Theory Nonimaging optics

1 theory

1.1 edge ray principle
1.2 design methods

1.2.1 flow-line design method

1.2.1.1 variations flow-line design method


1.2.2 simultaneous multiple surface (sms) design method
1.2.3 miñano design method using poisson brackets


1.3 conservation of etendue

theory

early academic research in nonimaging optical mathematics seeking closed form solutions first published in textbook form in 1978 book. modern textbook illustrating depth , breadth of research , engineering in area published in 2004. thorough introduction field published in 2008.

special applications of nonimaging optics such fresnel lenses solar concentration or solar concentration in general have been published, although last reference o gallagher describes work developed decades ago. other publications include book chapters.

imaging optics can concentrate sunlight to, @ most, same flux found @ surface of sun. nonimaging optics have been demonstrated concentrate sunlight 84,000 times ambient intensity of sunlight, exceeding flux found @ surface of sun, , approaching theoretical (2nd law of thermodynamics) limit of heating objects temperature of sun s surface.

the simplest way design nonimaging optics called method of strings , based on edge ray principle. other more advanced methods developed starting in 1990s can better handle extended light sources edge-ray method. these developed solve design problems related solid state automobile headlamps , complex illumination systems. 1 of these advanced design methods simultaneous multiple surface design method (sms). 2d sms design method (u.s. patent 6,639,733) described in detail in aforementioned textbooks. 3d sms design method (u.s. patent 7,460,985) developed in 2003 team of optical scientists @ light prescriptions innovators.

edge ray principle

in simple terms, edge ray principle states if light rays coming edges of source redirected towards edges of receiver, ensure light rays coming inner points in source end on receiver. there no condition on image formation, goal transfer light source target.

figure edge ray principle on right illustrates principle. lens collects light source s1s2 , redirects towards receiver r1r2.

edge ray principle

the lens has 2 optical surfaces and, therefore, possible design (using sms design method) light rays coming edge s1 of source redirected towards edge r1 of receiver, indicated blue rays. symmetry, rays coming edge s2 of source redirected towards edge r2 of receiver, indicated red rays. rays coming inner point s in source redirected towards target, not concentrated onto point and, therefore, no image formed.

actually, if consider point p on top surface of lens, ray coming s1 through p redirected towards r1. ray coming s2 through p redirected towards r2. ray coming through p inner point s in source redirected towards inner point of receiver. lens guarantees light source crossing redirected towards receiver. however, no image of source formed on target. imposing condition of image formation on receiver imply using more optical surfaces, making optic more complicated, not improve light transfer between source , target (since light transferred). reason nonimaging optics simpler , more efficient imaging optics in transferring radiation source target.

design methods

nonimaging optics devices obtained using different methods. important are: flow-line or winston-welford design method, sms or miñano-benitez design method , miñano design method using poisson brackets. first (flow-line) used, although second (sms) has proven versatile, resulting in wide variety of optics. third has remained in realm of theoretical optics , has not found real world application date. optimization used.

typically optics have refractive , reflective surfaces , light travels though media of different refractive indices crosses optic. in cases quantity called optical path length (opl) may defined



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index indicates different ray sections between successive deflections (refractions or reflections), ni refractive index , di distance in each section of ray path.

constant opl

the opl constant between wavefronts. can seen refraction in figure constant opl right. shows separation c(τ) between 2 media of refractive indices n1 , n2, c(τ) described parametric equation parameter τ. shown set of rays perpendicular wavefront w1 , traveling in medium of refractive index n1. these rays refract @ c(τ) medium of refractive index n2 in directions perpendicular wavefront w2. ray ra crosses c @ point c(τa) and, therefore, ray ra identified parameter τa on c. likewise, ray rb identified parameter τb on c. ray ra has optical path length s(τa) = n1d5 + n2d6. also, ray rb has optical path length s(τb) =n1d7 + n2d8. difference in optical path length rays ra , rb given by:

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{\displaystyle s(\tau _{b})-s(\tau _{a})=\int _{a}^{b}ds=\int _{\tau _{a}}^{\tau _{b}}{\frac {ds}{d\tau }}d\tau =\int _{\tau _{a}}^{\tau _{b}}{\frac {s(\tau +d\tau )-s(\tau )}{(\tau +d\tau )-\tau }}d\tau }

in order calculate value of integral, evaluate s(τ+dτ)-s(τ), again of same figure. have s(τ) = n1d1+n2(d3+d4) , s(τ+dτ) = n1(d1+d2)+n2d4. these expressions can rewritten s(τ) = n1d1+n2dc sinθ2+n2d4 , s(τ+dτ) = n1d1+n1dc sinθ1+n2d4. law of refraction n1sinθ1=n2sinθ2 , therefore s(τ+dτ) = s(τ), leading s(τa)=s(τb). since these may arbitrary rays crossing c, may concluded optical path length between w1 , w2 same rays perpendicular incoming wavefront w1 , outgoing wavefront w2.

similar conclusions may drawn case of reflection, in case n1=n2. relationship between rays , wavefronts valid in general.

flow-line design method

the flow-line (or winston-welford) design method typically leads optics guide light confining between 2 reflective surfaces. best known of these devices cpc (compound parabolic concentrator).

these types of optics may obtained, example, applying edge ray of nonimaging optics design of mirrored optics, show in figure cec on right. composed of 2 elliptical mirrors e1 foci s1 , r1 , symmetrical e2 foci s2 , r2.

cec

mirror e1 redirects rays coming edge s1 of source towards edge r1 of receiver and, symmetry, mirror e2 redirects rays coming edge s2 of source towards edge r2 of receiver. device not form image of source s1s2 on receiver r1r2 indicated green rays coming point s in source end on receiver not focused onto image point. mirror e2 starts @ edge r1 of receiver since leaving gap between mirror , receiver allow light escape between two. also, mirror e2 ends @ ray r connecting s1 , r2 since cutting short prevent capturing light possible, extending above r shade light coming s1 , neighboring points of source. resulting device called cec (compound elliptical concentrator).

cpc

a particular case of design happens when source s1s2 becomes infinitely large , moves infinite distance. rays coming s1 become parallel rays , same coming s2 , elliptical mirrors e1 , e2 converge parabolic mirrors p1 , p2. resulting device called cpc (compound parabolic concentrator), , shown in cpc figure on left. cpcs common seen nonimaging optics. used demonstrate difference between imaging optics , nonimaging optics.

when seen cpc, incoming radiation (emitted infinite source @ infinite distance) subtends angle ±θ (total angle 2θ). called acceptance angle of cpc. reason name can appreciated in figure rays showing acceptance angle on right. incoming ray r1 @ angle θ vertical (coming edge of infinite source) redirected cpc towards edge r1 of receiver.

rays showing acceptance angle

another ray r2 @ angle α<θ vertical (coming inner point of infinite source) redirected towards inner point of receiver. however, ray r3 @ angle β>θ vertical (coming point outside infinite source) bounces around inside cpc until rejected it. therefore, light inside acceptance angle ±θ captured optic; light outside rejected.

the ellipses of cec can obtained (pins and) string method, shown in figure string method on left. string of constant length attached edge point s1 of source , edge point r1 of receiver.

string method

the string kept stretched while moving pencil , down, drawing elliptical mirror e1. can consider wavefront w1 circle centered @ s1. wavefront perpendicular rays coming out of s1 , distance s1 w1 constant points. same valid wavefront w2 centered @ r1. distance w1 w2 constant light rays reflected @ e1 , these light rays perpendicular both, incoming wavefront w1 , outgoing wavefront w2.

optical path length (opl) constant between wavefronts. when applied nonimaging optics, result extends string method optics both refractive , reflective surfaces. figure dtirc (dielectric total internal reflection concentrator) on left shows 1 such example.

dtirc

the shape of top surface s prescribed, example, circle. lateral wall m1 calculated condition of constant optical path length s=d1+n d2+n d3 d1 distance between incoming wavefront w1 , point p on top surface s, d2 distance between p , q , d3 distance between q , outgoing wavefront w2, circular , centered @ r1. lateral wall m2 symmetrical m1. acceptance angle of device 2θ.

these optics called flow-line optics , reason illustrated in figure cpc flow-lines on right. shows cpc acceptance angle 2θ, highlighting 1 of inner points p.

cpc flow-lines

the light crossing point confined cone of angular aperture 2α. line f shown tangent @ point p bisects cone of light and, therefore, points in direction of light flow @ p. several other such lines shown in figure. bisect edge rays @ each point inside cpc and, reason, tangent @ each point points in direction of flow of light. these called flow-lines , cpc combination of flow line p1 starting @ r2 , p2 starting @ r1.

variations flow-line design method

there variations flow-line design method.

a variation multichannel or stepped flow-line optics in light split several channels , recombined again single output. aplanatic (a particular case of sms) versions of these designs have been developed. main application of method in design of ultra-compact optics.

another variation confinement of light caustics. instead of light being confined 2 reflective surfaces, confined reflective surface , caustic of edge rays. provides possibility add lossless non-optical surfaces optics.

simultaneous multiple surface (sms) design method

this section describes

a nonimaging optics design method known in field simultaneous multiple surface (sms) or miñano-benitez design method. abbreviation sms comes fact enables simultaneous design of multiple optical surfaces. original idea came miñano. design method developed in 2-d miñano , later benítez. first generalization 3-d geometry came benítez. further developed contributions of miñano , benítez. other people have worked miñano , later miñano , benítez on programming method.

the design procedure

is related algorithm used schulz in design of aspheric imaging lenses.

the sms (or miñano-benitez) design method versatile , many different types of optics have been designed using it. 2d version allows design of 2 (although more possible) aspheric surfaces simultaneously. 3d version allows design of optics freeform surfaces (also called anamorphic) surfaces may not have kind of symmetry.

sms optics calculated applying constant optical path length between wavefronts. figure sms chain on right illustrates how these optics calculated. in general, rays perpendicular incoming wavefront w1 coupled outgoing wavefront w4 , rays perpendicular incoming wavefront w2 coupled outgoing wavefront w3 , these wavefronts may shape. however, sake of simplicity, figure shows particular case or circular wavefronts. example shows lens of given refractive index n designed source s1s2 , receiver r1r2.

sms chain

the rays emitted edge s1 of source focused onto edge r1 of receiver , emitted edge s2 of source focused onto edge r2 of receiver. first choose point t0 , normal on top surface of lens. can take ray r1 coming s2 , refract @ t0. choosing optical path length s22 between s2 , r2 have 1 condition allows calculate point b1 on bottom surface of lens. normal @ b1 can calculated directions of incoming , outgoing rays @ point , refractive index of lens. can repeat process taking ray r2 coming r1 , refracting @ b1. choosing optical path length s11 between r1 , s1 have 1 condition allows calculate point t1 on top surface of lens. normal @ t1 can calculated directions of incoming , outgoing rays @ point , refractive index of lens. now, refracting @ t1 ray r3 coming s2 can calculate new point b3 , corresponding normal on bottom surface using same optical path length s22 between s2 , r2. refracting @ b3 ray r4 coming r1 can calculate new point t3 , corresponding normal on top surface using same optical path length s11 between r1 , s1. process continues calculating point b5 on bottom surface using edge ray r5, , on. sequence of points t0 b1 t1 b3 t3 b5 called sms chain.

another sms chain can constructed towards right starting @ point t0. ray s1 refracted @ t0 defines point , normal b2 on bottom surface, using constant optical path length s11 between s1 , r1. ray r2 refracted @ b2 defines new point , normal t2 on top surface, using constant optical path length s22 between s2 , r2. process continues more points added sms chain. in example shown in figure, optic has left-right symmetry and, therefore, points b2 t2 b4 t4 b6 can obtained symmetry vertical axis of lens.

now have sequence of spaced points on plane. figure sms skinning on left illustrates process used fill gaps between points, defining both optical surfaces.

sms skinning

we pick 2 points, b1 , b2, corresponding normals , interpolate curve c between them. pick point b12 , normal on c. ray r1 coming r1 , refracted @ b12 defines new point t01 , normal between t0 , t1 on top surface, applying same constant optical path length s11 between s1 , r1. ray r2 coming s2 , refracted @ t01 defines new point , normal on bottom surface, applying same constant optical path length s22 between s2 , r2. process continues rays r3 , r4 building new sms chain filling gaps between points. picking other points , corresponding normals on curve c gives more points in between other sms points calculated originally.

in general, 2 sms optical surfaces not need refractive. refractive surfaces noted r (from refraction) while reflective surfaces noted x (from spanish word reflexión). total internal reflection (tir) noted i. therefore, lens 2 refractive surfaces rr optic, while configuration reflective , refractive surface xr optic. configurations more optical surfaces possible and, example, if light first refracted (r), reflected (x) reflected again tir (i), optic called rxi.

the sms 3d similar sms 2d, calculations done in 3d space. figure sms 3d chain on right illustrates algorithm of sms 3d calculation.

sms 3d chain

the first step choose incoming wavefronts w1 , w2 , outgoing wavefronts w3 , w4 , optical path length s14 between w1 , w4 , optical path length s23 between w2 , w3. in example optic lens (an rr optic) 2 refractive surfaces, refractive index must specified. 1 difference between sms 2d , sms 3d on how choose initial point t0, on chosen 3d curve a. normal chosen point t0 must perpendicular curve a. process evolves sms 2d. ray r1 coming w1 refracted @ t0 and, optical path length s14, new point b2 , normal obtained on bottom surface. ray r2 coming w3 refracted @ b2 and, optical path length s 23, new point t2 , normal obtained on top surface. ray r3 new point b2 , normal obtained, ray r4 new point t4 , normal obtained, , on. process performed in 3d space , result 3d sms chain. sms 2d, set of points , normals left of t0 can obtained using same method. now, choosing point t0 on curve process can repeated , more points obtained on top , bottom surfaces of lens.

the power of sms method lies in fact incoming , outgoing wavefronts can free-form, giving method great flexibility. also, designing optics reflective surfaces or combinations of reflective , refractive surfaces, different configurations possible.

miñano design method using poisson brackets

this design method developed miñano , based on hamiltonian optics, hamiltonian formulation of geometrical optics shares of mathematical formulation hamiltonian mechanics. allows design of optics variable refractive index, , therefore solves nonimaging problems not solvable using other methods. however, manufacturing of variable refractive index optics still not possible , method, although potentially powerful, did not yet find practical application.

conservation of etendue

conservation of etendue central concept in nonimaging optics. in concentration optics, relates acceptance angle maximum concentration possible. conservation of etendue may seen constant volume moving in phase space.