Patent Publication Number: US-8983814-B1

Title: Laser device with a convergent reflector topology

Description:
CROSS REFERENCE 
     This application is a continuation of U.S. patent application Ser. No. 13/206,201, filed Aug. 9, 2011, now U.S. Pat. No. 8,630,825, issued Jan. 14, 2014, which, in turn, claims the benefit of U.S. Provisional Patent Application No. 61/425,433, filed Dec. 21, 2010, which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     Light Amplification by Stimulated Emission of Radiation (laser) provides for a spatially coherent, spatially narrow, spectrally narrow (monochromatic), low divergence beam of electromagnetic radiation. Lasers can be found in a number of modern-day applications, such as cutting tools, surgery, optical storage devices, and optical communications. A laser beam is produced by reflecting light back and forth using reflectors through an optical cavity containing a gain medium having an energy population inversion (i.e. more electrons in a relatively high energy state versus at a low energy state). The reflectors used for the reflection are typically flat, spherical, or parabolic. Additionally, once the laser beam is emitted through a partially transmitting reflector, the beam can be manipulated by optical elements such as reflectors, lenses, polarizers, optical fiber, and diffraction gratings. For some applications, such optical elements may need to be very precisely manufactured and may have a resulting high cost. Lenses, in particular may need to have a very precise shape and surface finish for some applications. The material properties of a lens, such as the index of refraction, vary with the wavelength of the laser beam. The uses of lenses to converge a beam of light may lead to various optical aberrations, including spherical and chromatic aberrations. 
     SUMMARY OF THE INVENTION 
     The invention relates to a laser device, having an optical cavity containing a gain medium, a total reflectance reflector positioned within the optical cavity, and a partial reflectance reflector positioned within the optical cavity in a juxtaposed relationship to the total reflectance reflector. The device comprises comprising: wherein the total reflectance reflector has a first surface topology; wherein the partial reflectance reflector has a second surface topology; wherein at least one of the first surface topology and the second surface topology is defined by a convergent reflector topology function that converges light emitted within the optical cavity to a laser beam that exits the optical cavity through the partial reflectance reflector by a series of reflections between the total reflectance reflector as defined by the first surface topology and the partial reflectance reflector as defined by the second surface topology. The laser beam emitted from the optical cavity has a predefined pattern for a given first surface topology and a second surface topology. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings: 
         FIG. 1  is a schematic diagram of a laser device on which an embodiment of the invention can be applied. 
         FIG. 2A  is a ray tracing diagram with a flat beam pattern applied to a reflector design known in the prior art. 
         FIG. 2B  is a ray tracing diagram with a cross beam pattern applied to a reflector design known in the prior art. 
         FIG. 2C  is a ray tracing diagram with a conjugate beam pattern applied to the reflector design known in the prior art. 
         FIG. 3A  is a ray tracing diagram with a flat beam pattern applied to a reflector design according to the invention. 
         FIG. 3B  is a ray tracing diagram with a cross beam pattern applied to the reflector design of  FIG. 3A . 
         FIG. 3C  is a ray tracing diagram with a conjugate beam pattern applied to the reflector design of  FIGS. 3A and 3B . 
         FIG. 4  is a flow diagram illustrating a method of determining a convergent reflector topology. 
         FIG. 5  is an example of a forcing function that provides a convergent reflector design. 
         FIG. 6  shows a first iteration of a reflector topology based upon the method of  FIG. 4 . 
         FIG. 7  shows a subsequent iteration of a reflector topology based upon the method of  FIG. 4 . 
         FIG. 8  is a flow diagram for determining a forcing function that will generate a convergent reflector topology when applied to the method of  FIG. 4 . 
         FIG. 9  is a fractal diagram of a forcing function that does not have non-convergent areas and will not generate a convergent reflector design when applied to the method of  FIG. 4 . 
         FIG. 10  is a fractal diagram of a forcing function determined according to the method of  FIG. 8  such that when the forcing function is applied to the method of  FIG. 4 , a convergent reflector topology results. 
         FIG. 11A  is a schematic representation of a laser beam pattern that can be generated using known reflectors in the prior art. 
         FIG. 11B  is a schematic representation of another laser beam pattern that can be generated using reflectors designed using the method described in  FIG. 4 . 
         FIG. 11C  is a schematic representation of yet another laser beam pattern that can be generated using reflectors designed using the method described in  FIG. 4 . 
         FIG. 11D  is a schematic representation of yet another laser beam pattern that can be generated using reflectors designed using the method described in  FIG. 4 . 
         FIG. 12A  is a schematic representation of a laser beam pattern that can be generated by using the method described in  FIG. 4  and then be modified to generated a single emission point. 
         FIG. 12B  is a schematic representation of a laser beam pattern with a single emission point by modifying the reflector design of  FIG. 12A . 
         FIG. 13  is a flow diagram illustrating a method of modifying a convergent reflector design to achieve a modified emission pattern. 
     
    
    
     DESCRIPTION OF AN EMBODIMENT OF THE INVENTION 
     The present invention relates generally to a method of determining the topology of reflectors that provide convergence of light that reflects upon it. More specifically, the topology of a convergent reflector is determined by using an iterative algorithm for converging upon the desired topology. The embodiments described herein can be applied to any convergent reflector design. 
     Although the convergent reflector designs are described in the context of optics and more particularly laser cavities, the convergent reflector designs can apply to any type of reflector used for reflecting any type of radiation or matter. 
       FIG. 1  shows a schematic diagram of a laser device  10  for producing a spatially coherent, spatially narrow, spectrally narrow (monochromatic), low divergence laser beam  18  of electromagnetic radiation as known in the prior art. A laser device  10  comprises an optical cavity  16  containing a gain medium  24  with a total reflectance reflector  12  on one end and a partial reflectance reflector  14  on the other end. The laser beam  18  is emitted through the partial reflectance reflector  14 . A lens  20  may be used to focus the laser beam  18  into a focused beam  22 . 
     The laser device  10  can be any one of various types, including, but not limited to solid state, gas, excimer, and semiconductor lasers. These various types of lasers differ in the type of gain medium  24  contained within the optical cavity, as well as, the methods of delivering power to the gain medium. For example, the gain medium in a gas laser contains a gas such as a mix of Helium and Neon and a solid state laser may have a gain medium comprising Neodymium-doped yttrium aluminum garnet (Nd:YAl 5 O 12  or Nd:YAG). To produce laser radiation, the gain medium  24  needs to be pumped, or supplied with energy to achieve a state called a population inversion, where there are more electrons at a higher energy level than a lower energy level. As electrons lose energy from the higher energy level to the lower energy level, the lost energy is emitted as a photon. If some of the light produced in the optical cavity is fed back into the cavity, by reflecting the light off of both the total reflectance reflector  12  and the partial reflectance reflector  14 , stimulated emission and coherence of the light is achieved. Therefore the reflectors  12  and  14  are critical to the efficient functioning of the laser device  10 . 
     The total reflectance reflector  12  reflects substantially all the light that is incident upon it. The partial reflectance reflector  14  reflects most of the light that is incident upon it and allows some of the light to transmit through the reflector  14 . For example, a partial reflectance reflector  14  may allow 3% of light to pass and 97% reflectance. Reflectors  12  and  14  enclosing the optical cavity  16  in the current art are typically flat, parabolic, or spherical shaped. Such conventional reflectors may result in reflections that do not converge the light. Reflectors  12  and  14  that converge the light rather than diverge the light within the optical cavity  16  may provide the advantage of greater lasing efficiency. Additionally, it is important to note that the use of convergent reflectors may pose the advantage of not requiring a lens  20  outside of the laser device  10  to shape the laser beam  18  emitted from the laser device  10 . Not requiring an external lens  20  or other optical elements can present a cost, form factor, and performance advantage in many systems and applications. 
       FIGS. 2A ,  2 B, and  2 C are ray tracing diagrams of flat, cross, and conjugate beam patterns, respectively with conventional reflector design for both the total reflectance reflector  50  and the partial reflectance reflector  52 .  FIG. 2A  shows that with a flat beam  56 , which may reflect back and forth multiple times between the two reflectors  50  and  52 , the start and end points are at the same distance away from the optical axis  54  and is emitted through the partial reflectance reflector  52  as beam  58 . With the cross beam pattern of  FIG. 2B , the same phenomenon arises, where the cross beam  60  starts and ends at the same distance away from the optical axis  54  after one or more reflections and is emitted through the partial reflectance reflector  52  as beam  62 . Again the beams  72  emitted through the partial reflectance reflector  52  are at the same distance away from the optical axis  54  as where the beam  70  started in the case of conjugate beam reflections of  FIG. 2C . Therefore, for conventional reflector designs, there is no convergence in the beam, since the start locations and the end locations of the beams are the same distance away from the optical axis  54  with symmetry around the optical axis  54 . In other words, with conventional reflector designs, the optical beam, if started at a particular distance away from the optical axis  54 , do not converge upon a point that is a different distance away from the optical axis  54  after multiple reflections by design. 
     With conventional reflectors, such as flat, spherical, and parabolic shapes, the emitted beam from the partially transmitting reflector is ideally the same distance away from the optical axis as where the optical beam started. However, there may be non-idealities such as surface imperfections on the reflector or alignment errors of the reflectors which may induce divergence of the beam after multiple reflections, where the optical beam is emitted at a distance from the optical axis that is greater than the distance where the optical beam started. As a result, in practical implementations of known reflector topologies  50  and  52 , the electromagnetic radiation emitted after multiple reflections between the two reflectors may result in divergence of the optical beam. 
     In contrast to the non-converging reflectors of  2 A-C, a convergent reflector behavior is shown in  FIGS. 3A-C , where a light beam&#39;s end distance relative to the optical axis  130  is less than the start position relative to the optical axis  130 . Therefore, the light converges to one or more points. 
       FIGS. 3A ,  3 B, and  3 C are ray tracing diagrams of flat, cross, and conjugate beam patterns, respectively of a convergent reflector design for both the total reflectance reflector  120  and the partial reflectance reflector  122 .  FIG. 3A  shows that with a flat beam  136 , which may reflect back and forth multiple times, such as at  128  between the two reflectors  120  and  122 , as multiple reflected beams, such as beam  132 , the start point may be at the start beam  136  and the end beam may be at the optical axis  130  and emitted through the partially reflecting reflector  122  as beam  138  and  140 . As a result the beam converges to a point relative to the optical axis  130  on the output of the optical cavity that is different from the start point relative to the optical axis  130 . With the cross beam pattern of  FIG. 3B , the same phenomenon as  FIG. 3A  arises, where the cross beam starts and ends at different points relative to the optical axis  130 . In the case of the cross beam for example a beam that starts at point  154  may converge at two points  154  and at the optical axis  130  and emitted as beams  156  and  158  through the partially transmitting reflector  122  after multiple reflections. Again the beams  164  and  166  emitted from the partial reflectance reflector  122  are at a different distance away from the optical axis  130  compared to where the beam  160  started in the case of conjugate beam reflections of  FIG. 3C . Therefore, for convergent reflector designs, there is convergence in the multiple reflected beams, since the start locations and the end locations of the beams are different distances away from the optical axis  130 . In other words, with convergent reflector design, the optical beam if started at a particular distance away from the optical axis  130  may converge upon a point that is a different distance away from the optical axis  130  after multiple reflections by design. 
     It is seen by comparing the ray tracing diagrams of  FIGS. 2A ,  2 B, and  2 C to  FIGS. 3A ,  3 B, and  3 C, the beams can be converged within the optical cavity by a series of reflections and can be emitted from the optical cavity in a predefined pattern without the use of optical elements outside of the optical cavity. A method is disclosed herein that can be used to derive the topology of the reflectors. In particular the method can be used to derive reflector topologies that result in convergence of a laser beam. Convergence within the optical cavity, instead of requiring optical elements outside of the optical cavity can pose several advantages including reduced cost, reduced form factor and greater efficiency and reliability. 
       FIG. 4  is a flow diagram illustrating the method of determining the surface topology of a reflector surface  200 . First a forcing function, f(x), is determined at  202 . The choice of forcing function is very important in the determination of the surface topology. To have a convergent design, the forcing function, f(x), must be one that has areas that do not converge in the x+jy space. This will be described in greater detail in conjunction with  FIGS. 8 ,  9 , and  10 . Once a forcing function is determined at  202 , the forcing function is applied to a convergent algorithm to generate y values of the reflector topology at  204 . The convergent algorithm can be any known algorithm for iterative determination of roots of a function. For example one such convergent algorithm can be the Newton-Raphson algorithm described as: 
     
       
         
           
             
               x 
               
                 n 
                 + 
                 1 
               
             
             = 
             
               
                 x 
                 n 
               
               - 
               
                 
                   f 
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
                 
                   
                     f 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
               
             
           
         
       
     
     where, x n+1  is the approximation of the value of a root of a function derived upon iterating the convergence algorithm one time beyond the approximation x n , f′(x) is the first derivative of the forcing function f(x), and n is a real integer. Although the Newton-Raphson algorithm is used throughout this specification, any known convergence algorithm may be used. 
     The set of x values, x p , generated by subjecting the forcing function f(x) to the convergent algorithm is the set of y-values in the x-y space in which the reflector topology is defined. In other words, the numerical values of {x 1 , x 2 , . . . , x p } in the x+jy space of the forcing function are numerically equivalent to y 1 , y 2 , . . . , y p  in the x-y space in which the surface topology of two reflector surfaces are defined as a set of points with p members. The reflector surface is defined as set of points in the x, y space, with p members. For purposes of discussion, the complete set of x values, {x 1 , x 2 , . . . , x p } in the x-y reflector topology space is defined as x P . Similarly, the complete set of y values, {y 1 , y 2 , . . . , y P } in the x-y reflector topology space is defined as y P . Therefore the reflector surface can be defined as a set of paired points {x P , y P } with p members. 
     At  204 , the x-values, x P , of the surface topology are known as a result of the size of the optical cavity. In other words, if the optical cavity has a length of λ, then the x-values may be either −λ/2 or λ/2 on the initial iteration. At  206 , the slope at each y P  is determined to generate a set of y values and slopes {y P , m P }, where m P  is a set of slope values, {m 1 , m 2 , . . . , m p } corresponding to each of the x-values, x P  and y-values, y P . The slope is calculated at any point, i, in the set of y P , by considering the values of x i , y i , x i−1 , y i−1 , x i+1 , and y i+1 . In other words, the slope at any point i, m i , is a function of the coordinates of that point (x i , y i ), as well as the coordinates of the previous (x i−1 , y i−1 ) and subsequent (x i+1 , y i+1 ) points in the set {x P , y P }. m i  is a real scalar that must satisfy a condition where a beam impingent on point (x i , y i ) with a trajectory defined by the direction from (x i−1 , y i−1 ) to (x i , y i ) must be reflected to the point (x i+1 , y i+1 ). The slope at any point (x i , y i ) can be determined as: 
     
       
         
           
             
               m 
               i 
             
             = 
             
               tan 
               ⁡ 
               
                 [ 
                 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
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     It should be noted that the derivation of the slope is just one known derivation. Any known algorithm or method may be used to determine the slope, m i , based on the reflections desired at that point (x i , y i ). 
     Next, at  208 , R′(x) or the first derivative of the surface topology is determined by fitting to the set of y-values and slopes, {y P , m P }. The fitting may be by any known method such as linear regression and least squares fit. Next R′(x) is integrated to generate the surface topology R(x) at  210  as:
 
 R ( x )=∫ R ′( x )* dx+K  
 
     where K is a constant of integration. If the system is centered at x=0, meaning the two reflectors are equidistant from the y-axis, then K=0. 
     Using the new R(x) function, new values of x are generated, x P , corresponding to the y-values, y P , at  212 . Next it is determined if the current iteration of the set x P  is different from the previous set of x P  by a predetermined value at  214 . If it is, then R(x) and therefore {x P , y P } of the reflector topology have not converged and therefore the method loops back to  206 , where the new x-values, x P , determined at  212  are used to determine new values of slope, m P , at each y-value at  206 . Once the y-values are determined based on the forcing function f(x) at  204 , the set of y-values do not change. Therefore, at each iteration, only the x-values change based on a revised set of slopes, m P , with the fixed set of y-values, y P . After a sufficient number of iterations with the feedback loop to  206 , the set of x-values, x P , eventually converge and the change in x P  from a previous x P  drops to less than a predetermined threshold at  214 . At this point, R(x), and the resulting {x P , R(x)} or {x P , y P } defines the reflector surface topology. 
     The sequence of steps depicted is for illustrative purposes only, and is not meant to limit the method  200  in any way as it is understood that the steps may proceed in a different logical or sequential order and different, additional, overlapping, or intervening steps may be included without detracting from the invention. 
     In this algorithm  200 , the more data points that are used, p, the more precise the reflector surface topology determination will be. Additionally, more iterations at  206  can result in greater precision and accuracy of the algorithm. The number of iterations can be increased by setting a lower predetermined threshold for the change in x P . 
     Due to the volume of calculations required in this method  200 , it is envisioned that the method  200  is executed on a computer with electronic memory running a mathematical software or with a software dedicated to the execution of this algorithm. Examples of commercial software on which the method  200  may be implemented include Microsoft Excel®, Maplesoft Maple®, or Mathworks Matlab®. 
     The use of the method of determining a surface topology of a reflector surface  200  will be better understood by example in conjunction with  FIGS. 5 ,  6 , and  7 . The top graph of  FIG. 5  shows a forcing function, represented as f(x+jy) in a complex space (x+jy, f(x+jy)). To determine a convergent reflector design, f(x+jy) must have areas in the x+jy space where f(x+jy) applied to a convergent algorithm does not converge to a root of f(x+jy). In this example, P1, representing (x 1+ jy 1 , f 1 (x 1 +jy 1 )) is a point within a non-convergent area and as a result in the non-convergent area in the x+jy space is applied to the convergence algorithm. This generates a series of points, P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , and P g  in the x+jy complex space. Each of the x values of each of the points are set as the y values in the real x-R(x) space of the reflector topology function. For simplicity, the method is depicted with only 8 points, or 4 points on each reflector. In the first iteration the left reflector topology is depicted as  230  and the right reflector topology is represented as  232 . As depicted in  FIG. 6 , in the first iteration, the reflector topology is flat, because the x values are one of two values for each reflector  230  and  232  on either side. Next the slopes at each of the (x, y) points is determined at  206  so that a reflection to the next (x, y) point is achieved along the reflection paths as depicted in  FIG. 6 . For example, at a certain slope at (x 2 , y 2 ), the beam incident upon that point from point (x 1 , y 1 ) will reflect from the surface at point (x 2 , y 2 ) and reach point (x 3 , y 3 ). In effect, slope at that point, m2 is determined such that the beam reflects from (x 1 , y 1 ) to (x 2 , y 2 ) and then to (x 3 , y 3 ). 
     Once the slope at each of the points, p, is determined, the complete set of x-values and slopes, {x P , m P } are fit to one of two different fitting functions R 1 ′(x) and R 2 ′(x) for the reflectors  230  and  232 , respectively. Each of the fitting functions R 1 ′(x) and R 2 ′(x) are then integrated to derive R 1 (x) and R 2 (x), respectively. New values of x, x P  are generated from R 1 (x) and R 2 (x) at each value of y P .  FIG. 7  shows the graphical representation of {x P , y P } with the new set of x P  values, where the x P  values are different from the x P  values in  FIG. 6 , and the y P  values are the same. 
     It is then determined if the x P  have converged by comparing the new x P  values to the old x P  values. There are many ways to determine if the functions have converged, including determining an error value of the average square root of the square of the difference by: 
     
       
         
           
             
               δ 
               = 
               
                 
                   1 
                   p 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     j 
                     p 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
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                             j 
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                           - 
                           
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                             prev 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
             , 
           
         
       
     
     where  6  is error value, p is the total number of points, j is an index, x j   new  is the new x value at the jth index, and x j   prev  is the previous x value at the jth index from the previous iteration. 
     This error value may be compared to a predetermined threshold to determine if the set of x P  have converged. As an alternative, the error value, δ, may be normalized by the mean of the set of x P , to determine a percentage error for comparison to a predetermined threshold of percentage error. For example the percentage error threshold may be 0.1%, meaning if the difference between the new x P  and previous x P  is less than 0.1%, then the functions R 1 (x) and R 2 (x) are deemed to be converged. 
     If it is determined that x P  has converged then the topology of  FIG. 7  represents the final topology of the reflectors  240  and  242 . If however, it is determined at  214  that the values of x P  have not converged, then the method repeats the previous steps starting from determining the slopes at each y-value, y P , using the new x P  values at  206 . 
     The reflector topologies determined by the method  200  can be used to fabricate reflectors  12  and  14  of the laser device  10  by any known method including, but not limited to machining or three-dimensional (3D) printing. The reflectors with the determined topologies therefore are an improvement over currently used reflectors as the new reflectors can result in greater lasing efficiency and can produce laser emissions with beam patterns that can be formed with a reduced number of, and in some cases no additional, optical elements outside of the laser device  10 . 
     From the discussion above, it is clear that the choice of the forcing function is important in determining a topology of the reflector that leads to convergence of light reflecting therefrom.  FIG. 8  is a flow diagram showing a method of determining an appropriate forcing function  230  that will produce a convergent reflector design when applied to the method for determining a reflector topology  200  of  FIG. 4 . In essence the required forcing function is one that produces areas in the x+jy space where the function does not converge when applied to a convergent algorithm. The method starts by picking a forcing function at  232 . The forcing function must be a third or higher order polynomial function and have at least one inflection point. An inflection point for this purpose is a point where the second derivative of the function is zero and has different signs on either side of the inflection point. The forcing function is more ideally a third order or higher polynomial. Such third and higher order polynomials are likely to produce forcing functions that are non-convergent over an area in the x+jy space. The next step is to repeatedly apply a convergent algorithm to f(x) at  234 . The convergent algorithm can be any known convergent algorithm, including the Newton-Raphson convergent algorithm as discussed above. The convergent algorithm is applied repeatedly by using a computer such as a micro-computer and the data points in the x+jy space are stored in an electronic memory of the computer. The convergent algorithm is applied to various portions of the x+jy space to test the convergence in all location. That means that for a function there are as many convergent areas as there are roots for a function. For example a fourth order polynomial function may have four roots, so in that case there may be four sets of areas that converge to four different roots. The areas associated with each of the roots can be represented as such on a plot of the x+jy space. The (x+jy, f(x+jy)) data is plotted as a Newton-Raphson fractal at  236 . It is then determined if the Newton-Raphson fractal has any non-convergent areas at  238 . If there are no non-convergent areas in the Newton-Raphson fractal, then the function is perturbed and the method loops back to  234  to apply the new function repeatedly to the convergent algorithm. If it is found that the function f(x) does have at least one non-convergent area at  238  then the forcing function, f(x) can be used to generate a convergent reflector topology and therefore an appropriate forcing function is identified at  242 . 
     An example of a non-convergent forcing function is:
 
ƒ( x )= x   4 +112
 
     Applying the above function to the method  200  of  FIG. 4  will generate a non-convergent reflector topology. 
     An example of a convergent surface topology function is:
 
ƒ( x )=− x   4 +4.29 x   2 +5.62
 
     Applying the above function to the method  200  of  FIG. 4  will generate a convergent reflector topology. Although in this example, f(x) is a fourth order polynomial, f(x) can be a polynomial function of any order, as well as, a sinusoidal, exponential, logarithmic, or any variety of known functions. 
     The perturbation of f(x) at  240  may in most cases be by changing the f(x)-offset of f(x). For example, consider a general function of the form: 
     
       
         
           
             
               
                 
                   f 
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
                 = 
                 
                   
                     ∑ 
                     q 
                     Q 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       K 
                       q 
                     
                     * 
                     
                       x 
                       q 
                     
                   
                 
               
                
             
             
               Q 
               ≥ 
               3 
             
           
         
       
     
     where q and Q are integers and K q  are real numbers. 
     If the Newton-Raphson fractal generated from this function f(x) does not indicate non-convergent areas at  238 , then the function may be perturbed by incrementing the value of K 0  by a predetermined amount. This has the effect of increasing the f(x)-offset of the function and for many polynomial functions, such an offset may lead to a new function with a non-convergent area in the x+jy space as indicated by areas of non-convergence in the associated Newton-Raphson fractal. Such functions can then be used in the method  200  of  FIG. 4  to determine a convergent reflector topology. 
     As a more specific example, consider a forcing function:
 
 f ( x )= ax   b   +cx   d   +e,  
 
     where b and d are integers and a, c, and e are real numbers. 
     If the Newton-Raphson fractal generated from this function f(x) does not indicate non-convergent areas at  238 , then the function may be perturbed by incrementing the value of e by a predetermined amount. This has the effect of increasing the f(x)-offset of the function and for many polynomial functions, such an offset may lead to a new function with a non-convergent area in the x+jy space as indicated by areas of non-convergence in the associated Newton-Raphson fractal. Such functions can then be used in the method  200  of  FIG. 4  to determine a convergent reflector topology. 
       FIG. 9  shows an example of a Newton-Raphson fractal without areas of non-convergence. As a result the function of this fractal is not an appropriate forcing function to generate a convergent reflector topology when applied to the method of  FIG. 4 . Each of the areas that converge to a particular root can be represented by a color (i.e., in the drawings, differences in “color” can be generally shown by differing grayscale shades). Each root has a different color associated with it. Therefore areas represented by  260 ,  262 ,  264 ,  266 ,  268 , and  270  converge to a first root, while areas  272 ,  274 ,  276 ,  278 , and  280  to a second root,  282 ,  284 ,  286 ,  288 ,  290 , and  292  converge to a third root, and finally, areas  294 ,  296 ,  298 , and  300  converge to a fourth root. The interfaces between two different areas corresponding to two different roots is infinitely small and cannot be construed an area in the x+jy space. In other words, there are no areas in the x+jy space that do not converge to one of the four roots of the function f(x). As a result, this function is not a forcing function that will produce a convergent reflector topology. 
     Unlike  FIG. 9 ,  FIG. 10  shows an example of a Newton-Raphson fractal of a function with areas of non-convergence. The areas defined by  360 ,  362 ,  364 ,  366 ,  368  are areas where all points converge to a first root when any point in those areas are applied to a convergence algorithm, such as a Newton-Raphson convergence algorithm. In other words, any point applied to a convergent algorithm in these areas will produce another point that converges to the same root and therefore that point is in an area designated by the same color as the areas  360 ,  362 ,  364 ,  366 , and  368 . The areas  370 ,  372 ,  374 ,  376  similarly contain points that converge to a second root when applied to a convergent algorithm. Points in the areas  380 ,  382 ,  384 ,  386 ,  388 ,  390 , and  392  do not converge to a root. Therefore there are areas that do not converge to a root for this function. As a result, this function can be used as a forcing function in the method for determining a convergent reflector topology of  FIG. 4 . 
       FIGS. 11A-D  show examples of various laser emissions.  FIG. 11A  shows a radiation emission using reflector topologies known in the prior art. The radiation emission  432  from the partial reflectance reflector  422  is distributed over the full surface of the reflector generating pattern  434 . A convergence of the radiation to specific regions is not observed in this prior art reflector design. On the other hand  FIGS. 11B-D  show examples of convergent laser emissions. Spatial convergence of the emitted radiation does not have to be at the optical axis, but can be anywhere along the topography of the reflector. Additionally, based on the forcing function that has been chosen, there may be multiple convergence points.  FIG. 11B  shows a radiation of emission through the partial reflectance reflector  422  with two convergent points  442  and  444 . As a result the emission pattern has an outside ring convergence  446  with an internal point convergence  448 .  FIG. 11C  shows another convergent emission pattern with an outside ring  466  and inside area of convergence  468  resulting from emissions  462  and  464  through the partial reflectance reflector  422 .  FIG. 11D  illustrates yet another convergent emission pattern where there is an annular region  486  of convergence with a round area  488  of convergence inside of the annular region  486  resulting from emissions defined by  482 ,  483 , and  484 . 
     All of the radiation patterns in  FIGS. 11B-D  have at least two points of convergence. In some cases, it may be advantageous to have a single point of convergence, such as a single center point. In such a case, one may design reflectors  420  and  422  to produce a radiation pattern as depicted in any of  FIGS. 11B-D  and then modify portions of the reflectors  420  and  422 , such that the rays  442 ,  462 ,  482 , and  483  that are not emitted from the origin point are reflected back in a manner so that they emit as a ray from the origin point. 
     Referring now to  FIGS. 12A and 12B , the concept of single point emission is described in greater detail. Using the method of determining a surface topology of a reflector surface  200 , as described above, total reflectance reflector  520  and partial reflectance reflector  522  can be designed to produce a beam pattern with a center emission  544 A and edge emissions  542  through the partial reflectance reflector  522 . Such an emission pattern is generated via multiple reflections within the cavity depicted as  540 A at multiple points P1-P9 on the total reflectance reflector  520  and partial reflectance reflector  522 . 
     The partial reflectance reflector  522  as defined by surface topology determined from method  200  can further be partitioned into a first section  524  and a second section  526 A. In the first section  522  does not have the undesired edge emissions  542  emitted therethrough and the second section  526 A does have the undesired edge emissions  542  emitted therethrough. As a next step, the slope in the second section  526 A is modified in a manner such that any beam incident upon it will be reflected back from a 90° incident surface so that the beam is reflected back to its origination point. By doing so, the second section  526 A is modified to second section  526 B ( FIG. 12B ) where there is no emission through the second section  526 B. Furthermore the reflected rays are modified to  540 B, different from  540 A, and all of the radiation reflected from the second section is ultimately emitted as part of center emission  544 B, which differs, at least in intensity from center emission  544 A. 
     The method  600  of generating a single convergent emission point is shown in  FIG. 13 . Initially, a convergent surface topology of the reflector surface must be determined using method  200 . Next the reflector surface must be divided into two or more sections at  602 . The sections must be non-overlapping and one section must contain the point from where emission is desired and another section must contain the point or region from where emission is not desired. Next, the slopes at the surface of at least one of the portioned sections are modified at  604 . The slopes at the surface of the modified section are set such that any light incident upon that surface will be reflected back to the source of the incident light. The modified topology with the modified section is then tested or simulated at  606  to determine if it produces the desired radiation pattern with a single convergence point. If it does not, the slopes in the modified section are further modified, so that incident light on any surface in that section is sufficiently reflected back to the origin of the light. If however it is determined that that the desired radiation pattern is achieved at  606 , then the final topology of the reflectors are known at  608 . 
     While the invention has been specifically described in connection with certain specific embodiments thereof, it is to be understood that this is by way of illustration and not of limitation. Reasonable variation and modification is possible within the scope of the forgoing disclosure and drawings without departing from the spirit of the invention which is defined in the appended claims.