opl ) conditions@xcite .
the design method can be utilized for numerous applications in imaging and nonimaging optics@xcite . +
zhang et al .
@xcite and shengqian et al .
@xcite solve the design problem by describing it in the form of a monge - ampre type pde , discretizing the equation by finite differences and then solving the resulting nonlinear equation system by the newton method .
the design method can be applied to a variety of wavefront shapes @xcite .
+ an alternative approach to construct freeform surfaces for irradiance and phase control is from ray mapping methods@xcite .
these methods are based on the seperation of the design process into two separate steps : the calculation of an _ integrable _ ray mapping between the source and the target irradiance and the subsequent construction of the _ continuous _ freeform surfacesfrom the mapping .
the integrability thereby ensures the continuity of the freeform surfaces and the mapping of the input irradiance onto the ouptut irradiance . since the integrability depends on the physical properties of the optical system it is in general a nontrivial task to find such a mapping . +
as shown in several publications , there is a strong relation between the inverse problem of nonimaging optics and optimal mass transport ( omt)@xcite .
the cost function , which has to be applied to a certain optical configuration , is thereby problem - specific .
for example , the mapping of two collimated beams with arbitrary irradiance onto each other with double freeform _ mirrors _ is described by a quadratic cost function @xcite and can be solved by corresponding numerical schemes@xcite . the same problem statement with double freeform _ lenses _ , which is considered in this work ,
is described by a different cost function , which depends on the opl between the freeform surfaces as it was shown by rubinstein and wolansky@xcite .
+ the investigations presented here are inspired by several publications @xcite , in which the authors applied the quadratic omt cost function to calculate a ray mapping to deal with the lens design problem . with this ray mapping
, designs have been demonstrated of both single freeform surfaces for irradiance control for collimated input beams and point sources @xcite , and that of double freeform surfaces for irradiance and phase control@xcite . as demonstrated for illumination control with single freeform surfaces in ref .
@xcite and for collimated beam shaping with double freeform surfaces in ref .
@xcite the design problems are thereby restricted to a paraxial approximation .
+ here we first investigate the design by ray - mapping methods of double freeform surfaces which map two collimated beams with arbitrary irradiance onto each other _ beyond _ the paraxial approximation . to overcome the restriction to the paraxial regime , which is necessary for the construction of compact systems
, the design problem will be modeled by two coupled pde s .
this involves on one hand the jacobian equation , expressing the local energy conservation , and on the other hand a ray mapping pde , enforcing the surface continuity and the constant opl .
the pdes will then be solved by an optimization scheme with the omt mapping from the quadratic cost function as the intial iterate , leading to a construction approach for the freeform surfaces .
+ to do so , the work is structured as follows . in section [ sec:2 ] , by using the law of refraction , the constant opl condition and a surface continuity condition , a pde for an integrable ray mapping , is derived . together with the jacobian equation it builds a system of pdes for the determination of the mapping components .
it is argued that the pde system is fulfilled within the paraxial approximation by the quadratic cost function omt map . in section [ sec:3 ] , a method for solving the pde system for general lens lens distances is presented .
it is based on discretizing the pdes with finite differences and solving the resulting system of nonlinear equations by a standard optimization scheme with the quadratic cost function omt map as the initial iterate .
a summary of the design algorithm and a detailed discussion of the implemenation is presented in section [ sec:4 ] , followed by the application of the presented method to the design of a single - lens and a two - lens system in section [ sec:5 ] .
finally , in section [ sec:6 ] , a short discussion of the results is presented .
in ref . @xcite
a design method was presented for the construction of a single freeform surface for a collimated input beam with irradiance @xmath0 and an arbitrary illumination pattern @xmath1 on a target plane .
it was shown that in the paraxial approximation the design process can be decoupled into two separate steps . in the first step
a raymapping @xmath2 is calculated from the theory of optimal mass transport , and in the second step the freeform surface is constructed from the mapping .
+ there are several basic physical principles that a ray mapping needs to fulfill .
firstly , to map the source irradiance @xmath0 onto the target irradiance @xmath1 , the ray mapping should be energy conserving .
the local energy conservation is expressed through the jacobian eq .
@xmath3 secondly , in case of freeform illumination optics , the mapping should allow the calculation of _ continuous _ freeform surfaces .
as shown in several publications , these so called _ integrable _ ray mappings are related to problem - specific cost functions representing different optical settings , where one has to distinguish between point sources and/or collimated beams , mirrors and/or lenses and so on@xcite .
the cost function defines a metric between the source distribution @xmath0 and the target illumination pattern @xmath1 and therefore represents an additional constraint to the underdetermined eq .
( [ eq:1 ] ) . in the case of a single freeform surface for the redistribution of collimated input beams , the quadratic cost function @xmath4 which is valid in the paraxial approximation , was studied by the authors@xcite . a key property there was the vanishing curl @xmath5 characterizing the quadratic cost function in eq . ( [ eq:2 ] ) @xcite .
+ as shown by rubinstein and wolansky , the cost function for collimated beam shaping with double freeform _ lenses _ takes a different form than eq .
( [ eq:2])@xcite .
the authors propose to minimize the corresponding cost function by a steepest descent algorithm to get the ray mapping@xcite , but unfortunatly a numerically stable implementation is a nontrivial problem .
+ due to its applicability in the paraxial approximation ( see below ) and the availability of numerous published stable numerical schemes for its calculation , the quadratic cost function omt mapping serves as an initial iterate for the optimization scheme presented below . it will therefore build the basis of the design approach presented in sections [ sec:3 ] and [ sec:4 ] .      [ cols="^ " , ]     the freeform calculation with the initial maps @xmath6 shows strong deviations from the predefined specifications , whereas the optimized maps @xmath7 show high quality illumination patterns and a wavefront uniformity beyond the diffraction limit .
remaining deviations from the ideal wavefront result ( besides fundamental numerical limitiations ) are from eq .
( [ eq:14b ] ) , which is not fulfilled exactly and therefore leads to an error accumulation along the integration path of eqs .
( [ eq:9 ] ) and ( [ eq:12 ] ) .
the precision of the illumination pattern on the other hand is mainly limited by the precision of @xmath6 .
the main deviations from the predefined distribution @xmath1 are resulting from steep gradients , which can be seen especially for the example  iap " in fig .
[ fig:6]d ) .
this is in agreement with fig . [
fig:3]b ) and observations which were made in ref . @xcite .
a design method for the calculation of compact continuous double freeform surfaces for collimated beam shaping with complex irradiance patterns was presented .
the method is based on the ray mapping condition ( [ eq:14b ] ) , which was derived from the law of refraction and the surface continuity condition in section [ sec:2 ] and builds together with the jacobian eq .
( [ eq:14a ] ) a system of nonlinear pdes for the unknown ray mapping @xmath8 .
+ due to the satisfaction of eq .
( [ eq:14 ] ) for infinite lens distances by the mapping from omt with the quadratic cost function ( [ eq:2 ] ) , this mapping serves as an ideal initial iterate for an optimization approach for solving the system of eq .
( [ eq:14 ] ) . as it was shown by approximating the eqs .
( [ eq:14 ] ) by finite differences and using a standard optimization scheme from matlab s optimization toolbox one can ensure a fast convergence to the solution of the eqs .
( [ eq:14 ] ) .
this was demonstrated by applying the presented method to two design examples with complex target distributions and validating the results by ray tracing .
the double freeform surfaces showed thereby high accuracy for the irradiance patterns and the wavefront , which was assessed by the calculation of the corresponding @xmath9 values of the normalized differences .
+ further improvements can be made by using omt methods for more complex boundaries of the source and target distributions , which requires the replacement of sulman s method ( with e.g. @xcite ) and the generalization of eq .
( [ eq:16 ] ) to more complex boundary shapes .
the scalability of the distance of the initial map @xmath6 by @xmath10 to the solution of the eqs .
( [ eq:14 ] ) suggests the application of e.g. the newton algorithm for a faster optimization .
+ in our future research , we want to generalize the presented approach to double freeform surfaces for wavefronts different from the plane case , like e.g. spherical wavefronts .
the authors want to thank ralf hambach and mateusz oleszko for valuable discussions , johannes stock for comments on the manuscript and david musick for a grammar and spelling check
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numerous applications require the simultaneous redistribution of the irradiance and phase of a laser beam . 
 the beam shape is thereby determined by the respective application . 
 an elegant way to control the irradiance and phase at the same time is from double freeform surfaces . in this work 
 the numerical design of continuous double freeform surfaces from ray mapping methods for collimated beam shaping with arbitrary irradiances is considered . 
 these methods consist of the calculation of a proper ray mapping between the source and the target irradiance and the subsequent construction of the freeform surfaces . by combining the law of refraction , the constant optical path length and the surface continuity condition , a partial differential equation ( pde ) for the ray mapping 
 is derived . 
 it is shown that the pde can be fulfilled in a small - angle approximation by a mapping derived from optimal mass transport with a quadratic cost function . to overcome the restriction to the paraxial regime we use this mapping as an initial iterate for the simultaneous solution of the jacobian equation and the ray mapping pde by an optimization . 
 the presented approach enables the efficient calculation of compact double freeform surfaces for complex target irradiances . 
 this is demonstrated by applying it to the design of a single - lens and a two - lens system .    * * *    2