Nonimaging Optics Winston Episode 7 docx

This cannot be done with a single sequential optical surface, asalready discussed, but it is possible if two surfaces are used.. In fact, Ong, Gordon, and Rabl 1996 showed that there are

y component of P2¥ T2> 0 Estimate an intermediate point P12= (P1+ P2)/2

and its normal N12= (N1+ N2)/2

3 Carry out the 3D ray-tracing on P12, associating it with the area of refractive

surface between P1and P2with the direction of impingement U(q2) to obtain

the value DA ap12 and thus calculate A ap2(q2) = A ap1(q2) + DA ap12, estimated angularresponse for the angle q2of the portion of lens up to P2

4 Repeat the steps from 2 to 3, iterating on the value of the parameter s (note that DA ap12 increases with s), until obtaining that |1 - A ap(q2) /A ap2(q2)|< e, e being

a preset margin of error

5 Carry out the 3D ray-tracing on the point P12resulting from the iteration 4,

associating it with the area of the refractive surface between P1and the point

P2resulting from the iteration 4, in order to obtain the function DA ap12(q) and

thus calculate A ap2 (q) = A ap1 (q) + DA ap12(q), estimated angular response of the

portion of lens up to P2

6 Increase the value q3= q2+ Dq and repeat the steps from 2 to 5, increasing thesubindices by one unit, until for a given angle qn the coordinate z of point P n

is negative

7 Repeat the steps 1–6, iterating on the abscissa of point P1until |1 - qn/qMAX| <

e ¢, e¢ being another preset margin of error

The design is finished The refractive surface is defined by the set of points culated in the process If required, it is possible to fit these points by a spline or

cal-a polynomical-al curve, which fcal-acilitcal-ates hcal-andling of the dcal-atcal-a

The design guarantees that the prescription is adjusted in the whole range

0 < q < qM, but the stepped transition to zero at q = qMis not (i.e., the prescriptionfor q > qM cannot be adjusted) because there are no degrees of freedom to makethe outer portion of the lens perform as a Cartesian oval (as done in Section 7.4.2for the linear case)

However, it should be emphasized that the design procedure uses rays ing on the receiver from nearly all possible directions (the whole field of view ofthe photodiode is covered) This situation is close to optimum in terms of maxi-

imping-mizing sensitivity—that is, making the constant A0(and k¢) as large as possible.

The optimum is equivalent to get isotropic illumination of all the points of thereceiver with rays from the specified range qMIN< q < qMAX

In the case that the active surface is not flat, as is common in the case of thesurface of LED or IRED emitters, the procedure described is applicable simply byconsidering the corresponding geometry of the active surface for the ray tracingsand directing the refracted ray tangent to that surface (as a generalized concept

of the point R) for the calculation of the normal at P k

The method can be easily generalized to include preset rotational sequentialsurfaces (either refractive or reflective), which deflect the ray trajectories This isthe case shown in Figure 7.9 shows the cross section of a lens designed for a cir-cular photodiode active area of silicon without antireflection coating The lens has

n = 1.49, and an encapsulating material of n¢ = 1.56 is assumed The surface separating both media is preset to a sphere The prescribed A ap(q) function is thelinear function of Eq (7.19)

The actual function A ap(q) function is finally calculated by ray tracing ing Fresnel losses at the air-lens and lens-silicon interfaces) is shown along withthe specifications in Figure 7.10a The procedure, if applied to other angular sen-

sitivity functions as A ap(q) = 1/cos(q) inside for 0 < q < 80° (and null outside thatrange) lead to another lens (whose profile is not shown here) that also producesthe specified sensitivity accurately, as shown in Figure 7.10b.

Figure 7.11 shows the results from M Hernández (2003) of ray trace on severallenses designed for a linear prescribed relative sensitivity with different values of

the z-coordinate of the on-axis point of the lens V (obtained from different tions of the z-coordinate of the point A1) All show very good agreement with the

selec-linear prescription (perhaps except near q = 0, especially for small V zvalues) The

very noticeable difference is that the smaller V zvalue, the smoother the transition

of the sensitivity at q = qM= 60° Note that, since by étendue conservation all curves

in Figure 7.11 fulfill the integral condition

y(mm)

x(mm)

Lens profile

Receiver (diammeter 3 mm)

Preset spherical profile

Figure 7.9 Cross-section of a lens designed to get a linear angular sensitivity function in

the range 0 £ q £ 60° (lens refractive index n = 1.49; encapsulant refractive index n¢ = 1.56)

In order to get the maximum possible value of the constant A0, the steppedtransition to zero at q = qM is needed (i.e., the null prescription for q > qMmustalso be adjusted) This cannot be done with a single sequential optical surface, asalready discussed, but it is possible if two surfaces are used This is done with theSMS design method presented in Chapter 8 Two complete surfaces are not needed

to solve this design problem For both refractive surfaces, one possible design isindicated in the next steps, which will refer to the points indicated in Figure 7.12:

1 Preset surface S Q from the center The last point Q T 2of the present portion of

surface S Qwill be calculated in step 3

2 Apply the procedure just described to achieve the prescribed intensity for the

calculation of refractive surface S P through the present surface S Qup to point

P T , which is the point such that the ray r¢ traced (inversely) from R¢ passing through P T (after the refraction on S R at point Q T1) exits the lens toward direc-tion q = qM

3 Calculate the point Q T2 as the point of S Q on which the ray r from R is refracted toward P T Note that, up to this point, the intensity prescription has beendesigned for 0 < q < q , which is the exit direction of ray r.

00.5

11.5

22.5

Figure 7.11 Effect of the lens size in the optical performance for the linear prescription of

rotational lenses (receiver diameter D = 3mm).

4 Calculate a new portion of surface S Pas the Cartesian oval that makes that

the rays r¢ traced (inversely) from R¢ and refracted at S Qat the portion between

Q T1 and Q T2 are refracted on the new points of S Ptoward direction q = qM

5 Apply the procedure just described to achieve the prescribed intensity for the

calculation of refractive surface S Q through the already known portion of

surface S Pcalculated in step 4

6 Repeat steps 4 and 5 up to convergence onto the line R–R¢.

The point source approximation of Section 7.3 can also be applied to the designproblem of a refractive sequential rotational surface for prescribed sensitivity Acomparison shows how important the finite dimension of the source is in a specificexample (Hernández, 2003)

The comparison of the performance for different lens sizes designed with the

point source approximation but ray-traced with the receiver of diameter D = 3mm

is shown in Figure 7.13 The linear prescription is well achieved only for large

sizes (V Z = 10D) For the size of this lens with practical interest, which is about

V z= 3mm, the point size model leads to a lens profile that performs far from the

specification, in contrast with the result for V z= 3mm already presented in Figure7.11

CYLINDRICAL OPTICS

Another particularly useful case is producing a constant irradiance on a distantplane from a cylindrical source of uniform brightness, such as a Lambertiansource As already mentioned, this was worked out by Ries and Winston (1994)

In fact, Ong, Gordon, and Rabl (1996) showed that there are four basic types of

f

A ap(q)/Aph

Figure 7.13 Effect of the lens size on the optical performance for the linear prescription of

the design obtained with the point source approximation (receiver diameter D = 3mm).

solutions for this type of problem Two classes derive from the fact that the tor curve can be diverging or converging—that is, the caustics formed can fallbehind or in front of the reflector These types have been referred to as compoundhyperbolic concentrator (CHC) or compound elliptical concentrator (CEC) The pos-sibilities are then doubled because the design can be done with the near edge orfar edge of the source being always illuminated The interested reader can findfurther information in the cited reference.

POINT SOURCES IN 3D

Freeform (without any prescribed symmetry) designs in 3D are not a simple sion from the 2D case These designs become much more difficult, and conse-quently, they are less developed than their 2D equivalents In this section weexamine overview 3D freeform design methods for point sources—that is, methodsthat use the point source approximation This means that the designs will perform

exten-as the theory foresees if the optical surfaces are far enough from the source (interms of source diameter) so it can be considered as a point At present only onemethod, which is currently being developed, is able to manage extended sources

in 3D geometry This method is the extension to 3D of the SMS method of Chapter

The basic equation governing the solution of this problem is a second ordernonlinear partial differential equation of Monge-Ampere type This was found in

1941 by Komissarov and Boldyrev (1994) Schruben (1972) created the equationgoverning the design of a luminary reflector that provides a prescribed irradiancepattern on a given plane when the reflector is illuminated by a nonisotropic punc-tual source

During the 1980s and 1990s a strong development of the method was aged by reflector antennas designers Wescott, Galindo, Graham, Zaporozhets,Mitra, Jervase, and (see References) others contributed to this field of antennareflector design The method starts with a procedure purely based in GeometricalOptics This is the part in which we are more interested for illumination applica-tions After the Geometrical Optics design, a Physical Optics analysis and syn-thesis procedure is necessary for a fine-tuning of the design At present there iscommercial software for designing these antenna reflectors based on this method(see, for instance, http://www.ticra.dk/)

encour-The method is particularly useful for satellite applications Satellite reflectorantennas must provide a given far-field (or intensity) pattern to fit, for instance,

a continent contour, in satellite-to-earth broadcasting applications And this should

be done efficiently In this case a single-shaped reflector is enough to solve theproblem The requirement is equivalent to saying that the amplitude of the field

at the aperture is prescribed In other cases it is required to achieve a prescribed7.7 Freeform Optical Designs for Point Sources in 3D 173

irradiance pattern at the antenna aperture (in general, this is required to reducethe side-lobes emissions) besides the specified far-field pattern In these cases, twoshaped reflectors are enough to solve the problem, and not only the output ampli-tude is controlled, but also the phase distribution at this aperture This secondproblem is very similar to the first, although it may look different.

The single reflect or designs for satellite applications do not differ stronglyfrom a parabola shape because the desired intensity pattern is highly collimated

in general This fact has allowed developing several approximate methods to solvethe Monge Ampere that worked well within these conditions

Beginning in the 1980s until the present, the subject has been of interest tomathematicians like Oliker, Caffarelli, Kochengin, Guan, Glimm, and Newman(see References) Conditions of existence and uniqueness of the solutions have beenfound as well as new design procedures have been proposed For instance, Glimmand Oliker (2003) have shown recently that the problem can also be solved as avariational problem in the framework of a Monge-Kantorovich mass transferproblem, which allows solving the problem numerically by techniques from linearprogramming The designs are not limited to reflectors but extend also to refrac-tive surfaces Already in the present decade, the subject has come back to the illu-mination field by Ries and Muschaweck (2002) In this reference, multigridnumerical techniques are efficiently used to solve the Monge-Ampere equation.The solutions are classified into four types depending on the location of the centers

of curvature of the output wavefronts to design: In two of these types, the surfaces

of curvature centers (each one corresponding to one of the two families of ture lines) are at one “side” of the optical surface, whereas in the remaining typesthe surfaces of curvature centers are at both sides of the optical surface

We shall restrict the explanations to the problem of designing a single opticalsurface (reflective or refractive) that transforms a given intensity pattern of thesource into another prescribed intensity pattern (Minˆano and Benítez, 2002) Let

rˆ be a unit vector characterizing an emitting direction of the source This unit vector can be determined with two parameters, u and v These two parameters

can be, for instance, the two angular coordinates (q, f) of the spherical coordinates

In this case, rˆ is given by (see Figure 7.14)

Let the unit vector sˆ define an outgoing direction of the rays after deflection on the optical surface Using spherical coordinates (a, b), sˆ can be written as

According to Herzberger (1958), if we have two surfaces defined by the vectors a

and a ¢ that are crossed by a one-parameter beam of rays and such that u is the

parameter, we have

(7.25)

where the unit vectors sˆ¢ and sˆ are pointing in the ray directions at each one of the surfaces, E is the optical path length from the surface defined by a¢ to the surface

defined by a , and n¢, n are, respectively, the refractive indices at each one of the

surfaces We can obtain both the equation of reflection and the equation from

Eq (7.25)

Assume that the one-parameter bundle of rays is passing through the

coordi-nate origin The surface defined by the vector a ¢ is just a point and thus a¢ u = 0

Let r  be a vector defining a reflective surface r is the vector a of Eq (7.25).

Now consider a two-parameter bundle of rays passing through the coordinate

origin The two parameters are u and v Then, application of Eq (7.25) gives (see

where it has been taken into account that E u = r u ; r is the modulus of r—that is,

r = r Eq (7.27) is derived from this definition of the modulus

(7.27)Combining Eqs (7.26) and (7.27) we get the reflection law in the form that we are

going to use (note that the vectors rˆ u and rˆ vare not a unit vectors)

(7.28)

We can apply Eq (7.25) to a refractive surface to obtain the refraction law in asimilar way as we got the reflection law The result is

(7.29)

that is, for our purposes, both laws can be summarized in Eq (7.29) taking n = 1

in the case of reflection

Eq (7.31) is the form that we will use for the energy conservation

Using Eq (7.24), Eq (7.31) can be written as

(7.32)

Where the sign ± takes into account that the trihedron sˆ - sˆ u - sˆ v may have two

possible orientations We have chosen rˆ, rˆ u , rˆ v to be in the positive orientation

(rˆ·rˆ u ¥ rˆ v > 0), but we don’t have the freedom to choose the orientation of sˆ - sˆ u - sˆ v

The dependence of sˆ with (u, v) is not totally free This is due to the Malus-Dupin

theorem, which states that a normal congruence remains like this after beingdeflected by a mirror or a lens surface For our particular case (a single reflective

or refractive surface and a punctual source) the Malus-Dupin theorem is nothingelse than the equality of the crossed derivatives of the function describing the

optical surface—that is, r uv = r vu(see Eq (7.26))

r s

n r s

r r

r s

r s

r r

which can also be written as

(7.34)

Eqs (7.29), (7.32), and (7.34) form a system of equations with unknown

vari-ables r(u, v) and sˆ(u, v) Variable r can be eliminated with Eqs (7.29) and (7.34),

resulting

(7.35)The system is now formed by Eqs (7.35) and (7.32), where the unknown function

is sˆ(u, v)—that is, we have to find a mapping of the unit sphere into itself ing Eqs (7.35) and (7.32) We can take (q, f) in Eq (7.21) as the parameters u, v and take a(u, v), b(u, v) (Eq (7.22)) as the unknown functions of this equation

satisfy-system

Eliminating sˆ and its derivatives from the equation system Eqs (7.29), (7.32),

and (7.41) leads to a single, second order partial differential equation of the MongeAmpere type, which can be found, for instance, in Schruben (1972) In this case

the unknown is the function r(u, v).

The previous development allows us to introduce easily the concept of dual opticalsurfaces (Miñano and Benítez, 2002) As seen in the previous section, the mathe-matical problem can be summarized in

(7.36)

assume that this equation system is solved—that we know the function sˆ(u, v)

sat-isfying Eq (7.36) with the contour conditions The calculation of the optical surface

can be done with Eq (7.28)—that is, by integration of r(u, v).

Note that the system of Eq (7.36) is the same that we would have if

(a) rˆ is the output unit vector.

(b) sˆ is a unit vector departing from the source.

(c) I(rˆ) is the required intensity distribution and E(sˆ) is the source intensity

s r

n r s

s s

If one of the systems is known, the other can be easily calculated with Eq (7.39)

One of the systems produces a pattern E(sˆ) when the point source radiates as given by I(rˆ), and the other (dual) system produces the pattern I(rˆ) when the source

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of edge rays that contain this subset In some special cases (linear symmetry withconstant refractive index distribution, for instance) the invariant imposed by thesymmetry allows one to calculate the trajectories of the rays from the trajectories

of their projections (on a plane normal to the axis of symmetry, in the linear case)

In this way, a full set of edge rays can be derived from the 2D subset of edge rays

In general, ray tracing is necessary to calculate the bundle of rays ted by the 3D concentrator This ray tracing will be the final step of the 3D design

transmit-if its result is sufficiently satisfactory

The reflectors joining entry and exit apertures’ edges are an essential part

of all concentrators designed with the flow-line design method Sometimes thesereflectors are inconvenient For instance, in optoelectronic applications, the exitaperture is the semiconductor surface A reflector close to this surface complicatesthe routing of the electrical contact In solar thermal applications the reflector may

be a source of thermal losses To avoid the reflector being close to the receiver,incorporation of cavities in the design of the reflector has been proposed This solu-tion allows a sizable gap between the receiver and the reflector, with a small reduc-tion in the concentration (Winston, 1980) For nonmaximal concentration, the exitaperture does not coincide with the receiver, and thus the reflectors do not touch

it If the receiver is circular, it is possible to design a set of nonmaximal trators that together give maximal concentration with reflectors not touching thereceiver but with complex reflector structure (Chaves and Collares-Pereira, 1999)

concen-In the SMS method there are no reflectors that join entry and exit apertures Thiswill require handling the edge rays in a slightly different way, as with the flow-

line method In the latter, some of the edge rays passing through the borders ofthe entry or exit apertures are not considered in the design procedure (see Appen-dix B) In the SMS method every edge ray must be considered.

Let Sibe the entry aperture of the optical system, and let Sobe the exit aperture.These apertures may be real or virtual Assume that both the entry and exit aper-

tures are on a z = constant plane Assume also that the rays coming from a source

and impinging on the entry aperture form the input bundle and that the rays minating any point of a receiver from the exit aperture form the output bundle.These assumptions simplify the following reasoning Figure 8.1 shows a 3D con-centrator with its entry and exit apertures, a source, and a receiver In our 2D

illu-problem we will be restricted to the plane x-z (see Figure 8.2).

Let n ibe the index of refraction of the medium between the source and the

entry aperture, and let n o be the index of refraction of the medium between the

exit aperture and the receiver Typically, n i = 1 and n o = 1, or it is in the range

ª 1.4 to 1.6 Let p be the optical direction cosine of a ray with respect to the x-axis For instance, p is n itimes cos(a) (a is the angle formed between the ray and the

x-axis) when p is calculated at a point of the ray-trajectory between the source and the entry aperture, and p is n ocos(a) at the point of the trajectory where the cal-

culations are done between the exit aperture and the receiver Let r be the optical direction cosine with respect to the z-axis Then p2+ r2= n2, n being the index of refraction of the point where p and r are calculated.

In 2D geometry, every ray reaching the entry aperture Sican be characterized

by two parameters These parameters can be, for instance, the coordinate x of the point of interception of the ray with the entry aperture and the coordinate p of

the ray direction at this point of interception Similarly, the rays issuing from theexit aperture Socan be characterized by another pair of parameters—for instance,

the coordinate x of the point of interception of the ray with the receiver and the optical direction cosine p of the ray at this point of interception A region of the phase space x-p represents the set of rays linking the source with the entry aper- ture We call this set of rays the input bundle, M i Similarly, the output bundle M o

is a region of the phase space x-p whose points represent the rays linking S owiththe receiver

The purpose of this section is to design an optical system such that the rays

of M i leave the system as rays of M o and the rays of M o, if reversed, leave the

system as rays of M i If this is the case, then the rays of M i and M oare the same

(M i = M o ), the only difference being that M iis the representation at Si and M o isthe representation at So We consider as a particular case when M o includes allpossible rays reaching the receiver This case is called maximal concentration

The requirement for the optical system is that M i = M o, regardless of the

par-ticular transformation of each one of the rays, just that M i and M o represent thesame bundle of rays at two different surfaces (at Siand at So) In general, an actual

optical system does not achieve this condition perfectly The bundle of rays M cnecting the source with the receiver through the optical system does not coincide

con-with M i , or with M o in the general case Obviously M c must be a subset of M i because the definition of M c includes the rays connecting source and receiver

through the optical system—that is, the rays of M cshould cross the entry

aper-ture (and also the exit aperaper-ture) Similarly, M c is a subset of M o

If M i and M o have to be the same set of rays (i.e., M i = M o), then necessarily

the étendue E of M i and M omust be the same—that is,

(8.1)

As any other design method for nonimaging concentrators, a key part of theprocedure is the edge-ray theorem (see Appendix B), which establishes that for

M i = M o it is enough that ∂M i = ∂M o , where ∂M i and ∂M oare, respectively, the sets

of edge rays of M i and M o, which are represented by the points of the borders of

the regions M i and M oin the phase space In other words, the optical system to be

designed must transform the rays of ∂M i into the rays of ∂M oand vice versa Again,

there are no requirements about which ray of ∂M ihas to be linked with a given