Nonimaging Optics Winston Episode 11 pps

Figure 11.13 Shape profile of star concentrator optimized to transfer flux from a sphere toa 30° disk.. Figure 11.16 Skewness distributions for spherical source, disk target, and flux tr

the curve C at some point P on C We also define the circle c that passes through the point P and is centered on the x-axis The line tangent to c at point P is designated as T¢ The angle between T and T¢ is designated as a A constant-angle

spiral is defined as any continuous curve C for which the angle a is a constant forall points on the curve

We define s to be the distance of a point P on a constant-angle spiral from the vertex of the cone We consider the differential change ds in the distance s pro- duced by a differential change df in the angle f The constant-angle criterion leads

to following relationship between ds and df:

where x0is the x-coordinate of the starting point on the spiral Substitution of Eq.

(11.60) into Eq (11.59) gives

(11.62)

Integrating both sides of Eq (11.62), we obtain an expression for s as a function

of f:

(11.63)where f0is the value of the angular coordinate f at the starting point of the spiral

Substitution of Eq (11.63) into Eq (11.60) gives the r-coordinate as a function of f:

(11.64)

Substitution of Eq (11.63) into Eq (11.61) gives the x-coordinate as a function of f:

(11.65)For the special case of b = 0°, Eqs (11.64) and (11.65) become

(11.66)and

(11.67)For the special case of b = 90°, Eqs (11.64) and (11.65) become

(11.68)and

(11.69)For the special case of b = 180°, Eqs (11.64) and (11.65) become

Each of the N segments of the curve is assumed to consist of two subsegments

having a-values which are equal in magnitude but opposite in sign The angularwidth of each subsegment is

(11.73)

We define the two indices m and n, which refer to the mthsubsegment of the nth

segment, where m = 0 or 1 and n = 0, 1, , N - 1 The central f-value of the mthsubsegment of the nthsegment is

(11.74)

It is also convenient to compute the value of m and n corresponding to any

par-ticular value of f between 0° and 360°:

(11.75)and

(f), we find that the periodic segmented spiral curve is specified by the formulas

(11.78)and

(11.79)

where the constant angle a has been multiplied by [2m(f) - 1] to produce the

desired alternation of the sign of the spiral angle for adjacent subsegments Similarly, for the special case of b = 0°, we obtain

f

ff( )= Ê

(11.81)For b = 90°, we find that

(11.82)and

(11.83)For b = 180°, we have

(11.84)and

(11.85)

By converting r0, x0, a, and b into the continuous functions r0(q), x0(q), a(q),

and b(q) of the parameter q, Eqs (11.78) through (11.85) then provide us with the desired specification of the entire reflector surface The specific forms of the

functions r0(q) and x0(q) are equivalent to the radial and axial coordinates for the

parameterization scheme described in the previous subsection for rotationallysymmetric reflectors The cone-angle function b(q) is defined as the slope angle

with respect to the x-axis of the local surface normal of the parametric curve defined by r0(q) and x0(q) We employ a finite-dimensional cubic-spline parameter-

ization to describe a as a function of q

We now present a design example in which the parameters used to control theshape deviation, truncation, acceptance angle, and axial source position shift werethe same 17 parameters used in the previous subsection In addition, 10 degrees

of freedom (knots) were allocated to the description of a as a function of q, for atotal of 27 design degrees of freedom in the optimization The lower and upperparameter ranges for the original 17 variables were the same as used in the pre-vious subsection The lower and upper parameter ranges for the knots used todefine a as a function of q were 0° and 35°, respectively When the shape devia-tions of the star lobes as a function of f are sufficiently small, the ray tracing can

be simplified by modeling the f-dependent shape perturbations purely as slopeperturbations imposed on a rotationally symmetric surface This approximationwas utilized in the ray tracing to obtain the results presented below

Global optimization was used to determine the form of 3D OSC that mizes the flux transferred from a 10-mm-diameter spherical Lambertian sourceinto an emergent conical beam subtending a 30° half angle The constraint thatthe source and the target must have equal étendue was enforced, which leads to

maxi-an aperture-to-source area ratio of 1/sin2

(30°) = 4 Thus, a target disk diameter of40mm was used The reflector surface was assumed to be loss-free and specular.The resulting shape of the 3D OSC design is shown in Figures 11.13 and 11.14

In contrast to the axisymmetric solution discussed in the previous subsection, thesource does not protrude behind the back of the reflector, and there exists a smallclearance gap between the source and the reflector The performance of this solu-tion was computed using 200,000 rays, which provides accuracy better than 0.1%

in efficiency and concentration The design produces a value of 84.9% for both

Figure 11.13 Shape profile of star concentrator optimized to transfer flux from a sphere to

a 30° disk.

Figure 11.14 3D cut-away view of star concentrator.

transfer efficiency and concentration The variation of the star-lobe angle alongthe arc-length of the reflector is depicted in Figure 11.15 The skewness distribu-tions for the sphere, the disk, and the rays output within the required 30° halfangle are depicted in Figure 11.16 Note that the output skewness distributionextends over the full skewness range of the disk target’s skewness distribution.The symmetry-breaking star cross section has allowed the original skewness distribution of the source to be transformed into an output skewness distribution

Figure 11.16 Skewness distributions for spherical source, disk target, and flux transferred

by 3D OSC to a disk within a 30° half angle.

providing a superior match to that of the target Figure 11.17 depicts the efficiency versus concentration curve for the 3D OSC in comparison with the curve for the truncated 30° involute CPC and the theoretical limiting curve forrotationally symmetrical systems As previously discussed, the efficiency versusconcentration curves for the two concentrators were generated by varying the

acceptance half angle of the disk target, while keeping all other source, target, andconcentrator characteristics fixed The large range of concentration values overwhich the efficiency of the optimized design exceeds the limit for axisymmetricdesigns is an indication of the robustness of the design relative to variations ineither efficiency or concentration from the nominal design point of equal étendue.

In Figure 11.18 we compare the far-field intensity profile produced by the 3D OSCwith that produced by the truncated 30° involute CPC It is of interest to note thatthe 3D OSC provides significantly improved far-field intensity uniformity relative

to the involute CPC design

Source and a Disk Target

We now consider the problem of designing a 3D OSC that maximizes the flux fer from a homogeneous cylindrical source to a coaxial disk target of equal étendue

trans-(Shatz, Bortz, Ries, and Winston, 1997) We choose a height-to-radius ratio H/R =

10, which is typical of an incandescent filament In this case the skewness match between the source and target is strongly pronounced The theoretical per-formance limit on efficiency for rotationally symmetric optics is 46.8% for theequal-étendue case, which means that this problem is more strongly affected byskewness mismatch than the case involving the spherical source

mis-We seek a star concentrator solution and state the design problem as one ofdetermining the form of a 3D OSC that maximizes the flux transferred from acylindrical Lambertian source into an emergent conical beam subtending a 30°half angle As in the previous subsection, we use a target disk diameter of 40mm

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Concentration (unitless) 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Figure 11.17 Efficiency versus concentration for 3D OSC optimized to transfer flux from a sphere to a disk within a 30° half angle.

The equal-étendue requirement, along with the chosen height-to-radius ratio,leads to a source height and radius of 22.36mm and 2.236mm, respectively Theoptimization was performed using the same parametrization scheme, number ofoptimization parameters, and parameter ranges as in the last subsection.The resulting optimized shape of the 3D OSC design is shown in Figures 11.19and 11.20 Note that the end of the source coincides precisely with the beginning

of the reflector The performance of this design was computed using 200,000 rays,

Figure 11.18 Far-field intensity versus emission angle for 3D OSC optimized to transfer flux from a sphere to a 30° disk.

-15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

x-coordinate (mm) -25

Figure 11.19 Shape profile of star concentrator, optimized to transfer flux from a cylinder

to a 30° disk.

Figure 11.20 3D cutaway view of star concentrator.

of the disk’s skewness distribution Figure 11.23 depicts the efficiency versus

-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

Skewness (mm) 0

Figure 11.22 Skewness distributions for cylindrical source, disk target, and flux ferred by 3D OSC to disk within its 30° acceptance half angle.

trans-0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Concentration (unitless) 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Figure 11.23 Efficiency versus concentration for 3D OSC optimized to transfer flux from a cylinder to a 30° disk.

concentration curve for the 3D OSC design in comparison with the theoretical limiting curve for rotationally symmetrical systems The efficiency versus concen-tration curve was generated by computing the efficiency and concentration for different values of the target’s acceptance half angle, while holding all othersystem characteristics constant It is of interest to note that the 3D OSC providesefficiency superior to the performance limit for axisymmetric systems over a largerange of concentration values Figure 11.24 depicts the efficiency versus TSER for the 3D OSC in relation to both the étendue limit, which is the upper limit fornonaxisymmetric optics, as well as the skewness limit, which is the upper limitfor axisymmetric optics.

Translational Symmetry

We now consider an example of a globally optimized design that uses structure ridges to overcome the performance limits imposed by translational symmetry (Bortz, Shatz, and Winston, 1997) The problem to be solved is that ofdesigning a north-south-oriented nontracking solar concentrator in a material ofunit refractive index, as discussed in Section 10.3.6.2 The latitudinal half angle

micro-of the solar radiation is 23.45° The target is assumed to be a 20-mm-diametercylindrical tube We consider the equal-étendue case, with longitudinal half angle

(11.86)Since the sun’s angular position changes by 15° every hour, this half angle corre-sponds to a total daily operation interval of 6hr, 40min We recall that the trans-

f0=50∞

TSER (unitless) 0.0

Figure 11.24 Efficiency versus target-to-source étendue ratio (TSER) for 3D OSC optimized

to transfer flux from a cylinder to a 30° disk.

lational skew invariant places an upper limit on efficiency and concentration of49.30% It can be shown that the involute CPC operates at this upper limit for thecase considered here, and therefore represents an optimal translationally sym-metric solution The equal-étendue assumption—and Eq (11.86)—gives a source-to-target area ratio of

(11.87)With this source-to-target area ratio, the involute CPC has a collection half angleof

(11.88)

To improve performance beyond the 49.30% limit, we use a type of perturbed lute CPC To break the symmetry, microstructure ridges are added to the surfacesuch that the ridge peaks and valleys are aligned with planes oriented perpen-dicular to the translational symmetry axis Rönnelid, Perers, and Karlsson (1994)considered the effect of constant tilt-angle microstructure ridges, superimposed on

invo-a conventioninvo-al CPC In the present cinvo-ase, however, the ridge tilt invo-angle is invo-a tion of position along the reflector’s shape profile The ridge slopes alternate insign as a function of position along the symmetry axis The ridges are assumed to

func-be sufficiently small that they can func-be considered to alter only the surface-normalvector as a function of position over the surface of the concentrator, without alter-ing the macroscopic shape In addition to the symmetry-breaking microstructure,the macroscopic shape profile itself is perturbed relative to the involute CPC Theparameterization scheme for the macroscopic shape perturbations is the same asthat described in Section 11.6.2 The microstructure tilt angle of the ridges as afunction of position along the shape profile is modeled as a cubic spline Theabsolute value of the tilt angle is constrained to be less than or equal to 30° Impo-sition of this angular limit improves manufacturability In addition, it simplifiesthe ray tracing and reduces reflection losses by preventing more than two reflec-tions from occurring for each ray incident on the region between two ridges.Global optimization was used to maximize the flux-transfer efficiency of thenontranslationally symmetric solar collector The computed efficiency and concen-tration of the optimized collector were found to be

(11.89)which represents a 47.4% performance improvement relative to the baseline trans-lationally symmetric concentrator The performance of this optimized design isindicated by the square marker on the efficiency versus concentration plot shown

in Fig 11.25 The shape profile of the optimized collector, with and without tracedrays, is shown in Fig 11.26 For comparison, the profile of the baseline unper-turbed involute CPC is depicted in Fig 11.26 as a dashed line A ray trace throughthe baseline concentrator is shown in Fig 11.27 The rays visible to the right

of the aperture in Figs 11.26 and 11.27 represent rays that have been rejected bythe concentrator As expected from the higher efficiency of the optimized concen-trator, fewer rejected rays are visible in Fig 11.26 than in Fig 11.27 A three-dimensional depiction of the optimized concentrator is shown in Fig 11.28 Toillustrate the microstructure geometry, the relative size of the symmetry-breakingridges has been magnified by a large factor in this figure The tilt angle of the

0 0.2 0.4 0.6 0.8 1 1.2 0

0.2 0.4 0.6 0.8 1 1.2

Figure 11.25 Plot of the efficiency limit as a function of the concentration limit for lationally symmetric nonimaging devices that transfer flux to a Lambertian target from a source having fixed longitudinal cutoffs parallel to symmetry axis with orthogonal fixed latitudinal cutoffs The latitudinal angular half width of the source is q 0 = 23.45° The performance achieved by the optimized nontranslationally symmetric concentrator and baseline concentrator for the case of f 0 = 50° is indicated by the square and diamond markers, respectively.

–100 –50 0 50 100

ridges as a function of the normalized position along the optimized concentrator’sshape profile is plotted in Fig 11.29

Plots of the transferred skewness distributions for the baseline and optimizedconcentrators are provided in Figs 11.30 and 11.31 As expected, the skewness dis-tribution for the translationally symmetric baseline design precisely matches theregion of overlap of the skewness distributions of the source and target The non-translationally symmetric optimized design, however, has produced a broadening

of the skewness distribution of the flux transferred to the target, thereby ing a better match to the target’s distribution

provid-The ridge microstructure used to break the symmetry of the optimized centrator can be thought of as a form of diffuser As such, it has the effect of intro-

con-0 50 100 150 200 250 300 –100

ducing a large number of small holes in the phase-space volume transferred to thetarget Since these holes are not filled with radiation from the source, their pres-ence reduces the achievable flux-transfer efficiency The introduction of holes intothe phase-space volume is analogous to the production of froth by injecting airbubbles into a liquid In the same way that the presence of froth reduces theamount of liquid that can be poured into a container of a given volume, the pres-ence of phase-space froth reduces the amount of étendue that can be transferredfrom a source to a target For this reason, 100% flux-transfer efficiency from a

–1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5

SourceTargetTransferredTranslational skewness (unitless)

is narrower than that of the source, this spreading of the skewness distributionwill only exacerbate the skewness mismatch between the source and the target,thereby reducing the achievable efficiency and concentration

A second nontranslationally symmetric north-south-oriented nontrackingsolar collector was designed for the equal-étendue case with a longitudinal half-angle of

(11.90)This half angle is representative of dawn-to-dusk operation of the concentrator.The computed flux-transfer efficiency and concentration for this second optimizeddesign were found to be

(11.91)which represents a 38.7% performance improvement relative to the optimal trans-lationally symmetric design

REFERENCES

Arnold, V I (1989) Mathematical Methods of Classical Mechanics Springer

Verlag, New York

Bortz, j., Shatz, N., and Winston, R (2001) Performance limitations of tionally symmetric nonimaging devices Proceedings of SPIE, Vol 4446,201–220

transla-Otten, R H J M., and van Ginneken, L P P P (1989) The Annealing Algorithm.

Kluwer Academic Publishers, Dordrecht

Ratschek, H., and Rokne, J (1998) New Computer Methods for Global tion Ellis Horwood and John Wiley, Hoboken.

Optimiza-Rönnelid, M., Perers, B., and Karlsson, B (1994) Optical properties of ing concentrators with corrugated reflectors Proceedings of SPIE, Vol 2255,595–602

nonimag-Rykowski, R., and Wooley, B (1997) Source modeling for illumination design ceedings of SPIE, Vol 3130, 204–208

Pro-Shatz, N., and Bortz, J (1995) An inverse engineering perspective on ing optical design Proceedings of SPIE, Vol 2538, 136–156

nonimag-hoptim=C optim= 68 4 %,

f0=90∞

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

SourceTargetTransferredTranslational skewness (unitless)