as examples of the application of the present invention to solid ur &# 39 ; s , mirrors of several aberrationless resonators of interest will be discussed , including a resonator with ideal spherical - wave modes , a half - symmetric resonator with a predistorted mode , whose output beam has identically zero spherical aberration after refraction through its output mirror and a self - collimated unstable resonator . consider a general ur illustrated in fig1 in which the shape of the surface of the left mirror 10 is specified parametrically . that surface is taken to be the locus points of ordered coordinate pairs x l ( z ), y l ( z ) , where x l and y l are known functions of a parameter z . assume that the fundamental mode of this resonator is described well by ray optics , in which the field at any point in the resonator can be characterized by giving the direction of propagation of a ray . this is generally the case in unstable resonators . by specifying the desired mode in terms of the direction of the ray emanating from position x l ( z ), y l ( z ), and denoting that direction as angle φ l ( z ), which is also a known function of z , it is possible to correct aberration of the ray by the output mirror 10 on the left by defining the shape of the end mirror 12 on the right . the reflected ray leaving the left mirror 10 at position x l ( z 0 ), y l ( z 0 ), and angle φ l ( z 0 ) will strike the right mirror 12 at a point x r ( z 0 ), y r ( z 0 ) and be reflected . parameterization of the right mirror 12 may be freely chosen so that a ray leaving the left mirror 10 from a position specified by parameter z 0 hits the right mirror 12 at a position specified by z 0 . the ray will eventually hit the left mirror 10 again at a position x l ( z 1 ), y l ( z 1 ) for some different parameter z 1 . for a given right mirror 12 , z 1 is a fixed function of both z 0 and the coordinates x r ( z 0 ), y r ( z 0 ). by thus treating x r and y r as functions of both z 0 and z 1 , then deriving a condition that gives the correct functional form of z ( z 0 ), the necessary shape of the right mirror 12 for aberration correction of the left mirror 10 is defined . there are seven basic equations relating the mirror shapes and the angles shown in fig1 . the angles θ l and θ r at which the left and right mirrors 10 and 12 locally deviate from the y axis , i . e ., the angles of inclination are related to the local gradient of the parametric mirror vector , the gradient taken with respect to the parameter . for the left mirror 10 , ## equ1 ## which is known since x l ( z 0 ) and y l ( z 0 ) are given . for the right mirror 12 , ## equ2 ## and since x r and y r are assumed to depend upon both z 0 and z 1 , the total derivatives d / dz in terms of partial derivatives α / αz is given by ## equ3 ## further examination of fig1 shows that the angles of inclination ( angles with respect to the y axis ) θ of the mirrors and the angles φ of the rays leaving each mirror are simply related ; for reflection at the left mirror 10 , finally , the angle of each ray can be related to the distances along the x axis separating the points at which the rays hit each mirror . for the ray leaving the left mirror 10 , ## equ4 ## while for the ray leaving the right mirror 12 , ## equ5 ## for the right mirror 12 , all of the variables with an r subscript are unknown ; for the left mirror 10 , all of the variables with an l subscript are given . the variable z 1 is also unknown ; however , it is possible to solve for its derivative in eq . ( 3 ) to construct a differential equation for z 1 : ## equ6 ## the difference between the numerator and the denominator of the right - hand side of eq . ( 8 ) is that in the numerator the partial derivatives α / αz are taken with respect to z 0 , and in the denominator they are taken with respect to z 1 . equation ( 8 ) will specify the function z 1 ( z 0 ) and will ultimately give parametric representation of the right mirror 12 . eqs . ( 4 ) and ( 5 ) can be solved for θ r ( z 0 , z 1 ) to yield it is useful to introduce two additional functions , ρ ( z ) and σ ( z ), to simplify algebra . define ## equ7 ## where the second part of eq . ( 11 ) follows from the definition in eq . ( 1 ). using these definitions and eq . ( 9 ), ## equ8 ## next , find formulae for x r and y r . solving eqs . ( 6 ) and ( 7 ), and using definitions ( 10 ) and ( 11 ) gives ## equ9 ## the remaining undefined quantity is tan θ r ( z 0 , z 1 ). now using eq . ( 4 ) and the definitions ( 10 ) and ( 11 ), write ## equ10 ## the remainder of the construction proceeds as follows : eq . ( 15 ) is inserted into eqs . ( 13 ) and ( 14 ); the partial derivatives with respect to z 0 and z 1 are taken ; and the resulting equations , plus eq . ( 12 ) are then used to construct the differential eq . ( 8 ). this first - order nonlinear differential equation may be numerically integrated to give the function z 1 ( z 0 ), which is then substituted into eqs . ( 13 ) and ( 14 ) to yield the final right mirror 12 configuration . it is not terribly useful to complete this construction in full generality , because the fully expanded derivatives cause the resulting expression for dz 1 / dz 0 to occupy several pages . this renders the expression unwieldy . however , the construction is easily carried out by a symbolic manipulation computer program , e . g ., mathematica , version 2 . 0 . the derivation can be considerably simplified by suitable choice of the left mirror and its parameterization . in the following section , discussion is restricted to the special case of a half - symmetric resonator shown in fig2 i . e ., a resonator in which the left output mirror 10 &# 39 ; is flat . if the left mirror 10 is taken to be flat , then the following simple choice of parameterization suggests itself . for the moment , ρ ( z ) in eq . ( 15 ) will be unspecified . eq . ( 15 ) reduces to and eqs . ( 12 )-( 14 ) consequently simplify to ## equ11 ## with this simplification , the task of explicitly obtaining the differential equation for z 1 , and calculating the partial derivatives is simplified as follows : ## equ12 ## if the function f ( z 0 , z 1 ) is defined as ## equ13 ## the equation for the function z 1 ( z 0 ) may be written in the pleasingly symmetric form , ## equ14 ## if the directions of the rays in fig1 were reversed , the roles of z 1 and z 0 would reverse ; the form of eq . ( 26 ) is therefore a consequence of this symmetry . these simplified expressions will now be used to work out two special cases as examples : first , when the mode inside the resonator is a perfect spherical wave ; and second , when the mode outside the resonator is a perfect spherical wave after passing through an aberrative refractive output coupler . the foregoing solutions for z 1 ( z 0 ), x r ( z 0 ), and y r ( z 0 ) should reduce to the conventional theory of urs for a spherically diverging mode and spherical mirrors in the paraxial approximation . to check the theory , choose a mode with a virtual source point at depth d = 1 ( i . e ., at position ( 1 , 0 )) inside the resonator . then substituting this definition into eq . ( 26 ) gives ( after some algebraic reduction ), ## equ15 ## this nonlinear equation may be integrated to yield ## equ16 ## m is a constant of integration . in the limit of z 1 & lt ;& lt ; 1 , z 0 & lt ;& lt ; 1 , eq . ( 29 ) reduces to from which it is recognized that m is simply the geometric magnification of the resonator . solving eq . ( 29 ) for z 1 ( z 0 ) gives ## equ17 ## analytic expressions for the right mirror shape can be found by noting that , according to eqs . ( 19 ) and ( 20 ), ## equ18 ## which gives an explicit parametric representation for the right mirror 12 &# 39 ;: ## equ19 ## by parametrically plotting x r ( z ) and y r ( z ) from eqs . ( 33 ) and ( 34 ), the mirror shape is plotted . in fig2 the shape of the right mirror 12 &# 39 ; for m = 3 is plotted along with some representative rays . ( only the upper half of the mirror 12 &# 39 ; is plotted ; the resonator is symmetric about y = 0 ). the shape of the right mirror 12 &# 39 ; is a hyperbola . if z is eliminated from equations ( 33 ) and ( 34 ), the equation for the right mirror is ## equ20 ## that the shape of the right mirror 12 &# 39 ; is a hyperbola could also be deduced from the knowledge that a hyperbola can refocus a spherical wave to another spherical wave , a property that is used in the secondary reflector of a cassegrainian telescope . in fact , for a virtual point source optical mode , the construction of fig2 is equivalent to the design of a cassegrain feed looking at itself in a planar mirror . although the mode of the resonator analyzed in the previous example is an aberrationless spherical wave inside the resonator , if the resonator is solid with an index of refraction greater than 1 , the output beam will be refracted as it exits the resonator and will acquire some spherical aberration . it is possible , however , to intentionally distort the right mirror 12 &# 39 ; opposite the output mirror 10 &# 39 ; functioning as a coupler and , consequently so distort the resonator mode that the aberrations of the mode are precisely canceled by the aberrations acquired by refraction through the output coupler . it is this possibility that is most relevant to high - index , high - numerical - aperture ur diode lasers . to implement this important case , the function φ ( z ) that yields an output with a virtual point source after refraction through the output reflector is calculated for use in the construction of the appropriate shape of the end reflector opposite the output reflector . from the geometric construction shown in fig3 it can be seen that for the output ray to have a virtual point source at depth d = 1 , it is necessary to have snell &# 39 ; s law gives the desired φ l ( z ): ## equ21 ## from which the parameterization is calculated , thus : ## equ22 ## this expression , and its derivative , are substituted into eqs . ( 25 ) and ( 26 ); the resulting differential equation , though complex , may be numerically integrated in a straightforward fashion . for the plots in fig4 and 6 , the fourth - fifth - order runge - kutta algorithm in the mathematica package rungs kutta , m was used . there is one complication , however . as in eq . ( 28 ), the differential equation for z 1 ( z 0 ) has a singularity at z 0 = 0 . in the numerical integration , this singularity may be avoided by beginning the integration at some nonzero value , z 0 = ε , and taking the initial condition to be z 1 ( ε )= mε . m plays the role of the constant of integration and is the geometric magnification of the resonator . the solution for z 1 ( z 0 ) may then be substituted into the expressions for x r ( z 1 , z 0 ) and y r ( z 1 , z 0 ) to obtain a parametric representation of the mirror shape . in fig4 the solution for a practical resonator is plotted with virtual source at depth d = 1 , magnification m = 3 , and n = 3 . 4 ( the index of refraction of algaas ) along with the trajectories of several rays , both inside and outside the resonator . the mirror opposite the output coupler on the left , i . e ., the right mirror , has been replotted in fig5 with the horizontal scale exaggerated to emphasize the unusual shape of the mirror required to eliminate all aberrations . it is neither circular , as the paraxial geometric theory presupposes , nor is it hyperbolic , as the cassegrainian virtual - point - source mode would specify . the modes of such a resonator are neither plane - wave nor spherical - wave ; however , because of the refractive output coupler , the beam outside the resonator possesses a perfect point source . the effects of refraction for solid unstable resonators in the paraxial approximation have been calculated in a paper by r . j . lang , titled &# 34 ; geometric formulation of unstable - resonator design and application to self - collimating unstable - resonator diode lasers ,&# 34 ; opt . lett ., 16 , no . 17 , pp . 1 - 3 , 1991 . the calculations showed that a self - collimating positive - branch resonator exists for a range of magnifications below a critical value . by using this analysis with a nonzero σ ( z ), it is possible to explicitly solve for the mirror shape for ideal self - collimation . if the left mirror is to be a defined semicircular mirror , it is possible to parameterize the mirror by setting if a collimated output is required , it is necessary to have ## equ23 ## where n is the index of refraction . substitution of these functions into eqs . ( 8 - 15 ) gives the equation of motion for z 1 . while this equation is too complex to present here , it is quite straightforward to integrate . the solution for n = 3 . 4 , m = 1 . 5 and some representative rays are plotted in fig6 . the constant of integration m is the geometric magnification of the resonator along the optical axis . however , the magnification is , in general , nonuniform across the resonator . consider a bundle of rays at the left - hand mirror that span the range z 0 to z 0 + δz . after one round trip , it will span the range z 1 ( z 0 )- z 1 ( z 0 + δz ). the local magnification is therefore ## equ24 ## it has been shown by ferguson and smithers , &# 34 ; optical resonators with nonuniform magnification ,&# 34 ; j . opt . soc . am . a1 , pp . 653 - 662 , ( 1984 ), that the intensity profiles of the modes of resonators that have nonuniform local magnification may be derived from the local magnification function in a paraxial analysis . an extension of their analysis to the more general nonparaxial case described here is straightforward . one of the interesting features of the resonators in the foregoing examples described above is that , since no paraxial approximations are being made , the results are exact even for light propagation nearly normal to the optical axis , although there are , of course , diffraction effects , which have not been included . while the examples have been based on relatively simple mirrors ( namely , flat and circular ), it is obviously possible to use this invention to construct a resonator with any shape mirror on the left side as an output coupler . some truly bizarre structures with well - behaved modes are possible . while an aberration - corrected resonator whose mirror is some arbitrary shape ( say , a gaussian ) may be just a curiosity , this invention has a practical application . one could construct a resonator with an aberrated mode and an aberrated output that is precisely compensates for the known aberrations of subsequent optics in the optical train , such as those of a collimating lens . such a design scenario for an unstable resonator diode laser 20 is illustrated in fig7 . it is desirable to have a flat output face 21 to allow butt - coupling to a cylindrical microlens 22 , which may be fabricated with an aspherical cross section to collimate the beam in the vertical direction . this lens is followed by a conventional cylindrical lens 23 to collimate the beam laterally . use of a conventional cylindrical lens for lateral collimation may introduce significant cylindrical aberration due to the high numerical aperture of the output beam . ( the larger lateral dimension of hundreds of μm presumably precludes use of an aberration - corrected microlens .) by the method described above , it is possible to raytrace a theoretically collimated output back onto and through the laser facet , thereby calculating the desired φ l ( z ) for the lasing mode . this φ l ( z ), which will precisely compensate for the cumulative system aberration , may be used to calculate the shape of the rear mirror 24 , which is then fabricated by lithographic methods and dry etching . the techniques for mirror definition and etching for unstable resonator diode lasers are still in the early stages of development , as are ur diode lasers themselves . nevertheless , as the ability to fabricate an arbitrary mirror shape of high optical quality becomes more readily available , it may be used to tailor the mirror shape to the requirements of the user . the traditional techniques of laser resonator design have been &# 34 ; specify a resonator ; calculate the resulting mode .&# 34 ; by using the present invention , one may specify the properties of the desired mode and construct the resonator to fit . consequently , the construction of unstable resonators as defined in the claims appended hereto with reference to mathematical equations constitutes construction of a resonator to fit the equations in a practical sense .