Abstract:
A beam shaping telescope includes two mirrors having rotationally symmetric curvature inclined to the optical axis of the telescope. By selecting an appropriate curvature, spacing, and inclination of the mirrors, the telescope can be used to transform an astigmatic laser beam having a non-circular cross section into a circular beam having essentially zero astigmatism.

Description:
PRIORITY 
       [0001]    This application claims priority to U.S. Provisional Application 60/693,820, filed Aug. 7, 2007, the disclosure of which is incorporated herein by reference. 
     
    
     TECHNICAL FIELD OF THE INVENTION 
       [0002]    The present invention relates in general to optical arrangements for projection of laser beams. The invention relates in particular to a telescope for transforming a non-circular astigmatic laser beam into a circular anastigmatic beam. 
       DISCUSSION OF BACKGROUND ART 
       [0003]    Certain laser systems such as amplified seed laser systems have a gain-medium medium that is astigmatic, i.e., an amplified beam delivered by the laser system is generally astigmatic. This astigmatism results in the output beam being generally elliptical, with a major-axis to minor-axis ratio that varies progressively with distance. In many applications of laser beams it is necessary to focus the beam to a minimum possible spot-size. If a conventional projection lens or telescope is used to focus an astigmatic beam it will be found that the beam can only be focused in only one of two planes perpendicular to each other. These planes are usually referred to as the tangential and sagittal plane by practitioners of the optical design art. There will be, in effect, different sagittal and tangential focal distances. As certain factors that lead to the astigmatism are ephemeral, the astigmatism can vary from laser to laser in manufacturing lasers that are nominally identical. 
         [0004]    It is possible to correct an astigmatic condition by using a beam-shaping optical arrangement to correct both the elliptical shape of beam and the astigmatism. Arrangements including cylindrical or anamorphic optical elements, i.e., optical elements having different optical power in two axes perpendicular to each other, are typically used for this purpose. Such elements include both reflective (catoptric) and refractive (dioptric) elements. 
         [0005]    Such cylindrical or anamorphic optical elements are relatively expensive to produce compared with rotationally symmetrical elements that have the same optical power in each axis. This is particularly true if surfaces are required to have some second or higher order components of curvature. Further, such beam-shaping optical arrangements may be intolerant of the above-discussed manufacturing variations in astigmatism unless designed to be tolerant. Such design adds to the cost and complexity of a system. Refractive optical elements are usually more expensive to manufacture than reflective elements, as two optical surfaces rather than only one must be generated. 
         [0006]    There is a need for a beam projecting arrangement that can transform a non-circular astigmatic beam into a circular anastigmatic beam using only a combination of rotationally symmetric optical elements. Preferably, the beam projecting arrangement should not include any refractive optical elements. 
       SUMMARY OF THE INVENTION 
       [0007]    The present invention is directed to a telescope having a longitudinal optical axis and arranged to receive an astigmatic, input laser-beam. In one aspect of the invention, the telescope comprises a plurality of reflective optical elements, each thereof having a reflective surface that has rotationally symmetric curvature and is inclined at an angle to the optical axis. The reflective surfaces are spaced apart on the optical axis. The curvature of the reflective surfaces, the angle of inclination of the reflective surfaces to the optical axis, and the spacing of the reflective surfaces along the optical axis is selected such that the astigmatic, input laser-beam received by the telescope is projected as an essentially anastigmatic output laser-beam. 
         [0008]    The reflective surfaces of at least two of the optical elements must have a non-zero curvature, i.e., a finite radius of curvature. In preferred embodiments of the present invention there are only two of the reflective optical elements. Both of the surfaces may have a concave curvature, or one may have a convex curvature and the other a concave curvature. While spherical surfaces of curvature are preferred for these elements for convenience of manufacture and telescope design, the use of aspheric surfaces, with second or higher order components of curvature is not precluded. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    The accompanying drawings, which are incorporated in and constitute a part of the specification, schematically illustrate a preferred embodiment of the present invention, and together with the general description given above and the detailed description of the preferred embodiment given below, serve to explain principles of the present invention. 
           [0010]      FIG. 1  schematically illustrates one preferred embodiment of a beam-shaping telescope in accordance with the present invention including first and second mirrors arranged spaced-apart, and tilted in the same plane with reference to a propagation axis of the telescope. 
           [0011]      FIG. 2  is a graph schematically illustrating graphical determination of mirror tilt angles and spacing between first and second mirrors in one example of the telescope having an arrangement similar to the arrangement of  FIG. 1 . 
           [0012]      FIG. 3  is a graph schematically illustrating graphical determination of mirror tilt angles and spacing between first and second mirrors in another example of the inventive telescope having an arrangement similar to the arrangement of  FIG. 1 . 
           [0013]      FIG. 4  is a graph schematically illustrating graphical determination of mirror tilt angles and spacing between first and second mirrors in yet another example of the inventive telescope wherein the first mirror is tilted in a first plane and the second mirror is tilted in a second plane at ninety degrees to the first plane. 
           [0014]      FIG. 5A  schematically illustrates the first and second mirror arrangement of the example of  FIG. 4 , seen in a sagittal plane of the second mirror. 
           [0015]      FIG. 5B  schematically illustrates the first and second mirror arrangement of the example of  FIG. 4 , seen in a tangential plane of the second mirror. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0016]    Turning now to the drawings, wherein like features are designated by like reference numerals.  FIG. 1  schematically illustrates one preferred embodiment  20  of a beam shaping telescope in accordance with the present invention. A laser beam  22  enters the telescope. Cartesian x- and y-axes (y, x 1 ) are depicted in graph (A) of  FIG. 1 . The axis system in graph (A) is rotated 90° about the x-axis to depict the input beam cross-section. In the optical portion of the drawing the y-axis is actually perpendicular to the plane of the drawing with the z-axis (propagation-axis) of the beam illustrated by a dashed line. 
         [0017]    Input beam  22  in this example is collimated in the x-z and y-z planes, but is asymmetric in cross-section. The plane of the drawing contains the two vectors (axes) x 1  (for input beam  22 ) and x 2  (for output beam  26 ). The cross-section of the input beam is depicted in graph (A) and indicates that the beam is “higher” in the y-axes than in the x-axis wide. The y-axis and x-axis are commonly referred to as the height and width directions, respectively, by practitioners of the optical art. 
         [0018]    Input beam  22  is incident first on a convex mirror M 1 . Reflection from mirror M 1  transforms the collimated beam into a diverging beam  24 . Mirror M 1  is tilted in the plane of the drawing (the tangential plane). Because of this, the mirror has a stronger effect in the plane of the drawing than in the plane perpendicular to the figure (the sagittal plane). Accordingly, the beam diverges more in the tangential plane than in the sagittal plane. This causes diverging beam  24  to grow rounder in cross-section with distance along the z-axis. At a distance d from mirror M 1 , a mirror M 2 , also tilted in the tangential plane, is used to collimate the beam. Mirror M 2  is concave, and focuses as a positive lens would. The collimated output beam  26  has a round cross-section as indicated in graph (B) of  FIG. 1 . Graph (C) depicts the tilt of the x 1 -axis (the x-axis of input beam  22 ) relative to the x 2 -axis (the x-axis of output beam  26 ) with the y-axis in the correct orientation. 
         [0019]    Because mirror M 2  is tilted in the tangential plane, the mirror has a stronger effect in the plane of the figure (called in this case the tangential plane) than in the plane perpendicular to the figure (called in this case the sagittal plane). As beam  24  is diverging faster in the tangential plane than in the sagittal plane, the tilt angle of mirror M 2  is selected to allow collimation of the beam in both the tangential and sagittal planes. This is achieved with a combination of the angle of incidence on the concave mirror as well as the distance d between the two mirrors. In practice, after the beam has been collimated in both planes by choosing the correct angle of incidence on the concave mirror as well as the correct distance d, the output beam may still be slightly asymmetric. In such a case, the tilt angle of mirror M 1  can be re-adjusted, and the tilt angle of mirror M 2  and distance d can be correspondingly re-adjusted. 
         [0020]    Generally, in practice, the astigmatism planes of the input beam may not correspond to the horizontal and vertical planes. In this case, the whole plane of reflections that is depicted in  FIG. 1  will need to be adjusted to coincide with the planes of astigmatism. The input beam may also have some divergence, and have a minimum size (beam-waist) which is located at two different locations, one in each plane of astigmatism. 
         [0021]    It should also be noted that the two-mirror arrangement of  FIG. 1  is not restricted to having a convex first mirror and concave second mirror. The two mirrors may both be concave, or the first mirror may be concave and the second mirror convex. In the latter case, the overall size of the beam is reduced as the beam cross-section is re-shaped. 
         [0022]    A particular geometry (tilt angles and distance) for any given mirror pair in the arrangement of  FIG. 1  may be calculated using a ray matrix calculation. In a ray-matrix calculation a mirror is represented by a 2×2 (A, B, C, D) matrix wherein: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           A 
                         
                         
                           B 
                         
                       
                       
                         
                           C 
                         
                         
                           D 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 γ 
                               
                               R 
                             
                           
                         
                         
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where R is the radius of curvature (positive for a concave mirror, negative for a convex mirror, infinite for a plane mirror) and γ is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     γ 
                     = 
                     
                       
                         1 
                         
                           cos 
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                        
                       
                           
                       
                        
                       for 
                        
                       
                         
                             
                         
                          
                         
                             
                         
                       
                        
                       the 
                        
                       
                           
                       
                        
                       tangential 
                        
                       
                           
                       
                        
                       beam 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     γ 
                     = 
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                        
                       
                           
                       
                        
                       for 
                        
                       
                           
                       
                        
                       the 
                        
                       
                           
                       
                        
                       sagittal 
                        
                       
                           
                       
                        
                       beam 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and where θ is the tilt angle of the mirror, i.e., the incidence angle of the z-axis on the mirror. 
         [0023]    The distance between the beam-waist (the location where the beam has a minimum size) in the first astigmatism plane and the mirror M 1  is defined as L 1 . This distance is positive if the beam-waist is located before M 1 , and is negative if the beam-waist is located after M 1 . The distance between the beam waist in the second astigmatism plane and the first mirror M 1  is defied as L 2 . Again, that distance is positive if the beam-waist is located before M 1 , and is negative if the beam-waist is located after M 1 . Ray matrices for both astigmatism planes: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             A 
                             1 
                           
                         
                         
                           
                             B 
                             1 
                           
                         
                       
                       
                         
                           
                             C 
                             1 
                           
                         
                         
                           
                             D 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                   
                                     γ 
                                     2 
                                   
                                 
                                 
                                   R 
                                   2 
                                 
                               
                             
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             d 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                   γ 
                                 
                                 
                                   R 
                                   1 
                                 
                               
                             
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             
                               L 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             A 
                             2 
                           
                         
                         
                           
                             B 
                             2 
                           
                         
                       
                       
                         
                           
                             C 
                             2 
                           
                         
                         
                           
                             D 
                             2 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 2 
                                 
                                   
                                     γ 
                                     2 
                                   
                                    
                                   
                                     R 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             d 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 2 
                                 
                                   
                                     γ 
                                     1 
                                   
                                    
                                   
                                     R 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             
                               L 
                               2 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where γ i  is the γ factor for mirror M i , equal to 
         [0000]    
       
         
           
             
                 
             
              
             
               1 
               
                 cos 
                  
                 
                   ( 
                   
                     θ 
                     i 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    if mirror M i  is tilted in the first plane of astigmatism. If mirror M i  is tilted in the second plane of astigmatism the γ factor for mirror M i , equal to cos(θi). In the arrangement of  FIG. 1 , the first and second planes of astigmatism are respectively the sagittal and tangential planes. Equations (3) and (4) can be developed to obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             A 
                             1 
                           
                         
                         
                           
                             B 
                             1 
                           
                         
                       
                       
                         
                           
                             C 
                             1 
                           
                         
                         
                           
                             D 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             1 
                             - 
                             
                               
                                 2 
                                  
                                 
                                   γ 
                                   1 
                                 
                                  
                                 d 
                               
                               
                                 R 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               L 
                               1 
                             
                             + 
                             
                               d 
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       L 
                                       1 
                                     
                                   
                                   
                                     R 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       2 
                                     
                                   
                                   
                                     R 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         γ 
                                         1 
                                       
                                        
                                       d 
                                     
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 2 
                                  
                                 
                                   γ 
                                   1 
                                 
                               
                               
                                 R 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       2 
                                     
                                   
                                   
                                     R 
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     L 
                                     1 
                                   
                                   + 
                                   
                                     d 
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         
                                           2 
                                            
                                           
                                             γ 
                                             1 
                                           
                                            
                                           
                                             L 
                                             1 
                                           
                                         
                                         
                                           R 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             + 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       L 
                                       1 
                                     
                                   
                                   
                                     R 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             A 
                             2 
                           
                         
                         
                           
                             B 
                             2 
                           
                         
                       
                       
                         
                           
                             C 
                             2 
                           
                         
                         
                           
                             D 
                             2 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             1 
                             - 
                             
                               
                                 2 
                                  
                                 d 
                               
                               
                                 
                                   γ 
                                   1 
                                 
                                  
                                 
                                   R 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               L 
                               2 
                             
                             + 
                             
                               d 
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       L 
                                       2 
                                     
                                   
                                   
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   2 
                                   
                                     
                                       γ 
                                       2 
                                     
                                      
                                     
                                       R 
                                       2 
                                     
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       d 
                                     
                                     
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         R 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               2 
                               
                                 
                                   γ 
                                   1 
                                 
                                  
                                 
                                   R 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   2 
                                   
                                     
                                       γ 
                                       2 
                                     
                                      
                                     
                                       R 
                                       2 
                                     
                                   
                                 
                                 [ 
                                 
                                   
                                     L 
                                     2 
                                   
                                   + 
                                   
                                     d 
                                      
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         
                                           
                                             2 
                                              
                                             
                                               L 
                                               2 
                                             
                                           
                                           
                                             
                                               γ 
                                               1 
                                             
                                              
                                             
                                               R 
                                               1 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       L 
                                       2 
                                     
                                   
                                   
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The complex {tilde over (q)} parameter for a laser beam is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     ~ 
                   
                   = 
                   
                     z 
                     + 
                     
                       j 
                        
                       
                         
                           π 
                            
                           
                               
                           
                            
                           
                             w 
                             o 
                             2 
                           
                         
                         λ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where w o   2  is the beam-waist size at the 1/e2 points, and z is the distance from that beam-waist, positive if the beam is past its minimum (and is therefore diverging), negative if the beam is before its minimum and is therefore focusing (converging); and λ is the wavelength of the laser beam. 
         [0024]    The complex {tilde over (q)} parameter after the ray matrix 
         [0000]    
       
         
           
               
             
               ( 
               
                 
                   
                     A 
                   
                   
                     B 
                   
                 
                 
                   
                     C 
                   
                   
                     D 
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    is given by the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       q 
                       ~ 
                     
                     out 
                   
                   = 
                   
                     
                       
                         A 
                          
                         
                           
                             q 
                             ~ 
                           
                           in 
                         
                       
                       + 
                       B 
                     
                     
                       
                         C 
                          
                         
                           
                             q 
                             ~ 
                           
                           in 
                         
                       
                       + 
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where {tilde over (q)} in  is the input parameter, and {tilde over (q)} out  is the output parameter. In the instant case, the complex {tilde over (q)} parameter is different in each astigmatism plane. Since in this case the input is located at exactly the beam-waist, it is known that 
         [0000]    
       
         
           
             
               
                 q 
                 ~ 
               
               
                 in 
                 , 
                 1 
               
             
             = 
             
               
                 j 
                  
                 
                   
                     π 
                      
                     
                         
                     
                      
                     
                       w 
                       1 
                       2 
                     
                   
                   λ 
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                   
                     q 
                     ~ 
                   
                   
                     in 
                     , 
                     2 
                   
                 
               
               = 
               
                 j 
                  
                 
                   
                     
                       π 
                        
                       
                           
                       
                        
                       
                         w 
                         2 
                         2 
                       
                     
                     λ 
                   
                   · 
                   
                     
                       q 
                       ~ 
                     
                     
                       in 
                       , 
                       1 
                     
                   
                 
               
             
           
         
       
     
         [0000]    will experience the ray matrix 
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     
                       A 
                       1 
                     
                   
                   
                     
                       B 
                       1 
                     
                   
                 
                 
                   
                     
                       C 
                       1 
                     
                   
                   
                     
                       D 
                       1 
                     
                   
                 
               
               ) 
             
             , 
           
         
       
     
         [0000]    and {tilde over (q)} in,2  will experience the ray matrix 
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     
                       A 
                       2 
                     
                   
                   
                     
                       B 
                       2 
                     
                   
                 
                 
                   
                     
                       C 
                       2 
                     
                   
                   
                     
                       D 
                       2 
                     
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0000]    The output complex {tilde over (q)} parameters after the telescope accordingly are: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       q 
                       ~ 
                     
                     
                       out 
                       , 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           A 
                           1 
                         
                          
                         
                           
                             q 
                             ~ 
                           
                           
                             in 
                             , 
                             1 
                           
                         
                       
                       + 
                       
                         B 
                         1 
                       
                     
                     
                       
                         
                           C 
                           1 
                         
                          
                         
                           
                             q 
                             ~ 
                           
                           
                             in 
                             , 
                             1 
                           
                         
                       
                       + 
                       
                         D 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       q 
                       ~ 
                     
                     
                       out 
                       , 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           A 
                           2 
                         
                          
                         
                           
                             q 
                             ~ 
                           
                           
                             in 
                             , 
                             2 
                           
                         
                       
                       + 
                       
                         B 
                         2 
                       
                     
                     
                       
                         
                           C 
                           2 
                         
                          
                         
                           
                             q 
                             ~ 
                           
                           
                             in 
                             , 
                             2 
                           
                         
                       
                       + 
                       
                         D 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    If the goal is to obtain a collimated and round beam at the output, using equation 7, three equations are provided as follows:
 
For a collimated output beam in first astigmatism plane:
 
         [0000]        Re ( {tilde over (q)}   out,1 )=0  (11) 
         [0000]    For a collimated output beam in second astigmatism plane: 
         [0000]        Re ( {tilde over (q)}   out,2 )=0  (12) 
         [0000]    For a round output beam: 
         [0000]    
       
         
           
             
               
                 
                   
                     Im 
                     ( 
                     
                       1 
                       
                         
                           q 
                           ~ 
                         
                         
                           out 
                           , 
                           1 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     Im 
                     ( 
                     
                       1 
                       
                         
                           q 
                           ~ 
                         
                         
                           out 
                           , 
                           2 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equation (13) can be rewritten as an equation between d and γ 1 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           2 
                         
                         [ 
                         
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         L 
                                         1 
                                       
                                     
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         L 
                                         2 
                                       
                                     
                                     
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         R 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       1 
                                     
                                   
                                   
                                     R 
                                     1 
                                   
                                 
                                 ) 
                               
                               2 
                             
                              
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   2 
                                   
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                              
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         d 
                         [ 
                         
                           
                             
                               2 
                                
                               
                                 
                                   L 
                                   1 
                                 
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         L 
                                         1 
                                       
                                     
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           - 
                           
                             
                               2 
                                
                               
                                 
                                   L 
                                   2 
                                 
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         L 
                                         2 
                                       
                                     
                                     
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         R 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           - 
                           
                             
                               
                                 4 
                                  
                                 
                                   γ 
                                   1 
                                 
                               
                               
                                 R 
                                 1 
                               
                             
                              
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           + 
                           
                             
                               4 
                               
                                 
                                   γ 
                                   1 
                                 
                                  
                                 
                                   R 
                                   1 
                                 
                               
                             
                              
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         Z 
                         
                           R 
                            
                           
                               
                           
                            
                           1 
                         
                       
                       - 
                       
                         Z 
                         
                           R 
                            
                           
                               
                           
                            
                           2 
                         
                       
                       + 
                       
                         
                           L 
                           1 
                           2 
                         
                         
                           Z 
                           
                             R 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       - 
                       
                         
                           L 
                           2 
                           2 
                         
                         
                           Z 
                           
                             R 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   where 
                    
                   
                     
 
                   
                    
                   
                     
                       Z 
                       
                         R 
                          
                         
                             
                         
                          
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               w 
                               
                                 o 
                                 1 
                               
                               2 
                             
                           
                           λ 
                         
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         
                           Z 
                           
                             R 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                       = 
                       
                         
                           π 
                            
                           
                               
                           
                            
                           
                             w 
                             
                               o 
                               2 
                             
                             2 
                           
                         
                         λ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    are the Rayleigh ranges of the beam in the first and the second astigmatism planes, respectively. Equation (14), is a simple second-order polynomial equation in d which can be solved easily for each value of γ 1 . Equation (11), in turn may be rewritten as a simple linear equation for γ 2  as a function of d and γ 1 , as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       2 
                        
                       
                         γ 
                         2 
                       
                     
                     
                       R 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     L 
                                     1 
                                   
                                   + 
                                   
                                     d 
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         
                                           2 
                                            
                                           
                                             γ 
                                             1 
                                           
                                            
                                           
                                             L 
                                             1 
                                           
                                         
                                         
                                           R 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         L 
                                         1 
                                       
                                     
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               
                                 2 
                                  
                                 
                                   γ 
                                   1 
                                 
                               
                               
                                 R 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       1 
                                     
                                      
                                     d 
                                   
                                   
                                     R 
                                     1 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           [ 
                           
                             
                               L 
                               1 
                             
                             + 
                             
                               d 
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       L 
                                       1 
                                     
                                   
                                   
                                     R 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                       + 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   2 
                                    
                                   
                                     γ 
                                     1 
                                   
                                    
                                   d 
                                 
                                 
                                   R 
                                   1 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           Z 
                           
                             R 
                              
                             
                                 
                             
                              
                             1 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equation (12) may be similarly treated to yield: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         2 
                       
                        
                       
                         γ 
                         2 
                       
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               L 
                               2 
                             
                             + 
                             
                               d 
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     
                                       L 
                                       2 
                                     
                                   
                                   
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                       + 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   2 
                                    
                                   d 
                                 
                                 
                                   
                                     γ 
                                     1 
                                   
                                    
                                   
                                     R 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           Z 
                           
                             R 
                              
                             
                                 
                             
                              
                             2 
                           
                           2 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     L 
                                     2 
                                   
                                   + 
                                   
                                     d 
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         
                                           2 
                                            
                                           
                                             L 
                                             2 
                                           
                                         
                                         
                                           
                                             γ 
                                             1 
                                           
                                            
                                           
                                             R 
                                             1 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       2 
                                        
                                       
                                         L 
                                         2 
                                       
                                     
                                     
                                       
                                         γ 
                                         1 
                                       
                                        
                                       
                                         R 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               2 
                               
                                 
                                   γ 
                                   1 
                                 
                                  
                                 
                                   R 
                                   1 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                      
                                     d 
                                   
                                   
                                     
                                       γ 
                                       1 
                                     
                                      
                                     
                                       R 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               Z 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0025]    Equations (14), (15), and (16) are somewhat complex, but can be solved numerically. A simple strategy to find a solution is to choose a value for γ 1  and then find d using equation (14). The solutions for γ 2  can then be calculated from equations (15) and (16). This procedure is iterated until the solutions for both equations (15) and (16) are equal. An exemplary solution for the arrangement of  FIG. 1  using this strategy is set forth below. 
         [0026]    Here, it is assumed that input beam  22  is collimated, i.e., L 1 =0 and L 2 =0, and has a beam waist at the input of w o     1   =500μ along the x 1 -direction, and w o     2   =1000 μm along the y-direction. It is assumed that the laser beam has a wavelength of 1000.0 nanometers (nm), i.e., 1.0 micrometers (μm), in which case the Rayleigh range in each plane will be Z R1 =78.54 centimeters (cm) and Z R2 =314.16 cm. Mirror M 1  is assumed to be convex with a radius of curvature R 1 =−10 cm, and mirror M 2  is assumed to be concave with a radius of curvature R 2 =+75 cm. 
         [0027]    For γ 1  from 0 to 1.414, both solutions of equation (14) are negative. A negative value of d, however, is not physically possible. For γ 1 &gt;1.414, one solution of equation (14) is positive. Solutions for γ 2  as a function of γ 1  from equations (15) and (16) are graphically depicted in  FIG. 2 . Here γ 1  ranges between 1.45 and 1.55. The two solutions are equal at γ 1 =1.49362, where γ 2 =1.05648 and d=32.153 cm. 
         [0028]    A value of γ 1 =1.49362 means that the mirror M 1  is tilted in the tangential plane as depicted in  FIG. 1  and that the tilt angle θ 1 =47.970°. A value of γ 2 =1.05648 means that the second mirror is also tilted in the tangential plane, and that the tilt angle θ 2 =18.819°. 
         [0029]    In another example of the arrangement of  FIG. 1  it is assumed that the input beam has two beam-waists w o     1    and w o     2    that are identical and equal to 800.0 μm, but are located at two different locations. The first beam waist is located right at the input of the telescope, i.e., L 1 =0. The second beam-waist second is located 2 meters before the telescope i.e., L 2 =200 cm. It is again assumed that the laser beam has a wavelength of 1000.0 nm. The Rayleigh range in each plane is identical and equal to 201.06 cm. Again, mirror M 1  is convex with a radius of curvature R 1 =−10 cm, and mirror M 2  is concave with a radius of curvature R 2 =+75 cm. For γ 1  from 0 to 1.196, both solutions of equation (14) are negative. For γ 1 &gt;1.196, one solution of equation (14) is positive. 
         [0030]    Solutions for γ 2  as a function of γ 1  from equations (15) and (16) are graphically depicted in  FIG. 3 . Here, γ 1  ranges between 1.20 and 1.25. The two solutions are equal at γ 1 =1.22866, where γ 2 =1.02677 and d=32.454 cm. A value of γ 1 =1.22866 means that mirror M 1  is tilted in the tangential plane and that θ 1 =35.522°. A value of γ 2 =1.02677 means that the mirror M 2  is also tilted in tangential plane, and that θ 2 =13.112°. 
         [0031]    In yet another example of the inventive telescope it is assumed that the input beam has both a severe astigmatism and asymmetry. It is assumed that the beam-waists of the input beam are w o     1   =50 μm along the x 1  direction, and w o     2   =1000 μm along the y direction. The beam-waists are assumed to be located at two different locations. The first beam-waist is located 13 cm before the input of the telescope, i.e., L 1 =13 cm. The second beam-waist is located at the input of the telescope, i.e., L 2 =0° It is yet again assumed that the laser beam has a wavelength of 1000 nm. The Rayleigh range is Z R1 =0.785 cm along the x 1  direction, and Z R1 =314.16 cm along the y direction. Yet again, mirror M 1  is assumed to be convex with a radius of curvature R 1 =−10 cm, and mirror M 2  is assumed to be concave with a radius of curvature R 2 =+75 cm. 
         [0032]    For values of γ 1  below 0.923, both solutions of equation 14 are negative. For γ 1 &gt;0.923, one solution of equation 14 is positive. Solutions for γ 2  as a function of γ 1  from equations (15) and (16) are graphically depicted in  FIG. 4 . Here, γ 1  ranges between 0.930 and about 0.945. The two solutions are equal at γ 1 =0.93612 with γ 2 =1.01195 and d=33.268 cm. A value of γ 1 =0.93612 means that mirror M 1  is tilted in the sagittal plane and that θ 1 =20.591°. A value of γ 2 =1.01195 means that mirror M 2  is tilted in tangential plane, and that θ 2 =8.814°. In this example, schematically depicted in  FIG. 5A  and  FIG. 5B  as telescope  20 A both mirrors M 1  and M 2  are tilted in different planes and accordingly the output beam does not remain co-planar with the input beam as in the arrangement of  FIG. 1 . 
         [0033]      FIG. 5A  is view of telescope  20 A in the sagittal plane of the output beam.  FIG. 5B  is view of telescope  20 A in the tangential plane of the output beam. The x-, y- and z-axes, graphs (A) and (C), are depicted in the correct location, i.e., the axes are not rotated to illustrate the beam cross-section. The input beam is designated as beam  22 A. The input beam is depicted as essentially collimated in the sagittal plane with a beam waist assumed to be on mirror M 1 . In the tangential plane there is a much narrower beam-waist ahead of mirror M 1  as discussed above with reference to the example of  FIG. 4 . The diverging beam between the mirrors is designated as beam  24 A; and the collimated (in both astigmatism planes) output beam is designated as beam  26 A. This geometry depicted in  FIGS. 5A and 5B  is not as desirable as the geometry of  FIG. 1 , as the output beam is out of plane with the input beam in both x and y directions when the beam exits the telescope and may need several reflections from different mirrors to be redirected as desired. 
         [0034]    An experimental example of the inventive beam-shaping and astigmatism correcting telescope was constructed to expand and correct a beam from a titanium-doped sapphire solid-state laser. The laser beam before expansion had astigmatism of 1.01 and asymmetry of 1.13. An ideal beam would have zero astigmatism and an asymmetry of 1.0. The astigmatism and asymmetry were measured on a commercially available M 2 -meter. 
         [0035]    In the experimental telescope, configured generally as depicted in  FIG. 1 , convex mirror M 1  has a radius of curvature R 1 =−10 cm, and concave mirror M 2  has a radius of curvature R 2 =+75 cm. Both mirrors were tilted in the plane of the laser. The angle of incidence θ 1  on convex mirror M 1  is about 35.0°. The angle of incidence θ 2  on concave mirror M 2  is about 12.6°. The mirror separation d was set around 32.6 cm. The resulting beam output beam had a diameter of about 6.5 millimeter (mm) diameter, and the astigmatism and asymmetry was greatly reduced compared with the astigmatism and asymmetry of the input beam. In a typical measurement, the output beam had astigmatism of about 0.03, i.e., close to zero, or essentially anastigmatic, and perfect asymmetry, i.e., an asymmetry of 1.0. 
         [0036]    It should be noted in examples of the inventive telescope discussed above only two mirrors, each thereof having a finite (positive or negative), rotationally symmetric radius of curvature, are used to achieve beam shaping and astigmatism correction. Those skilled in the art will recognize that similar results may be achieved using three of more mirrors having finite radius of curvature. The calculation of mirror radii and spacings may be anticipated to be somewhat more complex than in the case where only two such mirrors are used. 
         [0037]    In summary, the present invention is described above with reference to a preferred and other embodiments. The invention, however, is not limited to the embodiments described and depicted. Rather, the invention is limited only by the claims appended hereto.