Nonimaging Optics Winston Episode 2 pot

In order to do this we set up a Cartesian coordinate system Oxyz in the input medium and another, O¢x¢y¢z¢, in the output 2.7 The Generalized E´tendue or Lagrange Invariant 19 Figure 2.1

from obvious geometrical considerations q ¢ cannot exceed p/2, this suggests (p/2q)2

as a theoretical upper limit to the concentration

Unfortunately, this argument is invalid because the étendue as we havedefined it is essentially a paraxial quantity Thus, it is not necessarily an invari-ant for angles as large as p/2 In fact, the effect of aberrations in the optical system

is to ensure that the paraxial étendue is not an invariant outside the paraxialregion so that we have not found the correct upper limit to the concentration.There is, as it turns out, a suitable generalization of the étendue to rays atfinite angles to the axis, and we will now explain this The concept has been knownfor some time, but it has not been used to any extent in classical optical design,

so it is not described in many texts It applies to optical systems of any or no metry and of any structure—refracting, reflecting, or with continuously varyingrefractive index

sym-Let the system be bounded by homogeneous media of refractive indices n and n¢ as in Figure 2.16, and suppose we have a ray traced exactly between the points

P and P¢ in the respective input and output media We wish to consider the effect

of small displacements of P and of small changes in direction of the ray segment through P on the emergent ray so that these changes define a beam of rays of a

certain cross section and angular extent In order to do this we set up a Cartesian

coordinate system Oxyz in the input medium and another, O¢x¢y¢z¢, in the output

2.7 The Generalized E´tendue or Lagrange Invariant 19

Figure 2.15 The étendue for a multielement optical system with an internal aperture stop.

Figure 2.16 The generalized étendue.

6 It is necessary to note that the increments dL and dM are in direction cosines, not angles Thus, in Figure 2.17 the notation on the figure should be taken to mean not that dM is the angle

indicated, but merely that it is a measure of this angle.

medium The positions of the origins of these coordinate systems and the tions of their axes are quite arbitrary with respect to each other, to the directions

direc-of the ray segments, and, direc-of course, to the optical system We specify the input ray

segment by the coordinates of P(x, y, z), and by the direction cosines of the ray (L, M, N) The output segment is similarly specified We can now represent small displacements of P by increments dx and dy to its x and y coordinates, and we can represent small changes in the direction of the ray by increments dL and dM to the direction cosines for the x and y axes Thus, we have generated a beam of area dxdy and angular extent defined by dLdM This is indicated in Figure 2.17 for the

y section.6

Corresponding increments dx¢, dy¢, dL¢, and dM¢ will occur in the output

ray position and direction

Then the invariant quantity turns out to be n2dx dy dL dM—that is, we have

(2.8)The proof of this theorem depends on other concepts in geometrical optics that

we do not need in this book We have therefore given proof in Appendix A, wherereferences to other proofs of it can also be found

The physical meaning of Eq (2.8) is that it gives the changes in the rays of abeam of a certain size and angular extent as it passes through the system If thereare apertures in the input medium that produce this limited étendue, and if thereare no apertures elsewhere to cut off the beam, then the accepted light poweremerges in the output medium so that the étendue as defined is a correct measure

of the power transmitted along the beam It may seem at first remarkable thatthe choice of origin and direction of the coordinate systems is quite arbitrary.However, it is not very difficult to show that the generalized étendue or Lagrangeinvariant as calculated in one medium is independent of coordinate translationsand rotations This, of course, must be so if it is to be a meaningful physical quantity

The generalized étendue is sometimes written in terms of the optical direction

cosines p = nL, q = nM, when it takes the form

An étendue value is associated to any 4-parameter bundle of rays Each tion of the four parameters defines one single ray In the example of Figure 2.16,

combina-the four parameters are x, y, L, M (or x¢, y¢, L¢, M¢), but combina-there are many ocombina-ther

pos-sible sets of 4 parameters describing the same bundle For the cases in which the

rays are not described at a z = constant (or z¢ = constant planes), then the

follow-ing generalized expression can be used to calculate the differential of étendue of

the bundle dE:

(2.10)The total étendue is obtained by integration of all the rays of the bundles In what

follows we will assume that the bundle can be described at a z = constant plane.

In 2D geometry, when we only consider the rays contained in a plane, we canalso define an étendue for any 2-parameter bundle of rays If the plane in which

all the rays are contained is a x = constant plane, then the differential of étendue can be written as dE = n dy dM As in the 3D case, the étendue is an invariant of

the bundle, and the same result is obtained no matter where it is calculated For

instance, it can be calculated at z¢ = constant, and the result should be the same: n¢ dy¢ dM¢ = n dy dM, or, in terms of the optical direction cosines, dy¢ dq¢ = dy dq.

We can now use the étendue invariant to calculate the theoretical maximumconcentration ratios of concentrators Consider first a 2D design, as in Figure 2.18

We have for any ray bundle that transverses the system

(2.11)and integrating over y and M we obtain

(2.12)

so that the concentration ratio is

(2.13)

In this result a¢ is a dimension of the exit aperture large enough to permit

any ray that reaches it to pass, and q ¢ is the largest angle of all the emergent

a a

n n

¢=

¢sin ¢sin

qq

4nasinq =4n a¢ ¢sinq¢

n dy dM= ¢n dy dM¢ ¢

dE=dx dy dp dq+dy dz dq dr+dz dx dr dp

2.7 The Generalized E´tendue or Lagrange Invariant 21

Figure 2.18 The theoretical maximum concentration ratio for a 2D optical system.

rays Clearly q ¢ cannot exceed p/2, so the theoretical maximum concentration ratio is

(2.14)

Similarly, for the 3D case we can show that for an axisymmetric concentrator thetheoretical maximum is

(2.15)

where again q is the input semiangle

The results in Eqs (2.14) and (2.15) are maximum values, which may or maynot be attained We find in practice that if the exit aperture has the diameter given

by Eq (2.15), some of the rays within the incident collecting angle and aperture

do not pass it We sometimes also find in a number of the systems to be describedthat some of the incident rays are actually turned back by internal reflections andnever reach the exit aperture In addition, there are losses due to absorption,imperfect reflectivity, and so forth, but these do not represent fundamental limi-tations Thus, Eqs (2.14) and (2.15) give theoretical upper bounds on performance

of concentrators

Our results so far apply to linear concentrators [Eq (2.14)] with rectangularentrance and exit apertures and to rotational concentrators with circular entranceand exit apertures [Eq (2.15)] We ought, for completeness, to discuss briefly whathappens if the entrance aperture is not circular but the concentrator itself stillhas an axis of symmetry The difficulty with this case is that it depends on thedetails of the internal optics of the concentrator It may happen that the internaloptical system forms an image of the entrance aperture on the exit aperture—inwhich case it would be correct to make them similar in shape For an entry aper-ture of arbitrary shape but uniform entry angle ±qiall that can be said in general

is that for an ideal concentrator the area of the exit aperture must equal that ofthe entry aperture multiplied by sin2

qi We will see in Chapter 6 that such centrators can be designed

There is an invariant associated with the path of a skew ray through an

axisym-metric optical system Let S be the shortest distance between the ray and the

axis—that is, the length of the common perpendicular—and let g be the anglebetween the ray and the axis Then the quantity

(2.16)

is an invariant through the whole system If the medium has a continuously

varying refractive index, the invariant for a ray at any coordinate z1along the axis

is obtained by treating the tangent of the ray at the z value as the ray and using the refractive index value at the point where the ray cuts the transverse plane z1.The skew-invariant formula will be proved in Appendix C

If we use the dynamical analogy described in Appendix A, then h corresponds

to the angular momentum of a particle following the ray path, and the

skew-h=nSsing

a

n n

invariant theorem corresponds to conservation of angular momentum In terms ofthe Hamilton’s equations, the skew invariant is just a first integral that derivesfrom the symmetry condition.

CONCENTRATION RATIO

We now have some different definitions of concentration ratio It is desirable toclarify them by using different names First, in Section 2.7 we established upperlimits for the concentration ratio in 2D and 3D systems, given respectively by Eqs.(2.14) and (2.15) These upper limits depend only on the input angle and the inputand output refractive indices Clearly we can call either expression the theoreti-cal maximum concentration ratio

Second, an actual system will have entry and exit apertures of dimensions 2a and 2a¢ These can be width or diameter for linear or rotational systems, respec-

tively The exit aperture may or may not transmit all rays that reach it, but in any

case the ratios (a/a¢) or (a/a¢)2define a geometrical concentration ratio

Third, given an actual system, we can trace rays through it and determine theproportion of incident rays within the collecting angle that emerge from the exitaperture This process will yield an optical concentration ratio

Finally, we could make allowances for attenuation in the concentrator byreflection losses, scattering, manufacturing errors, and absorption in calculatingthe optical concentration ratio We could call the result the optical concentrationratio with allowance for losses The optical concentration ratio will always be lessthan or equal to the theoretical maximum concentration ratio The geometricalconcentration ratio can, of course, have any value

REFERENCE

Welford, W T (1986) “Aberrations of Optical Systems.” Hilger, Bristol, England

IMAGE-FORMING CONCENTRATORS

In order to fix our ideas we use the solar energy application to describe the mode

of action of our systems The simplest hypothetical image-forming concentratorwould then function as in Figure 3.1 The rays are coded to indicate that rays fromone direction from the sun are brought to a focus at one point in the exit aper-ture—that is, the concentrator images the sun (or other source) at the exit aper-ture If the exit medium is air, then the exit angle q ¢ must be p/2 for maximumconcentration Such a concentrator may in practice be constructed with glass orsome other medium of refractive index greater than unity forming the exit surface,

as in Figure 3.2 Also, the angle q ¢ in the glass would have to be such that sin q¢

= 1/n so that the emergent rays just fill the required p/2 angle For typical

mate-rials the angle q ¢ would be about 40°

Figure 3.2 brings out an important point about the objects of such a trator We have labeled the central or principal ray of the two extreme angle beams

concen-a concen-and b, respectively, concen-and concen-at the exit end these rconcen-ays hconcen-ave been drconcen-awn normconcen-al to

the exit face This would be essential if the concentrator were to be used with air

as the final medium, since, if rays a and b were not normal to the exit face, some

of the extreme angle rays would be totally internally reflected (see Section 2.2),and thus the concentration ratio would be reduced In fact, the condition that theexit principal rays should be normal to what, in ordinary lens design, is termed

the image plane is not usually fulfilled Such an optical system, called telecentric,needs to be specially designed, and the requirement imposes constraints thatwould certainly worsen the attainable performance of a concentrator We shall

therefore assume that when a concentrator ends in glass of index n, the absorber

or other means of utilizing the light energy is placed in optical contact with theglass in such a way as to avoid potential losses through total internal reflection

An alternative configuration for an image-forming concentrator would be as

in Figure 3.3 The concentrator collects rays over qmaxas before, but the internaloptics form an image of the entrance aperture at the exit aperture, as indicated

by the arrow coding of the rays This would be in optics terminology a telescopic

or afocal system Naturally, the same considerations about using glass or a similarmaterial as the final medium holds as for the system of Figures 3.1 and 3.2, andthere is no difference between the systems as far as external behavior is concerned

If the concentrator terminates in a medium of refractive index n, we can gain

in maximum concentration ratio by a factor n or n2, depending on whether it is a2D or 3D system, as can be seen from Eqs (2.13) and (2.14) This corresponds tohaving an extreme angle q ¢ = p/2 in this medium We then have to reinstate therequirement that the principal rays be normal to the exit aperture, and we alsohave to ensure that the absorber can utilize rays of such extreme angles

In practice there are problems in using extreme collection angles approaching

q ¢ = p/2 whether in air or a higher-index medium There has to be very good

match-Figure 3.1 An image-forming concentrator An image of the source at infinity is formed at the exit aperture of the concentrator.

Figure 3.2 In an image-forming concentrator of maximum theoretical concentration ratio

the final medium in the concentrator would have to have a refractive index n greater than unity The angle q ¢ in this medium would be arcsin (1/n), giving an angle p/2 in the air

outside.

ing at the interface between glass and absorber to avoid large reflection losses ofgrazing-incidence rays, and irregularities of the interface can cause losses throughshadowing Therefore, we may well be content with values of q ¢ of, say, 60° Thisrepresents only a small decrease from the theoretical maximum concentration, ascan be seen from Eqs (2.14) and (2.15).

Thus, in speaking of ideal concentrators we can also regard as ideal a systemthat brings all incident rays within qmaxout within q ¢maxand inside an exit aper-

ture a¢ given by Eq (2.12)—that is, a¢ = na sin qmax/n¢ sin q¢max Such a concentratorwill be ideal, but it will not have the theoretical maximum concentration.The concentrators sketched in Figures 3.1 and 3.2 clearly must contain some-

thing like a photographic objective with very large aperture (small f-number), or

perhaps a high-power microscope objective used in reverse The speed of a

photo-graphic objective is indicated by its f-number or aperture ratio Thus, an f/4

objec-tive has a focal length four times the diameter of its entrance aperture Thisdescription is not suitable for imaging systems in which the rays form large anglesapproaching p/2 with the optical axis for a variety of reasons It is found that indiscussing the resolving power of such systems the most useful measure of per-

formance is the numerical aperture or NA, a concept introduced by Ernst Abbe in

connection with the resolving power of microscopes Figure 3.4 shows an optical

system with entrance aperture of diameter 2a It forms an image of the axial object

3.2 Some General Properties of Ideal Image-Forming Concentrators 27

Figure 3.3 An alternative configuration of an image-forming concentrator The rays lected from an angle ±q form an image of the entrance aperture at the exit aperture.

col-Figure 3.4 The definition of the numerical aperture of an image forming system The NA

is n¢ sin a¢.

point at infinity and the semiangle of the cone of extreme rays is a ¢max Then thenumerical aperture is defined by

(3.1)

where n¢ is the refractive index of the medium in the image space We assume that

all the rays from the axial object point focus sharply at the image point—that is,there is (to use the terminology of Section 2.4) no spherical aberration Then Abbeshowed that off-axis object points will also be sharply imaged if the condition

(3.2)

is fulfilled for all the axial rays In this equation h is the distance from the axis of

the incoming ray, and a ¢ is the angle at which that ray meets the axis in the finalmedium Equation (3.2) is a form of the celebrated Abbe sine condition for goodimage formation It does not ensure perfect image formation for all off-axis objectpoints, but it ensures that aberrations that grow linearly with the off-axis angleare zero These aberrations are various kinds of coma The condition of freedomfrom spherical aberration and coma is called aplanatism

Clearly, a necessary condition for our image-forming concentrator to have thetheoretical maximum concentration—or even for it to be ideal as an image-formingsystem (but without theoretical maximum concentration)—is that the image for-mation should be aplanatic This is not, unfortunately, a sufficient condition.The constant in Eq (3.2) has the significance of a focal length The definition

of focal length for optical systems with media of different refractive indices in theobject and image spaces is more complicated than for the thin lenses discussed inChapter 2 In fact, it is necessary to define two focal lengths, one for the inputspace and one for the output space, where their magnitudes are in the ratio of therefractive indices of the two media In Eq (3.2) it turns out that the constant is

the input side focal length, which we shall denote by f.

From Eq (3.2) we have for the input semiaperture

(3.3)and also, from Eq (2.13),

(3.4)

By substituting from Eq (3.3) into Eq 3.4 we have

(3.5)where qmaxis the input semiangle To see the significance of this result we recallthat we showed that in an aplanatic system the focal length is a constant, inde-

pendent of the distance h of the ray from the axis used to define it Here we are

using the generalized sense of “focal length” meaning the constant in Eq (3.2),and aplanatism thus means that rays through all parts of the aperture of thesystem form images with the same magnification Thus, Eq (3.5) tells us that in

an imaging concentrator with maximum theoretical concentration the diameter ofthe exit aperture is proportional to the sine of the input angle This is true even

if the concentrator has a numerical exit aperture less than the theoretical

maximum, n¢, provided it is ideal in the sense just defined.

From the point of view of conventional lens optics, the result of Eq (3.3) iswell known It is simply another way of saying that the aplanatic lens with largest

aperture and with air as the exit medium is f/0.5, since Eq (3.5) tells us that a =

f The importance of Eq (3.5) is that it tells us something about one of the

shape-imaging aberrations required of the system—namely distortion A distortion-freelens imaging onto a flat field must obviously have an image height proportional totan q, so our concentrator lens system is required to have what is usually calledbarrel distortion This is illustrated in Figure 3.5

Our picture of an imaging concentrator is gradually taking shape, and we canbegin to see that certain requirements of conventional imaging can be relaxed.Thus, if we can get a sharp image at the edge of the exit aperture and if the diam-eter of the exit aperture fulfils the requirement of Eqs 3.3–3.5, we do not needperfect image formation for object points at angles smaller than qmax For example,the image field perhaps could be curved, provided we take the exit aperture in theplane of the circle of image points for the direction q , as in Figure 3.6 Also, the

3.2 Some General Properties of Ideal Image-Forming Concentrators 29

Figure 3.5 Distortion in image-forming systems The optical systems are assumed to have symmetry about an axis of rotation.

inner parts of the field could have point-imaging aberrations, provided these were

not so large as to spill rays outside the circle of radius a¢ Thus, we see that an

image-forming concentrator need not, in principle, be so difficult to design as animaging lens, since the aberrations need to be corrected only at the edge of thefield In practice this relaxation may not be very helpful because the outer part ofthe field is the most difficult to correct However, this leads us to a valuable prin-ciple for nonimaging concentrators Not only is it unnecessary to have good aber-ration correction except at the exit rim, but we do not even need point imaging atthe rim itself It is only necessary that rays entering at the extreme angle qmaxshould leave from some point at the rim and that the aberrations inside shouldnot be such as to push rays outside the rim of the exit aperture We shall return

to this edge-ray principle later in connection with nonimaging concentrators.The above arguments need only a little modification to apply to the alterna-tive configuration of imaging concentrator in Figure 3.3, in which the entranceaperture is imaged at the exit aperture Referring to Figure 3.7, we can imaginethat the optical components of the concentrator are forming an image at the exitaperture of an object at a considerable distance, rather than at infinity, and that

Figure 3.6 A curved image field with a plane exit aperture.

Figure 3.7 An afocal concentrator shown as two image-forming systems.

this object is the entrance aperture Alternatively, we can imagine that part of the

concentrator is a collimating lens of focal length f, shown in broken line in the

figure, and that this projects the entrance aperture to infinity with an angle

sub-tending 2a/f The same considerations as before then apply to the aberration

concentra-High-aperture camera lenses are made at about f/1.0, but these are complex

structures with many components Figure 3.8 shows a typical example with a focallength of 50 mm Such a system is by no means aberration-free, and the cost ofscaling it up to a size useful for solar work would be prohibitive Anyway, itsnumerical aperture is still only about 0.5 The only systems with numerical aper-tures approaching the theoretical limit are microscope objectives Figure 3.9 showsone of the simplest designs of microscope objective of numerical aperture about1.35, drawn in reverse and with one conjugate at infinity The image or exit spacehas a refractive index of 1.52, since it is an oil immersion objective Such systemshave good aberration correction only to about 3° from the axis Beyond this theaberrations increase rapidly, and also there is less light transmission because ofvignetting.1The collecting aperture would be about 4 mm in diameter Again, itwould be impracticable to scale up such a system to useful dimensions

Thus, a quick glance at the state of the art in conventional lens design gests that imaging concentrators in the form of lens systems will not be very effi-cient on a practical scale Nevertheless, it is interesting to see what might be donewith the classical imaging design techniques if practical limitations are ignored

sug-3.3 Can An Ideal Image-Forming Concentrator Be Designed? 31

Figure 3.8 A high-aperture camera objective The drawing is to scale for a 50-mm focal length The emerging cone of rays has a semiangle of 26° at the center of the field of view.

1 Vignetting is caused by rims of components at either end of a long system shearing against each other as the system is turned off-axis.

Roughly, the position seems to be that we cannot design an ideal tor—one with the theoretical maximum collection efficiency, using a finite number

concentra-of lens elements But, by increasing the number concentra-of elements sufficiently or by tulating sufficiently extreme optical properties, we can approach indefinitely close

pos-to the ideal Exceptions pos-to the preceding proposed rule occur in optical systemswith spherical symmetry It has been known since the time of Huygens that aspherical lens element images a concentric surface, as shown in Figure 3.10

The two conjugate surfaces have radii r/n and nr, respectively The

configura-tion is used in microscope objectives having a high numerical aperture, as inFigure 3.9 Unfortunately, one of the conjugates must always be virtual (the objectconjugate as the figure is drawn), so the system alone would not be very practical

as a concentrator It seems to be true, although this has not been proven, that nocombination of a finite number of concentric components can form an aberration-free real image of a real object However, as we shall see, this can be done withmedia of continuously varying refractive index The system of Figure 3.10 wouldclearly be useful as the last stage of an imaging concentrator It can easily beshown that the convergence angles are related by the equation

Figure 3.9 An oil-immersion microscope objective of high numerical aperture Such systems can have a convergence angle of up to 60° with an aberration-free field of about ±3° However, they can only be designed aberration free for focal lengths up to 2 mm—that is,

an actual field diameter of about 200 mm.

Figure 3.10 The aplanatic surfaces of a spherical refracting surface.

(3.6)Also, if there is a plane surface terminating in air, the final emergent angle a ≤ isgiven by

(3.7)Thus, the system could be used in conjunction with another system of relativelylow numerical aperture, as in Figure 3.11, to form a fairly well-corrected concen-trator This is, of course, merely a reinvention of the microscope objective of Figure

3.9, and the postulated additional system still needs to operate at about f/1 if

ordi-nary materials are used If we assume some extreme material qualities—say, arefractive index of 4 with adequate antireflection coating for the aplanatic com-

ponent—then the auxiliary system only needs to be f/8 to give p /2 emergent angles

3.3 Can An Ideal Image-Forming Concentrator Be Designed? 33

Figure 3.11 An image-forming concentrator with an aplantic component.

Figure 3.12 Use of an aplanatic component of high refractive index to produce a corrected optical system.