u . s . pat . no . 4 , 984 , 144 describes a dispersive linear prismatic film 1 that deviates a beam of light by refraction and total internal reflection , as shown in fig1 . the isosceles triangle micro prisms 2 have an apex angle α of 69 °. an incident ray 3 enters facet 4 at an entrance angle θ of about 75 °, where it is refracted . it is then totally internally reflected at facet 5 , and ray 7 exits perpendicular to the planar face 8 . the large entrance angle θ of about 75 ° was a requirement for this prismatic lens in its intended use in a high aspect ratio light fixture . fig2 shows the linear prismatic film 9 of the current invention , consisting of a series of isosceles triangle shaped micro prisms 10 with an apex angle α of 60 °. when an extended area collimated light beam 11 enters the film perpendicular to facet 12 at the specific entrance angle θ of 60 °, then the light rays 13 exit perpendicular to plano surface 14 with the deviation occurring entirely by total internal refraction at facet 15 . since there is no refraction at surfaces 12 or 14 , there is no dispersion and the ray deviation is independent of the refractive index of the material . moreover , since ray 11 of the entrance beam touches both the peak and valley of adjacent microgrooves , the entrance beam 11 completely fills the tir facet 15 , and there are no geometric losses or spurious ray deviations . for the 60 ° apex angle film , fig3 illustrates that two collimated incident beams 16a and 16b can be efficiently and spatially combined . the individual exiting ray bundles 17a and 17b from the left and right incident beams 16a and 16b are interlaced on a micro scale , such that the intensity of light over the total area of the film 9 is effectively doubled . for the linear prismatic beam - combining film 9 of the current : invention , the following conditions need to be satisfied for most efficient operation : 1 ) the incident light 16a and 16b must be collimated so that light rays from each lamp enter the entire prismatic film 9 at the same entrance angle θ ; 2 ) the preferred vertex angle α of the linear prismatic beam - combining film 9 is 60 ° plus or minus 2 °; and 3 ) the preferred entrance angle θ of the collimated light entering the linear beam combining film is 60 ° plus or minus 3 . 5 °. if the prism vertex angle α is greater than 62 °, less than 90 % of each reflecting facet is utilized . for example , for an acrylic plastic prismatic film with a refractive index n = 1 . 492 , a vertex angle α of 62 °, and an entrance angle θ of 63 . 5 °, only about 90 % of each reflecting facet is utilized . excessive underfilling of the reflecting facets causes the collimated exit beams 17a and 17b produced by each adjacent microprism to be spatially separated , and dark banding begins to appear on the illuminated projection screen . on the other hand , when the prism vertex angle α is less than 58 °, less than 90 % of the incident light rays 16a and 16b exit perpendicular to the film . for example , for an acrylic plastic prismatic film with a refractive index n = 1 . 492 , a vertex angle α of 58 °, and an entrance angle θ of 56 . 5 °, about 10 % of the incident rays miss each reflecting facet . the light missing the reflecting facet exits the film in an uncollimated and uncontrolled direction , and does not contribute to the illumination on the projection screen . table 1 below illustrates the fraction of beam filling of each facet ( bff ) for various vertex angles α of the film 9 . values of bff less than unity represent underfilling of the reflecting facet , while values of bff greater than unity represent overspilling of the reflecting facet . as explained above , more than 10 % of the incident light is spatially separated or wasted at vertex angles θ above 62 ° or below 58 °, upon exiting the film 9 . table 1______________________________________α θ bff______________________________________50 42 . 3 1 . 5558 56 . 5 1 . 1159 58 . 3 1 . 0560 60 . 0 1 . 061 61 . 7 0 . 94762 63 . 5 0 . 895 . 70 77 . 7 0 . 446______________________________________ it will be seen from table 1 that the entrance angle θ of the collimated light 16a and 16b generally changes in a similar fashion as the vertex angle α changes , but the relationship is not linear . this is because at entrance angles θ other than 60 ° the refractive index n of the material of the film has an effect . the exact relationship for the entrance angle θ necessary to produce maximum screen illumination at a given vertex angle α is given by the equation : ## equ1 ## wherein : θ = said angle of inclination of said collimated light fig4 shows a projection system 18 that efficiently integrates the output from two lamps 19a and 19b , providing uniform screen brightness when each lamp 19a or 19b is individually on , and doubling the screen brightness when both lamps 19a and 19b are on . light sources 19a and 19b are positioned at the focal point of rectangular - shaped fresnel lenses 20a and 20b , respectively , which collimate the light beams . each collimated beam fills the stage aperture 21 at an entrance angle θ of 60 °. the stage aperture 21 can be square , to accommodate full - size overhead transparencies , or a reduced size rectangular format , to accommodate lcd projection panels . near the stage aperture 21 is the 60 ° linear prismatic film 9 . the integrated and collimated light exiting from the 60 ° linear prismatic film 9 enters a circular fresnel lens 22 which focuses the light to the projection lens 23 . a glass platen 24 is usually placed above the fresnel lens 22 to supports the overhead transparency or lcd projection panel . if the projected facet widths of the linear prismatic film 9 are less than the resolving power of the eye at normal screen viewing distances , then each light source 19a and 19b appears to fully illuminate the entire screen . with both lamps 19a and 19b on , the screen brightness is effectively doubled over the brightness produced by a single lamp 19a or 19b . fig5 shows an alternate arrangement using folding mirrors 25a , 25b , and 26 , to combine the output of the two light sources 19a and 19b . an additional 60 ° beam combining linear prism 9a is required . fig6 shows a configuration that combines the output of four light sources 19a , 19b , 19c , and 19d . in this arrangement , the additional linear prismatic film element 9b , additional fresnel lens collimators 20c and 20d , and additional folding mirrors 25c and 25d are shown . this cascading process can be further extended to integrate the output of additional light sources . the beam shaping requirements to fill rectangular optical elements in this multiple lamp projection system are most easily achieved when these optical elements have an aspect ratio l / w that is close to unity , e . g . a square perimeter . to achieve this condition , it is preferable that the each additional optical element be oriented along the shorter side of the rectangular element preceding it . for the two lamp system shown in fig4 and illustrated again in more detail in fig7 with a rectangular linear prismatic element 9 having a length l 1 = 8 units , and a width w 1 = 6 units ( l 1 / w 1 = 1 . 33 ), if the fresnel lens collimators 20a and 20b ( only 20a is shown ) are oriented along the w 1 dimension , then the length l 2 of the collimator is l 2 = w 1 = 6 units , and the width w 2 of the collimator = l 1 / 2 = 4 units , giving an aspect ratio l 2 / w 2 = 6 / 4 = 1 . 5 . if the fresnel lens collimators were oriented along the l 1 dimension , then the collimator aspect ratio = l 2 / w 2 = 8 / 3 = 2 . 67 , and beam shaping is more difficult to achieve . similarly , for the four lamp system shown in fig6 with a square prismatic element 9 having a length l 1 = 12 units , and a width w 1 = 12 units , then the linear prismatic element 9a must have an aspect ratio l 2 / w 2 = 12 / 6 = 2 . if the fresnel lens collimators 20a and 20b are oriented along the w 2 dimension , then the aspect ratio of the collimators = l 3 / w 3 = 6 / 6 = 1 , which is the ideal beam shaping requirement . it is also important to note that for each additional level of multiple light sources , e . g . two lamps , four lamps , eight lamps , etc ., that the area of each additional linear prism element or fresnel collimator , is halved . this limits the achievable light collection and sets a practical limit on the number of lamps that can be integrated . fig9 shows a projection system that efficiently integrates the output of two lamps , as is also illustrated in fig4 again providing uniform screen brightness when each lamp is individually on , and doubling the screen brightness when both lamps are on . a polarization - modulating display 32 , such as a liquid crystal display , is positioned between fresnel lens 22 and the 60 ° linear prismatic film 9 to define an optical window through which light from the lamps is directed . this liquid crystal display panel 32 comprises a layer of liquid crystalline material which can be of a twisted nematic or a supertwisted nematic enclosed between two transparent substrates or plates . each of these plates may comprise a transparent control electrode which can be divided into a large number of columns and rows ; thus defining a large number of image elements in the display panel . these image elements are controlled by driving the electrodes , and the image display panel is referred to as passively controlled . alternatively , one of the substrates can be provided with an electrode while the other is provided with semi - conductor drive electronics . a device employing this type of control is referred to as an actively controlled image display panel . described below with respect to fig7 and further illustrated in fig8 is a specific arrangement that has been constructed for use as an lcd overhead projector . the rectangular stage aperture size has a length l 1 = 228 . 6 millimeters ( 9 . 0 inches ) and a width w 1 of 171 . 5 millimeters ( 6 . 75 inches ), giving an aspect ratio of l 1 / w 1 = 4 / 3 , a common ratio for many lcd projection panels . the light source filaments 26 ( only one is shown ) are positioned about 11 millimeters behind a pair of glass condensers 27 which collect and direct light to the fresnel lens collimator a . the light sources 26 are 400 watt , 36 volt flat mandrel capsule types , ansi code designation evd . a spherical reflector 28 having a diameter of 60 millimeters and a radius of curvature of 32 . 5 millimeters focuses the back rays in the forward direction . an aspheric symmetric pyrex condenser 29 , having an approximate focal length of 55 millimeters and a diameter of 60 millimeters , focuses the light from the lamps 26 into a light cone of circular cross - section . a square optical crown glass cylinder lens 30 having an approximate focal length of 175 millimeters , and dimensions of 82 by 82 millimeters , is placed in close proximity to the pyrex condenser 29 . this cylinder lens 30 further compresses this circular cone of light in one direction to form an elliptical shaped beam 31 . the single element fresnel lens collimators 20a have a focal length of 178 millimeters , operate between f / 0 . 75 and f / 1 . 0 , and are oriented symmetrically at 30 ° from the vertical . the height w 2 of the fresnel collimator 20a is half the length l 1 of the stage aperture , w 2 = 114 . 3 millimeters ( 4 . 5 inches ), and the width l 2 of the fresnel collimator 20a is the width of the stage aperture , l 2 = 171 . 5 millimeters ( 6 . 75 inches ). the aspect ratio l 2 / w 2 of the collimating fresnel lens is 1 . 5 . to efficiently fill the rectangular aperture of the collimating fresnel lens , the following relationship should be approximated : l 2 / w 2 = a / b , where a and b are the major and minor axes of the elliptic cross - section 31 of the light beam at the plane of the fresnel collimator as shown in fig8 . as illustrated in fig4 the 60 ° linear prismatic film 9 was fabricated in 2 millimeter thick acrylic plastic , with dimensions slightly larger than the stage aperture , and the width of each individual prismatic groove 10 was about 0 . 5 millimeters . a rectangular acrylic fresnel lens 22 , having a focal length of about 325 millimeters , groove widths between 0 . 5 and 0 . 125 millimeters , and approximately the same size as the linear prismatic film 9 , was placed between the linear prismatic film 9 and a glass platen 21 defining the stage aperture . a triplet projection lens 23 of 330 millimeter focal length projected an image of the stage to fill a 60 inch wide screen at approximately 6 . 7x magnification . with this configuration , an average screen illumination of 140 foot - candles was measured from each individual lamp , and an average screen illumination of 280 foot - candles with both lamps operating . this is equivalent to the brightness of a 7000 lumen square aperture overhead projector projecting a 60 inch square image . it can be appreciated by those skilled in the art , that by the use of additional folding mirrors in the configurations described , modifications of the arrangement of optical components can be achieved . these modifications can reduce the base height of the projector by variation of width and length of the projector base containing these components . it will also be appreciated that although specific examples of the invention have been illustrated , the invention is more generally applicable to any device which requires collimated light in the optical path .