Abstract:
In a light (optical) beam steering/sampling system, a matrix inversion control technique is used to decouple the operation of the actuators which drive the steering mirrors. The control technique uses two virtual variables, each having an associated independent feedback loop operating in a non-cross-coupled manner, each variable being associated with one of the two steering mirrors.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
   This application claims priority to U.S. provisional application No. 60/760,521 filed Jan. 20, 2006, incorporated herein by reference in its entirety. 

   FIELD OF THE INVENTION 
   This disclosure relates in general to the field of optics and more particularly to a method and apparatus for control of an optical system. 
   BACKGROUND 
   It is well known in the optics field to control electromagnetic beams, typically light beams. It is often necessary to sample a portion of a light beam for subsequent control purposes. This typically involves some sort of detector and a feedback loop. Beams are typically detected in terms of their displacement and angle. In the prior art, for instance, to control the beams the lenses present in such a system are sometimes used in conjunction with steering mirrors. Typically, for instance, there are two steering mirrors in the system and two detectors. In one known system it is arranged so that one detector only observes changes in the beam due to the tilt of the first steering mirror. But then it has been found that it is impossible to make the second detector output signal dependent only upon the tilt of the second steering mirror. In other words, this arrangement has undesirable in terms of feedback, making the feedback complicated and almost impossible to eliminate all cross coupling. For example, when a position or angle change occurs to the input light beam which is captured by a non-zero reading from the first detector with a reading of the second detector remaining unchanged, the first steering mirror will have to be moved to eliminate the non-zero reading. This leads to an angle change in the output beam which will be detected in the second detector leading to a correction signal applied to the second mirror. Even if the system is carefully tuned so as to be stable, changes in the relative locations of the steering mirrors and detectors require a complete re-tuning, and may even result in configurations for which no stable tuning is possible. In particular, the arrangement in which one detector only observes changes due to tilt of one steering mirror is only possible at one unique distance between steering mirror and detector, based on the focal length of an interposed lens. This is generally a complicated system and it has undesirably proven almost impossible to eliminate all its error, or to be re-configured in fielded applications which require variation in the optical layout. 
   SUMMARY 
   In accordance with this invention a matrix inversion control technique is used to decouple the operation of the actuators which drive the steering mirrors in a beam steering/sampling system. Decoupling of the steering mirror actuators allows further for a calibration technique to identify physical configurations and a reconfigurable method. The calibration further allows for a fixed sampling module which samples the position of the optical beam at locations arbitrarily positioned relative to the actuators. Thus, by using the matrix inversion to decouple the control, a system with the possibility of eliminating almost all error is provided both by factory adjustment, and later if needed, by calibration on site. 
   In accordance with the present invention two virtual variables are constructed for purposes of feedback control, each variable having an associated independent feedback loop operating in a non-cross-coupled manner. Hence, each of these variables is respectively identified with one and only one of the steering mirrors so that the changes in the state (e.g., tilt) of one steering mirror do not affect the other variable. Hence, each feedback loop can operate independently. The virtual variables do not in general, correspond to beam pointing and displacement, although they can be used to calculate the pointing and displacement. 
   This system is applicable, for instance, to semiconductor manufacturing lithography equipment which typically provides light in the form of ultraviolet to expose resist on a wafer. This is merely an exemplary application. The present system and method are applicable to manipulation of any type of collimated light including, for instance, laser (coherent) light but not so limited. The present method and apparatus are generally useful with optical systems having continuous or pulsed beams, ultraviolet to infrared wavelengths, large or small diameter light beams and varying system configurations. Exemplary applications include wavelength multiplexing and de-multiplexing, power splitting and monitoring, beam measurement and monitoring, laser cutting, machining or surgery, interferometry, and multi-port light management. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows in a block diagram an example of the present optical system. 
       FIG. 2  shows in a block diagram the feedback control method for the two feedback loops. 
       FIG. 3  shows the optical axis of the  FIG. 1  system. 
   

   DETAILED DESCRIPTION 
   The present beam steering sampling system, for one of the two controlled planes, is shown in  FIG. 1 . For instance, this depicts the system in the x-z plane. A control loop (not shown) including two additional actuators, one per steering mirror, and two additional detectors, is provided for similar control in the y-z plane. For simplicity, this description limits itself to one such plane, but extension to the other plane is routine and accomplished with the same method as described herein. All optical elements in  FIG. 1  are conventional and suitably mounted on an optical bench or other support. In one embodiment the detectors Det 1 , Det 2  are on a separate support from the other optical elements. In one embodiment Det 1 , Det 2  are derived from the four quadrants of a conventional split quadrant photodetector. A position sensing photodectector or other type may also be used. The steering mirrors, each driven by a suitable precision actuator A 1 , A 2  in the plane depicted in  FIG. 1 , are R 1 , R 2 . The input light beam (shown by parallel broken lines to depict its beam width) is at the input plane. There are provided beam splitters BS 1 , BS 2 . The main light beam (split off from the portion incident on the detectors) is supplied at the output plane. Mirror R 3  is located to direct the light to detector Det 2 . Given the distances A, B, C, D, E, F, G, and H defined in  FIG. 1  (where D is the distance between beam splitter BS 1  and focusing lens L 1  along the optic axis, and D+F is likewise the distance between beam splitter BS 1  and focusing lens L 2 ), the focal lengths of lens L 1  and lens L 2  f 1  and f 2  respectively, the mirror-angle to translation coupling coefficient T≡dz/dθ, and the angles of the steering mirrors R 1  and R 2 , θ 1  and θ 2 , the position and angle respectively of the beam relative to the optic axis, x out  and θ out , as a function of the beam input and beam angles, x in  and θ in , is given by: 
   
     
       
         
           
             [ 
             
               
                 
                   
                     x 
                     out 
                   
                 
               
               
                 
                   
                     θ 
                     out 
                   
                 
               
             
             ] 
           
           = 
           
             
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       
                         A 
                         + 
                         B 
                         + 
                         C 
                         + 
                         H 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         x 
                         in 
                       
                     
                   
                   
                     
                       
                         θ 
                         in 
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               
                 2 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           A 
                           + 
                           B 
                           - 
                           T 
                           + 
                           H 
                         
                       
                       
                         
                           B 
                           - 
                           T 
                           + 
                           H 
                         
                       
                     
                     
                       
                         1 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         θ 
                         1 
                       
                     
                   
                   
                     
                       
                         θ 
                         2 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
   
   Similarly, the beam position x 1 , x 2  at each of the two detectors&#39; active elements is given by: 
   
     
       
         
           
             
               [ 
               
                 
                   
                     
                       x 
                       1 
                     
                   
                 
                 
                   
                     
                       x 
                       2 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           α 
                           1 
                         
                       
                       
                         
                           β 
                           1 
                         
                       
                     
                     
                       
                         
                           α 
                           2 
                         
                       
                       
                         
                           β 
                           2 
                         
                       
                     
                   
                   ] 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           x 
                           in 
                         
                       
                     
                     
                       
                         
                           θ 
                           in 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   [ 
                   
                     
                       
                         
                           γ 
                           1 
                         
                       
                       
                         
                           δ 
                           1 
                         
                       
                     
                     
                       
                         
                           γ 
                           2 
                         
                       
                       
                         
                           δ 
                           2 
                         
                       
                     
                   
                   ] 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           θ 
                           1 
                         
                       
                     
                     
                       
                         
                           θ 
                           2 
                         
                       
                     
                   
                   ] 
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             where: 
           
         
       
     
     
       
         
           
             
               α 
               1 
             
             ≡ 
             
               1 
               - 
               
                 E 
                 
                   f 
                   1 
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               β 
               1 
             
             ≡ 
             
               A 
               + 
               B 
               + 
               C 
               + 
               D 
               + 
               
                 E 
                 ⁡ 
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         A 
                         + 
                         B 
                         + 
                         C 
                         + 
                         D 
                       
                       
                         f 
                         1 
                       
                     
                   
                   ) 
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               γ 
               1 
             
             ≡ 
             
               2 
               ⁡ 
               
                 [ 
                 
                   A 
                   + 
                   B 
                   - 
                   T 
                   + 
                   D 
                   + 
                   
                     E 
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             A 
                             + 
                             B 
                             - 
                             T 
                             + 
                             D 
                           
                           
                             f 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               δ 
               1 
             
             ≡ 
             
               2 
               ⁡ 
               
                 [ 
                 
                   B 
                   - 
                   T 
                   + 
                   D 
                   + 
                   
                     E 
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             B 
                             - 
                             T 
                             + 
                             D 
                           
                           
                             f 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             and: 
           
         
       
     
     
       
         
           
             
               α 
               2 
             
             ≡ 
             
               1 
               - 
               
                 G 
                 
                   f 
                   2 
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               β 
               2 
             
             ≡ 
             
               A 
               + 
               B 
               + 
               C 
               + 
               D 
               + 
               F 
               + 
               
                 G 
                 ⁡ 
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         A 
                         + 
                         B 
                         + 
                         C 
                         + 
                         D 
                         + 
                         F 
                       
                       
                         f 
                         2 
                       
                     
                   
                   ) 
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               γ 
               2 
             
             ≡ 
             
               2 
               ⁡ 
               
                 [ 
                 
                   A 
                   + 
                   B 
                   - 
                   T 
                   + 
                   D 
                   + 
                   F 
                   + 
                   
                     G 
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             A 
                             + 
                             B 
                             - 
                             T 
                             + 
                             D 
                             + 
                             F 
                           
                           
                             f 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               δ 
               2 
             
             ≡ 
             
               
                 2 
                 ⁡ 
                 
                   [ 
                   
                     B 
                     - 
                     T 
                     + 
                     D 
                     + 
                     F 
                     + 
                     
                       G 
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               B 
                               - 
                               T 
                               + 
                               D 
                               + 
                               F 
                             
                             
                               f 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               . 
             
           
         
       
     
   
   If one defines two new variables, u and v, such that: 
                   [         u           v         ]     =         1         γ   1     ⁢     δ   2       -       γ   2     ⁢     δ   1           ⁡     [           δ   2           -     δ   1                 -     γ   2             γ   1           ]       ⁡     [           x   1               x   2           ]                     =     M   ⁢           ⁢     1   ⁡     [           x   1               x   2           ]           ,                     then:                       [         u           v         ]     =           1         γ   1     ⁢     δ   2       -       γ   2     ⁢     δ   1           ⁡     [               α   1     ⁢     δ   2       -       α   2     ⁢     δ   1                   β   1     ⁢     δ   2       -       β   2     ⁢     δ   1                       -     α   1       ⁢     γ   2       +       α   2     ⁢     δ   1                   -     β   1       ⁢     γ   2       +       β   2     ⁢     γ   1               ]       ⁡     [           x   in               θ   in           ]       +       [         1       0           0       1         ]     ⁡     [           θ   1               θ   2           ]                     =         [           M   ⁢           ⁢   3   ⁢   A           M   ⁢           ⁢   3   ⁢   B               M   ⁢           ⁢   3   ⁢   C           M   ⁢           ⁢   3   ⁢   D           ]     ⁡     [           x   in               θ   in           ]       +       [         1       0           0       1         ]     ⁡     [           θ   1               θ   2           ]                     
and one can control u with θ 1  with no interference from θ 2 . Similarly, one can control v with θ 2  with no interference from θ 1 . Hence u, v are two virtual variables with no cross coupling for control by two steering mirrors.
 
   The equations above correspond to the control system block diagram shown in  FIG. 2 . The control system of  FIG. 2  receives as input the beam parameters x in , θ in . This control system may be conventionally embodied in analog electronic circuitry or digitally by a conventionally programmed microprocessor or microcontroller. Programming such a device is routine in light of this disclosure. Each block or node in  FIG. 2  represents a function with the nodes being summing nodes. The control output signals θ 1  and θ 2  are conventionally transmitted by the control system to drive the steering mirror actuators, thus providing closed-loop feedback control. For control loop gains G u (s) and G v (s) much greater than one, u=v≈0 and x out =θ out ≈0 for any x in  and θ in . 
   In another embodiment, the control loop is implemented using the following method. The reflecting mirror angle changes required to correct for the error in beam position are given by: 
               [           Δθ   1               Δθ   2           ]     =         -   M     ⁢           ⁢     1   ⁡     [           x   1               x   2           ]         +     [           u   offset               v   offset           ]         ,         
where the offset (setpoint) values are given by:
 
               [           u   offset               v   offset           ]     =         1     2   ⁢   A       ⁡     [           -   1           B   +   H             1         A   +   B   +   H           ]       ⁡     [           x   offset               θ   offset           ]         ,         
where x offset  and θ offset  are the desired beam position and pointing at the output plane. The maximum values of these offsets will be limited by the detectors&#39; usable sensing range. The mirror R 1 , and R 2  angle changes (tilt) are converted into an estimated number of actuator driving pulses as given by P#=kLΔθ/k#, where kL is a global gain constant that is used to speed up or slow down the loop, # is a placeholder for the appropriate actuator, and k# is that actuator&#39;s gain constant relating the actuator position to the signal applied to the actuator driver.
 
   The main sources of error in this beam sampling system are for example: shot- and Johnson-noise of the position-sensitive detectors Det 1 , Det 2 , quantization error in the conventional analog to digital converter (not shown) used to digitize the detectors&#39; output signals, physical misalignment of the beam sampling system as temperature changes, and the smallest incremental motion of the actuators A 1 , A 2  which are driving mirrors R 1 , R 2 . All of these error sources can be easily referred back to x 1  and x 2  where they will be injected into the u and v summing nodes with gains: 
   
     
       
         
           
             
               
                 ⅆ 
                 u 
               
               
                 ⅆ 
                 
                   x 
                   1 
                 
               
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   A 
                 
               
               ⁢ 
               
                 
                   B 
                   - 
                   T 
                   + 
                   D 
                   + 
                   F 
                   + 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           M 
                           2 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           f 
                           2 
                         
                         - 
                         A 
                         - 
                         B 
                         + 
                         T 
                         - 
                         D 
                         - 
                         F 
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       M 
                       2 
                     
                     ⁢ 
                     
                       f 
                       1 
                     
                   
                   - 
                   
                     
                       M 
                       1 
                     
                     ⁢ 
                     
                       f 
                       2 
                     
                   
                   + 
                   
                     
                       M 
                       1 
                     
                     ⁢ 
                     
                       
                         M 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             f 
                             1 
                           
                           - 
                           
                             f 
                             2 
                           
                           + 
                           F 
                         
                         ) 
                       
                     
                   
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               
                 ⅆ 
                 u 
               
               
                 ⅆ 
                 
                   x 
                   2 
                 
               
             
             = 
             
               
                 - 
                 
                   1 
                   
                     2 
                     ⁢ 
                     A 
                   
                 
               
               ⁢ 
               
                 
                   B 
                   - 
                   T 
                   + 
                   D 
                   + 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           M 
                           1 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           f 
                           1 
                         
                         - 
                         A 
                         - 
                         B 
                         + 
                         T 
                         - 
                         D 
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       M 
                       2 
                     
                     ⁢ 
                     
                       f 
                       1 
                     
                   
                   - 
                   
                     
                       M 
                       1 
                     
                     ⁢ 
                     
                       f 
                       2 
                     
                   
                   + 
                   
                     
                       M 
                       1 
                     
                     ⁢ 
                     
                       
                         M 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             f 
                             1 
                           
                           - 
                           
                             f 
                             2 
                           
                           + 
                           F 
                         
                         ) 
                       
                     
                   
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               
                 ⅆ 
                 v 
               
               
                 ⅆ 
                 
                   x 
                   1 
                 
               
             
             = 
             
               
                 - 
                 
                   1 
                   
                     2 
                     ⁢ 
                     A 
                   
                 
               
               ⁢ 
               
                 
                   A 
                   + 
                   B 
                   - 
                   T 
                   + 
                   D 
                   + 
                   F 
                   + 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           M 
                           2 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           f 
                           2 
                         
                         - 
                         A 
                         - 
                         B 
                         + 
                         T 
                         - 
                         D 
                         - 
                         F 
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       M 
                       2 
                     
                     ⁢ 
                     
                       f 
                       1 
                     
                   
                   - 
                   
                     
                       M 
                       1 
                     
                     ⁢ 
                     
                       f 
                       2 
                     
                   
                   + 
                   
                     
                       M 
                       1 
                     
                     ⁢ 
                     
                       
                         M 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             f 
                             1 
                           
                           - 
                           
                             f 
                             2 
                           
                           + 
                           F 
                         
                         ) 
                       
                     
                   
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               
                 ⅆ 
                 v 
               
               
                 ⅆ 
                 
                   x 
                   2 
                 
               
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   A 
                 
               
               ⁢ 
               
                 
                   
                     A 
                     + 
                     B 
                     - 
                     T 
                     + 
                     D 
                     + 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             M 
                             1 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             f 
                             1 
                           
                           - 
                           A 
                           - 
                           B 
                           + 
                           T 
                           - 
                           D 
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         M 
                         2 
                       
                       ⁢ 
                       
                         f 
                         1 
                       
                     
                     - 
                     
                       
                         M 
                         1 
                       
                       ⁢ 
                       
                         f 
                         2 
                       
                     
                     + 
                     
                       
                         M 
                         1 
                       
                       ⁢ 
                       
                         
                           M 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               f 
                               1 
                             
                             - 
                             
                               f 
                               2 
                             
                             + 
                             F 
                           
                           ) 
                         
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
   
   Assuming equal and independent fluctuation in x 1  and x 2 , δx, the fluctuations in the output beam position and angle, δx out  and δθ out , are given by: 
   
     
       
         
           
             
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 x 
                 out 
               
             
             = 
             
               2 
               ⁢ 
               
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     A 
                                     + 
                                     B 
                                     - 
                                     T 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅆ 
                                     u 
                                   
                                   
                                     ⅆ 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     B 
                                     - 
                                     T 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅆ 
                                     v 
                                   
                                   
                                     ⅆ 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   A 
                                   + 
                                   B 
                                   - 
                                   T 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   ⅆ 
                                   u 
                                 
                                 
                                   ⅆ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   B 
                                   - 
                                   T 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   ⅆ 
                                   v 
                                 
                                 
                                   ⅆ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                   
                 
               
               ⁢ 
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               x 
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               δθ 
               out 
             
             = 
             
               2 
               ⁢ 
               
                 
                   
                     
                       [ 
                       
                         
                           
                             ⅆ 
                             u 
                           
                           
                             ⅆ 
                             
                               x 
                               1 
                             
                           
                         
                         + 
                         
                           
                             ⅆ 
                             v 
                           
                           
                             ⅆ 
                             
                               x 
                               1 
                             
                           
                         
                       
                       ] 
                     
                     2 
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             ⅆ 
                             u 
                           
                           
                             ⅆ 
                             
                               x 
                               2 
                             
                           
                         
                         + 
                         
                           
                             ⅆ 
                             v 
                           
                           
                             ⅆ 
                             x 
                           
                         
                       
                       ] 
                     
                     2 
                   
                 
               
               ⁢ 
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 x 
                 . 
               
             
           
         
       
     
   
   The actuators&#39; A 1 , A 2  minimum step size, δz pico  leads to an output error given by: 
               δ   ⁢           ⁢     x   out       =     2   ⁢     (     A   +   B   -   T     )     ⁢       δ   ⁢           ⁢     z   pico         d   MM           ,     
     ⁢       δθ   out     =     2   ⁢     2     ⁢       δ   ⁢           ⁢     z   pico         d   MM           ,         
where d MM  is the lever arm between the actuator&#39;s screw and the center of the optic. The actuators are, e.g., screw driven such as the Picomotor™, a piezoelectric actuator sold by New Focus Inc. Finally, the output is sensitive to twisting and translation of components BS 1 , BS 2 , R 3 , L 1 , L 2 , Det 1 , and Det 2 . Assuming a uniform temperature of the beam sampling system, these errors will be negligible. The beam does, however, translate a distance dB S  by passing through each of beam splitters BS 1  and BS 2  given by:
 
               d   BS     =       t   BS     ⁢   sin   ⁢           ⁢     ϕ   (     1   -       cos   ⁢           ⁢   ϕ   ⁢           ⁢   sin   ⁢           ⁢   ϕ         n   FS     ⁢         n   FS   2     -       sin   2     ⁢   ϕ               )         ,         
where t BS  is the thickness of each beamsplitter, φ is the beam angle of incidence, and n FS  is the index of refraction of the material of the beam splitters. This translation changes as the ambient temperature changes by:
 
                 ⅆ     d   BS         ⅆ   T       =         d   BS     ⁢     α   FS       +           t   BS     ⁢   cos   ⁢           ⁢   ϕ   ⁢           ⁢     sin   2     ⁢   ϕ           n   FS   2     -       sin   2     ⁢   ϕ           ⁢     (       1     n   FS   2       +     1       n   FS   2     -       sin   2     ⁢   ϕ           )     ⁢       ⅆ     n   FS         ⅆ   T             ,         
where α FS  is the thermal expansion coefficient of the material of the beam splitters, leading to an error of:
 
               δ   ⁢           ⁢     x   out       =       [     1   +     2   ⁢     (     A   +   B   -   T     )     ⁢       ⅆ   u       ⅆ     x   2           +     2   ⁢     (     B   -   T     )     ⁢       ⅆ   v       ⅆ     x   2             ]     ⁢       ⅆ     d   BS         ⅆ   T       ⁢   Δ   ⁢           ⁢   T       ,     
     ⁢       δθ   out     =     2   ⁢     (         ⅆ   u       ⅆ     x   2         +       ⅆ   v       ⅆ     x   2           )     ⁢       ⅆ     d   BS         ⅆ   T       ⁢   Δ   ⁢           ⁢     T   .               
for a peak system temperature change ΔT.
 
   The matrix transformation that relates x 1  and x 2  to u and v can be set at the time of manufacture of the system, but even small variations in assembly will introduce large cross coupling between the feedback loops. Therefore, an in situ calibration procedure may be used but is not required. Calibration begins by zeroing both x 1  and x 2  (or at least verifying that the beam is in the linear range of the position detectors Det 1 , Det 2 ), and applying a given angle change to each steering mirror R 1 , R 2  respectively, Δθ 1  and Δθ 2 . The control system will record four quantities: Δx 11  the change in x 1  due to a change in θ 1 , Δx 21  the change in x 2  due to a change in θ 1 , Δx 12  the change in x 1  due to a change in θ 2 , Δx 22  the change in x 2  due to a change in θ 2 . Now the calibration matrix can be computed by noting that: 
               γ   1     =       Δ   ⁢           ⁢     x   11         Δθ   1         ,     
     ⁢       δ   1     =       Δ   ⁢           ⁢     x   12         Δθ   2         ,     
     ⁢       γ   2     =       Δ   ⁢           ⁢     x   21         Δθ   1         ,     
     ⁢       δ   2     =       Δ   ⁢           ⁢     x   22         Δθ   2         ,         
and as above:
 
   
     
       
         
           
             [ 
             
               
                 
                   u 
                 
               
               
                 
                   v 
                 
               
             
             ] 
           
           = 
           
             
               
                 
                   1 
                   
                     
                       
                         γ 
                         1 
                       
                       ⁢ 
                       
                         δ 
                         2 
                       
                     
                     - 
                     
                       
                         γ 
                         2 
                       
                       ⁢ 
                       
                         δ 
                         1 
                       
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           δ 
                           2 
                         
                       
                       
                         
                           - 
                           
                             δ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             γ 
                             2 
                           
                         
                       
                       
                         
                           γ 
                           1 
                         
                       
                     
                   
                   ] 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         x 
                         1 
                       
                     
                   
                   
                     
                       
                         x 
                         2 
                       
                     
                   
                 
                 ] 
               
             
             . 
           
         
       
     
   
   The calibration process can incorporate filtering, i.e. changing the angles multiple times and averaging the results, and recursion, i.e. using the feedback loop to zero u and v in between calibration attempts. 
   Once the system is installed and aligned, and with the position and angle offsets set to zero, the output beam will be driven toward the optical axis, x out =θ out =0. The optical axis is defined by the physical position of the detectors Det 1 , Det 2 , as imaged by the lenses L 1 , L 2 , and is shown in  FIG. 3 .  FIG. 3  shows the optical axis is defined as the line running through the image of each of the two detectors&#39; centers, I 1  and I 2 . Given an alignment tolerance Δx align  on the relative position of the lenses and detectors, the position of the beam images will be displaced by ΔI 1 =Δx align /M and ΔI 2 =Δx align /M, where it is assumed that magnification M=M 1 =M 2 . These displacements will lead to a maximum position and angle variation at the output plane, Δx axis  and Δθ axis , given by: 
   
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 x 
                 axis 
               
             
             = 
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     x 
                     align 
                   
                 
                 M 
               
               - 
               
                 
                   [ 
                   
                     A 
                     + 
                     B 
                     + 
                     C 
                     - 
                     
                       
                         f 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             1 
                             M 
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   Δθ 
                   axis 
                 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               Δθ 
               axis 
             
             = 
             
               - 
               
                 
                   
                     2 
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       x 
                       align 
                     
                   
                   
                     
                       ( 
                       
                         1 
                         + 
                         M 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           f 
                           1 
                         
                         - 
                         
                           f 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
   
   The present system and control signal processing result in two independent feedback loops that meet high performance requirements. The above field calibration can be performed after installation and periodically thereafter. 
   This disclosure covers control in two axes (one axis on each of two steering mirrors). The process underlying the third and fourth axes of a beam pointing and translation system (the second axis on each of the two steering mirrors) is identical. The overall effect is to generate two simultaneous control loops for u 1 , v 1  and u 2 , v 2  for both of the tip-tilt axes of the steering mirrors. In this disclosure the actuators A 1 , A 2  are shown as being arranged to be in parallel, but this is not limiting. The above calibration process and/or software control of the actuators can be employed to map the actuators (two, or four including those for the second axis of the steering mirrors) to each of the four control variables u 1 , v 1  and u 2 , v 2 . 
   This disclosure is illustrative but not limiting; further modifications will be apparent to one skilled in the art in light of this disclosure and are intended to fall within the scope of the appended claims.