Nonimaging Optics Winston Episode 10 potx

The upper limit on étendue that can be transferred fromthe source to the target by a translationally symmetric concentrator is computedusing the integral of Eq.. Figure 10.13 Plot of the

operating every day of the year from sunrise to sunset The formulas for the targetand source étendue are given in Eqs (10.71) and (10.81), respectively We assumethe total surface area of the target is

(10.86)

Eq (10.71) then allows us to calculate the target étendue:

(10.87)When the source étendue is set equal to the target étendue, Eqs (10.81) and (10.87)give the following value for the total surface area of the source:

(10.88)For this equal-étendue case, Figure 10.11 depicts the skewness distributions ofEqs (10.70) and (10.73) The upper limit on étendue that can be transferred fromthe source to the target by a translationally symmetric concentrator is computedusing the integral of Eq (10.50) For the equal-étendue case, this upper limit ontransferred étendue turns out to be

(10.89)With reference to Figure 10.11, this étendue limit is equal to that portion of theétendue region contained under the source’s skewness distribution that is inter-sected by the étendue region contained under the target’s skewness distribution

As indicated by Eqs (10.51) and (10.53), the upper limits on efficiency and centration are computed by dividing emaxby the total source and target étendue,respectively For the equal-étendue case, the efficiency and concentration limitshave the same value:

Figure 10.11 Translational skewness distributions in a unit-refractive-index medium for a Lambertian target and a source having fixed latitudinal cutoffs parallel to the symmetry axis The angular half width of the source is Q 0 = 23.45° The source étendue equals that of the target.

Using Eqs (10.51) and (10.53), we can also compute the efficiency and tion limits for source-to-target étendue ratios other than unity The resulting effi-ciency limit is plotted as a function of the concentration limit in Figure 10.12 Thediamond-shaped marker on this plot indicates the efficiency and concentrationlimits for the equal-étendue case It is also useful to plot the efficiency and con-centration limits as a function of the source-to-target étendue ratio itself, as shown

concentra-in Figure 10.13 Note that the equal-étendue case corresponds to the crossconcentra-ing poconcentra-int

of the two curves in this plot

trans-of the source is Q 0 = 23.45° The diamond-shaped marker indicates the performance limit for the equal-étendue case.

Figure 10.13 Plot of the efficiency and concentration limits as a function of the target étendue ratio for translationally symmetric nonimaging devices that transfer flux to

source-to-a Lsource-to-ambertisource-to-an tsource-to-arget from source-to-a source hsource-to-aving fixed lsource-to-atitudinsource-to-al cutoffs psource-to-arsource-to-allel to the try axis The angular half width of the source is Q 0 = 23.45° The equal-étendue case corre- sponds to the crossing point of the two curves.

symme-10.3.6.2 Flux Transfer to Lambertian Target from Source

Having Fixed Longitudinal Cutoffs Parallel to

Symmetry Axis with Orthogonal Fixed

Latitudinal Cutoffs

As a second and final example, we now consider a source of the type analyzed inSection 10.3.5.4 We set the latitudinal half angle equal to the latitudinal halfwidth of incident solar radiation for non-tracking solar concentrators:

(10.91)The refractive index is assumed to be unity for both the source and the target.With these choices, this case is representative of a north-south-oriented non-tracking solar concentrator For such a concentrator, the appropriate value of the longitudinal half angle f0depends on the daily hours of operation We assumethat

(10.92)which corresponds to daily operation from sunrise to sunset at an equatorial loca-tion As is apparent from Eq (10.84), the choice of f0 affects only the verticalscaling of the skewness distribution It therefore has no effect on the efficiency andconcentration limits as a function of étendue The formulas for the target andsource étendue are given in Eqs (10.71) and (10.85) As in Section 10.3.6.1, weassume the total surface area of the target is

(10.93)

so that the target étendue is

(10.94)When the source and target étendue are equal, Eqs (10.85) and (10.94) give thefollowing value for the total surface area of the source:

(10.95)For this equal-étendue case, the skewness distributions of Eqs (10.70) and (10.84)are as depicted in Figure 10.14 The upper limit on étendue that can be trans-ferred from the source to the target by a translationally symmetric concentrator

is computed using the integral of Eq (10.50) For the equal-étendue case, thisupper limit on transferred étendue turns out to be

(10.96)The upper limits on efficiency and concentration are computed by dividing emaxbythe total source and target étendue, respectively For the equal-étendue case, theefficiency and concentration limits have the same value:

Figure 10.14 Translational skewness distributions in a unit-refractive-index medium for a Lambertian target and 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 source étendue equals that of the target.

trans-tration limit in this figure never exceeds the 49.30% value of Eq (10.97) This isbecause the half width of the skewness distribution of the source is always sin(q0)

= 0.3979, independent of the value of the source-to-target étendue ratio Thus, nomatter what value of the source-to-target étendue ratio is used, no étendue can betransferred from the source to the target by a translationally symmetric non-

imaging device for skewness values satisfying 0.3979 < |S| £ 1

Figure 10.16 Plot of the efficiency and concentration limits as a function of the target étendue ratio for translationally symmetric nonimaging devices that transfer flux to

source-to-a Lsource-to-ambertisource-to-an tsource-to-arget from source-to-a source hsource-to-aving fixed longitudinsource-to-al cutoffs psource-to-arsource-to-allel to symmetry axis with orthogonal fixed latitudinal cutoffs The latitudinal angular half width of the source is q 0 = 23.45° The equal-étendue case corresponds to the crossing point of the two curves.

REFERENCES

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

Verlag, New York

Bortz, J., Shatz, N., and Ries, H (1997) Consequences of étendue and skewnessconservation for nonimaging devices with inhomogeneous sources and targets.Proceedings of SPIE, Vol 3139, 59–75

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

transla-Ries, H., Shatz, N., Bortz, J., and Spirkl, W (1997) Performance limitations of

rotationally symmetric nonimaging devices J Opt Soc Am A., Vol 14, 10,

2855–2862

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

Narkis Shatz and John C Bortz

Science Applications International Corporation, San Diego, CA

265

Many nonimaging optical design problems encountered in practice have no knownclosed-form solutions When this is the case, it is possible to obtain high-performance designs by means of global optimization When applied to the designproblem, this process is referred to as inverse engineering This computationallyintensive numerical design approach sequentially modifies the reflective and/orrefractive surfaces of the optical system within a given parameterization schemeand constraint set until performance objective global optimality, evaluated upon asystem radiometric model, is achieved Global optimization can be used to deter-mine reflector and lens configurations that achieve maximal flux transfer to agiven target or that produce a desired radiometric distribution, such as an irradi-ance or intensity pattern

This chapter provides an overview of the application of inverse engineering tothe problem of nonimaging optical design We begin with a brief summary of thebehavior of nonimaging optical systems in terms of properties of mappings Anunderstanding of this behavior is central to the design problem, since it affects the limits on system performance, the choice of parametrization schemes, and thechoice of the class of global-optimization algorithms to be used We review thevarious factors affecting the performance of nonimaging optical systems, andpresent generalized formulations of étendue limits for use with inhomogeneoussources and/or targets Following this preparatory material, we review the for-malism of inverse engineering and include references to global-optimization algo-rithms that are applicable in the domain of nonimaging optical design Finally, weprovide five case studies of globally optimized designs, including three designs thatuse symmetry breaking to overcome the limitations imposed by skewness mis-matches between the source and target

11.2 MATHEMATICAL PROPERTIES OF

MAPPINGS IN NONIMAGING OPTICS

In the geometrical optics approximation, the behavior of a nonimaging optical

system can be formulated and studied as a mapping g: S 2n

Let g be a differentiable mapping The mapping g is called canonical (Arnold, 1989) if g preserves the differential 2-form w2

generalized coordinate and p is the generalized momentum Applying the

Euler-Lagrange necessary condition to Fermat’s principle and then the Legendre formation, we obtain a canonical Hamiltonian system, which defines a vector field

trans-on a symplectic manifold (a closed ntrans-ondegenerate differential 2-form) A vector field

on a manifold determines a phase flow—that is, a one-parameter group of morphisms (transformations that are differentiable and also possess a differen-tiable inverse) The phase flow of a Hamiltonian vector field on a symplecticmanifold preserves the symplectic structure of phase space and consequently iscanonical

diffeo-The properties of these mappings can be summarized as follows:

1 The mappings from input phase space to output phase space are piecewise feomorphic Consequently they are one-to-one and onto

dif-2 The transformation of phase space induced by the phase flow is canonical—that is, it preserves the differential 2-form

3 The mappings preserve the integral invariants, known as the Poincaré-Cartaninvariants Geometrically, these invariants are the sums of the orientedvolumes of the projections onto the coordinate planes

4 The mappings preserve the phase-space volume element (étendue) Thevolume of gD is equal to the volume of D, for any region D

In order to uniquely determine a mapping, we require, in addition to Fermat’sprinciple, knowledge of material properties and the stipulation of boundary con-ditions, which constitute the geometry of the reflective and/or refractive surfaces.Determining these boundary conditions is the subject of the nonimaging opticaldesign problem

Because the mappings in nonimaging optics are diffeomorphic (i.e., smooth),the solution topology for nonimaging optical design problems is Lipschitz contin-uous This is an advantageous property that can be exploited as regards the selec-tion of a global optimization search algorithm Certain choices of parameterizationschemes, particularly in designs involving multiple optical components, may lead

to a finite number of discontinuities in the topology Since the optical source will,

in practice, be represented by a finite-sized ray set, the solution topology will bemildly stochastic

11.3 FACTORS AFFECTING PERFORMANCE

We now summarize the key factors affecting the flux-transfer performance of imaging optical designs—such as collection, projection or coupling optics—thatutilize three-dimensional sources and/or targets Performance limits are associ-ated with étendue matching, skewness matching, source and target inhomo-geneities, design constraints, design goals, and nonidealities The ultimateperformance of the nonimaging system will generally be driven by some combi-nation of these performance-limiting factors

Étendue matching is a dominant consideration in the design of nonimaging opticalsystems We define the target-to-source étendue ratio (TSER) as the total targetétendue divided by the total source étendue When the target étendue is smallerthan that of the source, the upper limit of the fraction of source étendue that can

be transferred to the target’s phase space is equal to the TSER When the target étendue is greater than that of the source, the upper limit on the fractionalétendue that can be transferred is unity, but the phase space of the target will

be diluted As a practical matter, the étendue of the source should be computedbased on an integration of experimental measurements or a valid source model,whereas the étendue of the target can usually be computed using an analyticalintegration

A skewness mismatch between the source and target may cause severe mance limitations A comparison of skewness mismatch between candidate sourcesand the target can be useful during the design selection process—for example, acommon dilemma is whether to select an on-axis or a transverse source orienta-tion Losses due to skewness mismatch may be recovered to a large extent byemploying an optimized nonrotationally symmetric design, which activelyattempts to match the skewness of the source to that of the target

Source and target inhomogeneities will affect the performance limits In order toassess these limits, weight functions need to be introduced based on the source’sspecific spatial and angular radiance distributions and the target’s preferentialcharacteristics A detailed method to accomplish this was introduced in the previ-ous chapter for nonimaging optical systems exhibiting rotational or translationalsymmetry An analogous method for the case of nonsymmetric systems is thesubject of Section 11.4

11.3.4 Design Constraints

Real world designs are often subject to constraints If constraints are active at theoptimal design point, then the performance of the system will be adverselyaffected Typical design constraints may include minimum source-to-reflectorclearance, reflector (or lens) diameter, and length constraints

We identify two main classes of design goals: maximum flux transfer and beamshaping Weighted combinations of these goals may also be constructed Flux-transfer performance limits may be adversely affected by the inclusion of beamshaping considerations

After a nominal design is achieved, off-nominal operating conditions should

be examined These may include design robustness to variations in TSER andskewness mismatch, as well as analysis of tolerance to optical-surface-figure errorsand component misalignments

of the source are captured from view angles over 4p steradians The capturedimages contain detailed information about the spatial and angular distributions

of light radiating from the source, enabling a detailed model of the source’s ance distribution to be developed From this model a radiometric ray set virtuallyrepresenting the three-dimensional physical source can be constructed With addi-tional processing to provide variance reduction, a stochastic ray-set model of the

radi-source can be readily prepared for insertion into a global optimization design loop.

A large variety of sources have been accurately characterized using this technique,including filament lamps, xenon arc lamps, light emitting diodes, compact fluo-rescent lamps, metal halide lamps, and so on

A target may also be inhomogeneous For example, a target could have anabsorptance function that decreases with increasing angle of incidence withrespect to the target’s local surface normal To achieve high-performance fluxtransfer to such a target, it would be preferable to transfer as much flux as pos-sible to the regions of phase space corresponding to smaller, rather than larger,angles of incidence In general, we refer to a target as inhomogeneous when a non-constant weight function has been defined over the phase space of the target inorder to specify the relative usefulness of flux transferred to different regions ofthe target’s phase space

When the source and/or target are inhomogeneous, the principle of étendueconservation still applies, but the computation of the upper limit on performance

is complicated by the fact that more importance is attached to some regions of thesource and/or target phase spaces than to others In this section we derive theupper limits on flux-transfer performance for nonimaging optical systems designedfor use with inhomogeneous sources and targets We consider the general case ofnonsymmetric optical systems, for which performance limitations due to the skewinvariant do not apply The special cases of performance limits for rotationally andtranslationally symmetric optics were considered in the previous chapter

to an Inhomogeneous Target

We consider an inhomogeneous source having a total emitted flux of P src,tot The

source radiance is represented by the function L src(x), where the vector x

repre-sents a point in the phase space S of the source For the target we define the weight

function Wtrg (x¢), where x¢ represents a point in the phase space S¢ of the target.

Our goal in designing a nonimaging system is to maximize the weighted flux

where L is the source-radiance threshold, and de(x) is the differential element of

the source’s phase-space volume The function esrc (L) represents the amount of

source phase-space volume associated with regions of the source’s phase space for

which the radiance is greater than the radiance threshold value L Since

increas-ing the threshold reduces the phase-space volume contributincreas-ing to the integral in

Eq (11.2), it is apparent that e (L) must be a monotonically decreasing function

of L for L min £ L £ L max , where L min and L maxare the minimum and maximum values

of the source radiance Specifically, esrc (L) must decrease monotonically from a

maximum value of esrc (L min ) at L = L min to a minimum value of 0 at L = L max It isalso apparent that

(11.3)where esrc,totis the total phase-space volume associated with the source

Because esrc (L) decreases monotonically with L, it can be inverted to obtain the monotonically decreasing inverse function L(e src ), which ranges from a value of L max

at esrc = 0 to L minat esrc = esrc,tot The inverse function L(e src) represents the sortedsource radiance as a function of the source étendue The sorting process has forcedthe largest radiance value to occur at esrc = 0, with decreasing values of radiance

as esrc is increased The integral of L(e src) over the range, 0 £ esrc£ esrc,totequals the

total source flux P src,tot

We now consider quantities related to the target In the phase space of

the target, the weight function W trg(x¢) plays a role analogous to that played by

the radiance function L src(x) in the phase space of the source We define the

cumu-lative sorted target étendue as

(11.4)

where W is the weight-function threshold The function e trg (W) is a monotonically decreasing function of W for W min £ W £ W max , where W min and W max are theminimum and maximum values of the target weight function This monotonicallydecreasing function ranges from a maximum value of etrg (W min ) at W = W minto a

minimum value of 0 at W = W max It is therefore possible to invert the function

etrg (W) to obtain the monotonically decreasing inverse function W(e trg), which

ranges from W maxat etrg = 0 to W minat etrg= etrg,tot, where etrg,tot= etrg (W min) is the total

target étendue The inverse function W(e trg) represents the sorted target weight as

a function of the target étendue The sorting process forces the largest weight value

to occur at etrg = 0, with decreasing weight values as etrgis increased

We can now write down an expression for the maximum possible value of the

weighted flux P wgt transferred from the source to the target for a nonsymmetricoptical system This maximum occurs when the source radiation is transferred tothe target in such a way that the largest source-radiance values preferentially fillthe regions of the target’s phase space having the largest weight values Thus,

referring to Eq (11.1), we find that the maximum value of P wgt can be expressed

in the form

(11.5)

where

(11.6)

is the lesser of the total target and source étendue values

Now that we have derived the upper limit on the transferred flux, we are in

a position to write down formulas for the upper limits on efficiency and tration For inhomogeneous sources and targets, the efficiency h is defined as theactual weighted flux transferred to the target divided by the weighted flux levelthat would be achieved if all of the source radiation could be transferred to the

concen-eupper=min(esrc tot, ,etrg tot, )

region of the target’s phase space for which the weight function has its maximum

value In other words, the efficiency h is defined as P wgt divided by the value of P max wgt that would be obtained by replacing W(e) and e upper by W maxand esrc,tot, respectively,

in Eq (11.5):

(11.7)where

(11.8)

is the total flux emitted by the source Analogously, the concentration C is defined

as the actual weighted flux transferred to the target divided by the weighted targetflux level that would be achieved if all of the target’s phase space could be filledwith radiation having radiance equal to the maximum radiance level of the source

Thus, the concentration C is defined as P wgt divided by the value of P max wgtthat would

be obtained by replacing L(e) and e upper by L maxand etrg , tot, respectively, in Eq (11.5):

(11.9)where

(11.10)

is the total weighted target étendue It should be noted that given the way h and

C have been defined, the values of both must always fall within the range from 0

to 1 The upper limits on the efficiency and concentration can now be obtained

simply by substitution of P max wgt for P wgtin Eqs (11.7) and (11.9), respectively:

(11.11)and

(11.13)

Since the weight function equals unity for all points x¢ in the phase space of the

target, the sorted target weight function W(e trg) must also equal unity over therange of allowed values of etrg:

(11.14)Thus, Eqs (11.1) and (11.5) can be rewritten in the form

=

e ,

hmax

max wgt max src tot

∫

e ,,

,

(11.16)The efficiency and concentration reduce to

(11.17)and

(11.18)

where we have used the facts that, for a homogeneous target, W max= 1 and ewgt trg,tot=

etrg,tot Similarly, the formulas for the maximum efficiency and concentration become

(11.19)and

where L0is the constant source radiance Since the source radiance is constant for

all points x in the source phase space, the sorted source radiance function L(e src)must also be constant over the range of allowed values of esrc:

(11.22)The expression for the weighted flux transferred to the target [Eq (11.1)] cannot

be simplified, since source homogeneity does not guarantee that the target

radi-ance L trg(x¢) will be homogeneous However, Eq (11.22) allows us to simplify

Eq (11.5) to obtain the following formula for the upper limit on the weighted flux transferred to the target:

(11.23)where

=

e , .

hmax

max wgt src tot

P P

=,

L

wgt max trg tot

=

e ,,

h = P

P

wgt src tot,

¢Œ ¢

x

The efficiency and concentration can then be expressed in the form

(11.25)and

(11.26)

where we have used the fact that P src,tot = L0esrc,tot for a homogeneous source Similarly, the formulas for the maximum efficiency and concentration become

(11.27)and

(11.28)

where Eq (11.23) has been used

a Homogeneous Target

This case is the same as the case discussed in the last section, except that theassumption of target homogeneity allows us to set the weight function equal tounity This allows us to simplify the expression for the weighted on-target flux by

setting W trg(x¢) = 1 in Eq (11.1) to give

(11.29)which is equivalent to Eq (11.15) Despite the homogeneity of the source, it doesnot follow that the radiance produced in the target’s phase space is necessarilyhomogeneous Therefore, it is not possible to further simplify the expression for

P wgt by setting L trg (x¢) equal to a constant in Eq (11.29) Setting W(e) equal to unity,

Eq (11.23) reduces to the following expression for the upper limit on the weightedflux transferred to the target:

(11.30)where Eq (11.24) has been used The formulas for the efficiency and concentra-tion reduce to

(11.31)and

(11.32)

The maximum efficiency and concentration are obtained by substitution of the

right-hand side of Eq (11.30) for P wgtin Eqs (11.31) and (11.32):

C P L

wgt trg tot

=0e ,

he

L

wgt src tot

P max wgt=L0eupper,

trg S

=Ú¢Œ ¢de x( ¢) (x¢)

C max max wgt

trg tot wgt

= e

e ,,

e

max

max wgt max src tot

W

=

,

C P L

wgt trg tot wgt

=0e ,,