Abstract:
Improved brightness and feedback multi-emitter laser diode modules and methods are provided. A plurality of laser diode emitters emit broad-area light beams in a beam direction. In cross-section, each beam is broad in its slow axis and narrow in its fast axis. Groups of downstream optical components collimate, shape, stack and direct the beams along a light path towards a beam spot (which may be fiber-coupled). After collimating, stacking and directing, the beams are Fourier transformed in the fast-axis through a lens feature having a fast-axis focal length less than about 3 millimeters. In some embodiments, the fast-axis focal length is between about 0.1 and 2.0 millimeters. Astigmatism may be introduced between the fast axis and the slow axis in the beams upstream of the lens feature and in accordance with the fast axis focal length of the lens feature. The emitters may receive feedback including wavelength locking feedback.

Description:
TECHNICAL FIELD 
       [0001]    The present invention relates to multi-emitter laser diode modules and more particularly, to improving the brightness of the beam generated by such modules at launch. 
       BACKGROUND OF THE INVENTION 
       [0002]    Maximizing brightness of laser diode modules is important for many applications, including pumping of fiber lasers and processing of materials directly with diode radiation. Even with many advances over recent years, fiber-coupled laser diode modules still do not reach their theoretically achievable brightness. This disclosure brings the brightness significantly closer to optimum. 
         [0003]    A typical prior-art high-power multi-emitter multimode-fiber-coupled laser diode module  10  is illustrated in  FIG. 1A  and described, for example, in U.S. Pat. No. 7,764,723 issued Jul. 27, 2010 to Ovtchinnikov et al. (“the &#39;723 patent”). At the most basic level, an array of diode emitters  12  output light beams  14  along a light path. In  FIG. 1A , each individual diode emitter  12  in the array is stacked on top of the other. Various optics  16 ,  18 ,  20  and  22  collimate and shape the beam  14  of each emitter such that each light beam  14  is concentrated and directed into a single multi-emitter beam  19 . The beam  19  is directed on the light path towards a fiber  30 . The beam  19  generates a beam spot  36  at the location where the fiber facet  31  of fiber  30  may be located. As some multi-emitter laser diode modules  10  may not be fiber-coupled, the beam spot  36  is the launch point of the module  10 . If the modules  10  are not fiber-coupled, then lens  22  is generally omitted. In a fiber-coupled module, as illustrated in  FIG. 1A , the beam spot  36  is located at the fiber facet  31 . As the fiber tip of fiber  30  is a flat surface in  FIG. 1A , the fiber tip of fiber  30  is also the fiber facet  31 . 
         [0004]    Referring to  FIG. 1A  in greater detail, each broad-area laser diode emitter  12  emits a non-circular beam  14  in the beam direction. Each beam  14  is broad (about 50 to about 200 microns wide) in its slow-axis and narrow (about 1 to about 2 microns) in its fast-axis. The fast and slow axes are transverse to the direction of propagation of the beam. In  FIG. 1A , the fast axis is in the x direction, the slow axis is in the y direction and the beam path is in the z direction. Each beam  14  is collimated and shaped by fast-axis collimator  16  and slow-axis collimator  18  to form a wide, vertically thin collimated beam  15 . Multiple beams  15  may be stacked in the fast-axis direction (vertically in the x direction in  FIG. 1A ) from the plurality of diode emitters  12  by a set of mirrors  20  that are slightly offset in the fast-axis direction. 
         [0005]    As a result, the multi-emitter laser diode module  10  launches into objective lens  22  a fast-axis-stacked set of thin, wide beams  19  that together fill a region of the objective lens  22 . The beams  19  are then focused by objective lens  22  on a beam spot  36  and may be coupled into an output fiber  30  through a fiber entrance facet  31 . 
         [0006]    A key benefit of the prior-art diode module  10  illustrated in  FIG. 1A  is that it can achieve relatively high brightness compared to other multimode-fiber-coupled diodes. Brightness is defined as the output power per output cross-sectional area per output solid angle and expressed as watts per square cm per steradian. The same information can also be conveyed by specifying the power in watts and the beam-parameter products (BPPs) in the two orthogonal dimensions transverse to the beam direction. The BPP is defined as the product of beam radius and half-divergence angle in a given transverse dimension and is expressed in mm-mrad. For a fiber  30  with a standard circular core (such as in  FIG. 1B ), the beam radii and the divergences in the two transverse dimensions equilibrate quickly, thus making the two BPPs equal. To maximize brightness for a given level of power, one needs to minimize the BPPs. 
         [0007]    Most multimode-fiber-coupled diodes have relatively poor brightness, i.e. high BPP, because broad-area diode emitters have very asymmetric BPPs in the two transverse dimensions. In the fast axis, these emitters are diffraction-limited, i.e. single-moded, which for a wavelength in the 0.9-1.0 micron range means a BPP of about 0.3; whereas in the slow axis, these emitters are highly multimode, with a width of typically 100 microns and a half-divergence of typically 0.1 radians, giving a BPP of about 5 in this dimension. In existing multimode fiber-coupled diode modules, the BPP is degraded because the modules must accommodate long tails in the near-field profile. Even though anamorphic optics such as prisms and cylinder lenses are able to alter spot sizes and divergences in one axis and not the other, the unfavorable BPPs do not change in either axis. 
         [0008]    As a result, regardless of the use of typical optics, the beam  19  launched into the fiber  30  still generally has very asymmetric BPPs. This means that the beam  19  is either asymmetric in the near field (the spatial distribution of the power at the beam spot  36  or fiber entrance facet  31  in the two dimensions), or asymmetric in the far field (the angular distribution in the two dimensions), or both. In the near field, the launched light will quickly spread out in both dimensions to fill the transverse size of the fiber core  32 , losing brightness corresponding to however much the facet  31  was initially under filled by the beam. In the far field, the angular distributions of the light in the two dimensions will rapidly mix and yield a net divergence that is a mean between the divergences in the two initial dimensions but is weighted toward the higher divergence. In a circular-core fiber  30  as illustrated in  FIG. 1B , if a beam with initially asymmetric BPPs is launched, regardless of whether the asymmetry is manifested in the near field or far field or both, the resultant equilibrated BPP will necessarily be worse (higher) than the average of the initial BPPs, and there will be a net loss of brightness. 
         [0009]    In order to achieve high brightness in a circular-core fiber  30 , it is important to have roughly equal BPP&#39;s in the two dimensions prior to launching into the fiber  30 , with a relatively symmetric beam in the two dimensions in both the near field and the far field; and the near-field spot  36  should be sized to fill the fiber core  32  as well as possible, as shown in  FIG. 1B . Diode module designs such as that illustrated in prior-art  FIG. 1A , or similar to the &#39;723 patent, achieve this through stacking a number of collimated beams  15  from the multiple emitters  12  in the vertical dimension (corresponding to the fast axis of the diodes  12 , and the x-axis in  FIG. 1A ) with the appropriate height per beam  15  so that the spot on the objective lens  22  is roughly square. As a result, after the beams  19  are focused by the lens  22  onto the beam spot  36  or into output fiber  30 , the light would have roughly equal divergence in the horizontal and vertical directions. Also, the magnifications in the two dimensions are chosen so that the beam spot  36  on the fiber facet  31  is roughly square and fills the fiber facet  31  as well as possible. Therefore, the BPPs in the two dimensions are roughly equal, and the amount of brightness that is lost after the light equilibrates in the fiber  30  is minimized. The result is that this type of design has higher brightness than most other fiber-coupled diode designs. 
         [0010]    However, in order to achieve the conditions necessary for even higher brightness, the design of these prior art diode modules is highly constrained because the number of emitters and their pitch are linked to the width of the emitters and the slow-axis divergence. 
         [0011]    Furthermore, designs similar to the &#39;723 patent still fall well short of the maximum theoretically achievable brightness. The reasons for decreased brightness include that the near-field spot  36  projected onto the fiber facet  31  is still rather poorly matched in both shape and power distribution to the fiber core  32 , resulting in an under-filled aperture and subsequent brightness loss. This problem can be illustrated by directing one&#39;s attention to what is happening between the objective lens  22  and the fiber facet  31 , inclusive, in region B of the prior-art design of  FIG. 1A . A typical, non-limiting numerical example will also be used concurrently to further illustrate the problem and the solutions proposed in this disclosure. 
         [0012]    Referring now to  FIG. 1B , a cross-section of the fiber facet  31  of output fiber  30  of  FIG. 1A  along the line A is illustrated. A typical circular fiber  30  has a fiber facet  31  comprising a flat surfaced face at the beginning of the core  32  at the end of the optical cable. A fiber tip (not illustrated in  FIG. 1A  or  1 B) may extend upstream of the fiber facet  31 . A cladding  34  surrounds all of the fiber  30  except for the tip (if any) and fiber facets at both ends of the fiber  30 . A typical circular fiber core  32  is 105 microns in diameter with a 125 micron cladding  34  and a numerical aperture (NA) of 0.22. 
         [0013]    To improve brightness, the output fiber  30  should be illuminated with as large a beam spot  36  from the multi-emitter laser diode module  10  as possible in the near field, and thereby as low as possible divergence of light, typically 0.15 radians (half-divergence) or less. This substantially under-fills the NA. Unlike in the near field, light in the far field in a fiber does not rapidly spread to fill the available aperture (0.22 NA in this case), so it is useful to launch with as small a divergence as possible (thus higher brightness) since this divergence will essentially be preserved and higher brightness will be present at the fiber output for the user&#39;s application. 
         [0014]      FIG. 2A  illustrates the beams  19  in the fast axis direction and shows near field intensity distribution  202  at the objective lens  22 , near field intensity distribution  204  at the fiber facet  31  and far field intensity distribution  206  at the fiber facet  31 . As with  FIG. 1A , there is no tip on fiber  30 , or alternatively, the fiber facet  31  is also the fiber tip. 
         [0015]    Starting at the beam spot  36  which is co-located with the fiber facet  31  in  FIG. 2A , the near-field intensity distribution  204  is a Gaussian-like distribution. The distribution is a magnified image of the near field of each laser diode emitter  12 . The images of each of the emitters  12  is practically identical, so the final distribution (comprising all beams  15  superimposed over top of each other to form beam  19 ) is essentially the same as the image from any one emitter  12 , barring any irregularities. A circular fiber core  32  having diameter of 105 microns can accommodate an inscribed square beam spot  36  of about 75 microns by about 75 microns. But in the fast-axis direction, because of the Gaussian-like distribution  204 , the actual full-width-at-half maximum (FWHM) size  208  of the beam  19  must be much less than 75 microns in order to accommodate the long tails or gradually sloping edges of the distribution. If these tails were not accommodated, a considerable amount of power would be clipped at the fiber facet  31 , resulting in poor power throughput as well as excessive heating and damage to the fiber. Typically, optics are chosen that provide an approximately 30 times magnification, so since the fast-axis beam height is typically about 1 micron at the emitter  12 , the distribution at the fiber facet  31  or beam spot  36  is about 30 microns FWHM, and it can be calculated that the available core aperture of at least 75 microns in the fast axis direction will then capture about 99% of the power in the beam  19 . 
         [0016]    Still looking at the beam spot  36  and fiber facet  31  in the fast direction, the far-field intensity distribution  206  is a scaled copy of the near-field distribution  202  entering the objective lens  22  because the lens  22  acts as a Fourier transformer in both axes. Since the near field entering the objective lens  22  is where the multiple collimated beams  15  from the individual diode emitters  12  have been stacked side-by-side vertically to form beam  19 , the summed intensity distribution  202  is a “top-hat” distribution with steep edges and a relatively flat top, as shown in  FIG. 2A  before the objective lens  22 . With a fast-axis height  210  of about 3 mm (from vertically stacking the beams  15  from each emitter  12 ) and an objective lens focal length  212  of about 10 mm in both axes, the resulting far-field half-divergence  214  is about 0.15 radians and is uniformly filled. 
         [0017]    Turning now to  FIG. 2B , beam  19  in the slow axis direction is illustrated including near field intensity distribution  220  at the objective lens  22 , near field intensity distribution  222  at the fiber facet  31  and far field intensity distribution  224  at the fiber facet  31 . The 10 mm fast-axis and slow axis focal lengths  212  of the objective lens  22  are the same as in  FIG. 2A . 
         [0018]    Again starting at the fiber facet  31 , in the slow-axis direction, the near-field intensity distribution  222  is an image of the wide axis of the emitters, which by the laterally multimode nature of the emitters tends naturally to be close to a steep edged top-hat distribution. Again, the wide-axis image of each emitter  12  is practically identical barring any irregularities, so the final distribution (comprising all beams  15  superimposed over top of each other to form beam  19 ) is essentially the same as the image from any one emitter  12 . The magnification can be chosen such that the near-field width  226  at the fiber facet  31  is about 75 microns so that it fits into the 105-micron core. The far-field distribution  224  at the fiber facet  31  also corresponds to that of the emitters  12  and is close to Gaussian with gradually sloped edged and long tails. The far-field half-divergence at half maximum  228  is about 0.075 radians for typical diode emitters. 
         [0019]    In the slow axis, it can be seen that the steep edged distribution  222  of the beam fills the available 75 micron aperture relatively uniformly and therefore efficiently, whereas in the fast axis, much of the 75 micron aperture is filled with the low-intensity tails and gradually sloped edges of the Gaussian distribution  224  of the beam and therefore the filling is about 50% efficient. Furthermore, even if the 75×75 micron inscribed square area was uniformly filled, this would still only fill 65% of the total aperture of the 105 micron circular fiber core  32  as illustrated in  FIG. 1B . As a result, the total fill in this numerical example is about 33% efficient. Immediately after launch into the fiber  30 , the light spreads out to fill the full aperture relatively uniformly, resulting in a drop in brightness of a factor of about 3. 
         [0020]    A further drawback of the prior art occurs in applications where it is desired to provide an external feedback signal to the diode emitters. Such feedback can be used, for example, to ensure that the laser light generated by module  10  is at a tightly controlled wavelength. Conventional Fabry-Perot cavity diode emitters generate light with a center wavelength that is typically controlled only within several nanometers at best. Applications such as optical pumping and optical wavelength multiplexing, for example, often require sub-nanometer wavelength control. As is well-known in the art, it is possible to wavelength-lock a diode emitter by providing feedback preferentially at a desired wavelength, whereby most or all of the light generated by the laser diode is at the desired wavelength. This locking is typically achieved at some cost in output power, corresponding roughly to the power used in the feedback signal. Various techniques exist for providing this feedback, including using structures such as Bragg gratings etched directly in the emitter chip and using wavelength-selective partially reflective optics such as volume holographic gratings adjacent to one of the emitter facets. Although desirable for reasons including cost, reliability, stability, and wavelength precision, it has to date not been practical to lock the wavelength of the emitters  12  using feedback from a wavelength-selective optic situated downstream of a fiber  30  on a multi-emitter module  10 . There are two reasons for this. First, as is well-known in the art, poor brightness performance of an optical assembly causes poor power efficiency in the reverse direction. Using the numbers in the above example, a drop in forward brightness by a factor of 3 implies, for backward-traveling power, a drop in power by 3×. Thus, if, for example, a wavelength-selective reflectivity of about 10% is required for reliable locking of a diode emitter, then a reflectivity of about 30% would be required in a wavelength-selective optic downstream of the fiber, resulting in an unattractively high cost of output power from the system. Second, when light is reflected back in the fiber of prior-art fiber-coupled modules and from there transmitted into individual emitters, it has been observed that the back-reflected power is non-uniformly distributed among the emitters, with the outlying emitters receiving the least amount of feedback. Since all of the emitters typically must be reliably locked, the required reflectivity of the downstream optic will be set by the weakest-locked emitter, and the surplus locking power provided to the other emitters will be wasted. The result is a further increase in the required reflectivity of the downstream optic. 
         [0021]    As will be described below, the present disclosure teaches a modified design that improves the brightness that can be achieved at the beam spot  36  and immediately after launch into a fiber  30 , if the multi-emitter laser diode module is fiber coupled. This design also enables, in fiber-coupled configurations, more efficient feedback to the emitters using downstream optics than is possible in prior-art designs. 
       SUMMARY OF THE INVENTION 
       [0022]    The present disclosure improves the brightness performance of multi-emitter laser diode modules by modifying the optics to provide a more steeply sloped near-field intensity distribution in the fast axis at the beam spot (or fiber facet if the module is fiber-coupled). The numerical example commenced above is continued below to illustrate that when the design changes described below are implemented, in some embodiments there may be a roughly two times improvement in brightness over prior art multi-emitter laser diode module  10 . 
         [0023]    According to the present disclosure, a short focal length Fourier transform in the fast-axis can be introduced to transform the fast-axis near-field intensity distribution at the beam spot or fiber facet. The slow axis optics may remain relatively unchanged, may compensate for the additional fast-axis optics or may be transformed as well. As a result of the additional Fourier transform, the fast axis presents power intensity distributions having a steep sloped top-hat distribution in the near field and a Gaussian distribution in the far field. The steep sloped top-hat distribution in the fast-axis near field reduces the long tails that must be accommodated within the fiber core, better fills the aperture and increases brightness. 
         [0024]    An embodiment of the present disclosure provides an apparatus comprising a plurality of laser diode emitters. Each emitter is provided for emitting a broad-area light beam in a beam direction. Each beam has a fast axis transverse to the beam direction and a slow axis transverse to both the beam direction and the fast axis. Each beam is broad in the slow axis and narrow in the fast axis. A group of optical components for each emitter are provided for collimating, shaping, stacking and directing the beams along a light path towards a beam spot. Each group is aligned downstream on the beam direction of the group&#39;s associated emitter. A lens feature is aligned on the light path downstream of the groups of optical components and upstream of the beam spot. The lens features has a fast axis focal length less than about 3 millimeters. 
         [0025]    A further embodiment of the present disclosure provides a method for increasing the brightness of a multi-emitter laser diode module. The method comprises emitting a broad-area light beam from each of a plurality of laser emitters. Each beam travels in a beam direction. Each beam has a fast axis transverse to the beam direction and a slow axis transverse to both the beam direction and the fast axis. Each beam is broad in the slow axis and narrow in the fast axis. 
         [0026]    The method includes collimating and shaping the beams, stacking the beams in the fast axis and directing the beams along a light path towards a downstream beam spot. After collimating, stacking and directing the beams, the method includes Fourier transforming the beams in the fast-axis through a lens feature having a fast-axis focal length less than about 3 millimeters. 
         [0027]    Another embodiment of the present disclosure provides a wavelength-locked fiber-coupled multi-emitter module. A plurality of laser diode emitters are provided. Each emitter emits a broad-area light beam in a beam direction. Each beam has a fast axis transverse to the beam direction and a slow axis transverse to both the beam direction and the fast axis. Each beam is broad in the slow axis and narrow in the fast axis. A group of optical components is provided for each emitter for collimating, shaping, stacking and directing the beams along a light path towards a beam spot. Each group is aligned downstream on the beam direction of the group&#39;s associated emitter. An output fiber is aligned on the light path downstream of the group of optical components having a tip aligned on the light path at the beam spot for receiving the beams. A two dimensional graded-index fiber lens is spliced on the tip having a fast axis focal length less than about 3 millimeters. A diffraction grating is aligned on the light path downstream of the lens with a reflector downstream of the diffraction grating. The diffraction grating and reflector provide for wavelength locking by providing a feedback signal upstream on the light path. 
         [0028]    In some embodiments, the fast-axis focal length is between about 0.1 and about 2.0 millimeters. In some embodiments, the lens feature is provided for performing a fast-axis Fourier transform of the beams such that the edges of a fast axis near-field intensity distribution of the beams at the beam spot increase in steepness. In some embodiments, an objective lens is provided aligned on the light path downstream of the groups of optical components. In some embodiments means for creating astigmatism are provided between the fast and slow axes in the beams upstream of the lens feature in accordance with the fast axis focal length of the lens feature. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0029]    Embodiments of the present disclosure are described with reference to the following figures wherein identical reference numerals identify identical or analogous elements: 
           [0030]      FIG. 1A  is a perspective schematic illustration of a prior art multi-emitter fiber-coupled laser diode module. 
           [0031]      FIG. 1B  is an exploded cross-sectional end view of the fiber facet  31  of  FIG. 1A  taken along the line A. 
           [0032]      FIG. 2A  is an exploded view of region B of  FIG. 1A  in the fast axis including graphs of near and far field light intensity distributions at various points. 
           [0033]      FIG. 2B  is an exploded view of region B of  FIG. 1A  in the slow axis including graphs of near and far field light intensity distributions at various points. 
           [0034]      FIG. 3A  illustrates an alternate configuration of region B of  FIG. 1A  in the fast axis according to the present disclosure including graphs of near and far field light intensity distributions at various points. 
           [0035]      FIG. 3B  illustrates the alternate configuration of  FIG. 3A  in the slow axis including graphs of near and far field light intensity distributions at various points. 
           [0036]      FIG. 4A  illustrates another alternate configuration of region B of  FIG. 1A  in the fast axis according to the present disclosure including graphs of near and far field light intensity distributions at various points. 
           [0037]      FIG. 4B  illustrates the alternate configuration of  FIG. 4A  in the slow axis including graphs of near and far field light intensity distributions at various points. 
           [0038]      FIG. 5A  illustrates yet another alternate configuration of region B of  FIG. 1A  in the fast axis according to the present disclosure including graphs of near and far field light intensity distributions at various points. 
           [0039]      FIG. 5B  illustrates the alternate configuration of  FIG. 5A  in the slow axis including graphs of near and far field light intensity distributions at various points. 
           [0040]      FIG. 6A  is a cross-sectional end view of a fiber according to the present disclosure. 
           [0041]      FIG. 6B  is a cross-sectional end view of another fiber according the present disclosure. 
           [0042]      FIG. 7  illustrates a fiber coupled system combining several multi-emitter diode modules according to the present disclosure. 
           [0043]      FIG. 8A  is a perspective schematic illustration of a fiber-coupled multi-emitter laser diode module according to the present disclosure for providing feedback to the module. 
           [0044]      FIG. 8B  illustrates a perspective schematic illustration of a fiber-coupled multi-emitter laser diode module according to the present disclosure for providing wavelength selective feedback to the module for wavelength locking. 
       
    
    
     DETAILED DESCRIPTION 
       [0045]    While preferred embodiments may be illustrated or described, they are not intended to limit the invention. Rather, numerous changes including alternatives, modifications and equivalents may be made as would be understood by the person skilled in the art. Ultimately, the invention is defined by the appended claims. 
         [0046]    The present disclosure improves the brightness performance of existing multi-emitter diode laser modules by modifying the fast-axis optics to include a short focal length fast axis Fourier transform before the beam spot (or fiber tip if the module is fiber-coupled). This improves how sharply or steeply the near field power intensity distribution in the fast axis slopes. Increasing the slope of the fast axis near-field intensity distribution is effected by an additional transforming lens feature performing an extra Fourier transform in the fast axis, so that a steep sloped top-hat distribution is presented in the fast-axis near field. Concurrently, a more gradually sloped or Gaussian distribution may be presented in the fast-axis far field, similar in power distribution to the near field output in the slow axis. 
         [0047]    In some embodiments, the additional Fourier transform can be achieved by greatly demagnifying the large, stacked, top-hat-shaped fast axis beam at the objective lens and applying the effects of an additional fast cylinder lens feature between the objective lens and the beam spot or fiber facet. Accordingly, a top-hat shaped distribution, similar to that of the fast-axis far-field, is now presented in the fast-axis near field while a Gaussian distribution, similar to that of the fast-axis near-field, is now presented in the fast-axis far field. This makes a top-hat image that is matched into the fiber. 
         [0048]    Because an additional Fourier transform is added in the fast axis dimension while passive propagation may occur in the slow axis direction, astigmatism between the fast and slow axis focal points may be introduced prior to the fast-axis Fourier transform so as to maintain focus on the beam spot or fiber tip (if any) after the additional Fourier transform. After the additional Fourier transform, the introduced astigmatism may be removed to return focus of the fast and slow axes to the beam spot. In some embodiments, an additional fast-axis Fourier transform can be achieved without astigmatism. 
         [0049]    Below, three non-limiting options are described implementing the above general principles while holding optical parameters in the slow axis, specifically the focal length of an optional objective lens, relatively constant. These three options are illustrated in  FIGS. 3A and 3B , in  FIGS. 4A and 4B  and in  FIGS. 5A and 5B . Instead of repeating all of the structure of known multi-emitter laser diode modules in each Figure,  FIGS. 3A through 5B  simply illustrate the modifications that would be performed to region B of the prior art multi-emitter laser diode module  10  described above and illustrated in  FIG. 1A  to implement improved brightness multi-emitter laser diode modules according to the present disclosure. 
         [0050]    Referring now to  FIGS. 3A and 3B , a first option  300  is illustrated in the fast axis and the slow axis respectively. An independent fast cylindrical lens  302  is inserted into the high-power multi-emitter multimode-fiber-coupled laser diode module  10  previously described in  FIG. 1A  to cause the desired additional fast-axis Fourier transform without changing the optics in the slow-axis. The objective lens  22  is replaced by a modified objective lens  303 . The fast cylindrical lens  302  is inserted between the modified objective lens  303  and the fiber facet  31  such that the modified objective lens  303 , fast cylindrical lens  302  and fiber facet  31  are aligned on the light path. The distance between the modified objective lens  303  and the front principal surface of cylindrical lens  302  is the sum of the fast-axis focal length  314  of the modified objective lens  303  and the fast-axis focal length  304  of the cylindrical lens  302 . The free propagating distance  305  between the back principal surface of the cylindrical lens  302  and the fiber facet  31  is the fast-axis focal length of the cylindrical lens  302 . 
         [0051]    In some embodiments, the cylinder lens  302  may have a focal length  304  of 0.25 mm. Such lenses are commercially available and are typically used as fast-axis collimators for laser diodes. In order to perform the additional Fourier transform properly, the 0.25 mm focal length fast cylindrical lens&#39; front principal surface should be located 0.25 mm downstream from the focus spot of the modified objective lens  303 , and the back principal surface should be located 0.25 mm upstream of the fiber facet  31  such that the beam spot  36  appears on the fiber facet  31 . In order to ensure stable alignment with the fiber facet  31 , this lens  302  may be actively aligned and mounted directly to a ferrule (not illustrated) holding the fiber  30 . This mounting is appropriate for fiber-coupled multi-emitter laser diodes; however, other mountings are also possible where there is no fiber because the multi-emitter laser diode implementing this embodiment is not necessarily fiber-coupled. 
         [0052]    Returning to the numerical example commenced in respect of  FIGS. 2A and 2B , the result of applying the 0.25 mm (250 micron) lens  302  is to convert the former 0.15-radian half-divergence  214  top hat far field distribution  206  to a 2×250×0.15=75-micron wide  306  top hat near field distribution  308 , and to convert the former 30-micron wide FWHM  208  Gaussian near field distribution  204  to a 0.5*30/250=0.06-radian FWHM  310  Gaussian far field distribution  312 . The brightness benefit of filling the near field  308  more uniformly is manifested in the improvement of the far field from a 0.15-radian top hat distribution  224  to a 0.06-radian FWHM Gaussian distribution  308  for a relative improvement in brightness of roughly a factor of 2. 
         [0053]    In order to accommodate the cylinder lens  302  in  FIG. 3A  and hold the slow-axis optical parameters the same as before, the modified objective lens  303  replaced the prior-art objective lens  22 . In the slow axis illustrated in  FIG. 3B , the focus of the beam  19  must still be directly at the beam spot  36  or fiber facet  31 , and, solely for easy of comparison of numerical examples, the slow-axis focal length of the objective lens  303  is held constant at 10 mm, the same as distance  212 . Other configurations are equally possible, but this one is described to illustrate the relative increase in brightness that is achieved according to the present disclosure over the prior art numerical example. 
         [0054]    Because the fast-axis cylinder lens  302  has been inserted, the objective lens&#39;s fast-axis focal point can be 0.50 mm in front of the fiber facet  31 , and the prior-art objective lens  22  can be modified (or other optical elements inserted) so that the focus is slightly astigmatic, i.e. the fast axis focal length  314  can be 9.50 mm while the slow axis focal length  212  remains 10 mm. In an alternative embodiment, the fast axis could remain at 10 mm focal length and the slow axis would then need to be extended to 10.5 mm focal length. This astigmatism can be accomplished in any number of ways; six non-limiting examples are provided below. 
         [0055]    (i) A separate, discrete, weak cylinder lens can be added to the module either before or immediately following the modified objective lens  303 . This could either be a positive cylinder lens oriented in the fast axis or a negative cylinder lens oriented in the slow axis. This technique, while adequate, would not be preferred since it adds two more optical surfaces to the module. 
         [0056]    (ii) Assuming that the modified objective lens  303  is plano-convex (spherical or possibly aspherical depending on tolerance to aberrations), the plano surface can be made into a weak cylinder lens. This technique would be straightforward and inexpensive to manufacture using glass-molding technology. 
         [0057]    (iii) One of the surfaces of the modified objective lens  303  could be made toroidal, i.e. having slightly different focal powers in the fast and slow axes. Similar to the previous technique, this technique would be straightforward and inexpensive to manufacture using glass-molding technology. 
         [0058]    (iv) A standard plano-convex objective lens  303  can be made slightly astigmatic by simply tilting it slightly. The required small amount of astigmatism can typically be achieved by a tilt of a few degrees in the plane of the fast axis. Tilting also induces other aberrations, notably coma, so it would be preferable in this technique to use a lens that is corrected for coma such as an aplanatic asphere. 
         [0059]    (v) A tilted plane-parallel plate can be inserted after the modified objective lens  303 ; the tilt induces an amount of astigmatism that depends on the plate thickness and tilt angle. 
         [0060]    (vi) Upstream anamorphic optics (e.g. the slow-axis collimators  18 ) can be defocused slightly so that the beam  19  entering the modified objective lens  303  is not precisely collimated in one or the other of the axes. 
         [0061]    The tilted lens and the tilted plate techniques have a benefit that the astigmatism can easily be varied, so that during the alignment of the system, precise focusing can be achieved in both axes independently—whereas with lenses that have fixed amounts of astigmatism, there is no adjustment with which to null out variations in the alignment or focal power of the fast cylinder lens  302 . Of course, it would also be possible to fine-tune the astigmatic lenses by tilting them slightly, an example of combining multiple of the above astigmatism methods. 
         [0062]    Referring now to  FIGS. 4A and 4B , a second option  400  is illustrated for introducing a short focal length Fourier transformation in the fast-axis. Instead of inserting an independent fast cylinder lens  302  as in  FIGS. 3A and 3B  (which introduces two additional optical surfaces), in the second option  400  a solid, coreless endcap  402  with a cylinder-lensed tip is spliced onto the output fiber  30  at the fiber facet  31 . The lens of the endcap  402  may have the same optical performance as described with respect to the fast cylinder lens  302  of the first option  300 . The distance between the modified objective lens  403  and the front principal surface of the endcap  402  is the sum of the fast-axis focal length  414  of the modified objective lens  403  and the fast-axis focal length  404  of the endcap  402 . For optimal performance, the free propagating distance  405  between the front principal surface of the endcap  402  and the fiber facet  31  is equal to the length  405  of the endcap  402  along the light path and this length should be the product of the cylinder-lensed endcap&#39;s fast-axis focal length and the refractive index of the material from which the endcap  402  is constructed. 
         [0063]    Returning to the numerical example, if the fast-axis focal length  404  of the endcap  402  is again chosen to be 0.25 mm, and the material of the endcap  402  is silica (having refractive index of 1.45 for silica fibers in the near-infrared) then the length of the endcap  402  and distance  405  is 0.25×1.45=0.36 mm. To maintain a slow-axis focal length of 10 mm between the modified objective lens  403  and the fiber facet  31  for comparison purposes, the modified objective lens  403  would require a fast-axis focal length  414  of 9.5 mm and distance  212  would increase to 10.11 mm. The difference between the distance  212  and the slow axis focal length occurs because the beam in the slow axis also experiences refraction as it passes through the endcap. Even though the endcap is 0.36 mm long, the optical length that the slow axis experiences is equivalent to 0.25 mm which is the same as the fast axis. Accordingly, the required astigmatism is again 0.5 mm and the same types of adjustments as describe with respect to the first option  300  may be performed to introduce the 0.50 mm astigmatism in the second option  400 . In all other respects, the near and far field intensity distributions of the second option  400  would be the same as those of the first option  300 , thus a roughly 2× greater brightness would be achieved over the prior art illustrated in  FIGS. 2A and 2B . 
         [0064]    Because the illuminated spot incident on the endcap  402  will be of somewhat greater size than the beam spot  36  launched into the core of the fiber  30  at fiber facet  31 , the endcap  402  can be of larger diameter than the output fiber  30  as illustrated in  FIGS. 4A and 4B . It is common to splice oversized endcaps, such as 250-micron diameter endcaps, onto smaller fibers such as 125-micron outer diameter fibers. One method of fabricating the fiber  30  and coreless cylinder-tipped endcap  402  is to splice a length of several centimeters of endcap rod onto the fiber and then use CO2-laser machining to cut the endcap rod to the right length and to shape the tip into the desired cylindrical lens (the tip could subsequently be antireflection coated). 
         [0065]    The second option  400  has advantages over the first option  300  because the first option  300  introduces new optical surfaces while the second option  400  does not; and because the first options  300  requires that the fast cylinder lens  302  be monolithically attached and aligned with respect to the fiber core  30  while the endcap  402  of the second option  400  can be more easily fabricated with the fiber  30 . However, the second option  400  may not be as easily implemented where the laser diode module is not fiber-coupled. 
         [0066]    Referring now to  FIGS. 5A and 5B , both axes of a third option  500  for a fast-axis Fourier transform are illustrated where a fast-axis fast cylinder lens is implemented directly in the fiber tip  502 . In some ways, the third option  500  is the simplest, but it involves a slight compromise in performance as well as some additional optical changes. Because the cylinder lens is implemented directly in the fiber tip  502  of the output fiber  30 , no endcapping process is necessary and the cylinder-lensed fiber tip  502  can be created directly on top of fiber facet  31  either by laser machining or mechanical polishing. However, doing so removes any distance between the back principle surface of the tip  502  and the fiber facet  31 . Because the free propagation distance  505  from the back of the cylinder-lensed fiber tip  502  to the fiber facet  31  is reduced to zero, the third option  500  does not induce a perfect Fourier transform, and it is necessary that the fast-axis focal beam spot  36  from the modified objective lens  503  be made smaller than in the other described options  300 ,  400  in order to achieve good performance. Similar to the second option  400 , it would be difficult to implement this brightness improvement in laser diode modules that are not fiber coupled. 
         [0067]    Returning again to the numerical example, if the fast-axis focal beam spot  36  is made smaller by a factor of 2 than the beam spot  36  described in respect of the first and second options  300 ,  400 , then using a cylinder-lensed fiber tip  502  having fast-axis focal length  504  of 0.1 mm, the modified objective lens  502  must have a fast-axis focal length  514  of 9.9 mm to maintain the 10 mm slow-axis focal length of objective lens  503 . Distance  212  also remains 10 mm. Accordingly, a near-field top hat distribution  506  having a width  508  of 75 microns and a far field distribution  512  having a half-divergence  510  of 0.075 radians can be achieved. This illustrates that the third option  500  is somewhat inferior to the other two options  300 ,  400 , but still provides a significant improvement over the prior art design  200 . In the slow-axis illustrated in  FIG. 5B , again, the slow-axis focal length of the modified objective lens  502  remains at 10 mm and the cylinder-lensed fiber tip  502  does not change the slow axis optics. 
         [0068]    In this third option, in order to achieve a meaningful brightness improvement over the prior art, the fast-axis spot  36  must be considerably smaller than the spot to which it is being transformed. In numerical modeling it was found that a factor of 2 is sufficient for meaningful improvement. 
         [0069]    In order to reduce the fast-axis focal beam spot size by a factor of 2, the beam  19  incident on the modified objective lens  503  can be doubled in fast-axis size  516  to 6 mm while that beam&#39;s slow-axis size remains unchanged. This necessitates changes to the fast-axis collimator  16  focal lengths, changes to the pitch at which the emitters are stacked vertically, and increases the susceptibility of the modified objective lens  503  to optical aberrations. It may also be necessary to modify the modified objective lens  503  so it can receive a 6 mm wide beam  19  in the fast-axis. It is also noteworthy that the third option  500  requires astigmatism between the fast axis and the slow axis of only 0.10 mm rather than 0.50 mm from the other two options  300 ,  400 . This smaller amount of astigmatism could be very easy to induce through a slight tilt of the modified objective lens  503 . 
         [0070]    As indicated above, three options  300 ,  400 ,  500  described in  FIGS. 3A and 3B ,  FIGS. 4A and 4B , and  FIGS. 5A and 5B  respectively are special cases of the general principle where the large, stacked, top-hat-shaped fast axis beam  19  at the entrance to a modified objective lens  303 ,  403 ,  503 , is demagnified by the combination of a modified objective lens  303 ,  403 ,  503  and fast-axis short-focal length transforming lens feature  302 ,  402 ,  502  to make a top-hat image that is matched into the beam spot  36  which may be co-located with a fiber facet  31  if the module is fiber-coupled. In some embodiments the fast-axis focal length of the lens features is less than about 3 millimeters. In some embodiments, the fast-axis focal length of the lens feature is between about 0.1 mm and about 2.0 mm. 
         [0071]    Optionally, astigmatism may be introduced to the beam  19  such that the fast-axis focal point from the objective lens aligns with the upstream fast-axis focal point of the fast axis Fourier transforming feature while the slow-axis focal point from the objective lens aligns with the beam spot. However, astigmatism is not necessary in all embodiments. For example, the lens feature may be a 2D gradient-index lens or a spherically polished endcap, which do not require astigmatism to perform a short focal length fast-axis Fourier transform. In these cases, the slow axis is transformed along with the fast axis, which may be acceptable or desirable in certain optical designs. 
         [0072]    The fast axis Fourier lens feature may remove the astigmatism such that the fast and slow axis focal points are the same. In some embodiments additional features are added to remove astigmatism after the additional Fourier transform. 
         [0073]    In the embodiments of the present disclosure, many different lens types are available and can be applied to effect the additional Fourier transform in the fast-axis, including aspheric cylinders, toroidal lenses, spherical lenses, aspherical lenses, axial-gradient-index lenses, and transverse-gradient-index lenses. 
         [0074]    In yet further embodiments of the present disclosure, the short-focal length lens (which introduces the additional fast-axis Fourier transform) may be a graded-index lens (GRIN), independently secured as cylindrical lens  302  or fabricated as a graded-index-core fiber of which a short segment can be fusion-spliced onto the tip of the fiber  30 . 
         [0075]    A one-dimensional GRIN lens could be very effective as the short-focal length lens: such a lens will have a precise parabolic index gradient (in order for a GRIN lens to have minimal aberrations) in one axis, and zero gradient in the other axis. However, it is challenging to fabricate such an optic with truly no gradient in the other axis and/or fabricate such an optic as an exteriorly round fiber segment to be fusion-spliced onto a fiber. 
         [0076]    A modified two-dimensional GRIN lens could also be used as the short-focal length lens. Such a lens could be designed to be Fourier-transforming in the fast axis and imaging, or nearly imaging, in the slow axis, by adjusting the focal strengths in the two axes. 
         [0077]    Referring back to  FIG. 1B , the resulting focal beam spot  36  at the fiber facet  31  generally has a square or rectangular shape rather than a circular or strongly rounded shape. This is especially true if any of the three options described above using a fast cylinder lens is applied to the laser diode module  10  because these options create very sharp-edged and uniformly-filled square or rectangular focal beam spots  36  (75×75 micron, using the numerical example). Therefore, in order best to maintain brightness, it is desirable to use a core shape that is well-matched to the illuminated spot  36 , which in the various options discussed above would be square or rectangular. This additional design change may result in up to about two times increase in brightness in addition to the two times increase in brightness effected by the additional Fourier transform. 
         [0078]    Referring now to  FIG. 6A , a fiber  600  is illustrated. The core  602  size could be 75×75 microns to match the focal beam spot size  604 , or it could be made slightly larger in order to account for aberrations and manufacturing deviations. The cladding  606  of the fiber  600  may be circular with a diameter sufficient to surround the core  602  as illustrated in  FIG. 6A  or any other shape, including square or rectangular to uniformly circumscribe the core  602 . 
         [0079]    In another embodiment illustrated in  FIG. 6B , a fiber  620  is illustrated having a non-square rectangular shaped core  622 , such as 75×150 microns. Cladding  626  may be similar in nature to cladding  606 . This shape of fiber  620  provides an extra degree of design freedom, effectively decoupling the fast and slow axes from their prior design constraints. For example, core  622  allows the use of 2× wider emitters  12 , thus providing higher output power, without altering the number of emitters, their vertical pitch, or the rest of the optical design. 
         [0080]    Square-core or rectangular-core fibers  600 ,  620  are relatively difficult to fabricate, although it is certainly possible. To date, they have seen significant use only in laser-delivery fibers, where it is desired to have a square or rectangular laser spot projected onto a work-piece to give more uniform illumination for materials processing. The potential applicability to brightness enhancement in fiber-coupled multi-emitter laser diode modules has not, to Applicants&#39; knowledge, been commercialized elsewhere. 
         [0081]    Using a 75×75 micron square core  602  results in 1.54× higher brightness than the brightness of a 75×75 micron spot  36  in a 105-micron circular core fiber  30  as in  FIG. 1B  because the area fill is only 65% of the circular core. A 75×150 micron spot  624  would barely fit into a 168-micron circular core, such that using the rectangular core  624  would result in 1.96× higher brightness. Coupling these brightness improvements with the approximately 2× improvement predicted using any implementation of the first design change (an additional fast-axis Fourier transform) gives a total improvement in brightness of a factor of approximately 3-4, which is very substantial and can be achieved for very little added cost. 
         [0082]    In some embodiments of the present disclosure, dichroic coatings and slanted fiber gratings as specified in previously identified U.S. Pat. No. 7,764,723 may also be applied to other optical surfaces disclosed herein than merely those specified in the &#39;723 patent. 
         [0083]    In some embodiments, the typical number of emitters  12  would be six or seven, with the possibility of more. In some embodiments, multiple sets of emitters  12  may be combined by another set of mirrors with the polarization of the stacked light beams rotated 90 degrees by, for example, a half-wave plate or polarization rotator. This rotated beam may then be merged into beam  19  by a polarizer located immediately in front of any of the modified objective lenses  303 ,  403 ,  503 . 
         [0084]    In yet further embodiments, spectral combining is also performed on the light before launching into the fiber  30 . This is illustrated in  FIG. 7 . System  700  contains an arrangement of multi-emitter modules  702  that are not fiber coupled. Each module  702  contains a plurality of laser diodes that are spatially and polarization-multiplexed into one collimated beam. For example, there may be 12 laser diodes in some of the modules  702 . Each module  702  is constructed according to any of the previously described embodiments of this disclosure that are not fiber coupled, with the following differences. The focusing or objective lens  22 ,  303 ,  403 ,  503  and fiber  30  of each module  702  has been removed and replaced by a simple window  704 , so that the collimated beam  19  emerges from each module  702 . Each module  702  emits a collimated beam  19  having a slightly different wavelength than any of the other modules  702  in the system  700 . The collimated beam  19  of each module  702 , except the first module, is incident to an adjacent dichroic mirror  706 . Each dichroic mirror  706  is chosen to transmit the wavelengths of its adjacent module  702  and reflect other wavelengths of light. The modules  702  and dichroic mirrors  706  are arranged such that the collimated beam  19  from the previous module  702  reflects off the dichroic mirror associated with the next module  702  along the same path as the collimated beam  19  emitted by the next module  702 . Accordingly, the different wavelength collimated beams  19  from each module  702  are successively combined together into a spectrally-combined beam  708 . The spectrally-combined beam  708  arrives at the final focus lens  703  and is focused into a fiber  30  through a connector  710 . Obviously, this is just one example of many schemes for spectrally combining the beams. Other schemes would include the use of diffraction gratings, volume phase gratings, other arrangements of dichroic mirrors, additional lenses between mirrors, etc. as is known in the art. Also the steps of spatial combining, polarization combining, and spectral combining could be done in different sequences. Furthermore, the optics may be separated into any number of modules, each having any number of emitters apiece. For example, instead of 9 modules  702  each having 12 laser diode emitters, all 108 emitters could be directly mounted onto one large coldplate. 
         [0085]    Referring now to  FIGS. 8A and 8B , example embodiments for providing feedback in a fiber-coupled multi-emitter module  800  are illustrated. As described in the various laser diode module embodiments above, a short-focal-length lens feature within a multi-emitter module improves the brightness efficiency of the module in the forward direction. This improvement correspondingly improves the backward power efficiency. In  FIGS. 8A and 8B , this feedback improvement is illustrated in fiber coupled laser diode modules; however it can also be achieved with any of the multi-emitter diode modules described in this disclosure. A further improvement achieved in these embodiments is a more even distribution of feedback to all of the emitters in the fiber-coupled multi-emitter module. 
         [0086]    In  FIG. 8A , feedback  801  to the emitters  12  of a fiber-coupled multi-emitter module  800  is provided using a reflector  802  downstream of the fiber facet  36 . The reflector  802  reflects at least one portion of the light beam  19  as feedback  801  back to the emitters  12  and may allow a second portion of the light beam  19  to exit the multi-emitter module  800  as an output beam. The reflector  802  may comprise a high reflector, a partially reflective reflector, a patterned reflector, a scraper-mirror reflector, a wavelength-dependent reflector, or multiple reflectors including any of these reflector types. The reflector  802  may be formed within the fiber  30 , within another fiber connected to fiber  30  or downstream of fiber  30 . In some embodiments, the reflector  802  is wavelength-selective or the reflector  802  comprises a reflector downstream of an wavelength selective optical element. In  FIG. 8A , the short-focal-length fast-axis lens feature  804  is adjacent to the fiber entrance facet  31  and illustrated as an endcap connected to the tip of fiber  30 ; however, any of the other lens feature configurations described in other embodiments may also be used. 
         [0087]    The feedback performance of the system of  FIG. 8A  surpasses that possible using prior art in two ways. First, as described above, the short-focal-length lens feature  804  improves the brightness efficiency of the system in the forward direction, correspondingly improving the backward power efficiency. If an improvement in brightness of 2× is achieved using any of the short-focal-length lens feature options described above, then the feedback efficiency is improved by 2× as well. 
         [0088]    Second, it has been found that the feedback signal  801  is dispersed more evenly to all of the emitters  12  by the short-focal-length lens feature  804 . More even distribution of the feedback signal  801  requires less power or brightness efficiency in the feedback signal  801  to provide sufficient feedback to the weakest-coupled emitter. In numerical simulations of a system providing feedback signal to seven 100-micron-wide emitters, it was found that without a short-focal-length lens feature  804 , the weakest-coupled emitter received about 30% as much feedback power as the strongest-coupled emitter; using the short-focal-length lens feature  804 , the weakest-coupled emitter received 72% as much feedback as the strongest-coupled emitter. Accordingly, the system illustrated in  FIG. 8A  provides a more evenly dispersed external feedback signal  801  across all of the emitters  12 . 
         [0089]    In some embodiments, the desired levels of such feedback signal  801  are generally at least 1% of the light beam  19  in order to discriminate strongly against residual reflections on the order of 0.01%-0.1% from the diode facet and other optics. In some embodiments, the percentage of light  19  reflected back in the feedback signal  801  by the reflector  802  is in the range of 5-20%. 
         [0090]    When feedback to the multi-emitter module  800  is used for wavelength locking, the reflected feedback signal  801  must be spectrally selective. This spectral selectivity is typically achieved in one of two ways: the reflector  802  can be made wavelength-dependent, or wavelength-dependent optics may be inserted upstream of the reflector  802 . 
         [0091]    Where the reflector  802  is wavelength-dependent, the reflector generally has maximum reflectivity at a desired emitter operating wavelength. Embodiments of a wavelength-dependent reflector include a fiber Bragg grating embedded in the fiber  30  or in another downstream fiber, a suitably designed thin-film filter, a Fabry-Perot etalon, a reflective diffraction grating in the Littrow configuration, a reflective volume holographic grating, or any other wavelength-dependent reflector located in free space at or beyond the distal end of the fiber  30 . 
         [0092]      FIG. 8B  illustrates the system of  FIG. 8A  further including a wavelength-dependent optical element  806  used in conjunction with a reflector  802  to provide wavelength selective feedback. The wavelength-dependent optical element  806  is inserted into the beam path upstream of the reflector  802  and downstream of the fiber entrance facet  31 . The wavelength-dependent optical element  806  causes the wavelengths of the beams  19  from the emitters  12  to be locked at a desired wavelength. The wavelength-dependent optical element serves to separate one or more wavelengths of light from the beams whereafter the separated wavelength is incident on the reflector  802 . In some embodiments, the wavelength-dependent optic  806  may comprise a thin-film filter, Fabry-Perot etalon, diffraction gratings, volume holographic gratings, a prism or combinations of those optics. 
         [0093]    Where examples, alternative embodiments and additional aspects of those embodiments have been described in the present disclosure, those examples embodiments and aspects may be combined in any manner within a single embodiment unless the present disclosure suggests otherwise. Where axis directions and orientations such as vertical and horizontal have been specified, it is understood that the orientation of these orthogonal directions may be modified within embodiments of the present disclosure. 
         [0094]    The values provided in the numerical examples are only a few examples of how the present disclosure may be implemented. The specific values provided were selected to enable ready comparison of the prior art and different embodiments of the present disclosure to demonstrate the invention. The specific values in the numerical examples should not be considered limiting.