WO2007081812A2 - Luminaire reflectors - Google Patents

Abstract

Reflectors (304, 702, 902, 1204, 1406, 1600, 1700, 1800, 1800, 2000, 2100) for use with compact sources have generatrices defined by second order differential equations. The reflectors are able to illuminate planes with controlled (e.g., substantially uniform) light distribution even though the compact sources emit light nonuniformly. Luminaires (300, 1000, 1100, 1200, 1300) including the reflectors are also shown and described.

Description

LUWIINAIRE REFLECTORS

FIELD OF THE INVENTION

[0001 ] The present invention relates in general to luminaires. More particularly, the present invention relates to luminaires with improved optics.

BACKGROUND OF THE INVENTION

[0002] In many technical applications, it is desirable to collect a high percentage of light that is emitted from a light source and to direct the collected light to an area (or object) to be illuminated. Examples of such applications include machines that use light in the processing of semiconductor wafers and image projectors (e.g., film projectors, LCD projectors, and micromirror array projectors).

[0003] For many applications it would be desirable to be able to collect a high percentage of light emitted by a light source, distribute the collected light within an area (or on an object) to be illuminated in a highly controlled manner (e.g., uniformly) and have the light incident on the illuminated area with a small local divergence.

BRIEF DESCRIPTION OF THE FIGURES

[0004] The present invention will be described by way of exemplary embodiments, but not limitations, illustrated in the accompanying drawings in which like references denote similar elements, and in which:

[0005] FIG. 1 is a view of compact arc lamp;

[0006] FIG. 2 is a plot of luminance weighted by the cosine of the elevation angle as a function of elevation angle for a compact arc lamp such as shown in FIG. 1;

[0007] FIG. 3 is a schematic diagram including a luminaire and an object plane that is illuminated by the luminaire according to certain embodiments of the invention;

[0008] FIG. 4 is a profile of a luminaire reflector according to an embodiment of the invention;

[0009] FlG. 5 is a plot of light intensity verses radial position on an object plane illuminated by a luminaire with the reflector shown in FIG. 4 and a lamp having the cosine weighted luminance distribution shown in FIG. 2; [001 0] FIG. 6 is a profile of a second Iuminaire reflector according to an embodiment of the invention;

[001 1 ] FIG. 7 is a plot of light intensity versus radial position on an object plane illuminated by a Iuminaire with the second Iuminaire reflector and a lamp having the radiance distribution shown in FIG. 2;

[001 2] FIG. 8 is a profile of a third Iuminaire reflector according to an embodiment of the invention;

[001 3] FIG. 9 is a plot of light intensity verses radial position on the object plane illuminated by a Iuminaire having the third Iuminaire reflector and the lamp having the cosine weighted luminance shown in FIG. 2;

[0014] FIG. 10 is a cross-sectional side view of a Iuminaire according to an embodiment of the invention;

[001 5] FIG. 11 is schematic diagram of a Iuminaire and an object plane that is illuminated by the Iuminaire according to an embodiment of the invention;

[001 6] FlG. 12 is a plot of light intensity versus radial position on the illuminated plane illuminated by the Iuminaire shown in FIG. 11- and the lamp having the cosine weighted luminance shown in FIG. 2;

[001 7] FIG. 13 is cross-sectional side view of a reflectorized discharge envelope integrated Iuminaire according to an embodiment of the invention;

[001 8] FIG. 14 is a schematic diagram of Iuminaire optics including a prism according to an embodiment of the invention;

[001 9] FIGs. 15-21 illustrate generatrices of types of Iuminaire reflectors according to additional embodiments of the invention;

[0020] FIG. 22 is a flowchart of a first method of manufacturing reflectors according to embodiments of the invention;

[0021 ] FIG. 23 is a flowchart of a second method of manufacturing reflectors according to embodiments of the invention;

[0022] FIG. 24 is a flowchart of an alternative beginning of the first or second methods shown in FIGs. 22-23; and [0023] FIG. 25 is a flowchart of a subprogram that is called by an optimization routine that is called in the flowchart shown in FIG. 24;

[0024] FIG. 26 is a block diagram of a projector system in which luminares having reflectors according to embodiments of the invention can be used.

MODES FOR CARRYING OUT INVENTION

[0025] As required, detailed embodiments of the present invention are disclosed herein; however, it is to be understood that the disclosed embodiments are merely exemplary of the invention, which can be embodied in various forms. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the present invention in virtually any appropriately detailed structure. Further, the terms and phrases used herein are not intended to be limiting; but rather, to provide an understandable description of the invention.

[0026] FIG. 1 is a view of compact arc lamp 100. The compact arc lamp 100 comprises a transparent envelope 102 which comprises a central discharge containing bulb portion 104, a first stem portion 106 extending from the bulb portion 104, and a second stem portion 108 that extends from the bulb portion 104 opposite the first stem portion 106. An anode 110 and a cathode 112 are located in the bulb portion 104 and are spaced from each other forming a discharge gap 114. The bulb portion 104 also contains a discharge fill (not visible) including one or more gases, metals and/or compounds. The anode 110 is connected to an anode terminal 116 located at a distal end of the first stem portion 106. The cathode 112 is connected to a cathode terminal 118 located at a distal end of the second stem portion 108. When energized by applying, power, from a power supply (not shown), across the anode terminal 116 and the cathode terminal 118 the discharge fill becomes ionized and emits light. The name of the compact arc lamp 100 derives from the relatively small gap 114 between the anode 110 and the cathode 112 (compared to other discharge lamps), and from the small size of the light emitting discharge that is formed in the gap 114 when the lamp 100 is powered. Compact arc lamps are considered quasi point sources. An X-Y coordinate system is drawn on the lamp 100 for reference. The Y-axis is aligned on a longitudinal axis of the lamp 100 about which the lamp 100 is substantially rotationally symmetric. The origin of the X-Y coordinate system is conveniently located in the discharge gap 114. Note that the compact arc lamp 100 is but one type of lamp that is to be used in the luminaires according to the present invention. Incandescent bulbs or other types of bulbs are alternatively used in luminaires according to the teachings described hereinbelow. However, the luminaire optics described herein are particularly suited to compact arc lamps because the luminaire optics described herein are able to collect a substation portion of light emitted by compact arc lamps notwithstanding the fact that compact arc lamps emit light over a substantial though not full range of elevation angle, and are able to distribute the light in a highly controlled manner.

[0027] FIG. 2 is a plot of luminance weighted by the cosine of the elevation angle φ as a function of elevation angle φ for a compact arc lamp of the general type shown in FIG. 1. In particular, the example data shown in FIG. 2 is from a 160 Watt DC "SHP" model Mercury Xenon lamp manufactured by Phoenix Electric of Hyogo Prefecture Japan. The plot shown in FIG. 2 was generated using light rays generated using a ProSource™ light source model software (version 6.2.4.) which is published by Radiant Imaging of Duvall Washington, in conjunction with a ProSource™ model. The ProSource™ model is based on measurements of the lamp using a photopic filter.

[0028] The coordinate system shown in FIG. 1 is retained in FIG. 2. The elevation angle denoted φ is measured, in the counterclockwise direction, from the positive X-axis. In as much as light radiated by the lamp 100 is not limited to a plane it is appropriate to consider a Z-axis that extends from the origin perpendicular to the X-Y plane (out of the plane of the drawing sheet). Given the addition of the Z-axis, it is noted that the elevation angle φ is, in fact, measured from the X-Z plane. The luminance shown in FIG. 2 is weighted by the cosine of the elevation angle φ because a factor of cos(φ) relates a differential of elevation angle Δ φ to a differential of solid angle, which is to say that the amount of light available within a differential of elevation angle, at a particular elevation angle is proportional to the product of the light intensity (e.g., radiance or luminance) at the particular elevation angle multiplied by cos(φ).

[0029] Although compact arc lamp lamps (e.g. 100) can exhibit some transient flicker, the time average light intensity distribution (radiance or luminance) is nominally rotationally symmetric about the longitudinal axes of compact arc lamps (e.g. 100), which coincides with the Y-axis shown in FIGs. 1-2. In other words the radiance is nominally uniform as a function of azimuth angle. On the other hand, as shown in FIG. 2, the light intensity distribution is not uniform as a function of elevation angle. For comparison, reference circle 202 shows what the light intensity distribution weighted by cos(φ) would be if the light intensity distribution were uniform as a function of elevation angle. Note that the actual light intensity distribution is not symmetric about the X-Z plane. Also, note the small peak 204 Note also that the light intensity distribution of a compact arc lamp (e.g. 100) is substantially confined to a range of elevation angle that extends upward and downward from the X-Z plane by less than ninety degrees. For the lamp on which FIG. 2 is based there is substantial light intensity between —45 degrees (ττ/4 radians) and +50 degrees (0.873 radians), and there is a non-negligible flank that extends up to about +75 degrees (1.31 radians). The light intensity along the optical axis (Y-axis) in both the positive and negative directions is substantially zero. The elevation angle is limited due to obstructions by the anode 110 and cathode 112. In FIG. 2 nominal lower and upper bounds of the elevation angle range in which the compact arc lamp 100 emits are denoted cp_A and <P_B respectively. There may be some minimal detectable light emitted outside the bounds.

[0030] Parts of the bulb portion 104 can be covered with a reflective coating in order to further restrict the angular range light emission and enhance the brightness of light emitted in the restricted angular range. Optics disclosed below can be used with bulbs having partial reflective coatings as well as bulbs without such coatings.

[0031 J FIG. 3 is a schematic diagram including a luminaϊre 300 and an area 302 that is illuminated by the first luminaire 300 according to embodiments of the invention. In FIG. 3 the full X-Y-Z, 3-space coordinate system alluded to earlier is shown. A lamp (e.g. 100) is schematically represented in FIG. 3 by a pair of opposed chevrons. The luminaire 300 comprises a reflector 304. (The chevrons representing the lamp are seen through the first reflector 304, as though in an X-ray view.) The reflector 304 distributes light emitted by the compact arc lamp 100 in a controlled manner, such that a predetermined desired radial light intensity distribution on the illuminated plane 302 is well approximated. The illuminated area 302 is a mathematically specified object plane that may or may not include an actual physical object (e.g., light modulator, semiconductor wafer, photosensitive coating, etc.) The luminaire 300 is able to achieve controlled distribution of light closely approximating specified desired distributions. FIG. 3 includes an idealized depiction of how light emitted within a differential of solid angle corresponding to a differential of elevation angle Δφ, at particular elevation angle φ is reflected into an annulus having a radius x and a differential width Δx. (Note that the differential of solid angle that maps into the annulus spans a complete 2π range of azimuth angle.)

The reflector 304 and other reflectors described hereinbelow can be surfaces of revolution, such that light emitted by a source (e.g., compact arc lamp 100) is distributed over a circular area of the illuminated plane 302. According to alternative embodiments only an off-axis portion of the first reflector 304 is used. The profile or generatrix of the reflector is given by the following second order differential equation defined in polar coordinates: EQU. 1

& i f ( d V _Λ f d ^v

— rr((ΦΦ)> == ^ -| [ --bDIISSTT[ 4 —rr((Φφ))J ~-£J⁾>IISSTTτT(Wφf ++DDIISSTT\ ^ ~r(φ) J cos(φ)² +_JD7Srr(φ)⁴ cos(φ)²

- 4 ^— r(φ)J sin(φ) r(φ) + Wr(φ)⁴ +3 ^— r(φ)J 7^-2 ^— r(φ)J r(φ)² cos(φ)

where,

<p is the domain variable of the domain in which equation 1 is defined and is the elevation angular coordinate of the generatrix of the reflector in the X-Y coordinate system; (cp is measured in a counterclockwise direction from the positive X-axis of the X- Y coordinate system.) r(φ) is a polar radial coordinate of the generatrix of the first reflector 304 (in the X-Y coordinate system r(φ) is equal to Jχ² + y );

Yt is equal to the Y coordinate of the object plane; and EQU. 2

where, Xt(φ) is the X-coordinate on the object plane 302 to which a ray emanating at an elevation angle φ (in the X-Y coordinate system) from the origin of the X-Y coordinate system would be reflected by the reflector 304, and is given by: EQU. 3

(Note that in using Xt it is assumed that the generatrix of the reflector is being derived in the X-Y coordinate system. In a cylindrical coordinate system coincident with the full X-

Y-Z coordinate system Xt is a cylindrical radial coordinate equal to (X²+Z²) "^Λ .)

Rad(φ) is the intensity of light (e.g., radiance or luminance) emitted by the lamp 100 of the luminaire 300 at elevation angle φ;

(Note as indicated above in reference to FIG. 2 that the luminance varies as a function of elevation angle φ)

Irr(Xt) is the desired irradiance at a given cylindrical radial coordinate on the illuminated plane 302;

Θj__R is an angle of incidence on the reflector 302 given by: EQU. 4

Γ_R(Θ_{J R}) is the reflectance of the reflector for light incident at angle of incidence θ_{i R} ;

(Note, that for reflectors with reflective surfaces that exhibit a small variation of reflectivity with incidence angle Γ_R(Θ_J__R) is suitably eliminated (set equal to 1), without substantially changing the shape of the reflector or incurring a significant penalty in the performance of the reflector)

X_MIN is the inner radius of an annulus of the object plane 302 to be illuminated by the luminaire 300;

(Note that if a circular area of the object plane 302 is to be illuminated X_MIN is equal to zero.)

XMAX is an outer radius of an annulus or circular area of the object plane 302 to be illuminated by the luminaire 300; φo is the lower limit of the elevation angle range subtended by the reflector 304; φn is the upper limit of the elevation angle range subtended by the reflector 304;

F is a normalization factor that compensates for reflection losses;

(Note if r_R(θ_{i R}) is eliminated by being set equal to one then F is also eliminated by setting F equal to 1. If r_R(θj__R) is not eliminated by setting equal to one then F is approximately equal to the inverse of the r_R(θj__R) for a typical angle of incidence θj__R

(e.g., 1/0.96 for silver). A more precise value of F is found by trial and error (e.g., by a numerical root finding method) such that Xt(cpn, r(φ_Q), 3r(φ)/5φ (at

where X_Ω is a chosen value of X (either X_M|_N or XM_AX) to which a ray emanating from the origin at angle

<PΩ is to be directed by the reflector); with initial conditions r(φ₀) set as described hereinbelow.

[0032] Rad(φ) is based on measurements of the lamp 100 or other light source to be used in the luminaire. For real world lamps Rad(φ) is nonuniform, which is to say Rad(φ) is a non-constant function of φ. Rad(φ) and r_R(θj__R) are suitably represented as a interpolating spline, such as a cubic spline. Irr(Xt) can be represented by a single mathematical function or a piecewise defined mathematical function such as a cubic spline.

[0033] Γ_R(Θ_LR) is based on measurements of the reflective material used to make the reflector. For certain materials Γ_R(Θ_J__R) can be calculated but calculated values should also be verified with measurements.

[0034] Reflectors defined by equation 1 are able to efficiently collect light from non-uniformly emitting sources and illuminate object planes with a relatively high degree of accuracy according to predetermined specified radial light intensity distributions. Note that despite that the fact that the source (e.g., lamp 100) does not in general maintain the same radiance or luminance over a large portion of the angular range over which the source emits φo and φo can be set wide apart to collect a high percentage of emitted light while still achieving relatively accurate approximation of a desired light intensity distribution. Examples given below illustrate achievable levels of accuracy. Note that equation 1 is not based on any assumptions as to the distance between the luminaire 300 and the object plane 302, and thus reflectors described by equation 1 are able to illuminate very close object planes. Additionally, reflectors described by equation 1 are able to distribute light in a highly controlled manner within illuminated areas that have a transverse dimension that is not so large that the transverse dimension of the reflector is negligible in comparison. In fact, in several examples shown below, the transverse dimension of the illuminated area is smaller than the transverse dimension of the reflector.

[0035] Note that there is either a plus or minus sign in front of the DIST subexpression. Equation 1 with the plus sign in front of the DIST subexpression specifies a reflector that (in the case of an ideal point source) reflects light such that as the elevation angle of a ray emanating from the source (e.g. 100) increases from φo to cpo, the object plane intercept of the ray, Xt decreases. On the other hand equation 1 with a minus sign in front of the DIST subexpression specifies a reflector that (in the case of an ideal point source) reflects light such that as the elevation angle of a ray emanating from the source (e.g. 100) is increased from φo to cpo, the object plane intercept, Xt increases. For real quasi point sources the relations are not strict. Reflectors described by equation 1 are able to produce low local divergence at the object plane. As will be describe below it is also possible to have a piecewise defined reflector in which the sign in front of DIST changes in each successive portion of the reflector.

[0036] Equation 1 is suitably integrated numerically in order to determine the shape of the generatrix of a reflector. A straight forward approach is to integrate equation 1 in the forward direction starting at a selected lowest value of the independent variable-the elevation angle φ which is φ₀. The initial polar radial coordinate r(φo) and the Y-coordinate of the object plane, Yt are then chosen. A wide range of initial polar radial coordinate values and Y coordinates of the object plane have been used successfully in integrating equation 1.

[0037] The initial condition r(φo) determines the transverse dimension of the reflector. Thus r(φo) is suitably selected small enough to fit within a space allowed for the reflector, however r(<po) should not be made so small that it approaches the dimension of the light source. Once r(φ₀) and ,Yt have been selected drfdφo is determined by: EQU. 5:

where X₀ is a chosen radial coordinate to which a ray emanating from the origin at angle cpo is to be reflected by the reflector. Xo is set equal to X_MAX or X_MIN-

[0038] Note that there is a singularity in DIST in equation 1 , when Xt is equal to zero. In the case that a circular area of the illuminated plane 306 is to be illuminated it may be necessary for certain numerical differential equation integrators to select Xmin equal to some small, physically insignificant, value, rather than zero in order to avoid the singularity causing difficulties for the integrator. For example selecting Xmin=0.001 mm as opposed to Xmin=0.0 can avoid problems caused by the singularity for certain numerical integrators..

[0039] Reflectors described by equation 1 are able to efficiently collect light from non-uniformly emitting sources and illuminate areas, including relatively small areas (e.g., areas the size of a projection image modulator) with a relatively high degree of accuracy according to predetermined specified radial intensity distributions lrr(Xt)

[0040] Equation 1 is suitably integrated numerically using a commercially or publicly available differential equation integrating program or application. To integrate equation 1 which is a second order differential equation, the following substitution is made in the equation 1 : EQU. 6:

[0041 ] Equation 1 (with the substitution) in combination with equation 6 itself, make up a system of two coupled first order differential equations that are equivalent to equation 1. The system of differential equations can be integrated using Ordinary Differential Equation (ODE) integrators that are included in computer algebra system (CAS), such as, for example, MAPLE V® by Maplesoft of Waterloo, Ontario, Canada, Mathematica ® by Wolfram Research, Inc. of Champaign, IL systems or using other commercially or publicly available ODE integrators. The inventor has used Maple V, and the FORTRAN Runge-Kutta ODE integrators included in the IMSL library published by Visual Numerics of Houston, Texas.

[0042] FIG. 4 is a generatrix of a first luminaire reflector 304 of the first luminaire 300 schematically depicted in FIG. 3 according to an embodiment of the invention. The generatrix is a solution of equation 1. The initial conditions and other design parameters associated with the reflector generatrix shown in FIG. 4 are given in Table I. In the examples described herein a constant reflectance r_R(9_{i R}) was assumed.

[0043] Table I

[0044] FIG. 5 is a plot 500 of light intensity verses radial position on the illuminated plane 302 when illuminated by the first luminaire 300 having the first reflector 304 generatrix shown in FIG. 4 and using the 160 Watt DC "SHP" model lamp. The light intensity profile shown in FIG. 5 was determined by ray tracing rays generated from the above mentioned Prosource™ light source model.

[0045] As shown in FIG. 5, the deviation of the light intensity from the desired uniform light intensity indicated by horizontal line 502, is relatively small. The average absolute value deviation from the specified distribution is 3.25%. (Note in calculating this figure equal weight was given to the deviation at each radius, (as opposed to weighting by annulus area) so as not to de-emphasize the effect of deviations closer to the optical axis). The maximum deviation is 14.8%. The area of high deviation is mainly limited to the vicinity of the X=O (optical axis), and the vicinity of Xmax. Assuming a reflectance of 0.95 the collection efficiency is 81%. Note that the diameter of the illuminated ^area 2^*Xmax is equal to the diagonal of a 0.9" (22.8 mm) micromirror array image modulator.

[0046] FSG. 6 is a generatrix of a second luminaire reflector 602 according to an embodiment of the invention. The reflector generatrix shown in FIG. 6 is another solution of equation 1. The initial conditions and other design parameters associated with the generatrix shown in FIG. 6 are given in Table II.

[0047] Table Il

[0048] As shown in Table II, the second luminaire reflector was intended to produce a light intensity profile that increased linearly from a minimum equal to 10% of the maximum intensity at the optical axis (Y-axis) to the maximum intensity at Xmax.

[0049] FIG. 7 is a plot 700 of light intensity verses radial position on the illuminated plane 302 when illuminated with the second reflector profile 602 shown in FIG. 6 and the 160 Watt DC "SHP" model lamp. The light intensity profile shown in FIG. 7 was also determined by ray tracing rays generated from the above mentioned Prosource™ light source model. The average absolute value deviation from the specified distribution is 3.08%. The maximum absolute value deviation which occurs near the optical axis is 9.26%. Assuming a reflector reflectance of 0.95 the collection efficiency is 83%.

[0050] As shown in FIG. 7, the deviation of the light intensity from the desired linearly increasing light intensity profile, which is indicated by dashed line 702 is quite small.

[0051 ] FIG. 8 is a generatrix of a third reflector 802 according to an embodiment of the invention. The reflector generatrix 802 shown in FIG. 8 is another a solution of equation 1. The initial conditions and other design parameters associated with the generatrix shown in FIG. 9 are given in Table III.

[0052] Table II!

[0053] As shown in Table III, the second reflector was intended to produce a light intensity profile that linearly decreased from a maximum intensity at the optical axis to 10% of the maximum intensity at Xmax.

[0054] FIG. 9 is a plot 900 of light intensity verses radial position on the illuminated plane 302 when illuminated by the third luminaire having the third reflector profile 802 shown in FIG. 8 and using the 160 Watt DC "SHP" model lamp. The light intensity profile shown in FIG. 8 was also determined by ray tracing based on the above mentioned Prosource™ light source model. The average absolute value deviation from the specified distribution is 2.15%. The maximum absolute value deviation which occurs near the optical axis is 9.5%. Assuming a reflector reflectance of 0.95 the collection efficiency is 91%.

[0055] FIG. 10 is a cross-sectional side view of a luminaire 1000 according to an embodiment of the invention. As shown in FIG. 10, the first luminaire 1000 comprises the arc lamp 100, which is positioned on the axis of symmetry of a reflector 1001. The reflector 1001 has a reflective surface 1002 having the profile that is a solution of equation 1. The reflector 1001 includes an upper flange 1004 and a lower flange 1006. Light reflected by the reflector 1001 exits through an opening 1007 (encircled by the lower flange 1006). A lamp support 1008 is coupled to upper flange 1004. The lamp support 1008 can be coupled to the upper flange 1004 by various methods or can be formed integrally with the upper flange 1004. The lamp support 1008 serves to support the arc lamp 100 relative to the reflector 1001. The lamp support 1008 may also serve as a heat sink for the lamp. The lamp support 1008 can be affixed to the arc lamp by various methods. Alternatively, a mechanism for adjusting the position of the arc lamp relative to the reflector is provided. Such a mechanism may be useful for adjusting the position of lamps if asymmetric erosion of the electrodes occurs. The particular mechanical arrangement for supporting the arc lamp 100 relative to the reflector 1002 that is shown here is merely exemplary. A cathode lead 1010 is coupled to the cathode terminal 118 and an anode lead 1012 is coupled to the anode terminal 116. In as much as the anode lead 1012 passes through light exiting the reflector 1001, the anode lead 1012 can be made flat and reflective and aligned so as minimize light blockage (aligned so that the optical axis is in the plane of the flat lead). This is particularly appropriate if it is necessary to make the cross section area of the anode lead 1012 large in order to carry a large current. Alternatively, multiple anode leads 1012 which may also be flat are provided to carry large currents. Current consumption varies from lamp to lamp. The details of the mechanical arrangement can vary widely from what is shown in FIG. 10 and are not the focus of the invention. For certain applications a nonuniformity in the light intensity distribution caused by an electrical lead obstructing light may be tolerable. Optionally, such a nonuniformity can be reduced by a diffuser located beyond the obstructing electrical lead. In an alternative embodiment, the anode lead 1012 is passed through a small hole (not shown) in the reflector 1001.

[0056] Note that although in the luminaire 1000 as shown in FIG. 10 the anode 110 is on the side of the reflector opening 1107 through which light passes from the reflector 1001, alternatively the orientation of the lamp 100 is reversed such that the cathode 112 is on the side of the reflector opening 1007. [0057] FIG. 11 is schematic diagram of a fourth luminaire 1100 and an object plane 1102 that is illuminated by the fourth luminaire 1100 according to a certain embodiments of the invention. The lamp 100 (or other light source, represented by opposed chevrons in FIG. 11) is positioned within a fourth luminaire reflector 1104. Light emanating from the lamp 100 is reflected by a reflective surface 1106 of the fourth reflector 1104 to the object plane 1102. In traversing from the reflective surface 1106 to the object plane 1102 the light passes through a transparent window 1108. For practical reasons it is often desirable to locate a transparent window 1108, in front of the reflector 1104 of the luminaire 1100. One reason is to contain fragments in the event that the lamp 100 of luminaire 1100 fails explosively. Another reason is to form a sealed environment within the reflector 1104 that protects the reflective surface 1106 of the reflector 1104 and lamp 100 from humidity and dust. Although as shown in FIG. 11, the transparent window 1108 is displaced from the reflector 1100 the window 1108 is alternatively located over a front aperture 1110 of the reflector 1100. Alternatively, a cylindrical housing part (not shown) is positioned between the transparent window 1108 and the front aperture 1110.

[0058] Reflectors conforming to equation 1 will, in general, reflect most rays to object planes at angles. (There may be a small subset of rays that do reach the illuminated plane at normal incidence.) As shown at 1112 in FIG. 11, rays passing through the transparent window 1108 at an angle will be offset by certain distance (denoted Δ) from their original path. Thus, if a transparent window (e.g., 1108) is used with reflectors conforming to equation 1, the irradiance profile will be disturbed. The degree of disturbance depends on the magnitude of the offsetting of rays which depends on the thickness and index of refraction of the transparent window 1108 and on the angle of incidence of light rays on the transparent window 1108. Generally, a larger difference between the size of the aperture 1110 and the illuminated area diameter, Xmax leads to rays passing through the transparent window 1108 at larger angles.

[0059] In certain cases it may be desirable to include two or more transparent windows having different indices of refraction between the reflector (e.g., 1104) and the illuminated plane (e.g., 1102). [0060] Equation 7, below describes a generatrix of the fourth reflector 1104, and more generally describes profiles of a class of reflectors that illuminate object planes through one or more with transparent objects (e.g., windows, prisms) with an light intensity distributions that closely approximate predetermined specified light intensity distributions. The transparent window 1108 or a plurality of transparent windows is a simple example of the aforementioned transparent objects. A prism that has an entrance face and an exit face, and one or more reflective faces is another example of a transparent object. One application of reflectors of the type described by equation 7 where an area is illuminated through a prism is the illumination of one or more micromirror array light modulator chips through a Total Internal Reflectance (TIR) prism in an image projector. EQU. 7

9φ' - sin(φ) — r(φ) cos(φ) +

where, capital N is the number of transparent objects (e.g., transparent windows or prisms) positioned between the reflector described by equation 7 and the illuminated plane; lower case n is an index that refers to the individual transparent objects; no is an index of refraction in the environment of the luminaire (e.g., nominally 1.0 for air) th_n is the thickness (measured along the optical axis of the reflector) of the n ,¹th transparent object (e.g., window 1108), npn is the index of refraction of the n^th transparent object; EQU. 8

where, EQU. 9

θ_{i n} is the angle of incidence of on the nth transparent object; t_n(θi_n) is the transmittance of the n^th transparent object; F, in this case is a normalization factor that compensates for reflection losses and transmission losses of the transparent objects, given by t_n(θ,__n).

[0061 ] Note, that in this case F is approximately equal to the inverse of the product of r_R(θj__R)*π t_n(θj__n) for typical values of the angle θj__R, θj__n. For example for a silver reflector and a single window having an index of refraction equal to 1.5 is used, F is approximately equal to 1.13=1/(0.96*0.96*0.96), 0.96 being the Fresnel loss at each surface of the window and 0.96 being the assumed nominal reflectance of silver. A more accurate value of F may be found as previously indicated.

[0062] Other variables in equation 7, are the same as defined above in reference to equation 1.

For use in integrating equation 7, the angle of incidence on each transparent object θj__n, is expressed in terms of the independent variable φ, polar radial coordinate r(φ), and the derivative of the polar radial coordinate drfdφ. The angle of rays reflected by the reflector relative the Y-axis (the optical axis) is given by:

EQU. 10

If light is incident on a transparent object from air and the surface of the transparent object is arranged perpendicular to the optical axis of the reflector (Y-axis) then the angle of incidence on the transparent object θj__n is equal to Θ_RR. If light is incident on a n^th transparent object from a preceding, abutting (n-1)^th transparent object, then angle of incidence on the nth transparent object denoted θj__n is according to Snell's law given by: EQU. 11

[0063] where, np_n-i is the index of refraction of the (n-1)^th transparent object. If multiple transparent objects are cemented together with optical cement, the reflectance of light passing between the multiple transparent object may be negligible. For an uncoated glass or optical plastic window Fresnel transmission formulas are used for t_n(θi__n). If one or more of the transparent objects are coated with single of multiplayer interference coatings known equations for transmittance given for example in A. Thelen, "Design of Optical Interference Coatings", McGraw Hill, 1989 can be used for t_n(θ_Ln). In cases in which t_n(θj__n) is not readily calculable, measurements of t_n(θj__n) can be made and fitted to a function or spline interpolant for use in DIST.

For common optical glasses and plastics the reflection losses do not vary significantly if the angle of incidence is confined to a range below Brewster's angle. If the distance of the reflector from the illuminated plane and the relative sizes of the reflector and the illuminated area are such that the angle of incidence is confined to a range in which the reflection losses do not vary significantly, then t_n(θj__n) can be eliminated (set equal to one) without substantially changing the shape of the reflector or incurring a significant penalty in the performance of the reflector. Also, if antireflection coatings that substantially eliminate reflection are used, then t_n(θj__n) can be eliminated. If, based on the considerations discussed above, Γ_R(Θ_J__R) is also eliminated, then F is also eliminated (set equal to one), leading to a simplified form of DIST as follows. EQU. 12

_DIST

[0064] Reflectors described, described by equation 7 are able to achieve a object plane light intensity distributions that approximate predetermined light intensity distributions specified by Irr(x) when illuminating the object plane through one or more transparent objects (e.g., window 1108).

Once the initial value of the independent variable φ₀ and the initial radial coordinate r(φ₀) have been chosen, in order to completely specify the initial conditions, the initial value of the derivative of the polar radial coordinate of the generatrix with respect to elevation angle:

must be determined. [0065] Because the one or more transparent object located between the reflectors of the type described by equation 7 and the object plane 1102 change ray paths, the calculation of the initial value of the derivative of the polar radial coordinate, differs from the calculation used for the initial value of the derivative of the polar radial coordinate given by equation 5. One approach to calculating the initial value of the derivate for equation 7, is to substitute chosen numerical values for all quantities (Yt, Xo, φo, r(φo), th_n, np_n, no, N) appearing in equation 9 and then numerically solve equation 9 for the initial value of the derivative of the polar radial coordinate. A chosen axial location of the object plane 1102 is substituted for Yt and a chosen coordinate X₀ (e.g., equal Xmax or Xmin) to which an ideal ray emanating from the origin at angle φo is to be reflected is substituted for Xt. The bisection method is one method that can be used to numerically solve equation 9 for the initial value of the derivative.

[0066] FIG. 12 is a plot 1200 of light intensity versus radial position on the object plane illuminated by the fourth luminaire reflector and a lamp having the cosine weighted luminance shown in FIG. 2. The average absolute value deviation from the specified uniform light distribution is 3.74%. The maximum absolute value deviation which occurs near Xmax is 10.5%. Assuming a reflector reflectance of 0.95, and Fresnel reflection losses of 8% the collection efficiency is about 76%. The initial conditions and other design parameters which were used to obtain the light intensity profile shown in FIG. 12 are given Table IV.

[0067] Table IV

[0068] Because the angle of incidence on the single transparent window 1108 was confined to a range less than brewsters angle, ti(θj_j) equal to 1.0 was used. Furthermore, a reflector for which the reflectance did not vary significantly within the range of angles of incidence Θ,__R was assumed. Accordingly for the example represented in Table IV F was also eliminated.

[0069] Although the reflectors with the profiles shown the figures are described by equation 1 and equation 7 with the predetermined light intensity distribution Irr(Xt) being uniform, linearly decreasing or linearly decreasing. It is noted that these are just three examples, and reflectors described by equation 1 and equation 7 are able to produce, to good approximation, a variety of nonlinear light intensity profiles. It is also noted that uniform intensity is often the desired intensity distribution.

[0070] FIG. 13 is a cross-sectional side view of a reflectorized discharge envelope integrated luminaire 1300 according to an embodiment of the invention. The luminaire 1300 includes a ceramic body 1302. An outside surface 1304 of the ceramic body 1302 is cylindrical and an inside surface 1306 of the ceramic body 1302 is formed in the shape of a rotationally symmetric reflector described by equation 7. A reflective coating 1307 (e.g., silver, aluminum) is suitably formed on the inside surface 1306. An anode support/heat sink 1308 abuts a back end 1310 of the ceramic body 1302. The anode support/heat sink 1308 is coupled to the ceramic body by a first sleeve 1312 that is located peripherally about the back end 1310 of the ceramic body 1302 and the anode support/heat sink 1308. The first sleeve 1312 is suitably Tungsten Inert Gas (TIG) welded to the anode support/heat sink 1308 and brazed to ceramic body 1302. [0071 ] A sapphire window 1314 is fitted at a front end 1316 of the ceramic body 1302. The window 1314 is attached to the ceramic body 1302 by a flange 1318, second sleeve 1320 and a set of spacers 1322. The set of spacers 1322 are suitably brazed to each other and to the flange 1318. The flange 1318 is suitably TIG welded to the second sleeve 1320 which is brazed to the ceramic body 1302.

[0072] An anode 1324 is supported by the anode support/heat sink 1308. A plurality of cathode support arms 1326 extend radially inward from one or more of the set of spacers 1322 to a centrally located cathode 1328. The anode 1324 and cathode 1328 are arranged on an axis of rotational symmetry of the inside surface 1306.

[0073] A tube 1330 fitted into the anode support/heat sink 1308 allows the luminaire 1300 to be evacuated and filled with a discharge fill such as Xenon gas.

[0074] Integrated luminaires have certain technical characteristics that make them preferable to conventional separate lamp luminares for certain applications. Using reflectors described by equation 7 allows controlled light intensity distributions to be obtained notwithstanding the presence of the window 1314. Construction details can vary considerably from the particular design shown. For example, a copper body can be used in lieu of ceramic. Also, the reflector can be separate part positioned within the body. In each case, the luminaire optics described herein may be used.

[0075] In as much as, in the case of integrated luminaires there is no bare lamp that can be measured, the question may arise as to how one can obtain the radiance data Rad(φ) for use in equation 7. A procedure that may be used to obtain the Rad(φ) is to measure the near field radiance in front of the window of an integrated luminaire that has a traditional elliptical or parabolic reflector and then use backward ray tracing to trace rays that have energies derived from the measured near field radiance back beyond points of reflection by the traditional reflector, then to trace the rays to an imaginary reference sphere and to bin the rays according to elevation angle at the reference sphere. After ray energies have been binned by elevation angle, a interpolant representing Rad(φ), e.g., a cubic spline interpolant can be fitted to the binned data.

[0076] FIG. 14 is a schematic diagram of luminaire optics 1400 according to an embodiment of the invention. The luminaire optics 1400 are similar to the luminaire optics 1100 shown in FIG. 11, but include a right-angle prism 1402 between the transparent window 1106 and the object plane 1102. The prism 1402 includes an entrance face 1404 facing the reflector 1104, an exit face 1406 facing the illuminated area, and a silvered reflective face 1408 tilted at forty five degrees. Note that the prism 1402 also turns the optical axis, labeled O. A., by ninety degrees. The reflective face 1408 can be set at another angle, such as an angle at which total internal reflection (TIR) occurs. Thus, in the case of the embodiment shown in FIG. 14 there are two transparent objects between the reflector 1104 and the object plane 1102. Although, as shown the object plane 1102 is located beyond the exit face 1406, alternatively, the object plane 1102 is located within the prism 1402.

[0077] FIG. 15 schematically illustrates the generatrix of the first luminaire reflector and FIGs. 16-21 illustrate generatrices of types of luminaire reflectors according to additional embodiments of the invention. FIGs. 16-20 show different embodiments of reflectors that are obtained from equation 1 by changing the sign in front of the expression DIST, changing the point X₀ at the object plane (used in calculating the initial condition) to which a ray emanating from the origin φ₀ is reflected and in the case of FIGs. 19-20 breaking the range φo-φo into two sub-ranges and integrating equation 1 for the sub-ranges to compute a generatrix that includes two parts that smoothly connect.

[0078] FIG. 16 shows a generatrix of a reflector 1600 that is given by equation 1 when X₀ is set to zero (or as discussed above to a small distance e.g., 0.001 mm) and there is a negative sign in front of the expression DIST. In the case of the reflector 1600 an ideal ray emanating from the origin at the elevation angle φ₀ is incident on an object plane 1602 at the optical axis (Y-axis) (or at a point removed by the small distance) and as the elevation angle increases the point of incidence moves out to X_MAX-

[0079] FIG. 17 shows a generatrix of a reflector 1700 that is given by equation 1 when X₀ is set to zero (or in this case to a small negative distance e.g., -0.001 mm). Strictly speaking, a positive sign in front of the DIST subexpression is used to obtain the generatrix of reflector 1700. However, in as much as the integral in the numerator of DIST is a measurement of light power (watts) or lumens, and may be precomputed and stored as a positive value, then it needs to be considered that for the reflector 1700 Xt appearing in the denominator will have a negative sign thus changing the sign of DIST. So, if the integral in the numerator is represented by a positive number, then the leading sign for DIST would be negative in the case of the reflector 1700 shown in FIG. 17. In the case of reflector 1700 an ideal ray emanating from the origin at elevation angle cpo is reflected to Xt=0.0 and as the elevation angle increase toward φ_Q the point of incidence moves towards negative Xmax.

[0080] FIG. 18 shows a generatrix of a reflector 1800 that is given by equation 1 when Xo is set to negative X_MAX- In the case of the reflector 1800, in the same strict sense discussed above, a negative sign is used in front of the DIST expression to obtain the generatrix of the reflector 1800. However, if the integral in the numerator of DIST is precomputed and stored as a positive number, then a positive sign is used in front of DIST to obtain the generatrix of the reflector 1800. In the case of the reflector 1800 an ideal ray emanating from the origin at the elevation angle φo is incident on an object plane 1802 at negative X_MAX and as the elevation angle increases toward φo the point of incidence moves towards Xt=O (the optical axis).

[0081 ] FIG. 19 shows a generatrix of a reflector 1900 that includes a lower part 1902 and an upper part 1904. The two parts 1902, 1904 join together at boundary 1906, that is located at an elevation angle, shown in FIG. 19 and referred to hereinbelow as φi . The generatrix of the reflector 1900 is obtained by two integrations of equation 1. In the first integration a chosen elevation angle of a lower edge 1908 is used as φo in equation 1 and for the initial condition of equation 1 and a chosen value of (pi is used as ΦΩ in equation 1. In the second integration the chosen value of (P₁ is used as φo in equation 1 and for the initial condition of equation 1, and a chosen elevation angle of an upper edge 1910 of the reflector 1900 is used as φ_Q in equation 1.

[0082] Note that each point of an object plane 1912 illuminated by the reflector 1900 is illuminated by the lower part 1902 and the upper part 1904. (Recall that what is shown in FIG. 19 is a generatrix, not a complete rotationally symmetric reflector). Thus, the desired irradiance Irr(x) which is approximately achieved by the reflector 1900 is the sum of a first contribution specified by Irn(x) due to the lower part 1902 and a second contribution specified by Irr₂(x) due to the upper part 1904. lrri(x) is used in DIST in integrating equation 1 to obtain the generatrix of the lower part 1902, and Irr₂(x) is used in DIST in integrating equation 1 to obtain the generatrix of the upper part 1904. The apportionment of irradiance or illuminance between the two parts of the reflector can be chosen in a variety ways. A simple equal apportionment would be Im(Xt) = lrr₂(Xt) = Vz lrr(x). For the latter apportionment one would need to calculate q>i such that: EQU. 13

[0083] Once Rad(φ) is represented by an interpolant, equation 13 can be solved numerically, e.g. by a bisection method.

[0084] Another apportionment that would not require solving equation 13, would be lrr₁(Xt)=lrr₂(Xt). To achieve such an apportionment one can select some value of φi and proceed as described above to compute the generatrices of the lower part 1902 and upper part 1904. For example one can chose the value of φi by inspecting a plot Rad(φ) an selecting a value that appears to approximately bisect the radiance into two equal parts.

[0085] Note that the lower part 1902 of the reflector 1900 is, by itself, a reflector of the type illustrated in FIG. 15 and the upper reflector 1904 is, by itself, a reflector of the type illustrated in FIG. 17.

[0086] Although in the case of the reflector 1900 the two parts 1902, 1904 of the reflector 1900 join together smoothly, this is not necessary. The two parts could be disjoint from each other.

[0087] FIG. 20 also shows a generatrix of a reflector 2000 that includes a lower part 2002 and an upper part 2004 that are joined at a boundary located at an elevation angle φ-i . As in the case of the reflector 1900 each reflector part 2002, 2004 has a generatrix obtained by a separate integration of equation 1. Note that the lower part 2002 is, by itself, a reflector of the type illustrated in FIG. 18 and the upper part 2004 is, by itself, a reflector of the type illustrated in FIG. 16. The considerations regarding apportionment of light energy that are discussed above with reference to FIG. 19 also apply to the reflector 2000.

[0088] Although the reflectors 1900, 2000 include two parts that distribute light energy, alternatives that have more than two parts that are defined by Equation 1 are also possible.

[0089] FIG. 21 shows a generatrix of a reflector 2100 of a luminaire according to another embodiment of the invention. The reflector 2100 is made up of four parts each of which is a solutions of equation 1. A first part 2102 ranges from elevation angle φ_O-i to elevation angle φ_Q-i. A second part 2104 ranges from elevation angle

to elation angle φo-2. A third part 2106 ranges from elevation angle φo-3=cpΩ-₂ to elevation angle φ_Q.₃. A fourth part 2108 ranges from elevation angle φo₄=φn-3 to elevation angle ΦΩ-4. The initial condition for the first part 2102 is set using XO equal to zero (or zero plus some small physically insignificant number, so as to avoid the aforementioned numerical difficulty for some integrators). Alternatively the initial condition is set using X₀ equal to some other value such as Xmax or some intermediate value. The initial conditions for each successive part 2104-2108 are set equal to the final values for the preceding part, so that the reflector 2100 is continuous and smooth. For ideal rays emanating from the origin in the angular ranges of the first part 2102 and third part. 2106, as the elevation angle increases, the X-coordinate of the intercept with an object plane 2110 increase. For the second part 2104 and fourth part 2108 as the elevation angle increases, the X value of the intercept decreases. The irradiance function Irr(X) for the parts 2102-2108 can be the same or different. For the embodiment shown in FIG. 21 a uniform irradiance was used for all parts 2102, 2104, 2106, 2108. In the case of the embodiment shown in FIG. 21, each part 2102, 2104, 2106, 2108 accounts for one-quarter of the total elevation angle range subtended by the reflector 2100. The elevation angle range is alternatively divided differently. For example, elevation angle range is alternatively divided into subranges that contain equal radiated light power.

[0090] The reflectors described above are able to collect a high percentage of light emitted by a compact arc lamp and distribute the light on an illuminated plane in a highly controlled manner. [0091 ] In the embodiment shown in FIG. 21 the full illuminated area is illuminated by each of the four parts of the reflector. This provides a degree of integration or averaging which makes the light intensity distribution at the object plane less susceptible to variations in the light source radiance Rad(φ) that may occur from unit to unit in a production run, or as a lamp ages. However, the benefit is obtained at the expense of increasing the etendue at the object plane. One way to control the tradeoff between making the light intensity distribution less susceptible to variations in Rad(φ) and controlling the etendue is to divide the object plane into a central circular area and one or more concentric annular areas. At least one of the resulting areas of the object plane is then illuminated with two or more parts of a reflector (each part being described by equation 1 or equation 7).

[0092] For example the object plane can be divided into a central circular area and two concentric annular areas, and the reflector can have nine parts, where the first three parts of the reflector (closest to the aperture) illuminate an outer annulus, the next three parts illuminate the inner annulus, and the last three parts illuminate the central circle. The elevation angle boundaries between the groups of three parts are to be chosen so that each group of three parts subtends an elevation angle range that includes light power in proportion to the integrated light power in the object plane area that the group of three parts is assigned to illuminate. If Xo for the first reflector part (closest to the aperture) equals Xmax of the whole area of the object plane to be illuminated, and the first, third, fourth, sixth, seventh, and ninth reflector parts have a leading positive sign for RIST₁ and the second, fifth and eighth reflector parts have a leading negative sign for DIST, the reflector will be smooth and continuous, and a balance will have been struck between making the reflector less susceptible to variations in Rad(φ) and controlling the etendue.

[0093] Although the embodiments shown in FIGs. 16-21 do not show transparent objects (e.g., windows, prisms) between the reflectors and the object planes, there are analogous embodiments based on equation 7 for the case that there are one or more transparent objects between the reflectors and the object planes.

[0094] FIG. 22 is a flowchart of a first method 2200 of manufacturing reflectors described by equation 1 or equation 7. In block 2202 the initial conditions of a system of coupled first order equations that is equivalent to equation 1 or equation 7 are set. In block 2204 the system of coupled first order equations is integrated to obtain an integrated solution. In block 2206 data that represents the integrated solution is entered into a computer numeric control (CNC) machine tool (e.g. a CNC lathe) and in block 2208 the CNC machine tool is used to machine a reflector according to the solution of the reflector equation. The reflector is suitably machined from a length of metal bar stock.

[0095] FIG. 23 is a flowchart of a second method 2300 of manufacturing reflectors described by equation 1 or equation 7. The first three blocks 2202-2206 in the second method 2300 are the same as in the first method 2200. In the second method, after the first three blocks 2202-2206, in block 2308 a machine tool is used to machine tooling for manufacturing a reflector according to the solution of the equation 1 or equation 7. The tooling suitably comprises, by way of example, a part of a mold that is used to mold reflectors or a mandrel used to electroform reflectors. In block 2310 the tooling is used to manufacture reflectors according the solution of the reflector equation that was obtained in block 2204. The tooling is suitably machined metal.

[0096] Rather than simply making judicious choices for the cpo and φo after reviewing the angular radiance distribution Rad(φ) and making judicious choices for the object plane location Yt and initial polar radial coordinate r(φo), improved accuracy of the achieved irradiance profile can be achieved by optimizing these design parameters or another set of parameters that determine these parameters.

FIG. 24 is a flowchart of an alternative beginning of the first or second methods shown in FIGs. 22-23. The alternative shown in FIG. 24 employs optimization to select design parameters. In block 2402 initial guesses and/or bounds for design parameters being optimized are set. Although r(φ₀) and Yr(φ₀) can be optimized, the inventor has chosen to use a different set of optimization parameters including X_R0 and Ψ_RAY. XRO is the cylindrical radial coordinate of the reflector corresponding to the polar coordinates (φo, ^r(Ψo)). so that r(φo)=X_Ro/COS((po). Ψ_RAY is the angle that the ray emanating at angle φ₀ and reflected to X coordinate Xo makes with the Y-axis after the ray has been reflected. (pRAY together with φ₀, r(φ₀) determine Yt by the following equation: EQU. 14 Yt=r(φo)*SIN(φ₀)-(r(φo)*COS(φo)-Xo)/TAN(φ_RAY)

[0097] Using Ψ_RAY as an optimization parameter allows direct control over, at least the initial value, of the angle of reflected rays relative to the Y-axis angle. In many cases, such as in the integrated profiles shown in FIGs. 4, 6, 8, 11 the initial value angle of reflected rays relative to the Y-axis is the maximum value of the angle of reflected rays relative to the Y-axis. For reflectors described by equation 7 that are intended to be used with a transparent object (e.g., window, prism) between the reflector and the object plane it is generally best to avoid values of Φ_RAY in excess of Brewster's angle so as to avoid excessive reflection losses from the surfaces of the transparent objects, however there may be some applications where this can be exceeded.

[0098] In addition to X_RO and Φ_RAY the set of parameters that is optimized can include φ₀ and φ_Q- (Alternatively the set of parameters φ₀, φo ,r(φo) and Yt can be optimized.) The ray tracing can use rays based on near field radiance measurements such as generated by Prosource™ light source models. The camera used to collect such measurements may not be centered exactly on the center of luminance, resulting in a Y-axis offset between the origin of the coordinate system in which the rays are defined and the center of luminance. (The center of luminance may not be known before measurements are taken.) Accordingly, a fifth optimization parameter that can be added is a Y coordinate shift, denoted ΔY_ray that is added to all ray origins used in ray tracing.

[0099] Certain optimization routines may require only bounds and certain optimization routines may require only initial guesses. In block 2404 an optimization routine is called. A known general purpose optimization routine that does not require derivative information is suitably used. Examples of general purpose optimization routines that can be used include the Simplex method, the Complex method and the Simulated Annealing method. The inventor has used the DBCPOL FORTRAN implementation of the complex method published by Visual Numerics of San Ramon, Ca. In block 2406 optimum values of the set of parameters are output. In block 2408 initial conditions and the Y-coordinate of the object plane Yt are calculated from the optimum values. If the optimization parameters are φ₀, q>n, X_RO Ψ_RAY then Yt, r(φ₀), and dr(φ)/3φ (at φo) need to be calculated at this point, unless they were stored during optimization. If the optimization parameters include r(φ₀) then it would not need to be calculated at this point. Connector 2410 branches to block 2204 in FIG. 22 or 23.

[001 00] In using a general purpose optimization routine one needs to provide a function to be optimized. The selection of values of parameters at which to evaluate the function to be optimized is handled by the general purpose optimization routine.

[00101 ] FIG. 25 is a flowchart of a subprogram 2500 that is called by an optimization routine that is called in the flowchart shown in FlG. 25. The subprogram 2500 is the function to be optimized. In block 2502 the initial conditions are calculated from the value of call parameters q>o, X_RO <P_RAY. Note that cpα, is not used in calculating the initial conditions. In block 2504 the integral of cos(φ) weighted light intensity Rad(φ) that appears in the denominator of DIST is calculated. Note that it is assumed that the integral involving Xt*lrr(Xt) from Xmin to Xmax which is independent of the parameters being optimized will have been precomputed and stored. (Note that one could allow for optimization of Xmax or a non zero Xmin within a small range in order optimize light spill at the illuminated area boundaries). In block 2506 the system of coupled differential equations defining the profile of the reflector is integrated to obtain an integrated reflector profile. In block 2508 a spline is fit to the integrated profile. In block 2510 ray tracing, using the spline fit of the integrated profile and a model of the source, e.g., a set of light rays from a Prosource™ model, is performed to determine an achieved irradiance profile, and optionally the collection efficiency. In block 2512 a cost function that depends on the mismatch between the achieved irradiance profile and the desired irradiance profile lrr(x) is evaluated. The cost function can, for example, comprise a sum of squares of the differences between Irr(x) and the achieved irradiance. The cost function, optionally, also depends on the coupling efficiency.

[001 02] FIG. 26 is a block diagram of a projector system 2600 in which the luminaires having the reflectors described above can be utilized. The projector system 2600 is an example of system that can utilize the reflectors described above. The projector system 2600 comprises a luminaire 2602 that includes a light source and reflector described by 1 or by equations 7. The luminaire is optically coupled (e.g., by simple free space propagation, one or more relay lenses, or a light guide) to an optional polarizer 2604. Polarized light is required for certain types of image modulators (e.g. liquid crystal based image modulators) but not others (e.g., DLP™ micromirror array based modulators). The polarizer 2604 is optically coupled to and may be tightly integrated with an optional polarization conversion/recovery device 2606, which is intended to avoid wasting light rejected by the polarizer 2604. The polarization conversion/recovery device 2606 is coupled to an optional beam shaping device 2608. The beam shaping device 2608 includes, for example an anamorphic beam expander or contractor and/or a light guide, or bundle of light guides that has a cross-section size and/or shape that changes along the length of the light guide. The beam shaping device 2608 is coupled to an optional beam smoothing device 2610. The beam smoothing device 2610 includes for example a holographic diffuser. Alternatively, a kinoform or Light Shaping Diffuser serves as both the beam shaping device 2608 and the beam smoothing device 2610. Light Shaping Diffusers are manufactured by Physical Optics Corporation of Torrance, CA. The beam smoothing device 2610 is coupled to an optional color separation apparatus 2612. The color separation apparatus includes 2612, for example, a static arrangement of dichroic mirrors that divide the light beam into a plurality (e.g., red, blue and green) separate light beams, or a dynamic filter arrangement, e.g., a rotating color wheel that filters the beam with different filters, or a rotating prism. The color separation apparatus 2612 is coupled to one or more imagewise light modulators 2614, such as transmissive or reflective liquid crystal modulators, or micromirror array modulators. (Typically, a single light modulator 2614 is used with a rotating filter wheel, and three image modulators 2614 are used with a static arrangement of dichroic mirrors.) In the case the case that multiple imagewise light modulators 2614 are used, the imagewise modulated light beams they produce are combined by an optional color channel recombination device 2616, e.g., a color combiner prism. (If a color wheel is used with a single image modulator, the color channel recombination device 2616 is not needed.) The color channel recombination device 2616 is coupled to a projection lens optics subsystem 2618. The projection optics subsystem suitably comprises a projection lens, reflective projection optics or a subsystem that combines lenses and reflective elements. The projection optics subsystem 2618 is coupled to a projection screen 2620 which can be a rear projection screen or a front projection screen. Note that the specified light intensity distribution Irr(x) can be set to compensate for any radial coordinate dependent losses of components of the system 2600. If the overall radial coordinate dependent losses are given by Loss(Xt) and uniform luminance on the projection screen is the goal, then Irr(Xt) can be set equal to 1/Loss(Xt). Note that losses may occur at radial coordinates that are not equal to Xt but are mapped from Xt by optical coupling (e.g., via lenses) within the projector system. It will be apparent to persons of skill in the art that order of the components represented in FIG. 26 can vary relative to the order shown in FIG. 26.

[00103] A small local divergence of light on the imagewise light modulator 2614 that the reflectors described above produce reduces geometric aberrations which tend to degrade the modulation transfer function (MTF) of the projection lens 2618. The reduction is due to the fact that such aberrations arise, in the first instance, from the light rays departing object points at different angles. If the range of angles is limited these geometric aberrations will be limited. Diffraction limits on the MTF and image distortion are a separate matter.

[001 04] According to alternative embodiments DIST has the following form which takes into account, an elevation angle dependence of the spectral energy distribution of the lamp, wavelength dependent light loss between the light source and the illuminated object and the spectral sensitivity of the illuminated object. EQU. 15

j M-Irr(M)dx

where, Rad(φ, λ) is the elevation angle dependent spectral radiance of the light source;

Γ_R(Θ_LR, λ) is the spectral reflectance of the reflector 304 which is also dependent on the angle of incidence Θ_J__R OΠ the reflector 304; SL(θ_Lτ, λ) is a factor that accounts for light loss (e.g., undesired reflectance, transmittance or absorption) at the illuminated object which is dependent on the angle of incidence θrr and the wavelength λ; tn(θj__n. λ) is the angle of incidence θj__n and wavelength dependent transmission of an n^th transparent object (e.g., prism, window) between the reflector and the illuminated area; S(A) is the spectral sensitivity of the illuminated object; λ₀ and AQ are lower and upper spectrum limits; and F is a constant factor.

[001 05] In DIST given by equation 15 Rad(φ) is replaced with an integral between limits A₀ and AQ of an integrand that is the product of the angular dependent spectral radiance, angle-of-incidence dependent spectral reflectance of the reflector, angle dependent spectral transmission of one or more transparent objects (e.g., prisms or windows) between the reflector and the illuminated object, a factor that accounts for light loss at the illuminated object, and the spectral sensitivity of the illuminated object.

[001 06] The limits of integration A₀ and AQ are suitably chosen to cover one or more ranges over which the integrand has non-negligible values. For example if the spectral radiance, reflectivity of the reflector, or sensitivity drop to negligible values beyond a particular wavelength, the upper limit AQ can be set equal to the particular wavelength.

[00107] Note that in the case of visible light applications, the spectral radiance may depend on the elevation angle if the lamp exhibits what is termed 'color separation'. Analogously, for ultraviolet or infrared applications the spectral radiance may also depend on elevation angle. On the other hand for certain types of lamps (e.g. xenon or high pressure mercury fill lamps, for example) that do not exhibit significant color separation, the elevation angle dependence of the spectral radiance may be ignored. Spectral radiance data can be obtained by measuring the light output of a light source with a spectrometer at each of a set of elevation angles. The data collected at the set of elevation angles can be represented by one or more interpolating splines in DIST given by equation 15. Near field, radiance can also be measured using optical bandpass filters, and the spectral radiance can be determined based on the near field radiance.

[001 08] In the case that the reflector includes a multilayer thin film reflecting surface, there may be significant angular dependence of the spectral reflectance. On the other hand, in the case that the reflector 304 includes a metal (e.g., aluminum or silver for example) reflecting surface, the angular dependence of the spectral reflectance may be negligible within a range of angle of incidence that is realized, and in such cases may be ignored. In practice if the variation of reflectivity of the reflector material varies with angle is not negligable, it is best to base r_R(θ_LR) on actual measurements, although a closed form expressions may be available for certain materials.

[001 09] For reflectors given by equation 7, that are to be used with one or more transparent objects (e.g., window, prism) between the reflector and the illuminated object, unless an Anti-Reflection (AR) coating that works at high incident angles is going to be used, it is better to set the initial conditions and object plane distance so as to avoid light rays incident on the transparent object(s) at angles larger than Brewster's angle, so as to avoid large reflection losses. For most common optical materials (e.g., glasses and plastics), for visible light, if the angles of incidence are less than Brewster's angle the variation of reflectance loss with angle is small and may be neglected. Additionally, for many common optical materials, the variation of Fresnel and absorption losses with wavelength within the visible spectrum is negligible. Accordingly, the factor(s) t_π(θj_n, λ) in DIST can be ignored for many visible light applications. It is noted that the factors t_n(θj__n, λ) in DIST are applicable to reflectors described by equation 7, but not to reflectors described by equation 1.

[001 1 0] The spectral sensitivity S(K) can be the photochemical sensitivity of a reaction that is to be driven by light reflected by the reflector. For projection applications in which the object that is illuminated by light collected by the reflector is an imagewise light modulator that is imaged by a projection lens onto a projection screen which is then viewed, the spectral sensitivity can include the photopic response of the human eye, or in a system that uses multiple luminaires for multiple color channels, the spectral sensitivity can include a tristimulus response curve, for example.

[001 1 1 ] In applications where a reflective image modulator is illuminated using the reflector SL(θj_τ, λ) is equal to the spectral reflectance of the modulator R(λ). In this case light loss is due to transmission and/or absorption. In applications where a transmissive image modulator is illuminated by the luminaire optics SL(θj__τ, λ) is equal the to spectral transmittance of the image modulator T(λ). In this case light loss is due to reflection and/or absorption. For applications where the desired effect of illumination depends on light being absorbed (e.g., ultraviolet photochemical curing) SL(θi_τ, λ) is equal to the spectral dependent absorption A(λ). In this case light loss is due to reflection and/or transmission.

[001 1 2] F is a normalization factor that compensates for angle of incidence dependent light loss from the reflector, transparent object(s) and/or illuminated object. F needs to be determined by trial and error (e.g., by a numerical root finding method) such that Xt(CpQ, r(φ_Q), dr(φ)/dφ(at (pα))=Xα where XQ is a chosen value of X (either X_MIN or X_MAX) to which a ray emanating from the origin at a selected angle cpo is to be reflected by the reflector. Determination of F is best started at an approximate value which is equal to the inverse of the spectrally weighted average of the product of t_n(θ, _„, λ), TR(SJ R, λ) and SL(Θ_J__L, λ), at some chosen angles of incidence (e.g., normal incidence or initial values of angles of incidence), where the spectral weighting is the product of Rad(φ, λ) and S(λ), with φ equal to some chosen value e.g., cpo or (<pQ-φ_o)/2. The exact angles of incidence used in determining the approximate value of F to start with is not critical.

If the index of refraction of the medium (e.g., air) between the lamp and the reflector is the same as the index of refraction of the medium above the illuminated object then the angle of incidence on the illuminated object θ_t_τ (assuming a typical case of the object being perpendicular to the Y-axis) is equal to the angle of the reflected ray Θ_RR given above. If the last (N^th) transparent object (which may be the only transparent object) is in contact with the illuminated object then the angle of incidence on the object θ,__τ is related to angle of reflection Θ_RR by Snell's law, i.e. EQU. 16

[001 1 3] If there is a transparent object in contact with the illuminated object then SL(θ,_τ, λ) will also be changed because the angle θι _τ, is changed and because there is a different change in index of refraction for light reaching the surface of the object. In any case, SL(θj_τ, λ) can still be determined by tests or calculation. In the case of reflectors given by equation 1 where there is no transparent object in contact with the illuminated object the angle of incidence on the object plane is given by: EQU. 17

[001 1 4] The angular dependence of one or more of the spectral radiance, the spectral reflectance of the reflector, the spectral transmission of the transparent objects (e.g., window, prism), and the spectral light loss at the illuminated object will for many cases be negligible, an in such cases can be ignored leading to simplifications in DIST given by equation 15. In certain embodiments it might be appropriate to leave in an angular dependence in one or more of the factors in DIST given by equation 15, but simplify to eliminate the spectral dependence. An example of the latter case is in the case that the illuminated object has a sharp spectral response at a particular wavelength (e.g., in the ultraviolet) and the reflectively of the reflector 304 at the particular wavelength is a strong function of incidence angle.

[001 1 5] If in a particular application it not necessary to consider spectral reflectance of the reflector r_R(θj__R, λ), the light loss at the illuminated object SL(θj_τ, λ), the spectral dependent transmission of an nth transparent object t_n(θi__n, λ) and the spectral sensitivity of the illuminated object S(λ) then these factors can be set equal to one in which case DIST given by equation 15 reduces to DIST given by equation 2.

[001 1 6] Note that in the case that Rad(φ) is based on measurements with a filter that matches S(λ), then Rad(φ) is equivalent to Rad(φ, λ) weighted by S(λ) and integrated from λ₀ to KQ. In other words Rad(φ) is equivalent to a convolution Rad(φ, λ) and S(K) integrated from A₀ to λ_Q. In this regard note that the cos(φ) weighted Rad(φ) shown in FIG. 2 is based on measurements with a photopic filter S(K) so that the cos(φ) weighted Rad(φ) shown in FIG. 2 is in photopic units. (Alternatively, radiometric units could be used.) [001 1 7] The reflectors described above can be complete surfaces of revolution or off-axis reflectors. The reflectors described above can be used in combination with other reflectors that subtend portions of the solid angle space about the light source. Examples of other types of reflectors with which the reflectors described above, include but are not limited to spherical reflectors that retroreflect portions of light back toward the light source, conical reflectors, and conic section (e.g., paraboloid, ellipsoid) reflectors.

[001 1 8] In addition to use in projection, and for illuminating and/or heating semiconductors, the reflectors described above can be used for any application that benefits from controlled illumination.

[001 1 9] The reflectors described above can be used with lamps that emit over a wide angular range, where some of the light is reflected by the reflector and some passes through the aperture of the reflector and reaches the illuminated area without reflection by the reflector. In this case the total light intensity at each position in the illuminated area will include a reflected light contribution specified by Irr(Xt) and a non- reflected (by the reflector) light contribution. In this case Irr(Xt) should be chosen in view of the non-reflected light contribution, so that the total light intensity is what is desired.

[001 20] While the preferred and other embodiments of the invention have been illustrated and described, it will be clear that the invention is not so limited. Numerous modifications, changes, variations, substitutions, and equivalents will occur to those of ordinary skill in the art without departing from the spirit and scope of the present invention as defined by the following claims.

[001 21 ] What is claimed is:

Claims

I claim

1. A luminaire comprising: a light source that emits light nonuniformly as a function of elevation angle over a substantial range of elevation angle; a reflector having a profiled reflector surface that is shaped to distribute light on an illuminated area substantially according ' to a predetermined radial intensity distribution Irr(x); wherein said reflector has transverse dimension that is larger than a transverse dimension of said illuminated area.

2. The luminaire according to claim 1 wherein said light source emits nonuniformly within said substantial range of elevation angle.

3. The luminaire according to claim 1 wherein said substantial range of elevation angle is at least 0.5 radians.

4. The luminaire according to claim 2 wherein said reflector subtends at least a substantial subrange of said substantial range of elevation angle, and said reflector collects at least a substantial portion of light emitted by said light source.

5. The luminaire according to claim 4 wherein said reflector collects at least 60% of light emitted by said light source.

6. A reflector having a generatrix that is substantially equal to at least one solution of a differential equation:

+ 2 (j|r(φ) JzrøTtfφ)² + 4 R (j^~i(Φ)J tfΦ)⁴ -4 *(Φ)³ sin(Φ) (yCΦ)J

"

^C0S<;φ)2'

where,

<p is an elevation angle coordinate measured in a clockwise direction from a positive X-axis in an X-Y coordinate system of the generatrix of the reflector, said X-Y coordinate system further comprising a Y-axis which is an optical axis of said reflector; r(φ) is a polar radial coordinate of the generatrix of the reflector in the X-Y coordinate system and is equal to:

Yt is equal to a Y coordinate of an object plane that is to be illuminated by light reflected by the reflector; and

DIST comprises a quotient comprising a numerator comprising Rad(φ) and a denominator comprising Irr(Xt), wherein:

Irr(Xt) is a predetermined light intensity at a given cylindrical radial coordinate Xt in the object plane;

Rad(φ) is an intensity of light emitted by a source, for which the reflector is designed, at elevation angle φ; and wherein:

7. The reflector according to claim 6 wherein: the generatrix is equal to the at least one solution of the differential equation.

8. The reflector according to claim 6 wherein: Irr(Xt) is equal to a constant; and Rad(cp) varies as a function of φ.

9. The reflector according to claim 1 wherein:

where,

XMIN is an inner radius of a portion of the object plane to be illuminated by light reflected by the reflector;

X_MAX is an outer radius of the portion of the object plane to be illuminated by light reflected by the reflector;

Φo is a lower limit of an elevation angle range subtended by the reflector; and q>Q is an upper limit of the elevation angle range subtended by the reflector.

10. The reflector according to claim 9 wherein: lrr(Xt) is equal to a constant for Xt from XMIN to X_MAX; and Rad(φ) varies as a function of φ.

11. The reflector according to claim 6 wherein:

where, Rad(φ, λ) is the intensity of light emitted by the source at elevation angle φ and at a wavelength λ;

ΓR(Θ_I__R, λ) is a reflectance of the reflector which is dependent on an angle of incidence ΘJ_R on the reflector and the wavelength λ, where θj__R is given by:

SL(A₁ θj_χ) is a factor accounting for light loss at an illuminated object that is dependent on an angle of incidence on the illuminated object θι_τ and the wavelength λ, wherein θj__j is given by: dr

Q . _τ = abs φ + 2 arctan —

S(λ) is a spectral sensitivity of the illuminated object; and λo is a lower spectral limit; AQ is an upper spectral limit; wherein, a convolution of Rad(φ, A) and S(A) integrated from A₀ to AQ in DIST is equal to Rad(φ).

XMIN is an inner radius of a portion of the object plane to be illuminated by light reflected by the reflector; X_MAX is an outer radius of the portion of the object plane to be illuminated by light reflected by the reflector;

<Po is a lower limit of an elevation angle range subtended by the reflector; and φo is an upper limit of the elevation angle range subtended by the reflector; and

F is a normalization factor that compensates for light loss losses at the reflector and illuminated object.

12. The reflector according to claim 6 wherein an initial value of a derivative of the polar radial coordinate of the reflector is given by:

wherein, φ₀ is an initial elevation angle of the reflector; r(Φo) is an initial polar radial coordinate of the reflector; X₀ is an initial value of Xt and is equal to a radial coordinate to which a ray emanating from an origin of the X-Y coordinate system is reflected by the reflector.

13. The reflector according to claim 12 wherein:

Xo is equal to an outer radius of a portion of the object plane to be illuminated by light reflected by the reflector; and the reflector reflects light such that as the elevation angle of rays φ increases, the X-coordinate to which rays are reflected decreases.

14. The reflector according to claim 6 wherein the generatrix is described by two solutions of the differential equation, said reflector comprising a first part described by a first solution of the differential equation; a second part described by a second solution of the differential equation.

15. The reflector according to claim 14 wherein said first part and said second part reflect light to a common area of said object plane.

16. The reflector according to claim 15 wherein the generatrix is described by at least six solutions of the differential equation, said reflector further comprising: a third part described by a third solution of the differential equation; a fourth part described by a fourth solution of the differential equation; a fifth part described by a fifth solution of the differential equation; a sixth part described by a sixth solution of the differential equation; wherein said first through sixth parts of said reflector join smoothly; wherein said common area is a central circular area; said first through third parts of said reflector illuminate said central circular area of said object plane; and said fourth through sixth parts of said reflector illuminate an annular area of said object plane surrounding said central circular area.

17. A luminaire comprising: a light source that emits light nonuniformly as a function of elevation angle wherein light intensity emitted by said light source as a function of elevation angle varies according to Rad(φ); and the reflector according to claim 6.

18. A projection system comprising: an imagewise light modulator; a projection optics subsystem; and the luminaire according to claim 17, wherein the luminaire is optically coupled to the imagewise light modulator, and the imagewise light modulator is optically coupled to the projection optics subsystem.

19. The luminaire according to claim 17 wherein the light emitted by the light source is substantially confined to an elevation angle range that is less than 180 degrees, whereby, the light source does not emit significant light along the optical axis.

20. The luminaire according to claim 19 wherein the light source comprises a compact arc lamp having a longitudinal axis aligned on the optical axis.

21. A reflector having a generatrix that is substantially equal to at least one solution of a differential:

where, φ is an elevation angle coordinate, measured in a clockwise direction from a positive X-axis in an X-Y coordinate system of the generatrix of the reflector, said X-Y coordinate system further comprising a Y-axis which is an optical axis of said reflector; r(φ) is a polar radial coordinate of the generatrix of the reflector in the X-Y coordinate system and is equal to:

Yt is equal to a Y coordinate of an object plane that is illuminated by the reflector; no is an index of refraction in an environment of the reflector; capital N is a number of one or more transparent objects positioned between the reflector and the object plane; lower case n is an index that refers to individual transparent objects; th_n is a thickness, measured along the optical axis of an n^th transparent object; np_n is the index of refraction of the n^th transparent object;

DIST comprises a quotient comprising a numerator comprising Rad(φ) and a denominator comprising Irr(Xt), wherein: lrr(Xt) is a predetermined light intensity at a given cylindrical radial coordinate Xt in the object plane;

Rad(φ) is an intensity of light emitted by a light source, for which the reflector is designed, at elevation angle φ; and

22. The reflector according to claim 21 wherein: the generatrix is equal to the solution of the differential equation.

23. The reflector according to claim 21 wherein an initial value of a derivative of the polar radial coordinate dr/dφ|φ=φ₀ and an initial polar radial coordinate of the reflector r(φ₀) are selected such that light emanating from an origin of the X-Y coordinate system at an lowest elevation angle of the reflector φ₀ is reflected through the one or more transparent objects to an outer radius of an area illuminated by the reflector and the reflector reflects light such that as the elevation angle of rays φ increases, the X- coordinate to which rays are reflected decreases.

24. The reflector according to claim 23 wherein: lrr(Xt) is equal to a constant for Xt from X_MIN to XMAX; and Rad(φ) varies as a function of φ.

25. The reflector according to claim 21 wherein:

_DIST

where,

X_MIN is an inner radius of a portion of the object plane illuminated by light reflected by the reflector;

X_MAX is an outer radius of the portion of the object plane illuminated by light reflected by the reflector; φo is a lower limit of an elevation angle range subtended by the reflector; and φ_Q is an upper limit of the elevation angle range subtended by the reflector.

26. The reflector according to claim 21 wherein: lrr(Xt) is equal to a constant for Xt from XMIN to X_MAX; and Rad(φ) varies as a function of φ.

7. The reflector according to claim 21 wherein:

where, Rad(φ, λ) is the intensity of light emitted by the source at elevation angle φ and a wavelength A;

ΓR(Θ,_R, λ) is a reflectance of the reflector which is dependent on an angle of incidence θ,__Ron the reflector and the wavelength λ, where θ_uR is given by:

SL(A, θj_τ) is a factor accounting for light loss at an illuminated object that is dependent on an angle of incidence on the illuminated object θj_τ and the wavelength λ;

t_n(θι_n, λ) is a transmission of an nth transparent object disposed between the reflector and the lens, which is dependent on the wavelength λ, and an angle of incidence θ_Ln on the nth transparent object, where θ,__n is given by:

where,

S(λ) is a spectral sensitivity of the illuminated object; and A₀ is a lower spectral limit;

AQ is an upper spectral limit; X_MIN is an inner radius of a portion of the object plane to be illuminated by light reflected by the reflector;

X_MAX is an outer radius of the portion of the object plane to be illuminated by light reflected by the reflector;

<Po is a lower limit of an elevation angle range subtended by the reflector; and

<pQ is an upper limit of the elevation angle range subtended by the reflector; and wherein, a convolution of Rad(φ, λ) and S(λ) integrated from λ₀ to AQ in DlST is equal to Rad(φ); and

F is a normalization factor that compensates for light loss at the reflector, one or more transparent objects, and illuminated object.

28. The reflector according to claim 21 wherein:

Irr(Xt) is equal to a constant for Xt from X_Mi_N to X_MAX; and Rad(φ) varies as a function of φ.

29. The reflector according to claim 21 wherein the generatrix is described by two solutions of the differential equation, said reflector comprising a first part described by a first solution of the differential equation; a second part described by a second solution of the differential equation.

30. The reflector according to claim 29 wherein said first part and said second part reflect light to a common area of said object plane.

31. The reflector according to claim 30 wherein the generatrix is described by at least six solutions of the differential equation, said reflector further comprising: a third part described by a third solution of the differential equation; a fourth part described by a fourth solution of the differential equation; a fifth part described by a fifth solution of the differential equation; a sixth part described by a sixth solution of the differential equation; wherein said first through sixth parts of said reflector join smoothly; wherein said common area is a central circular area; said first through third parts of said reflector illuminate said central circular area of said object plane; and said fourth through sixth parts of said reflector illuminate an annular area of said object plane surrounding said central circular area.

32. A luminaire comprising: a light source that emits light nonuniformly as a function of elevation angle according to Rad(φ); and the reflector according to claim 21.

33. The luminaire according to claim 32 wherein light emitted by the source is substantially confined to an elevation angle range that is less than 180 degrees, whereby the source does not emit significant light along the optical axis.

34. The luminaire according to claim 32 wherein the source comprises a compact arc lamp having a longitudinal axis aligned on the optical axis.

35. An integrated luminaire comprising the reflector according to 21 , wherein a first of said one or more transparent objects comprises a window of said integrated luminaire.

36. A projection system comprising: a imagewise light modulator; a projection optics subsystem; and the luminaire according to claim 34, wherein the luminaire is optically coupled to the imagewise light modulator, and the imagewise light modulator is optically coupled to the projection optics subsystem.

37. A method of manufacturing a luminaire reflector comprising: setting initial conditions for a system of coupled differential equations that describes a generatrix of a reflector, wherein the reflector distributes light substantially according to a predetermined distribution; integrating the system of equations to obtain an integrated solution; inputting data representing the integrated solution into a computer numeric control machine tool.

38. The method according to claim 37 further comprising: using the computer numeric control machine tools to machine tooling for manufacturing the reflector.

39. The method according to claim 37 wherein the system of coupled differential equations is equivalent to a second order differential equation:

;^φ)[F/-r(ψ}^-it;4)^sin(.φKJ?g<φ)J-<φ where, φ is an elevation angle coordinate measured in a clockwise direction from a positive X-axis in an X-Y coordinate system of the generatrix of the reflector, said X-Y coordinate system further comprising a Y-axis which is an optical axis of said reflector; r(φ) is a polar radial coordinate of the generatrix of the reflector in the X-Y coordinate system and is equal to:

J 2 2 Yt is equal to a Y coordinate of an object plane that is to be illuminated by light reflected by the reflector; and

DIST comprises a quotient comprising a numerator comprising Rad(φ) and a denominator comprising Irr(Xt), wherein:

Irr(Xt) is a predetermined light intensity at a given cylindrical radial coordinate Xt in the object plane;

Rad(φ) is an intensity of light emitted by a source, for which the reflector is designed, at elevation angle φ; and wherein:

40. The method according to claim 37 wherein the system of coupled differential equations is equivalent to a second order differential equation:

sin(φ) — r(φ) cos(φ) +

where, φ is an elevation angle coordinate, measured in a clockwise direction from a positive X-axis in an X-Y coordinate system of the generatrix of the reflector, said X-Y coordinate system further comprising a Y-axis which is an optical axis of said reflector; r(φ) is a polar radial coordinate of the generatrix of the reflector in the X-Y coordinate system and is equal to:

V. 2 2 x +y

Yt is equal to a Y coordinate of an object plane that is illuminated by the reflector; no is an index of refraction in an environment of the reflector; capital N is a number of transparent objects positioned between the reflector and the object plane; lower case n is an index that refers to individual transparent objects; th_n is a thickness, measured along the optical axis of an n^th transparent object; npn is the index of refraction of the n^th transparent object;

DIST comprises a quotient comprising a numerator comprising Rad(φ) and a denominator comprising lrr(Xt), wherein: lrr(Xt) is a predetermined light intensity at a given cylindrical radial coordinate Xt in the object plane;

Rad(φ) is an intensity of light emitted by a light source, for which the reflector is designed, at elevation angle φ; and