Patent Publication Number: US-2011058388-A1

Title: Rippled mixers for uniformity and color mixing

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 11/442,673, filed May 26, 2006, titled “RIPPLED MIXERS FOR UNIFORMITY AND COLOR MIXING,” which claims the benefit under 35 U.S.C. §119(e) of U.S. Provisional Application No. 60/703,808, filed Jul. 29, 2005, titled “RIPPLED MIXERS FOR UNIFORMITY AND COLOR MIXING;” the entire contents of each of the aforementioned applications are hereby incorporated by reference herein and made a part of this specification. 
    
    
     BACKGROUND 
     1. Field of the InventionS 
     Certain embodiments disclosed herein relate generally to structures for mixing light, such as for example mixing rods, and relate more specifically to light-mixing structures having multiple ridges and valleys. 
     2. Description of the Related Art 
     It is a common practice to improve the uniformity of light by coupling light sources with mixing rods. This practice can homogenize illuminance and color across a spatial region, as well as homogenize the intensity. However, known mixing rods suffer from various drawbacks including limitations in the level of homogenization achievable. 
     SUMMARY 
     In certain embodiments, an illumination system comprises a plurality of light emitting diodes and a mixer. The mixer comprises a light pipe having input and output ends and a central region therebetween. The mixer further comprises an optical path extending in a longitudinal direction from the input end through the central region to the output end with the plurality of light emitting diodes disposed in proximity to the input end. The central region of the light pipe comprises one or more rippled sidewalls having a plurality of elongate ridges and valleys and sloping surfaces therebetween. The ridges and valleys comprise vertices at least partially formed by a rounded surface or formed by two or more substantially nonorthogonal surfaces. Light from the plurality of light emitting diodes propagating along the optical path reflects from the sloping surfaces and is redirected at a different azimuthal direction toward the output end thereby mixing the light at the output end. 
     In certain embodiments, a lighting apparatus comprises a light source comprising a plurality of emitters and a mixer having an input and an output and a central region therebetween. The mixer comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. The peaks and valleys each comprise an apex at least partially formed by a rounded surface portion or by two or more substantially nonorthogonal surface portions. Light entering the input propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the mixer and uniformity at the output. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input and an output and a central region therebetween. The input has a different size or shape than the output. The elongate member comprises at least one rippled surface in the input, output, and central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output. 
     In certain embodiments, a mixing coupler comprises an elongate member having an input and an output and a central region therebetween. The elongate member comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output. The input comprises a plurality of separate input ports or the output comprises a plurality of separate output ports. 
     In certain embodiments, a lighting apparatus comprises a mixer having an input and an output and a central region therebetween. The mixer comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input propagating longitudinally along the central region of the mixer is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the mixer and uniformity at the output. The lighting apparatus further comprises a projection lens disposed to receive light from the mixer output and project the light to an enlarged spot. 
     In certain embodiments, a lighting apparatus comprises a first mixer comprising at least one rippled surface comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the first mixer is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the first mixer. The lighting apparatus further comprises a second mixer comprising at least one rippled surface comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the second mixer is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the second mixer. The lighting apparatus further comprises coupling optics coupling light output by the first mixer into the second mixer. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input and an output and a central region therebetween. The elongate member comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output. The rippled surface further comprises scatter features that diffuse light incident thereon. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input and an output and a central region therebetween. The central region comprises an elongate reflective region having a refractive index that varies azimuthally about a longitudinal axis through the elongate member so as to form alternating elongate high and low index portions with elongate gradient index portions therebetween. The elongate high and low index portions and the elongate gradient index portions extend longitudinally in the central region such that light entering the input propagating longitudinally along the central region is deflected by the elongate gradient index portions in an azimuthal direction thereby increasing mixing of the light and uniformity at the output. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input face and an output face and a central region therebetween. The input face or the output face is elliptical in shape. The elongate member comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input face propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output face. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input face and an output face and a central region therebetween. The input face or the output face has a shape other than circular or elliptical or square or rectangular. The elongate member comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input face propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output face. 
     In certain embodiments, a mixer for mixing light, the mixer comprises a rigid elongate member having an input face and an output face and a central region therebetween. The rigid elongate member is bent and comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input face propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output face. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input face and an output face and a central region therebetween. The elongate member comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input face propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output face. The mixer further comprises a non-imaging optical element at the input or the output. At least a portion of the peaks comprises rounded surfaces portions. 
     In certain embodiments, a lighting apparatus comprises a substrate and a plurality of mixers coupled with the substrate. One or more of the plurality of mixers comprises an input and an output and a central region therebetween. The one or more of the plurality of mixers comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input propagating longitudinally along the central region of the one or more of the plurality of mixers is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the one or more of the plurality of mixers and uniformity at the output. 
     In certain embodiments, a lighting apparatus comprises a mixer having an input and an output and a central region therebetween. The mixer comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. Light entering the input propagating longitudinally along the central region of the mixer is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the mixer and uniformity at the output. The lighting apparatus further comprises a diffuser disposed to receive light from the mixer output. 
     In certain embodiments, a mixer for mixing light comprises an elongate member having an input face and an output face and a central region therebetween. The elongate member comprises at least one rippled surface in the central region comprising a plurality of peaks and a plurality of valleys connected by sloping surface portions. Light entering the input face propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light and uniformity at the output face. A first portion of the plurality of peaks are formed by rounded sloping surface portions and a second portion of the plurality of peaks are formed by substantially planar sloping surface portions. 
     In certain embodiments, an illumination system comprises a plurality of light emitting diodes and a mixer. The mixer comprises a light pipe having input and output ends and a central region therebetween. The mixer further comprises an optical path extending in a longitudinal direction from the input end through the central region to the output end with the plurality of light emitting diodes disposed in proximity to the input end. The central region of the light pipe comprises one or more rippled sidewalls having a plurality of elongate ridges and valleys and sloping surfaces therebetween. At least one of the ridges or at least one of the valleys is formed by at least three surfaces. Light from the plurality of light emitting diodes propagating along the optical path reflects from the sloping surfaces and is redirected at a different azimuthal direction toward the output end thereby mixing the light at the output end. 
     In certain embodiments, a lighting apparatus comprises a light source comprising a plurality of emitters and a mixer having an input and an output and a central region therebetween. The mixer comprises at least one rippled surface in the central region comprising a plurality of peaks and valleys connected by sloping surface portions. At least one of the peaks and valleys is formed by three or more surfaces. Light entering the input propagating longitudinally along the central region is deflected by the sloping surface portions of the rippled surface in an azimuthal direction thereby increasing mixing of the light in the mixer and uniformity at the output. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments are depicted in the accompanying drawings for illustrative purposes, and should in no way be interpreted as limiting. 
         FIG. 1  is a schematic illustration of a mixer coupled with a light source. 
         FIG. 2  is a side elevation view of a light source optically coupled to a mixer via a reflector. 
         FIG. 3  is a perspective view of one embodiment of a mixer that comprises a rippled sidewall comprising a plurality of ridges and valleys. 
         FIG. 4  is a cross-sectional view of the mixer of  FIG. 3  showing sloping surfaces connecting the ridges and valleys. 
         FIG. 5A  is a plot of the illuminance distribution at an input face of one embodiment of a mixer. 
         FIG. 5B  is a histogram of the illuminance values displayed in the plot of  FIG. 5A . 
         FIG. 5C  is a plot of the illuminance distribution at an output face of one embodiment of a smooth mixer. 
         FIG. 5D  is a histogram of the illuminance values displayed in the plot of  FIG. 5C . 
         FIG. 5E  is a plot of the illuminance distribution at an output face of the mixer of  FIGS. 3 and 4  that comprises a rippled sidewall. 
         FIG. 5F  is a histogram of the illuminance values displayed in the plot of  FIG. 5E . 
         FIG. 6A  is a plot of the illuminance distribution at an input face of one embodiment of a mixer wherein the light source is off-center. 
         FIG. 6B  is a histogram of the illuminance values displayed in the plot of  FIG. 6A . 
         FIG. 6C  is a plot of the illuminance distribution at an output face of one embodiment of a smooth mixer wherein the light source is off-center. 
         FIG. 6D  is a histogram of the illuminance values displayed in the plot of  FIG. 6C . 
         FIG. 6E  is a plot of the illuminance distribution at an output face of the mixer of  FIGS. 3 and 4  that comprises a rippled sidewall. 
         FIG. 6F  is a histogram of the illuminance values displayed in the plot of  FIG. 6E . 
         FIG. 7A  is a partial cross-sectional view of one embodiment of a mixer wherein a rippled sidewall is formed by facets oriented by an angle α. 
         FIG. 7B  is a partial cross-sectional view of one embodiment of a mixer wherein a rippled sidewall has rounded ridges having a height and spacing that define an angle α. 
         FIG. 8A  is a side elevation view of a light source coupled with one embodiment of a mixer that comprises a rippled sidewall comprising a plurality of ridges and valleys. 
         FIG. 8B  is a cross-sectional view of the mixer of  FIG. 8A . 
         FIG. 8C  is a plot of the intensity distribution of the light source of  FIG. 8A , which is configured to have a central dark region. 
         FIG. 9  is a plot of the standard deviation, σ illuminance , of the illuminance distribution at the output face of a rippled mixer such as shown in  FIGS. 8A and 8B  as a function of the angle α of the ridges; plots for mixers having faceted and rounded rippled sidewalls are shown. 
         FIG. 10A  is a plot of the illuminance distribution at an output face of one embodiment of a mixer without rippled sidewalls. 
         FIG. 10B  is a plot of the illuminance distribution obtained by performing a virtual defocus of the light exiting the mixing rod back to the input face of the mixer without rippled sidewalls. 
         FIG. 10C  is a plot of the illuminance distribution at an output face of one embodiment of a mixer comprising a rippled sidewall. 
         FIG. 10D  is a plot of the illuminance distribution obtained by performing a virtual defocus of the light exiting the mixing rod back to the input face of the rippled mixer used with  FIG. 10C . 
         FIG. 10E  is a plot of the illuminance distribution at an output face of one embodiment of a mixer comprising a rippled sidewall. 
         FIG. 10F  is a plot of the illuminance distribution obtained by performing a virtual defocus of the light exiting the mixing rod back to the input face of the mixer used with  FIG. 10E . 
         FIG. 11  is a cross-sectional view of a square mixer that has rippled sidewalls having ridges and valleys. 
         FIG. 12A  is a plot of the illuminance distribution at an input face of one embodiment of a square mixer. 
         FIG. 12B  is a plot of the illuminance distribution at an output face of one embodiment of a square mixer without rippled sidewalls. 
         FIG. 12C  is a plot of the illuminance distribution obtained by performing a virtual defocus of the light exiting the mixing rod back to the input face of the square mixer without rippled sidewalls. 
         FIG. 12D  is a plot of the illuminance distribution at an output face of one embodiment of a square mixer with rippled sidewalls. 
         FIG. 12E  is a plot of the illuminance distribution obtained by performing a virtual defocus of the light exiting the mixing rod back to the input face of the square mixer with rippled sidewalls. 
         FIG. 12F  is a plot of the standard deviation of the illuminance distribution at the output face as a function of the angle α of the ridges for square mixers having faceted and rounded rippled sidewalls. 
         FIG. 13A  is a side elevation view of a light source coupled with one embodiment of a square mixer that comprises rippled reflective sidewalls. 
         FIG. 13B  is a plot of the intensity distribution of the light source of  FIG. 13A  which has a central dark region. 
         FIG. 13C  is a plot of the standard deviation of the illuminance distribution at an output face as a function of the angle α of the ridges for mixers such as shown in  FIG. 13A  having faceted and rounded rippled sidewalls. 
         FIG. 14A  is a cross-sectional view of one embodiment of an elliptically shaped mixer without rippled sidewalls. 
         FIG. 14B  is a plot of the illuminance distribution at an output face of the elliptical mixer of  FIG. 14A . 
         FIG. 14C  is a cross-sectional view of one embodiment of an elliptically shaped mixer comprising a rippled sidewall. 
         FIG. 14D  is a plot of the illuminance distribution at an output face of the elliptical mixer of  FIG. 14C . 
         FIG. 15  is a cross-sectional view of one embodiment of an irregularly shaped mixer that has rippled sidewalls. 
         FIG. 16A  is a perspective view of one embodiment of an angle-to-area converting mixer. 
         FIG. 16B  is a perspective view of one embodiment of an angle-to-area converting mixer that has rippled sidewalls. 
         FIG. 16C  is a perspective view of one embodiment of an angle-to-area converting rippled mixer that has an elliptically shaped input face and a rectangular output face. 
         FIG. 17A  is a front plan view of one embodiment of an angle-to-area converting mixer without rippled sidewalls coupled at an input end with an array of light emitting diodes. 
         FIG. 17B  is a perspective view of the mixer of  FIG. 17A . 
         FIG. 17C  is a plot of the illuminance distribution at an output face of the mixer of  FIGS. 17A and 17B  when only one of the light emitting diodes is illuminated. 
         FIG. 17D  is a histogram of the illuminance values displayed in the plot of  FIG. 17C . 
         FIG. 17E  is a perspective view of one embodiment of an angle-to-area converting mixer similar to that shown in  FIG. 17A  but with rippled sidewalls. 
         FIG. 17F  is a plot of the illuminance distribution at an output face of the rippled mixer of  FIG. 17E  when only one of the light emitting diodes is turned on. 
         FIG. 17G  is a histogram of the illuminance values displayed in the plot of  FIG. 17F . 
         FIG. 18A  is a perspective view of one embodiment of an angle-to-area converting mixer that can be coupled at an input end with a linear array of light emitting diodes. 
         FIG. 18B  is a partial perspective view of the mixer of  FIG. 18A  coupled at the input end with the linear array of light emitting diodes. 
         FIG. 19A  is a perspective view of one embodiment of a rippled angle-to-area converting mixer having a circular input face and a circular output face and comprising a rippled sidewall. 
         FIG. 19B  is a perspective view of one embodiment of a rippled angle-to-area converting mixer having a square input face and a square output face and rippled sidewalls. 
         FIG. 19C  is a perspective view of one embodiment of a composite angle-to-area converting mixer having a square input face, a circular output face, and rippled sidewalls, and that is formed by combining portions of the rippled mixers of  FIGS. 19A and 19B . 
         FIG. 20A  is a perspective view of one embodiment of a rippled mixer having circular input and output faces and a narrow region therebetween. 
         FIG. 20B  is a perspective view of one embodiment of a rippled mixer having square input and output faces. 
         FIG. 20C  is a perspective view of one embodiment of a rippled mixer that is formed by combining the rippled mixers of  FIGS. 20A and 20B . 
         FIG. 21A  is a perspective view of one embodiment of a mixer that comprises an input face, an output face, and ridges that decrease in height (as measured from the valleys) from the input face to the output face. 
         FIG. 21B  is a perspective view of one embodiment of a mixer that comprises an input face, an output face, and ridges that increase in height from the input face to the output face. 
         FIG. 21C  is a perspective view of one embodiment of a mixer that comprises an input face, an output face, and ridges that increase in height from the input face and the output face to the center of the mixer. 
         FIG. 21D  is a perspective view of one embodiment of a mixer that comprises an input face, an output face, and ridges that decrease in height from the input face and the output face to the center of the mixer. 
         FIG. 22A  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges that have rounded vertices and valleys that have sharp vertices. 
         FIG. 22B  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges that have sharp vertices and valleys that have rounded vertices. 
         FIG. 22C  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges and valleys that are highly rounded. 
         FIG. 22D  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges and valleys that have vertices connected by splines. 
         FIG. 22E  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges that have variable heights. 
         FIG. 22F  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges that have angles α that vary irregularly or randomly. 
         FIG. 22G  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges that are separated by a large gap. 
         FIG. 22H  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges and valleys with sharp vertices connected by planar sloping surfaces as well as rounded ridges and valleys connected by curved surfaces. 
         FIG. 22I  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges and valleys that are also rippled. 
         FIG. 23A  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges with rounded vertices and valleys with sharp vertices and curved surfaces therebetween. 
         FIG. 23B  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges and valleys with sharp vertices and planar faceted surfaces therebetween that approximate the rippled sidewall of  FIG. 23A . 
         FIG. 23C  is a cross-sectional view of one embodiment of a rippled sidewall comprising ridges and valleys having sharp vertices and planar faceted surfaces therebetween that approximate rounded valleys. 
         FIG. 24  is a cross-sectional view of a rippled sidewall comprising alternating rounded ridges and sharp ridges having an angle α of about 45 degrees. 
         FIG. 25A  is a top plan view of one embodiment of a rippled mixer that comprises ridges and valleys that are oriented at an angle β with respect to the length of the mixer. 
         FIG. 25B  is a perspective view of one embodiment of a rippled mixer that comprises ridges and valleys that are oriented at an angle β with respect to the length of the mixer. 
         FIG. 25C  is a top plan view of one embodiment of a rippled mixer that comprises ridges and valleys that are oriented at an angle γ with respect to the length of the mixer. 
         FIG. 25D  is a perspective view of one embodiment of a rippled mixer that comprises ridges and valleys that are oriented at an angle γ with respect to the length of the mixer. 
         FIG. 25E  is a perspective view of one embodiment of a composite mixer that is formed by combining the rippled mixers of  FIGS. 25B and 25D . 
         FIG. 26A  is a perspective view of one embodiment of a composite rippled mixer that is formed by combining a portion of a circular angular-to-area converting mixer that has ridges and valleys that are rotated by an angle, β, with respect to the length of the circular mixer with a portion of a square angle-to-area converting mixer that has ridges and valleys that are rotated by an angle, γ, with respect to the length of the square mixer. 
         FIG. 26B  is a plan view of the mixer of  FIG. 26A . 
         FIG. 27  is a perspective view of one embodiment of a mixer comprising ridges and valleys that curve with respect to the length of the mixer. 
         FIG. 28  is a perspective view of one embodiment of a solid mixer that comprises a reflective coating at an input end. 
         FIG. 29A  is a side elevation view of one embodiment of a mixer comprising ridges and valleys coupled with one embodiment of a projection lens. 
         FIG. 29B  is a cross-sectional view of one embodiment of a mixer comprising ridges and valleys coupled with one embodiment of a total internal reflection (TIR) collimator. 
         FIG. 30A  is a side elevation view of one embodiment of an optical system comprising a rippled mixer and a projection lens. 
         FIG. 30B  is a side elevation view of one embodiment of a zoom system formed from the optical system shown in  FIG. 30A  and a diffuser having a hole therein between the rippled mixer and the projection lens. 
         FIG. 30C  is a side elevation view of one embodiment of an optical system comprising a rippled mixer, an output lens, and a projection lens. 
         FIG. 30D  is a side elevation view of one embodiment of the optical system of  FIG. 30C  with the projection lens spaced away from the output lens to create a “spot” beam pattern. 
         FIG. 30E  is a side elevation view of one embodiment of the optical system of  FIG. 30C  with the projection lens in proximity to the output lens to create a “flood” beam pattern. 
         FIG. 31  is a schematic illustration of one embodiment of a fiber illuminator that comprises ridges and valleys. 
         FIG. 32A  is a perspective view of one embodiment of a mixer comprising bends. 
         FIG. 32B  is a plot of the illuminance distribution at an output face of the mixer of  FIG. 32A  that results when a relatively small light source is coupled with an input face of the mixer. 
         FIG. 32C  is a plot of the illuminance distribution at the output face of the mixer of  FIG. 32A  that results when a relatively large light source is coupled with the input face of the mixer. 
         FIG. 32D  is a perspective view of one embodiment of a mixer comprising bends and rippled sidewalls having ridges and valleys. 
         FIG. 32E  is a plot of the illuminance distribution at an output face of the mixer of  FIG. 32D  that results when a relatively small light source is coupled with an input face of the mixer. 
         FIG. 32F  is a plot of the illuminance distribution at the output face of the mixer of  FIG. 32D  that results when a relatively large light source is coupled with the input face of the mixer. 
         FIG. 33A  is a side elevation view of one embodiment of a mixer that comprises a rippled sidewall coupled with two nonimaging optical elements and a light source. 
         FIG. 33B  is a side elevation view of one embodiment of a mixer that comprises a rippled sidewall and two bends and is coupled with two nonimaging optical elements and a light source. 
         FIG. 33C  is a side elevation view of one embodiment of a mixer that comprises a rippled sidewall and four bends and is coupled with two nonimaging optical elements and a light source. 
         FIG. 34A  is a top plan view of one embodiment of a curved mixer comprising a rippled sidewall. 
         FIG. 34B  is a top plan view of one embodiment of a curved mixer comprising a rippled sidewall and two tapered ends. 
         FIG. 35  is a side elevation view of one embodiment of a mixer that comprises rippled sidewalls and a curved output face. 
         FIG. 36A  is a cross-sectional view of one embodiment of a mixer that comprises a solid rod and a film comprising ridges and valleys. 
         FIG. 36B  is a cross-sectional view of one embodiment of a mixer that comprises a hollow tube and an insert comprising ridges and valleys. 
         FIG. 37  is a cross-sectional view of one embodiment of a mixer comprising a rod and a hologram. 
         FIG. 38A  is a front plan view showing one embodiment of a circular mixer comprising a substantially optically transmissive outer layer having an index of refraction that varies around the perimeter of the mixer. 
         FIG. 38B  is a front plan view showing one embodiment of a square mixer comprising a substantially optically transmissive outer layer having an index of refraction that varies around the perimeter of the mixer. 
         FIG. 39A  is a cross-sectional view of one embodiment of a mixer comprising a core and clad that varies in thickness relative to a surface of the core. 
         FIG. 39B  is a cross-sectional view of one embodiment of a mixer comprising a core and clad comprising a substantially constant thickness relative to a surface of the core. 
         FIG. 40  schematically illustrates a partial perspective view of one embodiment of a rippled sidewall from which a collimated beam of light is reflected into a semicircular distribution pattern. 
         FIG. 41A  is a partial cross-sectional view of one embodiment of a sidewall comprising ridges and valleys having highly rounded vertices with sloping surfaces therebetween. 
         FIG. 41B  is a plot of the illuminance distribution obtained by reflecting a beam of collimated light from the sidewall of  FIG. 41A . 
         FIG. 41C  is a partial cross-sectional view of one embodiment of a sidewall comprising ridges having pointed vertices, valleys having rounded vertices, and sloping surfaces therebetween. 
         FIG. 41D  is a plot of the illuminance distribution obtained by reflecting a beam of collimated light from the sidewall of  FIG. 41C . 
         FIG. 41E  is a partial cross-sectional view of one embodiment of a sidewall comprising ridges having vertices that are more pointed than those shown in  FIG. 41C , valleys having vertices that are more gently rounded than those shown in  FIG. 41C , and sloping surfaces therebetween. 
         FIG. 41F  is a plot of the illuminance distribution obtained by reflecting a beam of collimated light from the sidewall of  FIG. 41E . 
         FIG. 42  is a schematic illustration of a system comprising two rippled mixers coupled with each other via transfer optics. 
         FIG. 43  is a schematic illustration of one embodiment of a rippled mixer coupled with a diffuser. 
         FIG. 44A  is a schematic illustration of a top plan view of one embodiment of a mixer array comprising a plurality of rippled mixers. 
         FIG. 44B  is a schematic illustration of a side elevation view of the mixer array of  FIG. 44A . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Many illumination designs benefit from homogenized light. Accordingly, efforts have been made to obtain substantially uniform illuminance distributions from light sources, such as light emitting diodes (LEDs), that produce non-uniform illuminance distributions. One known method of achieving this goal employs mixing rods. In many embodiments, flux from a light source is transferred to an input end of a mixing rod. The flux propagates through the mixing rod, typically reflecting from the sidewalls of the mixing rod one or more times. In certain embodiments, coupling a light source that produces a non-uniform illuminance distribution with the input end of the mixing rod produces a substantially uniform illuminance distribution at an output end of the mixing rod. 
     Certain mixing rod configurations are particularly effective in achieving substantially uniform illuminance distributions. For example, straight rods having rectangular or hexagonal cross-sections are known to work well. Such configurations produce rectangular and hexagonal beam patterns, respectively. However, circular beam patterns are preferred in many applications, such as flashlights, spotlights, fiber illuminators, and projection systems with circular pupils. Unfortunately, circular straight rods generally provide inferior spatial mixing as compared with rectangular or other faceted configurations. Accordingly, hexagonal mixing rods are often used in place of circular mixing rods in order to approximate a circular beam pattern while achieving the advantages of a mixing rod having planar sidewalls. Consequently, there is a need for mixing rods that produce circular beam patterns that have substantially uniform illuminance distributions. 
     Furthermore, known mixing rod configurations have various limitations. For example, even when the illuminance distribution at the output face of a mixing rod is substantially uniform, the distribution obtained by performing a virtual defocus of the light exiting the mixing rod back to the input face has non-uniformities. Similar non-uniformities exist in the angular distribution of the light exiting the mixing rod. In illumination systems that employ relays, for example, the input face of the mixing rod is often approximately conjugate to the pupil of the relay optics. As a result of multiple reflections of the light source in the mixing rod, the light distribution produced at the pupil plane generally contains a non-uniform illuminance distribution. This distribution may comprise multiple images of the light source, producing a regular pattern of bright and dark regions. This effect is similar to what is seen when looking into a kaleidoscope, hence such illuminance distributions are referred to herein as “kaleidoscope illuminance distributions.” There is a need for mixing rods that produce more uniform illuminance distributions at the pupil, and for mixing rods that produce more uniform intensity distributions. 
     In addition, the performance of square, hexagonal, and other known configurations of effective mixing rods tends to degrade as the length of the mixing rods is decreased. For example, uniformity of the kaleidoscope illuminance distribution from a mixing rod tends to diminish with decreasing length because fewer images of the non-uniform illuminance distribution at the input face are obtained. Furthermore, shorter mixing rods generally become increasingly sensitive to alignment of a light source with the input face. This phenomenon is particularly disadvantageous for systems that couple multiple LEDs with mixing rods. For example, some systems situate red, green, and blue LEDs at the input face of a mixing rod in order to produce white light. Because the LEDs are generally offset from a center point of the input face, the uniformity of the distribution of red, green, and blue light and the quality of the white light produced by the systems decreases as the mixing rod becomes shorter. 
     Certain embodiments disclosed herein reduce or eliminate the above-noted problems. In some instances, providing ridges and valleys along an exterior or interior perimeter of a mixing rod substantially improves the performance of the mixing rod. Other advantageous embodiments are also disclosed. 
       FIG. 1  schematically illustrates a light source  10  coupled with a mixer  20 . In the illustrated arrangement, the mixer  20  comprises an input face  21 , an output face  22 , and a central region  23  extending between the input face  21  and the output face  22 . 
     The light source  10  can comprise any suitable light-producing device, such as one or more fluorescent lamps, halogen lamps, incandescent lamps, discharge lamps, light emitting diodes, or laser diodes. In some embodiments, the light source  10  comprises the output of one or more fiber optic lines. In certain embodiments, the light source  10  is configured to generate multi-chromatic light (e.g., white light), while in other embodiments the light source  10  is capable of generating substantially monochromatic light at one or more selected wavelengths. The light source  10  can produce coherent or incoherent light. Other forms of light-producing devices can also be used. In some arrangements, the light source  10  comprises a combination of light-producing devices. In certain configurations, the light source  10  comprises a plurality of different color LEDs. In further configurations, the light source  10  comprises an array of red, green, and blue LEDs (an “RGB LED array”). In some embodiments, the light source  10  comprises one or more phosphor LEDs or LEDs (e.g., UV LEDs) with a phosphor. For example, phosphors can be used to convert the LED output to a more desired wavelength such as blue to yellow (using e.g., YAG). 
     The mixer  20  can have a variety of configurations. In many embodiments, the mixer  20  comprises a light pipe. In some arrangements, the mixer  20  comprises a hollow tube with one or more sidewalls. Accordingly, in some instances, a surface of the mixer  20  is lined with a substantially reflective film, coated with a substantially reflective material, or otherwise configured to reflect a substantial portion of light incident thereon. In some arrangements, a surface of the mixer  20  is lined with a substantially reflective film comprising metal or a dielectric multilayer coating. Other reflective coatings such as paint or pigment, e.g., diffuse white paint, may also be used. Still other arrangements for increasing the reflectivity of surfaces of the mixer  20  may also be used. In some embodiments, the hollow tube and accordingly the mixer is rigid, and in other embodiments, the hollow tube is flexible. 
     In other configurations, the mixer  20  comprises a solid light conduit. Accordingly, in some arrangements, the mixer  20  comprises a substantially optically transmissive material, such as glass or plastic or some other polymeric material. Other materials may also be used. In some embodiments, the material is rigid, and in other embodiments, the material is flexible. Light may propagate within the mixer  20  being reflected one or more times from the interface between the optically transmissive material and ambient medium, e.g., air, as a result of total internal reflection. In some embodiments, reflections within solid light conduits have smaller absorption losses than do reflections within hollow light conduits having reflective sidewalls. In certain embodiments, solid light conduits are thus more advantageous. 
     In many instances, the angle of light from the light source  10  is reduced when coupled into the solid mixer  20  as a result of refraction. Accordingly, some solid mixers  20  are longer than similarly sized and dimensioned hollow mixers  20  in order to achieve similar levels of mixing. In some instances, solid mixers  20  attenuate less energy than do hollow, reflective mixers  20 , and are therefore advantageous for certain applications. 
     As illustrated in  FIG. 1 , a longitudinal or optical axis  13  extends through the mixer  20  from the input surface  21  to the output surface  22 . The optical axis  13  shown in  FIG. 1  is parallel to the z-axis. In the embodiment shown, the mixer  20  is linear, although the mixer may bend and have different shapes as discussed more fully below. In some embodiments, the mixer  20  comprises a cylinder. 
     The shape of the cross-section of the mixer  20  that is orthogonal to the optical axis  13  can have a wide variety of shapes, including, for example, a circle, ellipse, oval, rectangle, pentagon, hexagon, or other polygon. Additionally, in some arrangements, the cross-sectional shape of the mixer  20  has a rippled perimeter that includes multiple elongate ridges and valleys. This rippled perimeter results from one or more rippled sidewalls of the mixers, which will be discussed more fully below. As used herein, these sidewalls are said to be ribbed as a result of the multiple elongate ridges and valleys. The terms ribbed and rippled, e.g., ribbed sidewalls and rippled sidewalls, are used interchangeably herein. 
     In some arrangements, the cross-sectional shape and area of the mixer  20  are constant along the full length of the mixer, as measured along the optical axis  13  or z axis. In such arrangements, the input face  21  and the output face  22  are the same shape and size. In other arrangements, the cross-sectional shape and/or area of the mixer  20  varies along the length thereof, as measured along the z axis. In certain of such configurations, the input face  21  and the output face  22  may vary with respect to each other in shape and/or size. In some instances, the input face  21  defines a rectangle or square and the output face  22  defines a circle or vice versa. Other arrangements are also possible. 
     The mixer  20  can have a wide variety of lengths. In some configurations, the length of the mixer  20  is larger than a width thereof (e.g., a measurement along the orthogonal x or y axis). In various configurations, the length-to-width ratio for a mixer  20  that comprises a hollow tube is about 11:1 or less, about 10:1 or less, 9:1 or less, about 8:1 or less, about 7:1 or less, about 6:1 or less, about 5:1 or less, about 4:1 or less, or about 3:1 or less. In other configurations, the length-to-width ratio for a mixer  20  that comprises a solid conduit is at least about 16:1 or less, about 15:1 or less, about 14:1 or less, about 13:1 or less, about 12:1 or less, about 11:1 or less, about 10:1 or less, about 9:1 or less, about 8:1 or less, about 7:1 or less, about 4:1 or less, or about 1.5:1 or less. Other sizes and configurations are also possible. 
       FIG. 2  schematically illustrates the light source  10  coupled with the mixer  20  via a reflector  12 . In some arrangements, the reflector  12  is an elliptical collector having an ellipsoidally shaped reflective surface. In certain of such arrangements, the light source  10  can be located at one focus of the reflector  12 , and an input face  21  of the mixer  20  can be located at the other focus of the reflector  12 . The reflector  12  can assume a variety of other shapes and configurations. Additional reflective devices or other optical elements (e.g., lenses, diffractive elements, etc.) can be used to couple the light source  10  with the mixer  20 . In certain configurations, no optical element is used. For example, in some arrangements, the light source  10  is placed at or adjacent to the input face  21  of the mixer  20 . Numerous other devices and methods can be employed to couple the light source  10  with the mixer  20 . 
       FIG. 3  schematically illustrates one embodiment of a mixer  120  comprising a rippled sidewall  15 . In certain embodiments, the rippled sidewall  15  is substantially curved and, in some embodiments, can form a cylinder. In some embodiments, the mixer  120  comprises a right circular cylinder. In many embodiments, light propagating within the mixer  120  is reflected more times than it would be within a mixer that has the same dimensions as the mixer  120  but does not have rippled sidewalls. In certain embodiments, the mixer  120  outputs a substantially circular beam at the output face  22 . 
     In some embodiments, the rippled sidewall  15  comprises a plurality of elongate ridges  24  and valleys  25  that extend along the length of the mixer. In the illustrated embodiment, the mixer  120  comprises a solid light conduit. The mixer  120  has a length, as measured along the z axis (defined in  FIG. 3 ), ten times greater than its diameter, as measured in the xy plane (defined in  FIG. 3 ). In the illustrated embodiment, the ridges  24  and the valleys  25  extend from the input face  21  to the output face  22  of the mixer  120  along the central region thereof  23 . Each of the ridges  24  and the valleys  25  has a constant height along the length of the mixer  120 . As shown, the ridges  24  and the valleys  25  run parallel to the length of the mixer  120 . As discussed below, numerous other configurations of the ridges  24  and the valleys  25  are possible. Accordingly, discussion of the illustrated embodiment is for illustrative purposes only, and should not be construed as limiting. 
       FIG. 4  schematically illustrates a cross-sectional view of the mixer  120  orthogonal to the optical axis  13  or z-direction. In the illustrated embodiment, the rippled sidewall  15  comprises ridges  24  and the valleys  25  around the full perimeter of the mixer  120 . The mixer  120  comprises thirty-two ridges  24  and thirty-two valleys  25 . Sixteen of the ridges  24  are formed by two substantially flat sloping surfaces  26  that meet at a vertex. The other sixteen ridges  24  are formed by curved sloping surfaces  26  that meet at a vertex. The two varieties of the ridges  24  alternate around the perimeter of the mixer  120 , e.g., each ridge  24  of one variety is between two ridges  24  of the other variety 
     In certain embodiments, the height of each ridge  24 , i.e., the shortest distance between the vertex of the ridge  24  and a straight line connecting the vertices of two adjacent valleys  25 , is between about 1 micron and about 1000 microns, between about 0.01 millimeters and 0.20 millimeters, between about 0.05 millimeters and about 0.15 millimeters, or between about 0.10 and about 0.13 millimeters. In other embodiments, the height is greater than about 0.01 millimeters, greater than about 0.05 millimeters, or greater than about 0.10 millimeters. In still other embodiments, the height is less than about 0.20 millimeters, less than about 0.15 millimeters, or less than about 0.13 millimeters. In some embodiments, the height is about 0.12 millimeters. 
     In various embodiments, the height-to-width ratio is greater than about 1:500, greater than about 1:300, greater than about 1:250, greater than about 1:200, greater than about 1:150, or greater than about 1:100. In some embodiments, the height-to-width ratio is less than about 1:100, less than about 1:150, less than about 1:200, less than about 1:250, less than about 1:300, or less than about 1:500. 
     In various embodiments, the width or spacing of each ridge  24 , e.g., the distance between the vertices of two adjacent valleys  25 , is between about 1 micron and about 1000 microns, between about 0.10 millimeters and 0.50 millimeters, between about 0.30 millimeters and 0.45 millimeters, or between about 0.35 millimeters and 0.40 millimeters. In some embodiments, the width is greater than about 0.25 millimeters, greater than about 0.30 millimeters, or greater than about 0.35 millimeters. In other embodiments, the width is less than about 0.50 millimeters, less than about 0.45 millimeters, or less than about 0.40 millimeters. In some embodiments, the width is about 0.38 millimeters. 
     In various embodiments, the ridge-width-to-mixer-width ratio is greater than about 1:100, greater than about 1:50, or greater than about 1:30. In some embodiments, the ridge-width-to-mixer-width ratio is less than about 1:30, less than about 1:50, or less than about 1:100. In some embodiments, the ridges  24  have an average spacing of between about 1 percent and about 30 percent of the width of the mixer  120 . 
     As described more fully below, the mixer  120  can comprise more or fewer ridges  24  and valleys  25 , and numerous heights, widths, and configurations of the ridges  24  and valleys  25  are possible. 
     Light input into the input face  21  propagates along an optical path down the mixer through the central region  23  between the input face and the output face  22 . Much of this light is reflected from the ribbed sidewalls and in particular reflects from the sloping surfaces forming the ridges  24  and valleys  25 . This light reflects from the sloping surfaces is redirected at a different azimuthal direction toward the output end thereby mixing said light at the output end. 
       FIGS. 5A-5F  demonstrate that circular mixers having rippled sidewalls  15 , such as the mixer  120  depicted in  FIGS. 3 and 4 , can be effective in creating a substantially uniform illuminance distribution at an output face thereof. Each of  FIGS. 5A-5F  corresponds with an arrangement such as that schematically depicted in  FIG. 2 , wherein the reflector  12  is an elliptical collector, the light source  10  is located at one focus of the reflector  12 , and the center of the input face  21  is located at the other focus of the reflector  12 , and wherein two distinct mixers are used separately for purposes of comparison, as detailed below. 
       FIGS. 5A and 5B  illustrate the illuminance distribution at the input face  21  of a mixer.  FIG. 5A  is a spatial plot representing illuminance values at the input face  21 , and  FIG. 5B  is a histogram of each illuminance value measured over the surface of the input face  21 . A shading scale corresponding with both  FIGS. 5A and 5B  is shown along the left-hand side of the histogram of  FIG. 5B . The shading scale generally indicates that the lowest intensities are shaded the darkest and the highest intensities are shaded the brightest, although the gradation between these two extremes is not uniform. 
       FIG. 5A  illustrates that the illuminance distribution at the input face  21  is non-uniform. The shaded spatial plot indicates that light intensity is high toward the center of the input face  21  and drops off toward the edges of the circular perimeter of the mixer.  FIG. 5B  also indicates that the illuminance distribution of the input face  21  is non-uniform due to the large range of illuminance values that are present in the histogram. 
     The plots shown in  FIGS. 5C and 5D  depict the illuminance distribution at the output face  22  of a smooth circular mixing rod (not shown). The plots are of the same variety described with respect to  FIGS. 5A and 5B , respectively. The mixing rod has a circular cross-section orthogonal to the optical axis or longitudinal (z) direction and a length-to-diameter ratio of about 10:1. This length-to-diameter ratio is useful for comparing the performance of the smooth circular mixing rod with that of a smooth mixing rod having a square cross-section orthogonal to the optical axis or z axis. Such smooth square mixing rods are known to produce substantially uniform illuminance distributions when the length-to-width ratio thereof is about 10:1 (for example, see  FIG. 12B ) and the numerical aperture of the coupled light is about 0.5. In certain embodiments, the length-to-width ratios are smaller. In some embodiments, substantially uniform illuminance distributions can be achieved with higher numerical apertures and shorter mixing rods or lower numerical apertures and longer mixing rods. As shown in  FIGS. 5A and 5B , the illuminance distribution is non-uniform, and is not much improved over the illuminance distribution at the input face  21 . 
     The plots shown in  FIGS. 5E and 5F  are of the same variety described with respect to  FIGS. 5A and 5B , respectively, but correspond to the output of the rippled mixer  120  shown in  FIGS. 3 and 4 , which comprises a rippled sidewall  15 . As mentioned above, the rippled mixer  120  has a length-to-diameter ratio of about 10:1. Accordingly, the only difference between the smooth circular mixing rod associated with  FIGS. 5C and 5D  and the rippled mixer  120  associated with  FIGS. 5E and 5F  is the ridges  24  and the valleys  25  disposed around the mixer  120 . 
       FIGS. 5E and 5F  depict the illuminance distribution at the output face  22  of the rippled mixer  120 . As shown, the illuminance distribution is substantially uniform.  FIG. 5E  demonstrates that the illuminance is fairly constant over the entire face  22 , and  FIG. 5F  comprises a much narrower and more concentrated range of illuminance values, as compared with  FIGS. 5B and 5D . Accordingly,  FIGS. 5A-5F  show that mixers comprising rippled sidewalls are able to create circular beam patterns having substantially uniform spatial illuminance distributions at the output face. In addition, in many embodiments, rippled sidewalls improve the uniformity of the angular distribution of light exiting the output face. Further, in many embodiments, rippled sidewalls improve uniformity of the color of the light at the output face. Accordingly, the foregoing and following discussion with respect to spatial illuminance distributions of rippled mixers is, in many embodiments, applicable to the angular and/or color distributions of the mixers. 
     The above-noted results satisfy a longstanding need. For example, in some instances it is desirable to couple a bundle of optical fibers with a mixing rod. Such bundles often are manufactured with round input ends. Accordingly, in some embodiments, a circular rippled mixer such as the mixer  120  can be used directly with a round bundle of fibers to provide each fiber within the bundle substantially the same flux density. 
       FIGS. 6A-6F  demonstrate that mixers having rippled sidewalls  15  can also reduce alignment sensitivity of the light source  10  with the input face  21 . Each of  FIGS. 6A-6F  corresponds with an arrangement similar to that schematically depicted in  FIG. 2 , wherein the center of the input face  21  is located at one focus of the reflector, and two distinct mixers are used separately for purposes of comparison, as detailed below; however, the light source  10  is shifted upward (i.e., along the y axis in the positive direction) within the reflector  12  and thus away from the other focus of the reflector. 
     The plots shown in  FIGS. 6A and 6B  depict the illuminance distribution at the input face  21 . As shown in  FIG. 6A , the illuminance distribution is shifted downward (i.e., along they axis in the negative direction) with respect to that shown in  FIG. 5A , and is thus off-centered. A region of high intensity is located below the center point of the input face  21  (e.g., below the optical axis), and the intensity drops off toward the edges of the mixer (e.g., the sidewalls). 
     The plots shown in  FIGS. 6C and 6D  were obtained with the smooth circular mixing rod described above with respect to  FIGS. 5C and 5D .  FIGS. 6C and 6D  depict the illuminance distribution at the output face  22  of the smooth circular mixing rod. As shown, the illuminance distribution is non-uniform, and is not much improved over the illuminance distribution shown in  FIGS. 6A and 6B . 
     The plots shown in  FIGS. 6E and 6F  were obtained with the rippled mixer  120 .  FIGS. 6E and 6F  depict the illuminance distribution at the output face  22  of the mixer  120 . As shown, the illuminance distribution is substantially uniform.  FIG. 6E  demonstrates that the illuminance is fairly constant over the entire face  22 , and  FIG. 6F  demonstrates a much narrower and more concentrated range of illuminance values, as compared with  FIGS. 6B and 6D . Furthermore, comparison of  FIGS. 6E and 6F  with  FIGS. 5E and 5F  reveals that performance of the mixer  120  is relatively unaffected by the shifted alignment of the light source  10  with respect to the input face  21 . Accordingly, providing mixers with rippled sidewalls can reduce alignment sensitivity of the mixers, which can increase the design and manufacturing tolerances and result in more robust designs. 
     While the above examples demonstrate the effectiveness of the mixer  120  schematically depicted in  FIGS. 3 and 4 , numerous other mixer and ripple configurations are possible.  FIGS. 7A and 7B  schematically illustrate two of many varieties of cross-sectional configurations of ripple sidewalls. Additional varieties are discussed in more detail below. 
       FIG. 7A  schematically illustrates one cross-sectional configuration of a rippled sidewall  15  comprising a plurality of ridges  24  and valleys  25 . Half of the ridges  24  resemble those shown in  FIG. 4 , as each is formed by two substantially planar sloping surfaces  26  that meet at a vertex. Rippled sidewalls  15  comprising substantially flat surfaces are referred to herein as “faceted.” 
     Although not always true (as detailed below), the respective shapes of ridges  24  and valleys  25  can be fairly complimentary to each other in some instances, thus the general shape of some rippled sidewalls  15  can be described in terms of ridges  24  alone. Accordingly, references to ridges  24  only are occasionally made herein when describing the general shape of certain rippled sidewalls  15 . However, such references should not be construed as limiting the principles and/or embodiments described herein to those instances where ridges  24  and valleys  25  are complimentary. 
     With continued reference to  FIG. 7A , each ridge  24  can be described in terms of an angle α defined by the sloping face  26  on either side of the ridge  24  and a line  27  extending between the vertices of the two closest valleys  25  on either side of the ridge  24 . The valleys  25  can similarly be described in terms of an angle α. Accordingly, the angle α depends on the spacing and height of the ridges  24  or the valleys  25 . For example, higher ridges  24  or closer spacing result in larger angle α. In the illustrated embodiment, each ridge  24  is symmetrical. As detailed below, other configurations are possible for faceted rippled sidewalls  15 . For example, the ridges  24  and/or valleys  25  of the rippled sidewalls  15  are not symmetrical and/or are formed by more than two substantially flat faces defining a ridge or valley. In some embodiments, the angle α is between about 15 degrees and about 65 degrees. In other embodiments, the angle α is between about 15 degrees and about 35 degrees or between about 10 degrees and about 25 degrees. Other ranges and values are possible. 
     In some embodiments, the ridges  24  or valleys  25  can be described in terms of a vertex angle ξ defined by the angle between two substantially planar sloping faces  26 . In some embodiments, the angle ξ is between about 15 degrees and about 65 degrees. Other ranges and values are possible. 
       FIG. 7B  schematically illustrates one cross-sectional configuration of a rippled sidewall  15  comprising ridges  24  and valleys  25  having rounded or curved vertices. In such configurations, the angle α can also generally describe an average angle of the ridges  24  or the valleys  25 . In the illustrated embodiment, the angle α is defined by the line  27  extending between the vertices of the two valleys  25  neighboring the ridge  24  and by a tangent line  28  extending between a vertex of a neighboring valley  25  and the vertex of the ridge  24 . As described below, numerous configurations are possible for rounded ridges  24 . For example, the ridges can be highly rounded, ellipsoidal, circular, parabolic, any combination thereof, or have any other curved shape. In various embodiments, the angle α is between about 20 degrees and about 65 degrees, between about 20 degrees and about 37 degrees, or between about 27 degrees and about 33 degrees. 
     The slope and curvature of the surfaces  26  forming the ridges  24  (and valleys) can affect the performance of the mixer  20 . However, in examples that follow, the angle α is generally associated with ridges  24  such as those shown in the illustrated embodiment. 
     With continued reference to  FIG. 7B , in some instances, the angle α is defined in terms of the height dH of the ridges  24  and the distance dC between successive ridges  24 . One particularly advantageous definition of the angle α is as follows: 
       α=arctan (2 dH/dC )
 
     In some embodiments, increasing or decreasing the number of ridges and valleys disposed around a mixer and thereby changing the distance dC, while adjusting the parameters dH such that α remains constant in the above equation, can often result in similar performances between the original mixer and the modified mixer. 
     However, in some arrangements, the number of ridges  24  disposed around a mixer affects the performance of the mixer. In certain embodiments, the number of ridges  24  is greater than 16. In other embodiments, the number of ridges  24  is greater than 50. In various other embodiments, the number of ridges  24  is greater than 100, 150, 200, 250, 300, 400, 500, 1000, 2000, 3000, or 4000. In some embodiments, the number of ridges  24  is greater than 5000. 
       FIG. 8A  schematically illustrates one embodiment of a mixer  130  coupled with the light source  10 . In the illustrated embodiment, the mixer  130  comprises a hollow tube that defines a generally cylindrical shape having a circular cross-section orthogonal to the optical axis (z-axis) therethrough. The hollow tube has a reflective coating or film covering its interior surface. The mixer  130  has a length-to-diameter ratio of about 6:1. The mixer  130  further comprises a rippled sidewall  15  having 128 ridges  24 , as schematically illustrated in  FIG. 8B . For purposes of comparison, as discussed below, in one configuration the ridges  24  are formed by planar sloping surfaces (as shown), and in another configuration the ridges  24  have rounded vertices and are formed by curved surfaces. 
     As shown in  FIG. 8A , the light source  10  is disposed at the center of the input face  21 . The light source  10  has a diameter that is one fifth the diameter of the input face  21 .  FIG. 8C  is a plot of the intensity distribution of the light source  10  of the illustrated embodiment. The light source  10  comprises a Lambertian source that is clipped such that the angular distribution for angles below about 9 degrees and above about 30 degrees is zero. As such, the light source  10  simulates the effect of a reflector that has a hole therein, such as can be found in certain flashlight configurations, and/or simulates the effect of the blockage caused by the light source itself being disposed within a reflector. Additionally, blocking low-angle rays removes light having a nearly uniform angular distribution that would otherwise pass through the mixer  20  without interacting with the sidewalls thereof, and thus may have only a small effect on the illuminance distribution at an output face  22  of the mixer  20 . 
     The following examples demonstrate that the angle α of the ridges  24  can affect the performance of the mixer  130 .  FIG. 9  is a plot of the standard deviation of the illuminance distribution, σ Illuminance , at the output face  22  as a function of the angle α for the system illustrated in  FIG. 8A . Smaller values of the standard deviation (i.e., smaller values along the y axis) correspond with greater uniformity of the illuminance distribution. For purposes of comparison, the plot comprises a curve  140  representing data obtained when the mixer  130  comprises ridges  24  having sharp vertices formed by planar facets (such as the ridges depicted in  FIG. 7A ), and a curve  141  representing data obtained when the mixer  130  comprises ridges  24  having rounded vertices formed by curved surfaces (such as the ridges depicted in  FIG. 7B ). The plot further comprises three illustrative points  142 - 144  that correspond with  FIGS. 10A and 10B ,  FIGS. 10C and 10D , and  FIGS. 10E and 10F , respectively. At the point  142 , the angle α is 0 degrees. Accordingly, the point  142  corresponds with a mixer  130  that does not comprise rippled sidewalls, and thus corresponds with a smooth circular mixing rod. Consequently, standard deviation values lower than the value at the point  142  represent illuminance distributions having greater uniformity than that produced by the smooth mixing rod. Conversely, higher standard deviation values represent less uniform illuminance distributions than that produced by the smooth circular mixing rod. 
     As shown by the curve  140 , the performance of the mixer  130  having sharp ridges  24  changes significantly with varying values of α. Rippled sidewalls  15  characterized by an angle α of about 45 degrees actually yield a less uniform illuminance distribution at the output face  22  than is obtained with a smooth circular mixing rod. As shown by the curve  141 , the mixer  20  having rounded ridges  24  is less sensitive to changing values of α. 
     As noted above,  FIGS. 10A and 10B  correspond with the point  142 .  FIG. 10A  is a spatial plot representing illuminance values at the output face  22 . As shown, the intensity is highest at the center of the output face  22  and drops off towards the edges thereof.  FIG. 10B  is a plot representing the kaleidoscope illuminance distribution obtained by defocusing the light exiting the mixer  130  back to the input face  21  of the mixer. As shown in  FIG. 10B , the kaleidoscope illuminance distribution is highly non-uniform and contains multiple bright and dark regions caused by multiple images of the light source  10 . 
       FIGS. 10C and 10D  correspond with the point  143 , which represents the lowest standard deviation value obtained with either of the mixers  130 , and hence represents the most uniform illuminance distribution. It is noted that the rippled circular mixer  130  having rounded ridges  24  produced the value at the point  143 . 
       FIG. 10C  is a spatial plot representing illuminance values at the output face  22 . As shown, the intensity is substantially uniform over the entire output face  22 .  FIG. 10D  is a plot representing the kaleidoscope illuminance distribution of the mixer  130 , which is much more uniform than that shown in  FIG. 10B . In  FIG. 10D , images of the light source  10  are substantially smeared, almost to the point of being indistinguishable from each other. Accordingly, mixers having rippled sidewalls  15  can dramatically improve the uniformity of kaleidoscope illuminance distributions. 
       FIGS. 10E and 10F  correspond with the point  144 , which represents the highest standard deviation value obtained with either of the mixers  130 , and hence represents the least uniform illuminance distribution. It is noted that the rippled circular mixer  130  having sharp ridges  24  formed by substantially flat surfaces and an angle α of about 45 degrees produced the value at the point  144 . In some embodiments, mixers having sharp ridges  24  characterized by an angle α of about 45 degrees act much like retroreflecting prisms. For example, in some embodiments, light incident on a ridge  24  is reflected from two surfaces of the ridge  24  such that light leaving the ridge  24  propagates along a path parallel to the path traced by the incident light, as viewed along the optical axis (e.g., the z axis defined in FIGS.  8 A and  10 A- 10 E) of the mixer  130 . 
       FIG. 10E  is a spatial plot representing illuminance values at the output face  22 . As shown, the intensity is highest at the center of the output face  22  and drops off towards the edges thereof. The illuminance distribution is even more peaked than that shown in  FIG. 10A .  FIG. 10F  is a plot representing the kaleidoscope illuminance distribution of the mixer  130 , which resembles the distribution shown in  FIG. 10B . 
     In certain embodiments, improved kaleidoscope illuminance is achieved with a diffuser at the output face of the mixer. In some embodiments, the diffuser is a low angle diffuser. For example, in certain embodiments the diffuser is configured to diffuse collimated light incident perpendicular to the diffuser into an angular distribution having a full-width half-maximum (FWHM) of about 30 degrees or less, about 6 degrees or less, or about 3 degrees or less. Other diffuser angles and diffuser distributions are also possible. In certain embodiments, the FWHM is approximately equal to the angular size of the input face of the mixer, as seen from the output face of the mixer. In some embodiments, the FWHM follows the equation: 
       FWHM≈ arctan (D/L)
 
     where “D” is the input size of the mixer and “L” is the length of the mixer. 
     In certain advantageous embodiments, a rippled mixing rod having a substantially uniform transverse cross section along a length thereof is coupled with a diffuser. In many embodiments, the diffuser homogenizes the light that does not hit the sidewalls of the rippled mixer. In some embodiments, the scatter angle of the diffuser is small compared to the numerical aperture of the flux coupled out of the mixer. In certain of such embodiments, the diffuser does not significantly increase the numerical aperture of the output light. As used herein, the term “diffuser” is a broad term used in its ordinary sense, and includes, without limitation, randomly textured surfaces and structured textured surfaces, such as lens arrays, holographic diffusers, and bulk scatter in a thin volume. In some embodiments, bulk scatter can be applied throughout the volume of the mixer or a portion thereof. 
     In various embodiments, a rippled mixer having a solid configuration, such as the mixer  120  or the mixer  130 , comprises a rigid material or a flexible material. In some embodiments, the mixer comprises optical grade acrylic. In further embodiments, the mixer comprises elongate ridges and valleys that extend along and run parallel to the full length of the mixer. In some embodiments, the ridges and the valleys have a substantially constant height of about 0.015 millimeters, about 0.017 millimeters, about 0.020 millimeters, or about 0.025 millimeters. In some embodiments, the height is greater than about 0.015 millimeters, greater than about 0.017 millimeters, greater than about 0.020 millimeters, or greater than about 0.025 millimeters. In other embodiments, the height is less than about 0.025 millimeters, less than about 0.020 millimeters, less than about 0.017 millimeters, or less than about 0.015 millimeters. In certain embodiments, the mixer comprises an input face and an output face that each has a diameter of about 3.0 millimeters. In other embodiments, the diameter is greater than about 3.0 millimeters. In further embodiments, the mixer has a length of about 9.0 millimeters, about 25.0 millimeters, or about 100 millimeters. In some embodiments, the mixer has a length greater than about 9.0 millimeters, greater than about 25.0 millimeters, or greater than about 100 millimeters. Other values are also possible. 
     In certain embodiments, a rippled mixer, such as the mixer  120 , comprises an optical fiber. In some embodiments, the fiber has a diameter (or a cross-sectional width) of at least 3.0 millimeters. In some embodiments, the fiber comprises a core. In certain embodiments, the core has a diameter (or a cross-sectional width) of at least 3.0 millimeters. 
     The above findings are presented herein primarily for illustrative purposes and should not be construed as limiting. As described above, in some instances circular mixers with elongate ridges and valleys can dramatically improve performance of the mixers. In some instances, such rippled circular mixers create illuminance distributions having uniformity comparable to or even better than that of square mixing rods having comparable dimensions. For example, in various embodiments, rippled circular mixers and square mixing rods having hollow configurations both provide substantially uniform illuminance distributions when the input light distribution has a numerical aperture of at least about 0.24 and when the length-to-width ratio of the mixer is at least about 12:1, when the numerical aperture is about at least 0.45 and the length-to-width ratio is at least about 6:1, and when the numerical aperture is at least about 0.7 and the length-to-width ratio is at least about 3:1. For various embodiments comprising a solid light conduit having an index of refraction of about 1.5, the same holds true when the numerical aperture is at least about 0.24 and the length-to-width ratio is at least about 18:1, when the numerical aperture is at least about 0.45 and the length-to-width ratio is at least about 9:1, when the numerical aperture is at least about 0.7 and the length-to-width ratio is at least about 4.5:1, and when the numerical aperture is at least about 1.0 and the length-to-width ratio is at least about 3:1. 
     In some embodiments, a rippled circular mixer having a solid configuration provides a substantially uniform illuminance distribution at its output face when its input face has a numerical aperture of about 0.5 and when the mixer has a length-to-width ratio of about 10:1. In some embodiments, similar results can be achieved when the numerical aperture is about 1.0 and the length-to-width ratio is about 3:1. In many embodiments, substantially uniform illuminance distributions can be achieved as the numerical aperture of a mixing rod is increased and the length-to-width ratio thereof is reduced. 
     Shorter mixers are also able to provide excellent uniformity, but the general trend is that shorter lengths become increasingly sensitive to the light distribution at the input face  21 . Nevertheless, the ability of rippled, circular mixers to produce a uniform, circular beam when the length-to-width ratio of the mixers is less than about 6:1 is a particularly advantageous result. Shorter mixers allow systems that are more compact, lighter, and potentially less expensive. Furthermore, in some instances, circular mixers have kaleidoscope illuminance distributions that are more uniform than those produced by square mixing rods. 
     As with the circular mixers  20  described above, the performance of mixers having square cross-sections can be improved by adding rippled sidewalls. Certain improvements effected by the addition of elongate ridges and valleys to the sidewalls of the square mixing rods are shown in the following specific examples, which are meant to be illustrative and in no way limiting. 
       FIG. 11  schematically illustrates the cross-section of one embodiment of a rippled square mixer  150 . In the illustrated embodiment, the mixer  150  comprises four rippled sidewalls  15 . In certain embodiments, one or more of the rippled sidewalls  15  are substantially planar. In some embodiments, the one or more rippled sidewalls  15  form a right rectangular prism or a right square prism. In many embodiments, light propagating within the mixer  150  is reflected more times than it would be within a mixer that has the same dimensions as the mixer  150  but does not have rippled sidewalls. In certain embodiments, the mixer  150  outputs a substantially rectangular or substantially square beam at the output face  22 . 
     In some embodiments, the one or more sidewalls  15  comprise a plurality of elongate ridges  24  and valleys  25 . In some embodiments, the ridges  24  and the valleys  25  extend along the length of the mixer  150 . In the illustrated embodiment, the cross-sectional shape and area of the mixer  150  is constant along the entire length of the mixer  150 . The height and width of the mixer  150  are each about 4 millimeters. The mixer  150  comprises thirty-two rounded ridges  24  around its perimeter that run parallel to the length of the mixer  150 . The vertices of the ridges  24  are spaced from each other by approximately 0.5 millimeters and have a constant height of about 120 microns along the length of the mixer  150 . Other sizes, dimensions, and configurations of the mixer  150  are possible. 
     Each of  FIGS. 12A-12F  corresponds with an arrangement such as that schematically depicted in  FIG. 2 , wherein the reflector  12  is an elliptical collector, the light source  10  is located at one focus of the reflector  12 , and the center of the input face  21  of a mixer is located at the other focus of the reflector  12 , and wherein two distinct mixers are used separately for purposes of comparison, as detailed below. 
       FIG. 12A  is a plot of the illuminance distribution at the input face  21  of a mixer in such an arrangement. As with the plot shown in  FIG. 5A , the illuminance is highest at the center of the input face  21 , and drops off toward the edges thereof. 
       FIGS. 12B and 12C  correspond with a smooth mixing rod (not shown) having a square cross-section orthogonal to the optical axis or longitudinal (z) direction. The smooth square mixing rod is a solid rod having an index of refraction of about 1.5 and a length-to-width ratio of about 10:1.  FIG. 12B  is a plot illustrating the illuminance distribution at the output face  22  of the smooth square mixing rod. As shown, the illuminance distribution is substantially uniform.  FIG. 12C  is a plot representing the kaleidoscope illuminance distribution obtained by defocusing the light exiting the mixing rod back to the input face  21  of the mixing rod. As shown in  FIG. 12C , the kaleidoscope illuminance distribution is highly non-uniform and contains multiple bright and dark regions produced by multiple images of the light source  10 . 
       FIGS. 12D and 12E  correspond with one embodiment of a rippled mixer similar to the rippled mixer  150 . The rippled mixer comprises a solid rod having an index of refraction of about 1.5 and a length-to-width ratio of about 10:1. The cross-sectional shape and area of the mixer is constant along the entire length thereof. The height and width of the mixer are each about 4 millimeters. The mixer comprises 128 rounded ridges  24  around its perimeter (thirty-two ridges per side) that run parallel to the length of the mixer. The vertices of the ridges  24  are spaced from each other by approximately 0.125 millimeters and have a constant height of about 30 microns along the length of the mixer. The only difference between the smooth square mixing rod associated with  FIGS. 12B and 12C  and the rippled square mixer associated with  FIGS. 12D and 12E  is the ridges  24  and the valleys  25  disposed around the rippled mixer. 
       FIG. 12D  is a plot illustrating the illuminance distribution at the output face  22  of the rippled mixer. As shown, the illuminance distribution is substantially uniform. This is a particularly advantageous and unexpected result. Whereas smooth square mixing rods are known to produce uniform illuminance distributions when the mixing rods are sufficiently long, the common practice has been to avoid rounding the corners of the mixing rods. In general, smooth square mixing rods with rounded corners do not perform as well as smooth square mixing rods with sharp corners. However, the addition of ridges to square mixing rods—even rounded ridges—yields illuminance distributions with uniformities comparable to those obtained with smooth square mixing rods that have sharp corners. Furthermore, various examples described below illustrate that rippled sidewalls can actually provide illuminance distributions having uniformities superior to those provided by smooth square mixing rods. 
     For example, adding ridges to square mixing rods can improve the uniformity of kaleidoscope illuminance distributions.  FIG. 12E  is a plot representing the kaleidoscope illuminance distribution obtained by defocusing the light exiting the rippled square mixer back to the input surface of the mixer. The kaleidoscope illuminance distribution is much more uniform than that shown in  FIG. 12C , which was obtained with a smooth square mixing rod. In  FIG. 12E , the images of the light source  10  are rotationally smeared. 
       FIG. 12F  is a plot of the standard deviation, σ, of the illuminance distribution at the output face  22  of the rippled square mixer as a function of the angle α. For purposes of comparison, the plot comprises a curve  160  representing data obtained when the rippled mixer comprises ridges  24  having sharp vertices formed by planar facets, such as the ridges  24  depicted in  FIG. 7A , and a curve  161  representing data obtained when the rippled mixer comprises ridges  24  having rounded vertices formed by curved surfaces, such as the ridges  24  depicted in  FIGS. 7B and 11 . When the angle α is 0 degrees, the rippled mixers correspond with a smooth square mixing rod. Consequently, standard deviation values lower than that at which α equals 0 degrees represent illuminance distributions having better uniformity than that produced by a smooth square mixing rod. Conversely, higher standard deviation values represent less uniform illuminance distributions than that produced by a smooth square mixing rod. 
     As shown by the curve  160 , the performance of the mixer comprising ridges  24  having sharp vertices formed by planar facets varies significantly with different values of α, and generally provides less uniform distributions than does a smooth square mixing rod. As with the circular mixer  130  comprising ridges  24  with sharp vertices above, the square mixer with sharp vertices creates the least uniform illuminance distribution at an angle α of about 45 degrees. As shown by the curve  161 , the mixer having ridges  24  with rounded vertices yields far more consistent results with changing values of α, and, at an angle α of about 30 degrees, actually provides more uniform distributions than does a smooth square mixing rod. 
       FIG. 13A  schematically illustrates one embodiment of a mixer  170  coupled with the light source  10 , which is similar to the arrangement described above with respect to  FIG. 8A . In the illustrated embodiment, the mixer  170  comprises a hollow square tube having a substantially reflective coating or film covering its interior surface. The mixer  170  has a height of about 2 millimeters, a width of about 2 millimeters, and a length-to-width ratio of about 6:1. In one embodiment, the mixer  170  comprises ridges  24  having sharp vertices formed by planar facets, and in another embodiment, the mixer  170  comprises ridges  24  having rounded vertices formed by curved surfaces. For the two embodiments just described, the mixer  170  comprises 128 rounded ridges  24  around its perimeter (thirty-two ridges per side) that run parallel to the length of the mixer. The vertices of the ridges  24  are spaced from each other by approximately 0.0625 millimeters and have a constant height along the length of the mixer. The light source  10  is approximately at the center of the input face  21  of the mixer  170  and has diameter that is one fifth the width of the input face.  FIG. 13B  is a plot of the intensity distribution of the light source  10  of the illustrated embodiment. The light source  10  comprises a Lambertian source similar to that described above with respect to  FIG. 8C . 
       FIG. 13C  is a plot of the standard deviation, σ, of the illuminance distribution at the output face  22  of the mixer  170  of  FIG. 13A  as a function of the angle α, and is similar to the plot shown in  FIG. 12F . The plot comprises a curve  180  representing data obtained when the mixer  170  comprises ridges  24  having sharp vertices formed by planar faces and a curve  181  representing data obtained when the mixer  170  comprises ridges  24  having rounded vertices formed by curved surfaces. When the angle α is 0 degrees, the mixers  170  correspond with a smooth square mixing rod. Accordingly, standard deviation values lower than that at which α equals 0 degrees represent illuminance distributions having improved uniformity over those produced by a smooth square mixing rod. Conversely, higher standard deviation values represent less uniform illuminance distributions. 
     As demonstrated by the curve  180 , illuminance distributions obtained from the mixer  170  comprising ridges  24  having sharp vertices formed by planar facets vary widely with varying values of α, and are generally less uniform than those obtained from a smooth square mixing rod. Planar facets with an angle α between 10 and 25 degrees can provide reasonable uniformity for this example. As with the circular mixer  130  comprising ridges  24  with sharp vertices and the square mixer  150  comprising ridges  24  with sharp vertices discussed above, the square mixer  170  comprising ridges  24  with sharp vertices creates the least uniform illuminance distribution at an angle α of about 45 degrees. 
     As demonstrated by the curve  181 , the square mixer  170  comprising ridges  24  having rounded vertices yields far more consistent results with changing values of α than does the square mixer  170  comprising ridges  24  with sharp vertices. Also, the square mixer  170  comprising ridges  24  with rounded vertices yields a more uniform illuminance distribution, as compared with a smooth square mixing rod, over a wider range of angles than does the square mixer  150  comprising ridges  24  with rounded vertices described above (with respect to  FIG. 12F ). 
     Rippled square mixers such as those just described generally have more uniform kaleidoscope illuminance distributions than do non-rippled square mixing rods. Rippled sidewalls tend to rotationally smear images of the non-uniform illuminance distribution of the light source, resulting in a less regular pattern of bright and dark regions. Furthermore, in some cases, rippled sidewalls comprising ridges with rounded vertices improve kaleidoscope illuminance distribution uniformity to a greater extent than do rippled sidewalls comprising ridges with sharp vertices. In certain instances, mixers comprising rippled sidewalls can be shorter than smooth mixers due to the improved kaleidoscope illuminance distributions that result. 
     While the above examples have focused on mixers comprising circular and square cross-sections and further comprising rippled sidewalls, it is noted that numerous other mixer configurations can benefit from rippled sidewalls. Some examples include mixers having rectangular, pentagonal, trapezoidal, cross-shaped cross-sections, although the possible embodiments should not be limited to these. In many instances, mixer configurations having round (e.g., non-faceted) cross-sections particularly benefit from rippled sidewalls, since mixers of this variety often produce highly non-uniform illumination distributions. 
     For example,  FIGS. 14A-14D  demonstrate the improvement in illuminance distribution resulting from adding rippled sidewalls to a smooth mixer having an oval-shaped cross-section.  FIG. 14A  schematically depicts the cross-section of a smooth mixer  191  without rippled sidewalls, and  FIG. 14B  is a plot of the illuminance distribution at an output face of the mixer  191 .  FIG. 14C  schematically depicts the cross-section of a rippled mixer  192  comprising a rippled sidewall  15  having ridges  24  and valleys  25 , and  FIG. 14D  is a plot of the illuminance distribution at an output face of the rippled mixer  192 . As shown, the rippled mixer  192  provides a more uniform illuminance distribution than does the smooth mixer  191 . In further embodiments, the mixer  192  comprises a substantially elliptical cross-section. 
       FIG. 15  schematically depicts the cross-section of one embodiment of a rippled mixer  195  comprising rippled sidewalls  15  having ridges  24  and valleys  25 . The mixer  20  comprises a first portion  197  that is substantially oval-shaped and a second portion  198  that is substantially pentagonal. Four sides of the pentagon are planar and the fifth side of the pentagon is replaced with the oval portion  197 . Accordingly, the mixer  195  is a composite of faceted and rounded sides and shapes. The rippled sidewalls  15  improve the illuminance distribution of the composite mixer  195 . Many composite mixer configurations can have improved illuminance distributions through the inclusion therein of rippled sidewalls  15 . 
       FIG. 16A  schematically illustrates one embodiment of a mixer  200  having smooth sidewalls and comprising an input face  21  and an output face  22  that have different areas. Accordingly, the cross-sectional area changes along the length of the mixer  200 . In some embodiments, the shape of the cross-section of the mixer  200  also changes along the length of the mixer  200 . In the illustrated embodiment, the input face  21  and the output face  22  are each rectangular, and the area at the input face  21  is smaller than the area at the output face  22 . In certain arrangements, the mixer  200  provides an angle-to-area conversion of the distribution of light entering the input face  21 . For example, in some instances, a change in the cross-sectional area along the length of the mixer  200  causes the angular distribution at the output face  22  to be narrower or wider than the angular distribution at the input face  21 . In certain embodiments, for example, etendue is conserved. Thus, the numerical aperture at the input face  21  may be larger than the numerical aperture at the output  22  end because the area at the input face is smaller than the area at the output face. In many instances, however, the mixer  200  does not provide uniform illuminance distributions at the output face  22 . This is especially true when the length of the mixer  200  is relatively short. 
       FIG. 16B  schematically illustrates a mixer  204  similar to the angle-to-area converting mixer  200  that further comprises rippled sidewalls  15  having ridges  24  and valleys  25 . In certain embodiments, the ridges  24  and the valleys  25  have a constant height and run parallel to the length of the mixer  204 . In many embodiments, the illuminance distribution of the rippled mixer  204  is more uniform than that of the mixer  200 . As describe above, both the areas and the size of the angular spread of light at the input and output are different. In certain preferred embodiments, for example, etendue is generally conserved. Thus, the small area of the input face is associated with a larger numerical aperture while the large area at the output face is associated with a small numerical aperture. Because of the inverse relationship between angle and area, these structures may be referred to herein as angle-to-area converters. 
       FIG. 16C  schematically illustrates a mixer  208  comprising an input face  21  and an output face  22  that have different shapes and sizes. The input face  21  is oval-shaped, and the output face  22  is rectangular and has a larger area than the input face  22 . The mixer includes a central region between the input face  21  and the output face  22  that transforms from an oval cross-section toward the input face to a rectangular cross-section toward the output face. In some embodiments, the mixer  208  provides input light with an angle-to-area conversion. In certain embodiments, for example, etendue is conserved. The mixer  208  comprises rippled sidewalls  15  having ridges  24  and valleys  25  that improve the illuminance distribution at the output face  22 . Numerous other arrangements are possible for the input face  21 , the output face  22 , the central region therebetween, and/or the rippled sidewalls  15 . 
     In certain advantageous embodiments, rippled angle-to-area converting mixers, such as the mixers  204 ,  208  just described, not only permit angle-to-area conversion of input light, but also homogenize the input light, as further discussed below. In some configurations, rippled sidewalls are advantageously included in angle-to-area converting mixers that are used to produce white light from RGB LED arrays. In certain arrangements, the rippled sidewalls reduce sensitivity of the mixers to alignment with the light source at the input face. The rippled sidewalls can also reduce the length required for the mixers to produce a substantially uniform illuminance distribution, as illustrated by certain embodiments described herein. 
       FIG. 17A  schematically depicts one embodiment of an angle-to-area converting mixer  210  coupled with the light source  10 . As noted above, the light source  10  can comprise a wide variety of light producing devices. In the illustrated embodiment, the light source  10  comprises an RGB LED array having four sources  221 - 224  disposed in two columns and two rows. In some arrangements, two of the sources  221 ,  222  are green LEDs, one source  223  is a red LED, and the other source  224  is a blue LED. In some embodiments, the array comprises one, two, three, four, five, or six rows of LEDs, and in further embodiments, the array comprises one, two three, four, five, or six columns of LEDs. Other numbers and combinations of sources are possible for the LED array. Accordingly, in some embodiments, the array comprises a plurality of rows and/or a plurality of columns of light emitting diodes. In some embodiments, the LED array comprises one or more phosphor LEDs. In certain embodiments, the sources  221 - 224  are substantially aligned in a plane. In some embodiments, the sources  221 - 224  are disposed on a common substrate and, in further embodiments, are situated in proximity (e.g., adjacent) to each other. In various other embodiments, the light source  10  comprises any number of LEDs and/or any other suitable light-producing devices. 
     In certain embodiments, the mixer  210  comprises an input face  21  and an output face  22 . In the illustrated embodiment, the input face  21  is substantially square and the output face  22  is substantially circular and has an area larger than the input face  21 . Accordingly, the mixer  210  can be symmetrical about two perpendicular axes, such as the x and y axes defined in  FIG. 17A . Such arrangements can be particularly suitable for planar RGB LED arrays. The mixer  210  can also be symmetrical about one axis only, and in other arrangements, the mixer  210  is asymmetrical. 
     In various arrangements, the input face  21  is substantially square and has a width between about 1.0 millimeters to about 12 millimeters, between about 0.5 millimeters to about 1.5 millimeters, between about 1.0 millimeters to about 2.0 millimeters, between about 2.0 millimeters to about 3.0 millimeters, or between about 3.0 millimeters to about 4.0 millimeters. In other arrangements, the width is less than about 2.5 millimeters, less than about 2.0 millimeters, less than about 1.5 millimeters, less than about 1.0 millimeters, less than about 0.5 millimeters, or less than about 0.25 millimeters. In other arrangements, the width is greater than about 0.25 millimeters, greater than about 0.5 millimeters, greater than about 1.0 millimeters, greater than about 2.0 millimeters, or greater than about 3.0 millimeters. In some arrangements, the width is about 0.5 millimeters, about 1.0 millimeters, about 1.5 millimeters, about 2.0 millimeters, about 2.1 millimeters, about 2.2 millimeters, about 2.5 millimeters, or about 3.0 millimeters. 
     In other arrangements, the input face  21  is substantially square and has a width of from about 1.0 centimeters to about 10 centimeters, from about 1.0 centimeters to about 5.0 centimeters, or from about 5.0 centimeters to about 10.0 centimeters. In other arrangements, the width is less than about 10.0 centimeters, less than about 5.0 centimeters, or less than about 1.0 centimeters. In still other arrangements, the width is greater than about 1.0 centimeters, greater than about 5.0 centimeters, or greater than about 10.0 centimeters. In some arrangements, the width is about 1.0 centimeters, about 1.5 centimeters, about 2.0 centimeters, about 2.5 centimeters, about 3.0 centimeters, about 5.0 centimeters, or about 10.0 centimeters. Other sizes and configurations for the input face  21  are possible. 
     In certain embodiments, the output face  22  is substantially circular and has a diameter of from about 0.5 millimeters to about 3.0 millimeters, from about 1.0 millimeter to about 5.0 millimeters, from about 2.0 millimeters to about 7.0 millimeters, from about 2.5 millimeters to about 6.0 millimeters, or from about 3.0 millimeters to about 5.0 millimeters. In other arrangements, the diameter is less than about 7.0 millimeters, less than about 5.0 millimeters, less than about 2.0 millimeters, less than about 1.0 millimeters, and less than about 0.5 millimeters. In other arrangements, the diameter is greater than about 0.5 millimeters, greater than about 1.0 millimeters, greater than about 2.0 millimeters, greater than about 2.5 millimeters, or greater than about 3.0 millimeters. In some arrangements, the diameter is about 0.7 millimeters, 1.0 millimeters, 2.0 millimeters, 3.0 millimeters, about 4.0 millimeters, about 4.5 millimeters, about 4.7 millimeters, about 5.0 millimeters, or about 5.5 millimeters. 
     In other embodiments, the output face  22  has a diameter of from about 1.0 centimeters to about 10.0 centimeters, from about 2.5 centimeters to about 6.0 centimeters, or from about 3.0 centimeters to about 5.0 centimeters. In other arrangements, the diameter is less than about 10.0 centimeters, less than about 5.0 centimeters, or less than about 1.0 centimeters, and in other arrangements, the diameter is greater than about 1.0 centimeters, greater than about 5.0 centimeters, or greater than about 10.0 centimeters. In some arrangements, the diameter is about 3.0 centimeters, about 4.0 centimeters, about 4.5 centimeters, about 5.0 centimeters, about 5.5 centimeters, or about 6.0 centimeters. Other sizes and configurations for the output face  22  are possible. 
     As schematically illustrated in  FIG. 17B , the mixer  210  comprises an exterior surface that slopes outwardly from the input face  21  to the output face  22 , smoothly joining the two faces  21 ,  22 . The mixer  210  includes a central region between the input face  21  and the output face  22  that transforms from a square cross-section toward the input face to a circular cross-section toward the output face. The area of the cross-section also increases from the input face  21  through the central region to the output face  22 . 
     In some arrangements, the mixer  210  comprises a solid piece of substantially transmissive material. In various embodiments, the index of refraction of the material is between about 1.3 and about 2.2, between about 1.4 and about 1.6, or between about 1.45 and about 1.55. In some embodiments, the index of refraction is greater than about 1.3, greater than about 1.4, or greater than about 1.45. In other embodiments, the index of refraction is less than about 2.2, less than about 1.7, less than about 1.6, or less than about 1.55. In some embodiments, the index of refraction is about 1.40, about 1.45, about 1.50, about 1.55, or about 1.6. In other arrangements, the mixer  210  comprises a hollow tube with a substantially reflective interior and/or exterior surface. 
     In various embodiments, the length of the mixer  210 , as measured along the z axis defined in  FIG. 17B , is from about 0.1 centimeters to about 10.0 centimeters, from about 0.2 centimeters to about 3.5 centimeters, from about 1.0 centimeters to about 5.0 centimeters, or from about 1.5 centimeters to about 3.0 centimeters. In some embodiments, the length is less than about 10.0 centimeters, less than about 5.0 centimeters, less than about 3.0 centimeters, less than about 1.0 centimeters, or less than about 0.5 centimeters. In other embodiments, the length is greater than about 0.5 centimeters, greater than about 1.0 centimeter, greater than about 2.0 centimeters, greater than about 5.0 centimeters, or greater than about 10.0 centimeters. In some embodiments, the length is about 1.0 centimeters, about 1.5 centimeters, or about 2.0 centimeters. Other lengths are possible. 
     In some respects, the plots shown in  FIGS. 17C and 17D  are similar to those described above with respect to  FIGS. 5A and 5B , respectively. The plots illustrate the illuminance distribution at the output face  22  of the mixer  210  when only one of the sources  221 - 224  is illuminated.  FIGS. 17C and 17D  represent plots obtained from a mixer  210  comprising an input face  21  having a width of about 2.1 millimeters, an output face  22  having a diameter of about 4.7 millimeters, and a length of about 15 millimeters. The mixer  210  comprises a solid material having an index of refraction of about 1.5. As shown in  FIG. 17C , the illuminance distribution at the output face is highly non-uniform. A region of high intensity is located in the top left quadrant of the output face  22 , which is correlated with the location of the illuminated LED at the input face  21 . A region of lower intensity is located in the bottom right quadrant. In  FIG. 17D , the histogram comprises a large range of illuminance values, also indicating poor uniformity. 
       FIG. 17E  schematically illustrates one embodiment of a rippled angle-to-area converting mixer  230  that is coupled at an input face  21  with the sources  221 - 224  (not shown) described above. In certain embodiments, the mixer  230  resembles the mixer  210  in various respects. In some embodiments, the width of the input face, the diameter of an output face  22 , the length of the mixer  230 , and the index of refraction of the mixer  230  are equal to any of the respective values or ranges of values set forth above with respect to certain embodiments of the mixer  210 . In certain embodiments, etendue is conserved. 
     In some embodiments, the rippled mixer  230  comprises rippled sidewalls  15  having ridges  24  and valleys  25 . In some embodiments, the ridges  24  have rounded vertices, and in other embodiments, the ridges have sharp vertices. In various arrangements, the mixer  20  comprises between about 15 and about 25, between about 35 and about 45, or between about 70 and about 90 ridges  24 . In other arrangements, the mixer  230  comprises greater than about 15, greater than about 30, greater than about 70, greater than about 100, greater than about 500, greater than about 1000, or greater than about 5000 ridges  24 . In some arrangements, the mixer  230  comprises about 20, about 40, or about 80 ridges  24 . 
     In some embodiments, the ridges  24  have a constant height along the length of the mixer  230 . In certain embodiments, the height is from about 5.0 microns to about 50 microns. In some embodiments the height is less than about 50 microns, less than about 40 microns, less than about 30 microns, less than about 20 microns, less than about 10 microns, or less than about 5.0 microns. In other embodiments, the height is greater than about 30 microns, greater than about 40 microns, or greater than about 50 microns. Other heights are possible. In some embodiments, the height is from about 10 to about 100 times larger than the wavelength or wavelengths for which mixing is desired. 
     In other embodiments, the ridges  24  have a constant angle α along the length of the mixer  230 . In various arrangements, the angle α of the ridges  24  is between about 20 degrees and about 40 degrees, between about 33 degrees and about 37 degrees, between about 53 degrees and about 73 degrees, or between about 60 degrees and about 66 degrees along the length of the mixer  20 . In some embodiments, the angle α is about 26 degrees, about 33 degrees, about 35 degrees, or about 63 degrees. Other angles are possible. 
     In some respects, the plots shown in  FIGS. 17F and 17G  are similar to those described above with respect to  FIGS. 17C and 17D , respectively. The plots illustrate the illuminance distribution at the output face  22  of the rippled mixer  230  when only one of the sources  221 - 224  is illuminated.  FIGS. 17F and 17G  represent plots obtained from one embodiment of the mixer  230  comprising an input face  21  having a width of about 2.1 millimeters, an output face  22  having a diameter of about 4.7 millimeters, and a length of about 15 millimeters. The mixer  230  comprises 80 ridges  24  around its perimeter that run along the length of the mixer  230  and have a constant height of about 0.077 millimeters. The angle α of the ridges  24  varies along the length of the mixer  230 . The mixer  230  comprises a solid material having an index of refraction of about 1.5. The embodiment just described is illustrative and should not be interpreted as limiting. Other configurations are possible. In certain embodiments, the angle α of the ridges  24  is constant along the length of the mixer  230  and the height of the ridges  24  varies along the length of the mixer  230 . 
     As shown in  FIG. 17F , the illuminance distribution at the output face of the mixer  230  is highly uniform. Similarly,  FIG. 17G  shows a concentrated range of illuminance values, which also indicates a high degree of uniformity. 
     In certain embodiments, each of the sources  221 - 224  produces similar results when separately illuminated. Accordingly, in some embodiments, the mixer  230  produces highly uniform white light when all of the sources  221 - 224  are illuminated simultaneously. Advantageously, the circular output face  22  can produce a highly uniform circular beam that can be used in applications such as illuminating a circular bundle of optical fibers. Additional applications include flashlights and other portable lights, spot lights, and projection systems with circular pupils. Other applications are also possible. 
     In certain embodiments, the mixer  230  has many advantages over the smooth mixer  210 . In some instances, the smooth mixer  210  is capable of producing uniform white light, but only when it is relatively long. The mixer  230  is effective at shorter lengths. Accordingly, the mixer  230  can comprise less material than the mixer  210 , and can thus be more lightweight and more compact. 
     In addition, in certain embodiments, the mixer  230  is relatively insensitive to alignment of the light source  10  with the input face  21 , which permits more robust performance of the mixer  230 . For example, the mixer  230  can be easier to manufacture and have greater longevity since it will work well even if the sources  221 - 224  are moved out of alignment due to manufacturing constraints, manufacturing defects, user error, or other factors. Furthermore, some arrangements are configured such that fewer than four LEDs are illuminated to produce white light. Accordingly, one or more of the LEDs can serve as a backup in case one of the other LEDs fails. In some embodiments, more than four LEDs are used, and in further embodiments, some of the LEDs serve as dedicated backups. In still other embodiments, the mixer  230  comprises four or more LEDs, with one or more of the LEDs configured to produce wavelengths in the visible spectrum and one or more LEDs configured to produce other wavelengths, such as infrared or ultraviolet. In certain of such embodiments, a plurality of the LEDs produce white light, and one or more of the remaining LEDs produce a pattern imperceptible to the unaided human eye. In some embodiments a phosphor is used to convert the invisible light another wavelength such as a visible wavelength. In other embodiments, the light from the invisible light emitter may be infrared or ultraviolet and may be detected with an IR or UV detector. In certain embodiments, the one or more of the LEDs are pulsed. In some embodiments, the LEDs are pulsed simultaneously, and can be used in such applications as emergency vehicle lights or a camera flash. In some embodiments, the pulse rates vary among the LEDs. 
     In addition, insensitivity to light source alignment permits certain embodiments of the mixer  230  to be used effectively with various light source arrangements and coupling methods. As discussed above, in some embodiments the mixer  230  comprises a planar RGB LED array that is positioned adjacent to the input face  21  and that completely fills the area thereof. In various other embodiments, the array covers a smaller area than the input face  21  or comprises fewer LEDs. In certain embodiments, underfilling the input face  21  has little effect on the uniformity of the illuminance distribution at the output face  22  of the mixer  230 . In still other embodiments, light sources other than LEDs are coupled with the input face  21 . In some arrangements, the light source  10  is coupled with the input face  21  via a reflector, such as the reflector  12  described above. Any other suitable method or device can be used to couple the light source  10  with the mixer  230 . 
     In certain embodiments, the mixer  230  provides substantially uniform distributions when the light source  10  is centered with respect to the input face  21 . In other embodiments, the mixer  230  comprises ridges  24  with relatively large angles α, and can achieve comparable levels of uniformity as arrangements wherein the light source  10  is centered. Accordingly, in some embodiments, ridges  24  having larger angles α can be used to compensate for an off-centered light source. 
       FIG. 18A  schematically illustrates one embodiment of an angle-to-area converting mixer  240  comprising sidewalls  15  having ridges  24  and valleys  25 . The mixer  240  resembles the mixer  230  shown in  FIG. 17E  in many respects, but can differ in manners such as those now described. In certain embodiments, the mixer  240  comprises a rectangular input face  21 . The mixer  210  includes a central region between the input face  21  and the output face  22  that transforms from a rectangular cross-section toward the input face to a circular cross-section toward the output face. The area of the cross-sections also increase from the input face  21  through the central region to the output face  22 . In the embodiment shown in  FIG. 17E , the shape of the cross-section midway between the input face  21  and the output face  22  includes portions of the circular cross-section and portions of the rectangular cross-section. Cross-sections closer to the input face include more of the rectangular cross-section portions and cross-sections closer to the output face include more of the circular cross-section portions. 
     In certain embodiments, the width of the input face  21 , as measured along the x axis (defined in  FIG. 18A ) is about 4.3 millimeters. In various embodiments, the height of the input face  21 , as measured along the y axis (defined in  FIG. 18A ) is about 1.0 millimeters. In some embodiments, the ridges  24  have a height of 0.012 millimeters to 0.028 millimeters. Other configurations are possible. 
     As schematically illustrated in  FIG. 18B , the mixer  240  can be coupled with an RGB LED array having four sources  221 - 224 . In the illustrated embodiment, the sources  221 - 224  are aligned in a single column along the input face  21 . Accordingly, in some embodiments, the plurality of LEDs are arranged in a linear array. In various other embodiments, the array comprises more or fewer LEDs, and can be arranged in multiple columns and rows. 
     In some embodiments, the mixer  240  comprises a solid, substantially transmissive material having an index of about 1.5. The mixer  240  has a length of about 15 millimeters, an output face  22  having a diameter of about 4.7 millimeters, and an input face  21  having a width of about 1.0 millimeter and a height of about 4.2 millimeters. The mixer  240  comprises 80 ridges  24  along the full length thereof, each having a constant height of about 0.024 millimeters. When the sources  221 - 224  are all illuminated, the mixer  240  produces white light that is superior to that produced by a mixer without rippled sidewalls of comparable composition and dimensions. 
     As noted above, numerous configurations are possible for the ridges  24  and the valleys  25 . Accordingly, the configurations discussed above are nonexclusive. The various ridge and valley configurations, including those described elsewhere herein, as well as others yet to be devised, can be incorporated in these and other embodiments. 
       FIG. 19A  schematically illustrates one embodiment of a rippled angle-to-area converting mixer  250  comprising a circular input face  21  and a circular output face  22 . In the illustrated embodiment, the mixer  250  comprises ridges  24  and valleys  25  that have a constant height and extend along the full length of the mixer. The mixer is conical in shape, and includes a central region between the input face  21  and the output face  22 . Cross-sections through the central region are also circular. The cross-sections increase in size (e.g., area, diameter, etc.) from the input face  21  to the output face  22 . 
     In some embodiments, the input face  21  of the mixer  250  has a diameter of about 3.0 millimeters, the output face  22  has a diameter of about 6.0 millimeters, and the mixer  250  has a length of about 18.0 millimeters. In some embodiments, the ridges  24  and the valleys  25  vary in height and/or width along the length of the mixer  250 . 
       FIG. 19B  schematically illustrates one embodiment of a rippled angle-to-area converting mixer  252  comprising a square input face  21  and a square output face  22 . In the illustrated embodiment, mixer  210  includes a central region between the input face  21  and the output face  22 . Cross-sections through the central region are also square. The cross-sections increase in size (e.g., area, width, etc.) from the input face  21  to the output face  22 . The circular input face  21  of the mixer  250  has a diameter that is approximately equal to the width of the square input face  21  of the mixer  252 , and the circular output face  22  of the mixer  250  has a diameter that is approximately equal to the diagonal measurement of the square output face  22  of the mixer  252 . The mixer  252  comprises ridges  24  and valleys  25  that have a constant angle α and extend along the full length of the mixer. 
     In some embodiments, the input face  21  of the mixer  252  has a width of about 3.0 millimeters, the output face  22  has a width of about 4.2 millimeters, and the mixer  252  has a length of about 18.0 millimeters. In some embodiments, the ridges  24  and the valleys  25  vary in height and/or width along the length of the mixer  252 . 
       FIG. 19C  schematically illustrates one embodiment of a composite rippled mixer  255  that results when portions of the illustrated embodiments of the mixer  250  and the mixer  252  are combined. The composite mixer  255  comprises a square input face  21  that is identical to the input face  22  of the mixer  252 , and comprises a circular output face  22  that is identical to the output face  22  of the mixer  250 . The mixer  210  includes a central region between the input face  21  and the output face  22  that transforms from a square cross-section toward the input face to a circular cross-section toward the output face. The area of the cross-sections also increase from the input face  21  through the central region to the output face  22 . In the embodiment shown in  FIG. 19C , the shape of the cross-section midway between the input face  21  and the output face  22  includes portions of the square cross-section and portions of the circular cross-section. Cross-sections closer to the input face include more of the square cross-section portions and cross-sections closer to the output face include more of the circular cross-section portions. The mixer  255  also comprises ridges  24  and valleys  25  that represent a combination of portions of the respective ridges  24  and valleys  25  of the constituent mixers  250 ,  252 . 
     In some embodiments, each constituent mixer  250 ,  252  comprises the same number of ridges  24  and valleys  25 . Accordingly, in some embodiments, the ridges  24  and the valleys  25  of the composite mixer  255  are continuous and extend along the full length of the mixer. In some embodiments, the height of the ridges  24  and the valleys  25  of the composite mixer  255  are constant along the length of the mixer, and in other embodiments, the height changes along the length of the mixer. Similarly, the width of the ridges  24  and the valleys  25  can also remain constant or change along the length of the mixer  255 . 
     In other embodiments, each constituent mixer  250 ,  252  comprises a different number of ridges  24  and valleys  25 . Accordingly, the composite mixer  255  can comprise a different number of ridges at the input face  21  and at the output face  22 . In certain of such embodiments, neither set of ridges  24  and valleys  25  from the constituent mixers  250 ,  252  extends along the full length of the mixer. Consequently, in some instances, the height of the ridges  24  and of the valleys  25  of the composite mixer  255  changes along the length of the mixer. Similarly, the width of the ridges  24  and the valleys  25  can also change along the length of the mixer. 
     In certain embodiments, such as the embodiment illustrated in  FIG. 19C , Boolean operations can be used to combine portions of the constituent mixers  250 ,  252  into the composite mixer  255 . Other operations can also be used to combine portions of the constituent mixers  255 . In some embodiments, portions of more than two constituent mixers  250 ,  252  are combined to form the composite mixer  255 . 
       FIG. 20A  schematically illustrates one embodiment of a rippled mixer  260  comprising ridges  24  and valleys  25 . The mixer  260  resembles the mixer  120  described above with respect to  FIGS. 3 and 4 , except that the mixer  260  narrows toward the center thereof. A circular input face  21  and a circular output face  22  of the mixer  20  are substantially equally sized. 
       FIG. 20B  schematically illustrates one embodiment of a rippled mixer  162  comprising ridges  24  and valleys  25 . The mixer  262  resembles the square mixers  150 ,  170  described above in some respects. In certain embodiments, square input face  21  and a square output face  22  are substantially equally sized, and have a diagonal that is equal to the diameter of the circular input face  21  and the circular output face  22  of the mixer  260 . 
       FIG. 20C  schematically illustrates one embodiment of a composite rippled mixer  265  that results when portions of the mixer  260  and of the mixer  262  are combined. The composite mixer  265  comprises a circular input face  21  and a circular output face  22  that are identical to those of the mixer  260 . In the embodiment shown in  FIG. 20C , the shape of the cross-section midway between the input face  21  and the output face  22  includes portions of the square cross-section and portions of the circular cross-section. For example, the four corners of the square cross-section of the mixer shown in  FIG. 20B  are included midway between the input and output faces  21 ,  22 . Also at the midway location are curved side portions corresponding to portions of the narrowed circular cross-section of the mixer shown in  FIG. 20A . In the embodiment shown in  FIG. 20C , cross-sections closer to the input face and output faces include more of the circular cross-section portions and less of the square portions. The mixer  265  comprises ridges  24  and valleys  25  that represent the combination of portions of the respective ridges  24  and valleys  25  of the constituent mixers  260 ,  262 . In the illustrated embodiment, portions of the constituent mixers  260 ,  262  are combined via Boolean operations. In some embodiments, the constituent mixers  260 ,  262  are combined via the “union” Boolean operation, such as when the composite mixer  265  comprises the larger profile of the constituent mixers  260 ,  262  for each cross section along the length of the mixer  265 . In other embodiments, the constituent mixers  260 ,  262  are combined via the “intersection” Boolean operation. 
       FIG. 21A  schematically illustrates one embodiment of a rippled mixer  270  comprising ridges  24  and valleys  25  that decrease in height from an input face  21  to an output face  22 . In some embodiments, the ridges  24  and the valleys  25  do not extend along the full length of the mixer  270 . For example, in some embodiments, a portion of the mixer  270  near the output face  22  is smooth.  FIG. 21B  schematically illustrates one embodiment of a rippled mixer  272  comprising ridges  24  and valleys  25  that increase in height from an input face  21  to an output face  22 . In some embodiments, the ridges  24  and the valleys  25  do not extend along the full length of the mixer  272 . For example, in some embodiments, a portion of the mixer  272  near the input face  22  is smooth. In various embodiments, the ridges  24  and the valleys  25  have an average length of between about 50 percent and about 100 percent of the length of the mixers  270 ,  272 . In some embodiments, the ridges  24  and the valleys  25  have an average length of between about 0.1 centimeter and about 10 centimeters. Other ranges and values are possible. 
     In some embodiments, the increase or decrease in height of the ridges  24  and the valleys  25  of the mixers  270 ,  272  is linear (e.g., as viewed from a plane parallel to that of the yz plane defined in  FIGS. 21A and 21B , wherein the top and/or bottom of either of the mixers  270 ,  272  defines a straight line). In other embodiments, the increase or decrease in the height of the ridges  24  and valleys  25  is nonlinear. 
       FIG. 21C  schematically illustrates one embodiment of a rippled mixer  274  comprising ridges  24  and valleys  25  that increase in height from both an input face  21  and an output face  22  toward the center of the mixer  274 .  FIG. 21D  schematically illustrates one embodiment of a rippled mixer  276  comprising ridges  24  and valleys  25  that decrease in height from both an input face  21  and an output face  22  toward the center of the mixer  276 . In various other embodiments, the ridges  24  and the valleys  25  increase and decrease in height toward a portion of the mixers  274 ,  276  other than the center thereof. In some embodiments, the ridges  24  and the valleys  25  extend along less than the full length of the mixers  274 ,  276 . In various embodiments, the ridges  24  and the valleys  25  have an average length of between about 50 percent and about 100 percent of the length of the mixers  274 ,  276 . In some embodiments, the ridges  24  and the valleys  25  have an average length of between about 0.1 centimeter and about 10 centimeters. Other ranges and values are possible. 
     In some embodiments, the ridges  24  and the valleys  25  increase and decrease in height multiple times between the input face  21  and the output face  22 . In some embodiments, the ridges  24  and the valleys  25  increase and decrease in height in a regular pattern along the length of the mixers  274 ,  276 . For example, in some embodiments, the ridges  24  and the valleys  25  undulate along the length of the mixers  274 ,  276  in a sinusoidal pattern. As discussed further below, in some embodiments, ridges  24  and valleys  25  that increase and/or decrease in height along a length of a mixer can scatter light incident thereon. 
     In some embodiments, the width of the ridges  24  and/or the valleys  25  varies in manners similar to those just described. In other embodiments, both the width and the height of the ridges  24  and the valleys  25  vary along the length of the mixers  270 ,  272 ,  274 ,  276 . 
     In certain embodiments, each of the mixers  270 ,  272 ,  274 ,  276  depicted in  FIGS. 21A-21D  provides a more uniform illuminance distribution at the output face  22  than does a smooth circular mixing rod of equal length and comprising input and output faces of like area. In some instances, such as for mixers that have a constant cross-sectional profile along the full length thereof, ridges and valleys that have constant widths and constant heights are preferred. In many instances, such mixers perform as well as or better than alternative embodiments that comprise ridges and valleys having varying widths and heights. Furthermore, mixers having constant ridge and valley widths and heights can be easier to manufacture. 
     In some arrangements, the height and/or width of the ridges  24  and the valleys  25  can be varied to accommodate an unbalanced or off-centered illuminance distribution at the input face  21  and/or to create a desired illuminance distribution at the output face  22 . In some cases the ripples are different on different sides of the mixer. Such variation may also be useful in dealing with unbalanced or off-centered light sources and illuminance distributions. In some instances, the performance for such arrangements can be determined using illumination simulations with commercially available software tools, such as LightTools®, available from Optical Research Associates of Pasadena, Calif. 
     With reference again to  FIGS. 7A and 7B , two cross-sectional profiles of ridges and valleys are schematically illustrated. In  FIG. 7A , each faceted ridge  24  comprises two substantially planar faces  26 . In  FIG. 7B , the rounded ridges  24  and rounded valleys  25  have smoothed vertices. In various embodiments, mixers comprise any alteration and/or combination of faceted and rounded profiles, such as the examples now described. 
       FIGS. 22A-22I  schematically illustrate the cross-sectional profiles of illustrative embodiments of rippled sidewalls. 
       FIG. 22A  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  that have rounded vertices and valleys  25  that have sharp vertices. 
       FIG. 22B  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  that have sharp vertices and valleys  25  that have rounded vertices. 
       FIG. 22C  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  and valleys  25  that are highly rounded, as indicated by dashed circles  280  which have a diameter that is a substantial portion of the spacing between the ridges or valleys. For example, the diameter may be between about 30 to 40% of the spacing between the ridge or valleys. The diameter may be larger than 40 or 50% in some embodiments as well. 
       FIG. 22D  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  and valleys  25  having vertices that are connected by splines. In some embodiments, the profile is sinusoidal. In other embodiments, the profile is composed of parabolic arcs with ridges  24  and valleys  25  having opposite concavities. In still other embodiments, the profile is composed of a faceted profile, such as the profile illustrated in  FIG. 7A , that is smoothed into a rounded shape by using a cubic spline. Other spline profiles are also possible. 
       FIG. 22E  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  that have variable heights dH and/or variable angles α. In some embodiments, one or more of the ridges  24  comprise substantially planar sloping surfaces  26  that meet at a vertex. In other embodiments, one or more of the ridges  24  comprise rounded vertices. 
       FIG. 22F  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  that have angles α that vary irregularly or randomly. 
       FIG. 22G  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  that are separated by a large gap  282 . The result is a duty cycle that is different from 50%. 
       FIG. 22H  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  and valleys  25  with sharp vertices connected by planar sloping surfaces as well as ridges  24  and valleys  25  with rounded vertices connected by curved sloping surfaces. In some embodiments, the ridges  24  and the valleys  25  with sharp vertices and the ridges  24  and the valleys  25  with rounded vertices comprise various heights dH and widths w. In the embodiment shown, different ridges  24  and valleys  25  have different heights and widths. 
       FIG. 22I  schematically illustrates one embodiment of a rippled sidewall  15  comprising ridges  24  and valleys  25  that are rippled. In some embodiments, ridges  24  such as those illustrated in  FIG. 22I  comprise one periodic structure with another structure having a smaller period superposed thereon. 
       FIG. 23A  schematically illustrates the cross-sectional profile of one embodiment of a rippled sidewall  15  comprising ridges  24  with rounded vertices and valleys  25  with sharp vertices and curved surfaces therebetween. As discussed above with respect to  FIGS. 9 ,  12 F, and  13 C, in some instances, mixers that comprise ridges  24  with rounded vertices provide more uniform illuminance distributions than do similarly configured mixers that comprise ridges  24  with sharp vertices. However, in some instances, it is preferable to use ridges  24  with sharp or angled vertices with mixers. For example, mixers comprising angled ridges  24  can be easier to manufacture. Accordingly, in some arrangements, the ridges  24  comprise multiple facets in order to approximate a rounded profile. 
       FIG. 23B  schematically illustrates the cross-sectional profile of one embodiment of a rippled sidewall  15  comprising ridges  24  and valleys  25  with sharp vertices and planar surfaces therebetween that approximate the rippled sidewall  15  depicted in  FIG. 23A . Because these sloping surfaces comprise a plurality of planar portions, these sloping surfaces therefore said to be faceted. Each ridge  24  and valley  25  comprises six substantially planar faces or facets  26  (three for each sloping surface), although more or fewer faces or facets could be used. 
       FIG. 23C  schematically illustrates the cross-sectional profile of one embodiment of a rippled sidewall  15  comprising ridges  24  and valleys  25  having sharp vertices and faceted sloping surfaces therebetween that approximate rounded valleys. Each ridge  24  and each valley  25  comprises four substantially planar faces  26  (two on each sloping surface), although more or fewer faces could be used. 
     While in many instances it is desirable to have an illuminance distribution that uniformly covers the output face of a mixer, in some instances, is desirable to have a “peaked” illuminance distribution that has illuminance values that are high at the center of the output face and drop toward the edges thereof. In other instances, it is desirable to have a “dipped” illuminance distribution that has illuminance values that are low in the center of the output face and rise toward the edges thereof. In some instances, light source alignment with the mixer affects the illuminance distribution at the output face of the mixer. In certain arrangements, off-centering the light source helps to create a dipped illuminance distribution. 
     Rippled sidewalls can be configured to produce any one of, or any combination of, substantially uniform, substantially peaked, or substantially dipped illuminance distributions. For example, with reference again to  FIGS. 9 ,  10 E,  12 F, and  13 C, in some embodiments, ridges with sharp vertices having planar surfaces that define an angle α of 45 degrees produce highly non-uniform illuminance distributions that, in many instances, are peaked. This is especially true with certain circular arrangements, as illustrated by the point  144  in  FIG. 9  and the plot in  FIG. 10E . Accordingly, in some embodiments, rippled sidewalls comprise sharp ridges having an angle α of 45 degrees in combination with other ridge forms to control the illuminance distribution at the output face of a mixer. 
       FIG. 24  schematically illustrates a cross-sectional view of a portion of one embodiment of a mixer  290  comprising a rippled sidewall  15  having sharp ridges  24  that define an angle α of about 45 degrees and that are disposed between ridges  24  that have rounded vertices. In various embodiments, the mixer  290  produces a highly peaked or mildly peaked illuminance distribution, depending on the angle α used for the sharp ridges  24 . 
     In certain embodiments, such as those depicted in  FIGS. 3 ,  8 A, and  13 A, ridges run substantially parallel to the length of the mixer. In other embodiments, the ridges are angled with respect to the length of the mixer. For example,  FIG. 25A  schematically illustrates one embodiment of a mixer  300  comprising a rippled sidewall  15  having elongate ridges  24  and valleys  25  that are oriented at an angle with respect to the length of the mixer. As shown, the mixer  300  runs substantially parallel with the z axis, as defined by a dashed line in  FIG. 25A , and the ridges  24  and the valleys  25  are oriented at a positive angle β with respect to the z axis. In various embodiments, the angle β is less than about 40 degrees, less than about 30 degrees, less than about 20 degrees, less than about 10 degrees, less than about 5 degrees, or less than about 3 degrees. In various other embodiments, the angle β is between about 5 degrees and about 40 degrees, between about 7.5 degrees and about 30 degrees, or between about 10 degrees and about 30 degrees. Values outside these ranges are also possible. In some embodiments, the angle β becomes smaller as the number of ridges  24  and valleys  25  increases.  FIG. 25B  schematically illustrates one embodiment of a mixer  305  comprising a rippled sidewall  15  having ridges  24  and valleys  25  that have a small angle β. The angled ridges  24  and valleys  25  appear to twist about the circular mixer. 
       FIG. 25C  schematically illustrates one embodiment of a mixer  310  comprising a rippled sidewall  15  having ridges  24  and valleys  25  that are oriented at a negative angle γ with respect to the z axis, as defined by a dashed line in  FIG. 25C . The absolute value of the angle γ can equal any of the values described above with respect to the angle β.  FIG. 25D  schematically illustrates one embodiment of a mixer  315  comprising rippled sidewall  15  having ridges  24  that have a small angle γ. 
       FIG. 25E  schematically illustrates one embodiment of a composite mixer  320  comprising a rippled sidewall  15  having ridges  24  and valleys  25  such as those of the mixer  305  superposed on the ridges  24  and the valleys  25  of the mixer  315  (or vice versa). Accordingly, the composite mixer  320  of  FIG. 25E  has two sets of ridges  24  and valleys  25 . In some arrangements, small values of the angles β and γ are preferred in order to avoid etendue increases. In various embodiments, the ridges  24  and the valleys  25  are oriented at angles β and γ, and the value of the angle β plus the absolute value of the angle γ is less than about 80 degrees, less than about 60 degrees, less than about 40 degrees, less than about 20 degrees, less than about 10 degrees, or less than about 6 degrees. In various other embodiments, the angle β is between about 5 degrees and about 80 degrees, between about 5 degrees and about 45 degrees, or between about 10 degrees and about 30 degrees. Additionally, in various other embodiments, the angle γ is between about 5 degrees and about 80 degrees, between about 5 degrees and about 45 degrees, or between about 10 degrees and about 30 degrees. Other values outside these ranges are possible. 
       FIGS. 26A and 26B  schematically illustrate one embodiment of a composite mixer  330  that results from the combination of portions of two angle-to-area converting mixers. In certain embodiments, the composite mixer  330  is similar to the mixer  255  described above with respect to  FIG. 19C , but varies in manners such as those now described. In some embodiments, the composite mixer  330  has a circular input face  21 , a square output face  22 , and a central region  23  extending therebetween. The central region  23  can comprise a first portion  331  and a second portion  332 . 
     In certain embodiments, the circular input face  21  and the first portion  331  correspond with a constituent mixer that is a circular and angle-to-area converting, such as the mixer  250  described above with respect to  FIG. 19A . However, unlike the mixer  250 , the circular constituent mixer comprises a rippled sidewall  15   a  having ridges  24   a  and valleys  25   a  that are oriented at an angle β of 20 degrees with respect to the z axis. In some embodiments, the square output face  22  and the second portion  332  correspond with a constituent mixer that is a square and angle-to-area converting, such as the mixer  252  described above with respect to  FIG. 19B . However, unlike the mixer  252 , the square constituent mixer  20  comprises one or more rippled sidewalls  15   b  having ridges  24   b  and valleys  25   b  that are oriented at an angle γ of −20 degrees with respect to the z axis. Accordingly, in some embodiments, the composite mixer  20  comprises two sets of rippled sidewalls  15   a, b  with respective ridges  24   a,b  and valleys  25   a,b  that run in opposite directions along a length of the mixer. For example, the first set of ridges and valleys  24   a ,  25   a  are oriented at an angle β of 20 degrees with respect to the z axis and the second set of ridges and valleys  24   b ,  25   b  are oriented at an angle γ of −20. Other angles are possible. 
     In some embodiments, Boolean operations are used to combine portions of the constituent mixers into the composite mixer  330 . In other embodiments, processes other than Boolean operations are used to form the composite mixer  330 . In further embodiments, portions of more than two mixers can be combined into the composite mixer  330 . Additionally, various embodiments of the composite mixer  330  comprise combinations of mixers that have ridges and valleys that parallel the length of the mixers, that have one or more positive angles β, and/or that have one or more negative angles γ. 
       FIG. 27  schematically illustrates one embodiment of a mixer  340  comprising a rippled sidewall  15  having elongate ridges  24  and valleys  25  that curve with respect to the length of the mixer. Accordingly, in some embodiments, the value of the angle β and/or the angle γ changes over length of the mixer  340 . In the illustrated embodiment, the ridges  24  and the valleys  25  are oriented at a positive angle β at an input face  21  of the mixer  20 , and are oriented at a negative angle γ at an output face  22  of the mixer  20 . The values of β and/or the angle γ decrease away from the input and output faces  21 ,  22  where these angles are highest. The elongate ridges  24  and valleys  25  may have different shapes, caused for example, by different changes in the values of β and/or the angle γ with respect to the length of the mixer. 
       FIG. 28  schematically illustrates one embodiment of a rippled angle-to-area converting mixer  400  comprising a solid, optically transmissive material. The mixer  400  comprises an input face  21  with which an LED array or other light source can be coupled. In certain embodiments, the mixer  400  comprises a coating  405  at an input end thereof. The coating  405  can comprise any suitable reflective material, such as metal. In some arrangements, the coating  405  comprises a film, and in others, the coating  405  comprises paint. 
     In certain embodiments, the reflective coating  405  may be useful, for example, when an LED array is coupled with the mixer  400  in such a manner that reduces the amount of light that is total internally reflected at the sidewalls at the input end. For example, in some embodiments, the LED array may be situated adjacent the input face  21  such that there is no air gap therebetween, and thus no reduction of the angle of input light at an air/transmissive material interface as a result of refraction. More light rays are incident at higher angles, e.g. more normal to, the sidewalls. These lights rays are less likely to be total internally reflected. The coating  405  can be used to reflect these rays. Thus, the coating  405  can reflect light into the mixer  400  that would otherwise escape the mixer. The coating  405  can be applied to any mixer configuration. 
     In still other embodiments, the angle α can vary along the length so that loss of light caused by flux that does not totally internally reflect at the mixer surfaces is minimized. For example, in some embodiments, faceted ridges and/or faceted valleys characterized by an angle α of about 45 degrees near the input end of the mixer are followed by rounded ridges and/or rounded valleys characterized by an angle α of about 30 degrees over portions of the mixer where the cross-sectional area of the mixer  400  is larger (e.g., near the output end of the mixer  400 ). 
       FIG. 29A  schematically illustrates one embodiment of a rippled mixer  410  coupled with a projection lens  420 . The mixer  410  can comprise any suitable rippled sidewall configuration such as those disclosed herein. In various embodiments, the projection lens  420  is shaped and sized to collimate light exiting the mixer  410 . In certain embodiments, the light exiting the projection lens  420  forms a highly collimated, highly uniform “spot” with a sharp cutoff. In certain embodiments, the “spot” size is larger than the beam exiting the mixer  410 . In other embodiments, the light is not collimated by the projection lens  420 . For example, the lens  420  may focus the beam (e.g., spot lighting) or may spread the beam (e.g., flood lighting). 
     In some embodiments, the mixer  410  and the projection lens  420  comprise separate pieces. In other embodiments, the mixer  410  and the projection lens  420  are integrally formed of a unitary piece of material. In the shown embodiment, the mixer  410  and/or the projection lens  420  may comprise portions of substantially optically transmissive material, such as glass or plastic that are shaped to provide ribs to the mixer and a curved front surface to the projection lens. The projection lens also includes a tapered region  424  that couples to the mixer  410 . In certain embodiments, the tapered region  424  comprises a rippled sidewall  15  having ridges  24  and valleys  25 . 
     In certain embodiments, light propagates through the mixer  410 , through the tapered region  424 , and out a curved output face  422  of the projection lens. Certain embodiments that comprise a unitary piece of material can be particularly useful in, for example, flashlights and other portable lights, uplights, downlights, spot lights, architectural lights, or other types of lights. Some advantageous embodiments can be integrated into optical systems that comprise more than just the illuminator, such as, for example, machine vision, lithography, dental curing, fiber illuminator, or other systems. In such systems, the desired illuminance distribution may also be produced at an internal surface of the system. Other applications are also possible. 
     In certain embodiments, the mixer  410  is similar to the mixers  230 ,  240  described above. Advantageously, in some embodiments, the cross-sectional area of the mixer  410  increases along the length thereof such that the angular output of the mixer  410  matches the numerical aperture of the projection lens  420 . 
       FIG. 29B  schematically illustrates a cross-sectional view of one embodiment of a rippled mixer  411  coupled with a total internal reflection (TIR) collimator  425 . The mixer  411  can comprise any suitable rippled sidewall configuration such as those disclosed herein. The TIR collimator  425  can comprise any suitable configuration, such as those known in the art as well as those yet to be devised. Some TIR collimators are described in U.S. Pat. Nos. 2,215,900 and 2,254,961. In certain embodiments, the TIR collimator  425  comprises an input surface  426 , an outer surface  427 , and an output surface  428 . In some embodiments, light propagates through the mixer  411  and through the input surface  426  of the TIR collimator  425 , is reflected at the outer surface  427  via total internal reflection, and propagates through the output surface  428 . Some of the light propagates through the center of the TIR collimator without being totally internally reflected. In many embodiments, the light exiting the output surface  428  is collimated. 
       FIG. 30A  schematically illustrates one embodiment of a zoom system  430  comprising a rippled mixer  412  coupled with the projection lens  420 . The mixer  412  can comprise any suitable configuration, such as those disclosed herein. In certain embodiments, the projection lens  420  comprises an input face  421  and an output face  422 . In some arrangements, the projection lens  420  is moved along the z axis, as defined in  FIG. 30A , in order to alter the beam pattern exiting the output face  422 . In other embodiments, the mixer  412  is moved along the z axis to achieve the same effect. The result may be to increase or decrease the size of the beam, for example, from “spot” mode, wherein the beam pattern is highly collimated, to “flood” mode, wherein the beam pattern covers a larger area. In certain embodiments, the “spot” size of the light exiting the projection lens  420  is larger than an output face of the mixer  412  or larger than a beam exiting the output face of the mixer  412 . In some embodiments, the projection lens  420  and/or the mixer  412  is moved via a translator, such as a manually operated translator, a stepper motor, or a piezo translation device. 
       FIG. 30B  schematically illustrates one embodiment of a zoom system  431  that is similar to the system  430  in some respects. In certain embodiments, the zoom system  431  comprises a rippled mixer  413 , the projection lens  420 , and a diffuser  432 . The mixer  413  can comprise any suitable configuration, such as those disclosed herein. In some embodiments, the diffuser  432  has a hole  433  therethrough and is positioned between the mixer  413  and the projection lens  420 . In some arrangements, the diffuser  432  is moved along the z axis, as defined in  FIG. 30B , in order to alter the beam pattern exiting the output face  422  of the projection lens  420 . The diffuser  432  can be moved via a translator. In certain embodiments, the hole  433  causes the beam pattern to be highly collimated, and in other embodiments, the diffuser  435  causes the beam pattern to “flood” a larger area. In certain embodiments, the “spot” size of the light exiting the projection lens  420  is larger than an output face of the mixer  413  or larger than a beam exiting the output face of the mixer  413 . Other variations of the zoom system  431  are possible. See, e.g., U.S. patent application Ser. No. 11/210,275, entitled “Lighting Systems for Producing Different Beam Patterns” filed Aug. 23, 2005 and published as U.S. Patent Application Publication No. 2006/0039160 (Attorney Docket No. OPTRES.043A) on Feb. 23, 2006, the entire contents of which are hereby incorporated by reference herein and made a part of this specification. 
       FIG. 30C  schematically illustrates one embodiment of a zoom system  434  that is similar to the system  430  in some respects. In certain embodiments, the zoom system  434  comprises a rippled mixer  414 , an output lens  435 , and the projection lens  420 . The mixer  414  can comprise any suitable configuration, such as those disclosed herein, and can comprise an input face  415  and an output face  416 . In some embodiments, the output lens  435  is a hemispherical lens. Other lens configurations are also possible. In some embodiments, the projection lens  420  is a Fresnel lens, an aspherical condensing lens, a catadioptric collimator (such as, for example, the TIR collimator  425 ), or any other suitable device, such as, for example, one or more of the devices disclosed in U.S. patent application Ser. No. 10/658,613, entitled “Internally Reflective Ellipsoidal Collector with Projection Lens,” filed Sep. 8, 2003 and issued as U.S. Pat. No. 6,819,505 on Nov. 16, 2004, the entire contents of which are hereby incorporated by reference herein and made a part of this specification. 
     In certain embodiments, the output lens  435  is in proximity to the output face  416  of the mixer  414 . In some embodiments, a surface of the output lens  435  is adjacent the output face  416 . In many embodiments, the projection lens  420  is moved along the z axis, as defined in  FIG. 30C , in order to alter the beam pattern exiting the output face  422  of the projection lens  420 . In other embodiments, the mixer  414  and the output lens  435  are moved along the z axis to achieve the same effect. The result may be to increase or decrease the size of the beam, for example, from “spot” mode to “flood” mode. 
       FIG. 30D  is a cross-sectional view schematically illustrating one embodiment of the zoom system  434  wherein the projection lens  420  is spaced away from the output lens  435 . A series of lines  436  is provided to illustrate different paths that light could take, demonstrating that, in the shown embodiment, the output face  416  of the mixer  414  is imaged a distance from the output lens (for example, much longer than the focal length of the lens) by the zoom system  434 . In certain embodiments, the illustrated orientation of the projection lens  420  with respect to the mixer  414  and the output lens  435  results in a highly collimated, highly uniform “spot” with a sharp cutoff 
       FIG. 30E  is a cross-sectional view schematically illustrating one embodiment of the zoom system  434  wherein the projection lens  420  is in close proximity to the output lens  435 . A series of lines  437  is provided to illustrate different paths that light could take, demonstrating that, in the shown embodiment, the zoom system  434  produces a blurred image of a plane  438  within the mixer  414 . In certain embodiments, the illustrated orientation of the projection lens  420  with respect to the mixer  414  and the output lens  435  results in a larger beam pattern than that resulting from the orientation shown in  FIG. 30D . In some embodiments, the length of the mixer is such that the mixer  414  provides a substantially uniform illuminance distribution at the plane  438 . Accordingly, in some embodiments, an increase in the length of the mixer  414  permits the projection lens  420  to be positioned near the mixer  414  and still produce a substantially uniform beam pattern. In certain embodiments, projection optics (e.g., output lenses and projection lenses) having longer back focal lengths provide larger changes in the size of the beam pattern than do projection optics having shorter back focal lengths, as compared over the same range of movement of the optics along the z axis. 
     In many embodiments, the system  434  has advantages over systems that do not comprise both an output lens and a projection lens, such as the system  430 . In some embodiments, the output lens  435  reduces Fresnel losses of light exiting the mixer  414 . In further embodiments, the output lens  435  reduces the size of the system  434 , such as by making the system shorter as measured along the z-axis. In certain embodiments, the output lens  435  reduces the angle of light exiting the output face  416  of the mixer  414 . Accordingly, in some embodiments, light can be introduced to the mixer  414  at higher angles than is desirable for systems lacking an output lens, which can result in better mixing of the light. Additionally, in some embodiments, the output lens  435  increases the apparent size of the mixing rod. In certain embodiments, the output lens  435  is hemispherical and increases the apparent size by a factor about equal to the index of refraction of the output lens  435 . 
       FIG. 31  schematically illustrates one embodiment of a fiber illuminator  440  comprising a mixer portion  442  and a plurality of illumination branches  444 . The mixer portion  442  can comprise an input face  21  and an output face  22 , and each illumination branch can extend from the output face to an output end  446 . The mixer portion  442  can comprise a variety of suitable mixer configuration such as described herein. Accordingly, in some embodiments, the mixer portion  442  comprises a hollow tube, and in other embodiments, the mixer portion  442  comprises a solid piece of optically transmissive material. In some arrangements, each illumination branch  444  is integrally formed with an output end of the mixer portion  442 . The fiber illuminator  440  can comprise any number of illumination branches  444 . In some embodiments, the output end  446  of each illumination branch  444  is configured to couple with an optical fiber. 
     The fiber illuminator  440  can comprise one or more rippled sidewalls  15  having ridges  24  and valleys  25 . In some embodiments, the ridges  24  and the valleys  25  extend from the input face  21  to the output face  22 . In other embodiments, the ridges  24  and the valleys  25  extend from the output face  22  to the output ends  446 . In still other embodiments, the ridges  24  and the valleys  25  extend from the input face  21  to the output ends  446 . In some arrangements, the ridges  24  and the valleys  25  assume the same configuration on the mixer portion  442  and on the illumination branches  444 . In other arrangements, the ridges  24  and the valleys  25  vary between the mixer portion  442  and any of the illumination branches  444 . 
     The fiber illuminator  440  can be used to substantially uniformly illuminate a bundle of optical fibers in a manner superior to conventional methods for doing so. Often, a bundle of fibers is placed at the output end of a mixing rod in order to illuminate the fibers. Consequently, light is lost to the regions between the fibers. In certain embodiments, the fiber illuminator  440  decreases these losses by coupling light into each fiber via the illumination branches  444 . Furthermore, in some embodiments, the one or more rippled sidewalls  15  having ridges  24  and valleys  25  ensure that the light is substantially uniformly distributed into the different fibers. The one or more rippled sidewalls  15  can also allow the mixer portion  442  to be more compact, since the length thereof can be reduced without substantially altering the mixing capacities of the mixer portion  442 . In some embodiments, the fiber illuminator  440  operates in reverse, such that light enters the output face  22  of the mixer portion  442  from one or more of the plurality of illumination branches  444 . 
       FIG. 32A  illustrates one embodiment of a smooth mixer  460  comprising an input face  21  and an output face  22 . In the illustrated embodiment, the input face  21  and the output face  22  comprise squares having approximately equal areas. In certain embodiments, the mixer  460  comprises a smooth square mixing rod having one or more bends  462  along the length thereof in one or more planes. In some embodiments, the mixer  460  comprises one or more bends  462  parallel to the yz plane and/or the xz plane, as defined in  FIG. 32A . In the illustrated embodiment, the mixer  460  comprises two gradual, oppositely-directed bends  462  parallel to the yz plane and two gradual, oppositely-directed bends  462  along the xz plane. In other embodiments, the bends are in different directions. Nevertheless, in many embodiments, the illuminance distribution obtained at the output face  22  of the mixer  460  is less uniform than the distribution obtained at the output face of a square mixing rod that has the same length as the mixer  460  but is not bent. 
       FIGS. 32B and 32C  are spatial plots representing the illuminance distribution at the output face  22  of one embodiment of the mixer  460 .  FIG. 32B  illustrates the illuminance distribution when a source is very small in comparison with the input face  21 , and thus does not fill the input face  21  with light. As shown in  FIG. 32B , the illuminance distribution can be highly non-uniform. 
       FIG. 32C  illustrates the illuminance distribution when a source is large in comparison with the input face  21 , and substantially fills the input face  21  with light. As shown in  FIG. 32C , the illuminance distribution from the larger source can be non-uniform. However, in some embodiments, the distribution is much more uniform than that obtained from the smaller source (e.g., in comparison with  FIG. 32B ). 
       FIG. 32D  illustrates one embodiment of a rippled mixer  470  comprising an input face  21  and an output face  22 . In the illustrated embodiment, the input face  21  and the output face  22  comprise squares having approximately equal areas. In certain embodiments, the mixer  470  further comprises one or more rippled sidewalls  15  having ridges  24  and valleys  25 . The rippled sidewalls  15  can comprise any suitable configurations. In certain embodiments, the mixer  470  comprises one or more bends  472  along the length thereof. In some embodiments, the mixer  470  comprises one or more bends  472  along the yz plane and/or the xz plane, as defined in  FIG. 32D . In the illustrated embodiment, the mixer  470  comprises two gradual, oppositely-directed bends  472  along the yz plane and two gradual, oppositely-directed bends  472  along the xz plane. In some embodiments, the ridges  24  and the valleys  25  also comprise bends  472  along the length of the mixer  470 , and can comprise the same contour as the mixer  470 . In the illustrated embodiments, the only difference between the mixer  460  and the mixer  470  is the ribbed sidewalls  15  of the mixer  470 . In many embodiments, the illuminance distribution obtained at the output face  22  of the mixer  470  is more uniform than the distribution obtained at the output face  22  of the mixer  460 . 
       FIGS. 32E and 32F  are spatial plots representing the illuminance distribution at the output face  22  of one embodiment of the mixer  470 .  FIG. 32E  illustrates the illuminance distribution when a source is very small in comparison with the input face  21 , and thus does not fill the input face  21  with light. As shown in  FIG. 32E , the illuminance distribution can be much more uniform than that obtained from the smooth mixer  460  having either a smaller or larger source at the input face  21  (e.g., in comparison with  FIGS. 32B and 32C ). 
       FIG. 32F  illustrates the illuminance distribution when a source is large in comparison with the input face  21 , and substantially fills the input face  21  with light. As shown in  FIG. 32F , the illuminance distribution from the larger source can be much more uniform than that obtained from the smooth mixer  460  having either a smaller or larger source at the input face  21  (i.e., in comparison with  FIGS. 32B and 32C ), and can be more uniform than, or about as uniform as, that obtained from a smaller source at the input face  21  (e.g., in comparison with  FIG. 32E ). 
     Accordingly, in some embodiments, the mixer  470  comprising one or more rippled sidewalls  15  can comprise one or more bends  472  along a length thereof without significantly degrading the uniformity of the illuminance distribution at the output face  22 . This result can be particularly advantageous since the one or more bends  472  can aid in the mixing of light that enters the input face  21  at high angles and/or low angles. For example, such bends can be used to affect light emitted normal from the light source that would otherwise pass straight through a mixer that is not bent. The same result can also be achieved with certain embodiments of a circular mixer comprising one or more bends along a length thereof and rippled sidewalls or mixers having other shapes. 
       FIG. 33A  illustrates one embodiment of a rippled mixer  480  coupled with a nonimaging optical element  481  at an input face  21  of the mixer  480 . In certain embodiments, a light source  10  is coupled to an input face  482  of the nonimaging optical element  481 . In further embodiments, the mixer  480  is coupled with a nonimaging optical element  483  at an output face  22  of the mixer  480 . The nonimaging optical element  483  can comprise an output face  484 . 
     In various embodiments, the mixer  480  comprises one or more rippled sidewalls  15  having ridges  24  and valleys  25 . The mixer  480  can comprise any suitable configurations such as those disclosed herein. In further embodiments, one or more of the nonimaging optical elements  481 ,  483  comprise rippled sidewalls having ridges and valleys (not shown). Such non-imaging optical elements may comprise, for example, compound parabolic collectors (CPC) or reflector with elongate ridges  24  and valleys  25 . 
       FIG. 33B  illustrates one embodiment of a rippled mixer  490  coupled with the nonimaging optical elements  481 ,  483  described above. In certain embodiments, the mixer  490  comprises one or more bends  492  along the length thereof along one or more planes. In some embodiments, the mixer  490  comprises one or more bends  492  along the yz plane and/or the xz plane, as defined in  FIG. 33B . As described above, the bends may be in different directions as well. In the illustrated embodiment, the mixer  490  comprises two gradual, oppositely-directed bends  492  along the yz plane. In various embodiments, the mixer  490  comprises one or more rippled sidewalls  15  having ridges  24  and valleys  25 . In some embodiments, the ridges  24  and the valleys  25  also comprise bends  492  along the length of the mixer  490 , and can comprise the same or different bends as the mixer  470 . 
       FIG. 33C  illustrates one embodiment of a rippled mixer  500  comprising one or more rippled sidewalls  15 . The mixer  500  is coupled with the nonimaging optical elements  481 ,  483  described above. The rippled mixer  500  resembles the rippled mixer  490  in many respects, but can differ in manners such as now described. In some embodiments, the mixer  500  comprises four gradual bends  502  along the yz plane. In the illustrated embodiment, the bends  502  alternate directions to form an undulating structure. In further embodiments, any number of bends  502  along any combination of planes is possible. 
     In many embodiments, the illuminance distribution at the output face  22  of any of the mixers  480 ,  490 ,  500  described above and/or the output face  484  of the nonimaging optical element  483  is much more uniform than the distribution that could be obtained with a mixer of identical geometry and dimensions, but having smooth sidewalls. In further embodiments, the illuminance distribution at the output faces  22 ,  484  is more uniform for configurations wherein one or more of the nonimaging optical elements  481 ,  483  comprise rippled sidewalls than for configurations wherein one or more of the nonimaging optical elements  481 ,  483  have smooth sidewalls. 
       FIG. 34A  schematically illustrates one embodiment of a curved mixer  510 . In certain embodiments, the mixer  510  comprises an input face  21  and an output face  22 . In further embodiments, the mixer  510  comprises one or more rippled sidewalls  15  having ridges  24  and valleys  25 . The rippled sidewalls can comprise any suitable configuration such as described herein. In some embodiments, the ridges  24  and the valleys  25  vary in depth along the length and/or around the circumference of the mixer  510 . 
       FIG. 34B  schematically illustrates another embodiment of a curved mixer  520 . In certain embodiments, the mixer  520  comprises an input face  21  and an output face  22 . In some embodiments, the mixer  520  tapers inward from the input face  21  toward a smaller cross sectional area of a center section  522 , and tapers outward from the smaller cross sectional area of the center section  522  to the output face  22 . In further embodiments, the mixer  520  comprises one or more rippled sidewalls  15  having ridges  24  and valleys  25 . The rippled sidewalls can comprise any suitable configuration such as those disclosed herein. In some embodiments, the ridges  24  and the valleys  25  vary in depth along the length and/or around the circumference of the curved mixer  520 . 
     In many embodiments, the curved mixers  510 ,  520  advantageously redirect light propagating in one direction (e.g., perpendicular to the input face  21 ) toward another direction (e.g., perpendicular to the output face  22 ). In some embodiments, the mixers  510 ,  520  having rippled sidewalls  15  produces an illuminance distribution at the output face  22  that is more uniform than that which can be produced by certain embodiments having smooth sidewalls. 
       FIG. 35  schematically illustrates one embodiment of a mixer  525 . In certain embodiments, the mixer  525  comprises an input face  21  and an output face  22 . In some embodiments, the mixer  525  tapers gradually from the input face  21  to the output face  22 . In some embodiments, the taper is linear. In other embodiments, the taper is more gradual near the input face  21  than it is near the output face  22 . In some embodiments the output face is curved with respect to a plane perpendicular to the length of the mixer  525 . Such curvature may provide optical power such that the output face  22  forms a lens. In some embodiments, the mixer  525  comprises a gradual taper from the input face  21  toward the output face  22  and further comprises a curved output face  22  that acts as a field lens. 
     In some embodiments, the mixer  525  comprises one or more rippled sidewalls  15  having ridges  24  and valleys  25 . The rippled sidewalls can comprise any suitable configuration such as those disclosed herein. In some embodiments, the ridges  24  and the valleys  25  vary in depth along the length and/or around the perimeter of the mixer  525 . 
     In certain embodiments, the rippled sidewalls  15  provide a substantially uniform illuminance distribution at the output face  22  of the mixer  525 . In further embodiments, the curvature of the output face  22  provides an improved angular distribution as compared with a rippled mixer having a planar output face. In some embodiments, the angular distribution is substantially uniform. 
     Many methods are possible for forming certain embodiments of rippled sidewalls having ridges and valleys such as those disclosed herein. In various embodiments, the ridges and the valleys are formed by injection molding or embossing. In other embodiments, the ridges and the valleys are formed by extrusion. In certain of such embodiments, a long extruded piece of material is cut into smaller pieces to form mixers. 
     In some embodiments, as schematically depicted in  FIG. 36A , a film  530  comprising ridges  24  and valleys  25  is applied to a solid rod  535 . In certain embodiments the film  530  comprises a flexible material capable of conforming to the shape of a curved surface. In other embodiments, the film  530  can be substantially rigid. In many embodiments, the film  530  comprises the ridges  24  and the valleys  25  on one side thereof, and a substantially smooth surface on an opposite side. In further embodiments, the rod  535  comprises a substantially smooth exterior surface. In some embodiments, the substantially smooth surfaces of the film  530  and the rod  535  face each other, as illustrated, and in other embodiments, the ridges  24  and the valleys  25  face the substantially smooth exterior surface of the rod  535 . In many embodiments, the film  530  and the rod  535  comprise substantially the same index of refraction and, in further embodiments, are joined by an index matching adhesive or other material. 
     In some embodiments, as schematically illustrated in  FIG. 36B , an insert  540  comprising ridges  24  and valleys  25  is placed in a hollow tube  545 . In certain embodiments, the insert  540  comprises a substantially transparent solid material, such as plastic or glass. In some embodiments, the insert  540  is substantially rigid, and in others, the insert  540  is flexible. In some embodiments, the ridges  24  and the valleys  25  are formed on an interior surface of the insert  540 , and in other embodiments, the ridges  24  and the valleys  25  are formed on an exterior surface of the insert  540 . In many embodiments, the tube  545  is reflective. In some embodiments light is refracted by the ridges  24  and valleys  25  and sloping sidewalls therebetween that are disposed on the rippled surface of the insert  540 . The effect is the same as if the light is reflected from a rippled reflective sidewall. Many of the refracted rays are redirected at a different azimuthal direction toward the output end thereby mixing the light at the output end. 
     In certain embodiments, the film  530  or the insert  540  comprises between about 20 and about 5000, between about 1000 and about 5000, or between about 2000 and about 4000 ridges  24  and valleys  25 . In other embodiments, the film  530  or the insert  540  comprises greater than about 15, greater than about 20, greater than about 50, greater than about 100, greater than about 200, greater than about 500, greater than about 1000, greater than about 2000, greater than about 3000, greater than about 4000, or greater than about 5000 ridges  24  and valleys  25 . In still other embodiments, the film  530  or the insert  540  comprises fewer than about 5000, fewer than about 4000, fewer than about 3000, fewer than about 2000, fewer than bout  1000 , fewer than about 500, fewer than about 200, fewer than about 100, fewer than about 50, or fewer than about 20 ridges  24  and valleys  25 . 
     Some advantageous embodiments comprise relatively large numbers of ridges  24  and valleys  25 . In some embodiments, as the number of ridges  24  and valleys  25  increases, the size of the ridges  24  and the valleys  25  becomes smaller relative the surface shape of the mixer to which the ridges  24  and the valleys  25  are applied. This result can be particularly advantageous, such as, for example, in systems for which substantially circular or substantially square beam patterns are desirable. 
     In certain embodiments, relatively small ridges  24  and valleys  25  result in diffraction effects. In some embodiments, the diffraction effects are beneficial to creating substantially uniform illuminance distributions. In other embodiments, the diffraction effects limit the uniformity of illuminance distributions. In some embodiments, ridges  24  and valleys  25  that are relatively small and that produce diffractive effects can produce illuminance distributions similar to larger ridges  24  and valleys  25  and/or other macroscopic features of a mixer. 
     As schematically illustrated in  FIG. 37 , in some embodiments, a mixer  550  comprises a rod  552  and a hologram  554 . In some embodiments the rod  552  comprises a solid, substantially optically transmissive material. In other embodiments, the rod  552  comprises a hollow tube. In some embodiments, the hologram  554  comprises a flexible sheet of material and is placed on an exterior surface of the rod  552 , and in other embodiments, is placed on an interior surface of the rod  552 . In some embodiments, the hologram  554  is fabricated directly on a surface of the rod  552 . In certain embodiments the hologram comprises a transmissive optical element and is included as an insert in a reflective tube such as shown in  FIG. 35B . In various embodiments, the hologram  554  is configured to produce illuminance distributions such as for example those produced by rippled sidewall configurations disclosed herein. The hologram may be optically generated, e.g., via optical interference within the tube, or computer generated using, for example, processes well known in the art. 
     With reference again to  FIGS. 3 and 4 , in some embodiments, the mixer  120  (or another type of mixer such as discloses elsewhere herein) comprises a solid light conduit. Accordingly, in some arrangements, the mixer  120  comprises a substantially optically transmissive material, such as glass or plastic or some other polymeric material. Other materials may also be used. In many embodiments, the mixer  120  comprises a substantially constant index of refraction along a cross-section thereof. In many embodiments, the substantially optically transmissive material has negligible absorption for wavelengths that are desired at the output end  22 . In some embodiments, the mixer  120  is configured to shield the output  22  from undesired flux entering the input end  21 , such as infrared radiation or ultraviolet radiation, and to transmit desired flux, such as visible light. In some embodiments, the substantially optically transmissive material is configured to absorb radiation having undesired wavelengths, and in further embodiments, is configured to re-radiate the absorbed radiation at one or more desired wavelengths. In such embodiments, the substantially optical transmissive material may comprise a material that passes certain (e.g. visible) wavelength and absorbs other (e.g., visible) wavelengths. In such embodiments, for example, the wavelength spectrum at the output may be a particular color different from the input. For example, white light could be input into the mixer while the light output from the mixer could be red, blue, green or another color. The optically transmissive material may fluoresce so as to re-radiate light. In some embodiments the optically transmissive material is doped with constituents that case absorption and/or fluorescence. 
       FIG. 38A  schematically illustrates a front plan view of one embodiment of a circular mixer  560  that comprises a substantially optically transmissive outer layer  561  over a core  562 . In various embodiments, the core  562  comprises a vacuum, one or more gases, or a substantially optically transmissive material. Other materials are also possible. In certain embodiments, the outer layer  561  comprises indices of refraction that vary between lower and higher values around the perimeter of the mixer  560 , as indicated by lighter and darker shading, respectively. In some embodiments, the gradation from lower to higher indices of refraction, or from higher to lower indices of refraction, is gradual. In many embodiments, index variation along the longitudinal direction is nominal. In certain embodiments, the index of refraction variations comprise high and low refractive index portions that extend longitudinally along the length of the mixer  560  in a fashion similar to the elongate ridges and valleys in embodiments described above. The index of refraction variations refract light within the mixer in a similar manner as the elongate ridges and valleys described above. 
     In some embodiments, the indices of refraction vary in a random or irregular manner around the perimeter of the mixer  560 . In other embodiments, the indices of refraction vary in a regular pattern around the perimeter of the mixer  560 , and in some embodiments, vary sinusoidally. In certain embodiments, the mixer  560  comprises a smooth outer surface. In various embodiments, the outer layer  561  comprises a uniform thickness or a variable thickness around the perimeter of the mixer  560 . In further embodiments, the mixer  560  comprises a cladding over the outer layer  561 . Other configurations are also possible. 
     In many embodiments, the mixer  560  produces illuminance distribution patterns at an output face (not shown) that are more uniform than those produced by a mixing rod that comprises the same size and shape but does not comprise an outer layer with varying indices of refraction. In many embodiments, the mixer  560  produces illuminance distribution patterns that are more uniform than those produced by smooth circular mixers comprising the same size and shape. 
       FIG. 38B  schematically illustrates a front plan view of one embodiment of a square mixer  565  that comprises a substantially optically transmissive outer layer  566 , such as the outer layer  561  described above, over a core  567 , such as the core  562  described above. 
     With reference to  FIGS. 4 ,  39 A, and  39 B, in some embodiments, the mixer  120  (or any other suitable mixer such as those disclosed herein) comprises a core  570 . In further embodiments, the mixer  120  comprises a cladding  572  for protecting the surface of the core  570 , for assisting in confining light within the mixer, or for other suitable purposes. As illustrated in  FIG. 39A , in some embodiments, the cladding  572  comprises a thickness that varies relative to the surface of the core  570 . In the embodiment shown in  FIG. 39A  therefore, the exterior surface of the cladding  572  is smooth. As illustrated in  FIG. 39B , in other embodiments, the cladding  572  comprises a substantially constant thickness relative to the surface of the core  570 . In this embodiment, as shown, the exterior surface of the cladding  572  may likewise be rippled. Other configurations are also possible. In some embodiments, the cladding  572  comprises a material that has a lower index of refraction than that of the material of the core  570 . In certain of such embodiments, the cladding  572  does not significantly alter the illuminance distribution at the output end  22  of the mixer  120 , as compared with a mixer  120  that does not comprise the cladding  572 . In certain of such embodiments, such as, for example, when the cladding  572  is sufficiently thin, the cladding  572  does not significantly alter the illuminance distribution at the output end  22  of the mixer  120 , as compared with a mixer  120  that does not comprise the cladding  572 . In other embodiments, a material that has an index of refraction substantially equal to or higher than that of the material of the core  570  is disposed about the core. 
       FIG. 40  schematically illustrates a partial perspective view of one embodiment of a rippled sidewall  15  comprising a plurality of elongate ridges  24  and valleys  25 . In the illustrated embodiment, the ridges  24  and the valleys  25  extend parallel to the z axis, as defined in  FIG. 40 . In certain embodiments, a method for determining spreading from the ridges  24  and the valleys  25  comprises illuminating the rippled sidewall  15  with a beam  580  of collimated light to produce an illuminance distribution  582 . In some embodiments, the beam  580  is introduced in a plane of incidence that runs substantially parallel to the ridges  24  and substantially perpendicular to the sidewall  15  (e.g., parallel to the yz plane defined in  FIG. 40 ). In further embodiments, the plane of incidence runs parallel to the ridges  24  and perpendicular to a plane that is substantially tangent to the sidewall  15 . 
     In some embodiments, at least a portion of the sidewall  15  does not comprise ridges  24  and valleys  25 . In some embodiments, the portion of the sidewall  15  is substantially planar and comprises a substantially specularly reflective material. For the case of specular reflection from a planar section, the illuminance distribution  582  is merely a spot. 
     In other embodiments, the rippled sidewall  15  comprises ridges  24  and valleys  25 . As noted, numerous cross-sectional profiles are possible for the ridges  24  and the valleys  25 . In certain embodiments, the rippled sidewall  15  comprises a substantially specularly reflective material. In many of such embodiments, light reflected from the sidewall  15  is distributed in a semicircular arc, as schematically illustrated by lines  583  in  FIG. 40 . In some embodiments, substantially uniform illuminance distributions are preferred. In other embodiments, non-uniform and/or patterned illuminance distributions are preferred. In certain embodiments, the method for determining spreading comprises comparing the illuminance distributions  582  obtained from two or more rippled sidewalls  15  having different ripple profiles. 
     The incident beam  580  can be oriented at an angle δ with respect to a plane (e.g., the xz plane defined in  FIG. 40 ) running substantially parallel to the sidewall  15 . In various embodiments, the distributions produced by two or more angles δ are compared. 
       FIG. 41A  schematically illustrates a partial cross-section of one embodiment of a sidewall  15  comprising ridges  24  and valleys  25 . The ridges  24  and the valleys  25  comprise highly rounded vertices with sloping surfaces therebetween.  FIG. 41B  is a plot of the illuminance distribution obtained by reflecting a beam of collimated light from the sidewall  15  in a manner such as described above, wherein the beam  580  is in a plane parallel to the yz plane defined in  FIG. 40 , and is oriented at an angle δ with respect to the xz plane. As shown, the illuminance is concentrated toward the top of the semicircular distribution. 
       FIG. 41C  schematically illustrates a partial cross-section of another embodiment of a sidewall  15  comprising ridges  24  and valleys  25 . In the illustrated embodiment, the ridges  24  comprise pointed vertices, the valleys  25  comprise rounded vertices, and the sidewall  15  comprises curved sloping surfaces between the ridges  24  and the valleys  25 .  FIG. 41D  is a plot of the illuminance distribution obtained by reflecting a beam of collimated light from the sidewall  15  in a manner such as described above with respect to  FIG. 41B . As shown, the illuminance is more uniformly distributed along the semicircular distribution. 
       FIG. 41E  schematically illustrates a partial cross-section of yet another embodiment of a sidewall  15  comprising ridges  24  and valleys  25 . In the illustrated embodiment, the ridges  24  comprise more highly pointed vertices, the valleys  25  comprise more gently rounded vertices, and the sidewall  15  comprises curved sloping surfaces between the ridges  24  and the valleys  25 .  FIG. 41F  is a plot of the illuminance distribution obtained by reflecting a beam of collimated light from the sidewall  15  in a manner such as described above with respect to  FIG. 41B . As shown, the illuminance is even more uniformly distributed along the semicircular distribution. 
     In certain embodiments, the sidewalls include scatter features (e.g., surface roughness) that scatter light incident thereon. These scatter features may be included on the ridges, valleys, and/or sloping sidewalls therebetween. In various embodiments, these scatter features increase the uniformity of the output of the mixer. In some embodiments, the scatter features may increase the thickness of an illuminance distribution such as the illuminance distributions shown in  FIGS. 41B ,  41 D, and  41 F. 
     In certain embodiments, the ridges  24  and the valleys  25  vary in height along a length of a mixer so as to introduce scatter effects. In some embodiments, the variations in height comprise multiple undulations along the length of the mixer. The undulations can, for example, have regular, irregular, or random intervals or periods, or any combination thereof. 
     With reference again to  FIG. 40 , in some embodiments, the ridges  24  and the valleys  25  have surface portions that extend at an angle ∈ with respect to a plane running substantially tangent to and/or parallel with a plane comprising the ridges  24  and the valleys  25  (e.g., the xz plane). In some embodiments, the angle ∈ is constant along the length of the mixer. In other embodiments, the angle ∈ varies along the length of the mixer, such as, for example, when the ridges  24  and the valleys  25  vary (e.g., undulate) along the length of the mixer. In various embodiments, it is preferable that the ridges  24  and the valleys  25  have a maximum absolute value of ∈ that is no greater than about 7 degrees, no greater than about 5 degrees, no greater than about 3 degrees, or no greater than about 1 degree along the length of the mixer. 
     In many embodiments, light propagating through a rippled mixer having ridges and valleys is scattered by the surface portions inclined at an angle ∈, which may vary. Such scatter may further mix light propagating through the mixer. In some embodiments, this scattering improves the uniformity of the illuminance distribution at the output face of the mixer. In some embodiments, the scatter features may increase the thickness of an illuminance distribution such as those shown in  FIGS. 41B ,  41 D, and  41 F. In some embodiments, the scattering increases the angle at which light exits the output face of the mixer. In further embodiments wherein the ridges and the valleys comprise variable, undulating heights, the angle at which some of the light is oriented with respect to the optical axis of the mixer can cumulatively increase with successive reflections. Accordingly, in some embodiments, relatively smaller angles ∈ are preferred. 
     In certain embodiments, scattering such as just described can be produced by a diffuser or a hologram (e.g., a holographic diffuser). In some embodiments, a rippled mixer comprises ridges and valleys that have a constant height along the length of the mixer (e.g., the mixers  120 ,  130 ,  170 ). In certain embodiments, a reflective hologram or diffuser is wrapped around a solid mixer or a transmissive optical element is disposed inside a hollow mixer in a manner such as schematically depicted in  FIG. 37 . Alternatively, surface features that form the diffuser or hologram may be included in and/or on the sidewall. In some embodiments this diffuser may comprise an elliptical or asymmetric diffuser. For example, in some embodiments, the diffuser or hologram scatters light at a low angle in one direction and at a large angle in another direction. In some embodiments, the hologram produces a 60°×0.1° or a 60°×5° elliptical distribution. 
       FIG. 42  schematically illustrates one embodiment of a system  600  comprising a first rippled mixer  610 , a second rippled mixer  620 , and transfer optics  630 . In some embodiments, the first mixer  610  is a circular mixer comprising ridges  24  and valleys  25  and the second mixer  620  is a square mixer comprising ridges  24  and valleys  25 . In some embodiments, the transfer optics  620  comprise one or more lenses or other suitable optical elements for transferring spatial distributions to angular distributions and vice versa. Such transfer optics may, for example, convert the uniform spatial illuminance distribution output from the first mixer  610  into a uniform angular distribution input into the second mixer  620 . Other combinations and configurations of either of the mixers  610 ,  620 , the transfer optics  630 , or the system  600  are possible. 
     In some embodiments, the system  600  provides highly uniform light at an output face  22  of the second mixer  620 . In many embodiments, both the angular distribution and the spatial distribution of the light are uniform. In some embodiments, the spatial distribution exiting the first mixer  610  is substantially uniform, but the angular distribution is not. In further embodiments, the transfer optics  630  direct light exiting the first mixer  610  such that the angular distribution exiting the first mixer  610  is transferred to the input face  21  of the second mixer  620  as a spatial distribution, and the spatial distribution exiting the first mixer  610  is transferred to the input face  21  of the second mixer  620  as an angular distribution. For example, in some embodiments, the transfer optics  630  comprises a lens that satisfies the Abbe sign condition, wherein light exiting the mixer  610  at an angle θ would focus at a distance F×sin(θ) above the optical axis, where F is the focal length of the transfer optics  630 . In some embodiments, the distribution at the input face of the mixer  620  is telecentric. In some embodiments, the second mixer  620  produces a substantially uniform spatial distribution from the non-uniform distribution introduced at the input face  21 , while the substantially uniform angular distribution introduced at the input face  21  remains substantially uniform at the output face  22 . 
     The transfer optics  630  can be configured to produce a variety of angular distribution patterns from uniform illuminance distributions that, in some embodiments, exit the first mixer  610 . For example, in some embodiments, the transfer optics  630  produces distributions following the equation 1/cos ̂3. 
       FIG. 43  schematically illustrates one embodiment of a mixer  650 , such as the mixers disclosed herein, coupled with a diffuser  660 . In certain embodiments, the diffuser  660  is a low angle diffuser. For example, in certain embodiments the diffuser  660  is configured to diffuse collimated light incident perpendicular to the diffuser into an angular distribution having a full-width half-maximum (FWHM) of about 30 degrees or less, about 6 degrees or less, or about 3 degrees or less. Other diffuser angles and diffuser distributions are also possible. In certain advantageous embodiments, the diffuser  660  homogenizes the light that does not hit the sidewalls of the rippled mixer. In some embodiments, the scatter angle of the diffuser is small compared to the numerical aperture of the flux coupled out of the mixer. In certain of such embodiments, the diffuser does not significantly increase the numerical aperture of the output light. A wide variety of different types of diffusers may be used. Surface diffuses, volume diffusers, holographic diffusers are some non-limiting examples. Other diffusers are possible. In some embodiments, the end of the mixer includes a surface that is diffusing (e.g., the output face) or include volume features therein that are diffusing. Still other configurations are possible. 
       FIG. 44A  schematically illustrates a top plan view of one embodiment of a mixer array  700 . In certain embodiments, the array  700  comprises a plurality of mixers  710 , such as any of the mixers disclosed herein. Accordingly, in some embodiments, one or more of the mixers  710  comprises a rippled sidewall  15 . In some embodiments, each of the mixers  710  comprises a rippled sidewall  15 . In the illustrated embodiment, the mixers  710  comprise generally circular cross-sections. In some embodiments, mixers with hexagonal, rectangular, or other shapes that tile to cover a substrate  730  are used. Other cross-sectional configurations are also possible. 
     In some embodiments, the mixers  710  are arranged in rows  722  and columns  724 . In other embodiments, the mixers  710  are spaced from each other by irregular or random distances. In still other embodiments, the mixers  710  are grouped or bunched together, and can be in close proximity (e.g., adjacent) to each other. In some embodiments, the rippled sidewalls  15  of two or more adjacent mixers  710  are complementary to each other. In certain of such embodiments, adjacent mixers  710  the mixers  710  are separated by a thin void that may allow leakage from one mixer to another. In some embodiments, optical axes of two or more of the mixers  710  are substantially parallel. In other embodiments, some optical axes are angled with respect to each other. 
     In certain embodiments, two or more of the mixers  710  are mounted on the substrate  730 . In some embodiments, the substrate  730  comprises glass, plastic, or polarizing materials. Other materials are possible. The mixers  710  can be mounted to the substrate in any suitable manner, such as by an optically transmissive and/or index-matching adhesive. In other embodiments, the mixers  710  are integrally formed with the substrate  730 . In some embodiments, the mixers  710  and the substrate  730  are formed in a common mold. 
       FIG. 44B  schematically illustrates a side elevation view of the mixer array  700  depicted in  FIG. 44A . In certain embodiments, the cross-sectional area of one or more of the mixers  710  is constant along the length of the mixers  710 . In some embodiments, one or more of the mixers  710  are tapered from a larger input end  21  to a smaller output end  22 . In other embodiments, one or more of the mixers  710  are tapered from a smaller input end  21  to a larger output end  22 . In various embodiments, one or more of the mixers  710  comprise a draft angle of about 1 degree, about 2 degrees, about 3 degrees, or about 5 degrees; greater than about 1 degree, greater than about 2 degrees, greater than about 3 degrees, or greater than about 5 degrees; or less than about 5 degrees, less than about 3 degrees, less than about 2 degrees, or less than about 1 degree. 
     Other shapes are possible. In some embodiments, for example, the rippled mixers comprise rippled compound parabolic collectors. Other designs may also be used. 
     In some embodiments, the input faces  21  are spaced away from the substrate  730 , and in other embodiments, the output faces  22  are spaced away from the substrate  730 . In some embodiments, input faces  21  of two or more of the mixers  710  are substantially coplanar. In other embodiments, output faces  22  of two or more of the mixers  710  are substantially coplanar. In some embodiments, the mixers have different lengths or are not coplanar. Other configurations are possible. 
     In certain embodiments, a light source (not shown), such as the light source  10  is coupled with the array  700 . In some embodiments, the light source is configured to provide a single beam to the mixers  710 . In some embodiments, the single beam is coupled with one of the mixers  710 , a plurality of the mixers  710 , or all of the mixers  710 . In other embodiments, two or more light sources are coupled with the array  700 . In some embodiments, one or more light sources are individually coupled with one or more of the mixers  710 . In further embodiments, one or more light sources are individually coupled with each of the mixers  710 . In some embodiments, one or more light sources are coupled with the array  700  such that light propagates through the substrate and into one or more of the mixers  710 , and in other embodiments, the light propagates through one or more of the mixers  710  and into the substrate. Other configurations and designs are possible. 
     In some embodiments, the array of rippled mixers may provide mixing without loss in etendue. Performance is generally less sensitive to the angular distribution exiting the light pipe when ripples are included. In some embodiments, the area between individual mixers is reflective so that light that does not enter the input face of the individual mixers is reflected back toward the source and can therefore be recycled. 
     In certain embodiments, mixers comprising rippled sidewalls, such as those disclosed herein, produce a substantially uniform illuminance distribution at an output face thereof. In some embodiments, the standard deviation of the illuminance distribution (σ Illuminance ) is less than 5 percent. 
     In certain embodiments, the rippled mixers produce a substantially uniform color at an output face thereof. One common specification for characterizing color uniformity is as follows: 
       delta —   u′   —   v ′=(( u′−u′ _avg′) 2 +( v′−v′ _avg′) 2 ) 1/2  
 
     where u′_avg and v′_avg are the average color point, and u′ and v′ are the test point. In some embodiments, the mixers produce a color having a delta_u′_v′ less than 0.02. In further embodiments, the delta_u′_v′ is less than 0.009. In still further embodiments, the delta_u′_v′ is less than 0.003. 
     Certain embodiments disclosed herein may be used in numerous applications. For example, mixers comprising ribbed or rippled sidewalls may be used to substantially uniformly illuminate a spatial light modulator (e.g., DLP, LCOS, GLV, LCD) in a display, such as a micro-display, a front/rear projector, a heads-up device, a television, etc. In addition, certain embodiments enable spectrally desired or controlled solid state, incandescent, and fluorescent lighting, which can be beneficial in commercial and residential applications. Embodiments may be used for architectural and landscape lighting, display case lighting, stage and studio lighting, which often use spot lighting and flood lighting, signage, and other lighting applications. In particular, many lighting and lighting replacements may benefit from certain embodiments, including portable lights such as flashlight, downlights, spot lights, flood lights, lighting, and wall washing lights. Other areas that can benefit from certain embodiments include lithography, biomedical sensing, and industries where beam patterns are regulated, such the automotive and aviation industries. Military and medical applications are also possible. 
     As described above, a wide variety of configurations are possible. For example, in different embodiments, features may be added, excluded, rearranged, or configured differently. In some embodiments, for example, the shapes, angles, dimensions, and/or number of features can be different than those disclosed herein. 
     Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures or characteristics of any embodiment described above may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments. 
     Similarly, it should be appreciated that in the above description of embodiments, various features of the inventions are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure and aiding in the understanding of one or more of the various inventive aspects. This method of disclosure, however, is not to be interpreted as reflecting an intention that any claim require more features than are expressly recited in that claim. Rather, as the following claims reflect, inventive aspects lie in a combination of fewer than all features of any single foregoing disclosed embodiment. Thus, the claims following the Detailed Description are hereby expressly incorporated into this Detailed Description, with each claim standing on its own as a separate embodiment.