Abstract:
A nonimaging illumination or concentration optical device. An optical device is provided having a light source, a light reflecting surface with an opening and positioned partially around the light source which is opposite the opening of the light reflecting surface. The light reflecting surface is disposed to produce a substantially uniform intensity output with the reflecting surface defined in terms of a radius vector R i  in conjunction with an angle φ i  between R i , a direction from the source and an angle θ i  between direct forward illumination and the light ray reflected once from the reflecting surface. R i  varies as the exponential of tan (φ i  -θ i )/2 integrated over φ i .

Description:
The present invention is directed generally to a method and apparatus for providing user selected nonimaging optical outputs from electromagnetic energy sources of finite but small extent. More particularly, the invention is directed to a method and apparatus wherein the design profile of an optical apparatus for small, finite optical sources can be a variable of the acceptance angle of reflection of the source ray from the optical surface. By permitting such a functional dependence, the nonimaging output can be well controlled. 
     Methods and apparatus concerning illumination by light sources of finite extent are set forth in a number of U.S. Pat. Nos. including 3,957,031; 4,240,692; 4,359,265; 4,387,961; 4,483,007; 4,114,592; 4,130,107; 4,237,332; 4,230,095; 3,923,381; 4,002,499; 4,045,246; 4,912,614 and 4,003,638 all of which are incorporated by reference herein. In one of these patents the nonimaging illumination performance was enhanced by requiring the optical design to have the reflector constrained to begin on the emitting surface of the optical source. However, in practice such a design was impractical to implement due to the very high temperatures developed by optical sources, such as infrared lamps, and because of the thick protective layers or glass envelopes required on the optical source. In other designs it is required that the optical source be separated substantial distances from the reflector. In addition, when the optical source is small compared to other parameters of the problem, the prior art methods which use the approach designed for finite size sources provide a nonimaging output which is not well controlled; and this results in less than ideal illumination. Substantial difficulties therefore arise when the optical design involves situations such as: (1) the source size is much less than the closest distance of approach to any reflective or refractive component or (2) the angle subtended by the source at any reflective or refractive component is much smaller than the angular divergence of an optical beam. 
     It is therefore an object of the invention to provide an improved method and apparatus for producing a user selected nonimaging optical output. 
     It is another object of the invention to provide a novel method and apparatus for providing user selected nonimaging optical output of electromagnetic energy from optical designs using small, but finite, electromagnetic energy sources. 
     It is a further object of the invention to provide an improved optical apparatus and method of design wherein the optical acceptance angle for an electromagnetic ray is a function of the profile parameters of both two and three dimensional optical devices. 
     It is a further object of the invention to provide an improved optical apparatus and method of design for radiation collection. It is yet another object of the invention to provide a novel optical device and method for producing a user selected intensity output over an angular range of interest. 
     It is still an additional object of the invention to provide an improved method and apparatus for providing a nonimaging optical illumination system which generates a substantially uniform optical output over a wide range of output angles. 
     Other objects, features and advantages of the present invention will be apparent from the following description of the preferred embodiments thereof, taken in conjunction with the accompanying drawings described below wherein like elements have like numerals throughout the several views. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows a two-dimensional optical device for providing nonimaging output; 
     FIG. 2 illustrates a portion of the optical device of FIG. 1 associated with the optical source and immediate reflecting surface of the device. 
     FIG. 3A illustrates a bottom portion of an optical system and FIG. 3B shows the involute portion of the reflecting surface with selected critical design dimensions and angular design parameters associated with the source; 
     FIG. 4A shows a perspective view of a three-dimensional optical system for nonimaging illumination and FIG. 4BA, 4BB and 4BC illustrate portions and views of the optical system of FIG. 4A; and 
     FIG. 5A shows such intensity contours for an embodiment of the invention and FIG. 5B illustrates nonimaging intensity output contours from a prior art optical design. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     In the design of optical systems for providing nonimaging illumination using optical sources which are small relative to other system parameters, one should consider the limiting case where the source has no extent. This is in a sense the opposite of the usual nonimaging problem where the finite size and specific shape of the source is critical in determining the design. In any practical situation, a source of finite, but small, extent can better be accommodated by the small-source nonimaging design described herein rather than by the existing prior art finite-source designs. 
     We can idealize a source by a line or point with negligible diameter and seek a one-reflection solution in analogy with the conventional &#34;edge-ray methods&#34; of nonimaging optics (see, for example, W. T. Welford and R. Winston &#34;High Collection Nonimaging Optics,&#34; Academic Press. New York, N.Y. (1989)). Polar coordinates R,Φ are used with the source as origin and θ for the angle of the reflected ray as shown in FIG. 3. The geometry in FIG. 3 shows that the following relation between source angle and reflected angle applies: 
     
         d/dΦ(logR)=tanα,                                 (1) 
    
     where α is the angle of incidence with respect to the normal. Therefore, 
     
         α=(Φ-θ)/2                                  (2) 
    
     Equation (1) is readily integrated to yield, 
     
         log(R)-∫tanαdΦ+const.                       (3) 
    
     so that, 
     
         R=const. exp(∫tanαdΦ                        (4) 
    
     This equation (4) determines the reflector profile R(Φ) for any desired functional dependence θ(Φ). 
     Suppose we wish to radiate power (P) with a particular angular distribution P(θ) from a line source which we assume to be axially symmetric. For example, P(θ)=const. from θ=0 to θ 1  and P(θ) ˜0 outside this angular range. By conservation of energy P(θ)dθ=P(Φ)dΦ (neglecting material reflection loss) we need only ensure that 
     
         do/do=P(Φ)/P(θ)                                  (5) 
    
     to obtain the desire radiated beam profile. To illustrate the method, consider the above example of a constant P(θ) for a line source. By rotational symmetry of the line source, P/do=a constant so that, according to Equation (4) we want θ to be a linear function of Φ such as, θ=aΦ. Then the solution of Equation (3) is 
     
         R=R.sub.0 /cos.sup.k (Φ/k)                             (6) 
    
     where, 
     
         k=2/(1-a),                                                 (7) 
    
     and R 0  is the value of R at Φ=0. We note that the case a=0(k=2) gives the parabola in polar form, 
     
         R=R.sub.0 /cos.sup.2 (Φ/2),                            (8) 
    
     while the case θ=constant=θ 1  gives the off-axis parabola, 
     R=R 0  cos 2  (θ 1  /2)/cos 2  [Φ-θ 0 )/2](9) 
     Suppose we desire instead to illuminate a plane with a particular intensity distribution. Then we correlate position on the plane with angle θ and proceed as above. 
     Turning next to a spherically symmetric point source, we consider the case of a constant P(Ω) where Ω is the radiated solid angle. Now we have by energy conservation, 
     
         P(Ω)dΩ=P(Ω.sub.0)dΩ.sub.0          (10) 
    
     where Ω 0  is the solid angle radiated by the source. By spherical symmetry of the point source, P(Ω 0 )=constant. Moreover, we have dΩ=(2π)dcosθ and dΩ 0  =(2π)dcosΦ; therefore, we need to make cosθ a linear function of cosΦ, 
     
         cosθ=a cosΦ+b                                    (11).sub.1 
    
     With the boundary conditions that θ=0 at Φ=0, at Φ=θ, θ=θ.sub. at Φ=Φ 0 , we obtain, 
     
         a=(1-cosθ.sub.1)/(1-cosΦ.sub.0)                  (12) 
    
     
         b=(cosθ.sub.1 -cosΦ.sub.0)/(1-cosΦ.sub.0)    (13) 
    
     [For example, for θ 1  &lt;&lt;1 and Φ 0  ˜π2/2 we have, θ˜√2θ 0  sin(1/2Φ).] 
     This functional dependence is applied to Equation (4) which is then integrated, such as by conventional numerical methods. 
     A useful way to describe the reflector profile R(Φ) is in terms of the envelope (or caustic) of the reflected rays r(Φ). This is most simply given in terms of the direction of the reflected ray t=(-sinθ, cosθ). Since r(Φ) lies along a reflected ray, it has the form, 
     
         r=R+Lt.                                                    (14) 
    
     where R=R(sinΦ 1  -cosΦ). Moreover, 
     
         RdΦ=Ldθ                                          (15) 
    
     which is a consequence of the law of reflection. Therefore, 
     
         r=R+t/(dθ/dΦ)                                    (16) 
    
     In the previously cited case where θ is the linear function aΦ, the caustic curve is particularly simple, 
     
         r=R+t/a                                                    (17) 
    
     In terms of the caustic, we may view the reflector profile R as the locus of a taut string; the string unwraps from the caustic r while one end is fixed at the origin. 
     In any practical design the small but finite size of the source will smear by a small amount the &#34;point-like&#34; or &#34;line-like&#34; angular distributions derived above. To prevent radiation from returning to the source, one may wish to &#34;begin&#34; the solution in the vicinity of θ=0 with an involute to a virtual source. Thus, the reflector design should be involute to the &#34;ice cream cone&#34; virtual source. It is well known in the art how to execute this result (see, for example, R. Winston, &#34;Appl. Optics,&#34; Vol. 17, p. 166 (1978)). Also, see U.S. Pat. No. 4,230,095 which is incorporated by reference herein. Similarly, the finite size of the source may be better accommodated by considering rays from the source to originate not from the center but from the periphery in the manner of the &#34;edge rays&#34; of nonimaging designs. This method can be implemented and a profile calculated using the computer program of the Appendix (and see FIG. 2) and an example of a line source and profile is illustrated in FIG. 1. Also, in case the beam pattern and/or source is not rotationally symmetric, one can use crossed two-dimensional reflectors in analogy with conventional crossed parabolic shaped reflecting surfaces. In any case, the present methods are most useful when the sources are small compared to the other parameters of the problem. 
     Various practical optical sources can include a long arc source which can be approximated by an axially symmetric line source. We then can utilize the reflector profile R(Φ) determined hereinbefore as explained in expressions (5) to (9) and the accompanying text. This analysis applies equally to two and three dimensional reflecting surface profiles of the optical device. 
     Another practical optical source is a short arc source which can be approximated by a spherically symmetric point source. The details of determining the optical profile are shown in Equations (10) through (13). 
     A preferred form of nonimaging optical system 20 is shown in FIG. 4A with a representative nonimaging output illustrated in FIG. 5A. Such an output can typically be obtained using conventional infrared optical sources 22 (see FIG. 4A), for example high intensity arc lamps or graphite glow bars. Reflecting side walls 24 and 26 collect the infrared radiation emitted from the optical source 22 and reflect the radiation into the optical far field from the reflecting side walls 24 and 26. An ideal infrared generator concentrates the radiation from the optical source 22 within a particular angular range (typically a cone of about ±15 degrees) or in an asymmetric field of ±20 degrees in the horizontal plane by ±6 degrees in the vertical plane. As shown from the contours of FIG. 5B, the prior art paraboloidal reflector systems (not shown) provide a nonuniform intensity output, whereas the optical system 20 provides a substantially uniform intensity output as shown in FIG. 5A. Note the excellent improvement in intensity profile from the prior art compound parabolic concentrator (CPC) design. The improvements are summarized in tabular form in Table I below: 
     
                       TABLE I______________________________________Comparison of CPC With Improved Design                CPC  New Design______________________________________Ratio of Peak to On Axis Radiant Intensity                  1.58   1.09Ratio of Azimuth Edge to On Axis                  0.70   0.68Ratio of Elevation Edge to On Axis                  0.63   0.87Ratio of Corner to On Axis                  0.33   0.52Percent of Radiation Inside Useful Angles                  0.80   0.78Normalized Mouth Area  1.00   1.02______________________________________ 
    
     In a preferred embodiment designing an actual optical profile involves specification of four parameters. For example, in the case of a concentrator design, these parameters are: 
     1. a=the radius of a circular absorber; 
     2. b=the size of the gap; 
     3. c=the constant of proportionality between θ and Φ-Φ 0  in the equation θ=c(Φ-Φ 0 ); 
     4. h=the maximum height. 
     A computer program has been used to carry out the calculations, and these values are obtained from the user (see lines six and thirteen of the program which is attached as a computer software Appendix included as part of the specification). 
     From Φ=0 to φ=Φ 0  in FIG. 3B the reflector profile is an involute of a circle with its distance of closest approach equal to b. The parametric equations for this curve are parameterized by the angle α (see FIG. 3A). As can be seen in FIG. 3B, as Φ varies from 0 to Φ 0 , α varies from α 0  to ninety degrees The angle α 0  depends on a and b, and is calculated in line fourteen of the computer software program. Between lines fifteen and one hundred and one, fifty points of the involute are calculated in polar coordinates by stepping through these parametric equations. The (r,θ) points are read to arrays r(i), and theta(i), respectively. 
     For values of Φ greater than Φ 0 , the profile is the solution to the differential equation: 
     
         d(lnr)/dΦ=tan{Φ-θ+arc sin(a/r]} 
    
     where θ is a function of φ. This makes the profile r(φ) a functional of θ. In the sample calculation performed, θ is taken to be a linear function of Φ as in step 4. Other functional forms are described in the specification. It is desired to obtain one hundred fifty (r,theta) points in this region. In addition, the profile must be truncated to have the maximum height, h. We do not know the (r,theta) point which corresponds to this height, and thus, we must solve the above equation by increasing phi beyond Φ 0  until the maximum height condition is met. This is carried out using the conventional fourth order Runga-Kutta numerical integration method between lines one hundred two and one hundred and fifteen. The maximum height condition is checked between lines one hundred sixteen and one hundred twenty. 
     Once the (r,theta) point at the maximum height is known, we can set our step sizes to calculate exactly one hundred fifty (r,theta) points between Φ 0  and the point of maximum height. This is done between lines two hundred one and three hundred using the same numerical integration procedure. Again, the points are read into arrays r(i), theta(i). 
     In the end, we are left with two arrays: r(i) and theta(i), each with two hundred components specifying two hundred (r,theta) points of the reflector surface. These arrays can then be used for design specifications, and ray trace applications. 
     In the case of a uniform beam design profile, (P(θ)=constant), a typical set of parameters is (also see FIG. 1): 
     
         a=0.055 in. 
    
     
         b=0.100 in. 
    
     
         h=12.36 in. 
    
     
         c=0.05136 
    
     
         for θ(Φ)=c(Φ-Φ.sub.o) 
    
     In the case of an exponential beam profile, generally, (P(θ)=Ae -B θ where, 
     
         θ(φ)=-C.sub.2 ln(1-C.sub.1 φ) 
    
     where A, B, C 1 , and C 2  are constants. 
     In the case of a Gaussian beam profile: 
     [P(A)=Aexp(-Bθ 2 )]; 
     
         θ(φ)=C.sub.1 Erf.sup.-1 (C.sub.2 φ+1/2) 
    
     where Erf -1  is the inverse error function given by the expression, Erf(x)=(1/√2π) ∫exp(-x 2  /2)dx, integrated from minus infinity to x; 
     In the case of a cosine beam profile; 
     
         P(θ)=AcosBθ; 
    
     
         θ=C.sub.1 sin.sup.-1 (C.sub.2 φ) 
    
     where A, B, C 1  and C 2  are constants. 
     A ray trace of the uniform beam profile for the optical device of FIG. 1 is shown in a tabular form of output in Table II below: 
     
                                           TABLE II__________________________________________________________________________ELEVATION   114      202         309            368               422                  434                     424                        608                           457                              448                                 400                                    402                                       315                                          229                                             103   145      295         398            455               490                  576                     615                        699                           559                              568                                 511                                    478                                       389                                          298                                             126   153      334         386            465               515                  572                     552                        622                           597                              571                                 540                                    479                                       396                                          306                                             190   202      352         393            452               502                  521                     544                        616                           629                              486                                 520                                    432                                       423                                          352                                             230   197      362         409            496               496                  514                     576                        511                           549                              508                                 476                                    432                                       455                                          335                                             201   241      377         419            438               489                  480                     557                        567                           494                              474                                 482                                    459                                       421                                          379                                             230   251      364         434            444               487                  550                     503                        558                           567                              514                                 500                                    438                                       426                                          358                                             231   243      376         441            436               510                  526                     520                        540                           540                              482                                 506                                    429                                       447                                          378                                             234   233      389         452            430               489                  519                     541                        547                           517                              500                                 476                                    427                                       442                                          344                                             230   228      369         416            490               522                  501                     539                        546                           527                              481                                 499                                    431                                       416                                          347                                             227   224      359         424            466               493                  560                     575                        553                           521                              527                                 526                                    413                                       417                                          320                                             205   181      378         392            489               485                  504                     603                        583                           563                              530                                 512                                    422                                       358                                          308                                             194   150      326         407            435               506                  567                     602                        648                           581                              535                                 491                                    453                                       414                                          324                                             179   135      265         382            450               541                  611                     567                        654                           611                              522                                 568                                    446                                       389                                          300                                             130   129      213         295            364               396                  404                     420                        557                           469                              435                                 447                                    351                                       287                                          206                                             146   AZIMUTH__________________________________________________________________________ ##SPC1##