Jiakun Liu and Xu-Jia Wang

1 School of Mathematicsand Applied Statistics,University of Wollongong,Wollongong,NSW 2522,Australia;

2 Centrefor Mathematicsand Its Applications,Australian National University,Canberra,ACT 0200,Australia.

Abstract.In this paper,w e show that the near f ield light refraction can be formulated as a nonlinear optimisation problem,both for a point light source and for a parallel light source.

Key words:Refraction,nonlinear optimisation,Monge-Amp`ere equation.

1 Introduction

Geometric optics such as light ref lection and refraction problems have attracted much attention in recent years,due to various practical applications.For instance,an exciting connection with theresearch of artif icial intelligencehasbeen discovered in[18].For light ref lection problems,in[24]it was showed that the far f ield case is an optimal transportation problem,and thus a linear optimisation problem.Later on,in[19]it was showed that the near f ield case can be formulated into a nonlinear optimisation problem.The various regularity results for ref lection problems have been obtained by many people.We refer the readers to[1,5,14,21,23]and references therein.In particular,the work of[23]inspired the discovery of Ma-Trudinger-Wang condition in optimal transportation[22],see also[2,3]for some recent results on associated Monge-Amp`ere equations.The regularity theory was then extended to more general Monge-Amp`eretypeequations and generated Jacobian equations as well,see,e.g.,[6,11,12].Similar to light ref lection,light refraction problems are also divided into the far f ield case and the near f ield case depending on the location of thetarget.Thefar f ield refraction problem has been studied by Guti′errez-Huang[7]using mass transport,and Karakhanyan[13]for the regularity of weak solutions.

In this paper,we shall focus on the near f ield case and formulate it into a class of nonlinear optimisation problems.Depending on the type of light source,the refraction system could be of point light source or parallel light source.The near f ield refraction problem with a point source can be described as follows:Suppose the light emits from the originOsurrounded by mediumIwith positive intensityfon??Sn.There is a surface R,separates two homogeneous and isotropic mediaIandII,such that all rays refracted by R into mediumIIilluminate a target hypersurface??in Rn+1with positive intensitygon??.Letn1,n2be the indices of refraction of mediaI,II,respectively,andκ=n2/n1.Whenκ<1,the refracted raystend to bent away from the normal,while whenκ>1,the refracted rays tend to bent towards the normal.

In the case ofκ<1,let(u,v)∈C(?)×C(??)and def ine the constraint functionφ:?×??×R×R→R by

where

Denote the restriction setKby

Assume that the intensity functionsf,gsatisfy the energy conservation condition

andf,gare bounded from below and above,namely for some positive constantc0,

Denote the restriction set T by

Note thatS∈T satisf ies a stronger condition than that in optimal transportation,since a measurereserving mapping may not beinvertible in general.However,our model arises from geometric optics,where the light ref lection and refraction processes are invertible,namely if a light beam from a given sourcexpasses a target pointythrough any number of mirrors(or surfaces between different media),then a light source at positionywill producea light beam that passes through positionx.

Consider thenonlinear optimisation problem of maximising the functional

among all pairs(u,v)∈Kand the mappingT∈T,def ined in(1.3)and(1.8),respectively.Denote

such thatφ(x,y,u,v)=u+?(x,y,v).Note that for anyT∈T and(u,v)∈K,one has

and thus

where the last inequality is due to(u,v)∈K.Hence,supK,TI≤0.

Our main result is the following:

Theorem 1.1.Assumethatκ<1,f,g satisfy(1.4)–(1.5)and(R1)holds.Then thereexistsadual maximising pair(u,v)∈K and a mapping T∈Tsuch that0=I(u,v,T)=supK,TI.Moreover,ρ=eu isasolution of therefractor problem with theinterfacesurfaceRρgiven by theradial graph of functionρ,namelyRρ={xρ(x):x∈?}.

We remark that the constraint function(1.1)follows from the geometry of special interfacesurface S between mediaIandII,called Cartesian oval[8],that refracts all rays emitting from the originOinto a pointy.In the polar coordinates,represent

S={xρo(x):x∈Sn}.

In the caseκ<1,by the Snell law of refraction one has

wherepisthefocal parameter,and?(x,y,v)isdef ined in(1.2).For non-degenerate Cartesian ovals,there are some physical constraints for refraction

We refer the readers to[8]for more physical interpretations and detailed calculations.

In the case ofκ>1,the physical constraint becomes|y|

with

Let(u,v)∈C(?)×C(??)and def ine the constraint functionφ:?×??×R×R→R by

Note that the physical constraint implies that|y|

Consider the nonlinear optimisation problems of maximising the functional among all pairs(u,v)∈Kand the mappingT∈T,def ined in(1.3)and(1.8),respectively.Analogously to Theorem 1.1 we have the following existenceresult.Noting that the negative sign in the solutionρ=e?u,which is due to the nature of supporting oval satisfying that atx∈?,ρ(x)=ρo(x,y,p)for somey∈??and parameterp∈R,whileρ(x′)≥ρo(x′,y,p),?x′∈?.(In the case ofκ<1,the supporting oval satisf iesρ(x′)≤ρo(x′,y,p),?x′∈?.)

Corollary 1.1.Assumethatκ>1,f,g satisfy(1.4)–(1.5)and(R?1)holds.Then thereexistsadual maximising pair(u,v)∈K and a mapping T∈Tsuch that0=I(u,v,T)=supK,TI.Moreover,ρ=e?u is a solution of the refractor problem with the interface surfaceRρgiven by the radial graph of functionρ,namelyRρ={xρ(x):x∈?}.

Another model of light refraction is of parallel source,which can be described as follows:Suppose that a parallel light emits from??Rn×{0}alongen+1=(0,···,0,1)with positive intensityf∈L1(?),??is a hypersurface in Rn+1,which is referred to as the target domain.Suppose that?and??are surrounded by two homogeneous and isotropic mediaIandII,respectively.One seeks an optical surface R interface between mediaIandII,such that all rays refracted by R into mediumIIare received at the surface??with the prescribed intensityg∈L1(??).Assume the intensitiesf,gsatisfy the energy conservation condition(1.4)and thebounded condition(1.5).

Letn1,n2be the indices of refraction of mediaI,II,respectively,andκ=n1/n2.We assume that mediaIIis denser than mediaI,that is,κ<1.The case whenκ>1 can be treated in a similar way but the geometry of surface changes[7,9].For simplicity,we assume that???{yn+1=h}for a constanth>0,and denoteY=(y,h)for points on??andX=(x,0)for points on?.

From the geometry of special surface“inverse”ellipsoid of revolutionwith the uniform refracting property,namely all raysfrom theparallel source?alongen+1will berefracted to a focal pointy,we def ine the constraint function

Similarly as in(1.3)and(1.8)onehas the constraint setsKand T,respectively.Def inethe functional

and the corresponding optimisation problem is to maximise the functionalIwithin the constraint setsKand T.

Theorem 1.2.Assume that f,g satisfy(1.4)–(1.5)and(A1)holds.Then there exists a dual maximising pair(u,v)∈K and amapping T∈Tsuch that0=I(u,v,T)=supK,TI.Moreover,u is asolution of therefractor problem with theinterfacesurfaceRu given by thegraph of function u,namelyRu={(x,u(x)):x∈?}.

Note that the solutions need to be understood as generalised solutions,as in[16,17]for ref lection problems.The existence result was previously obtained by Guti′errez-Tournier in[9],where they f irst considered the discrete case when the target is a set of points,then used an approximation to obtain the existence in the general case.Under suitable conditions on domains?,??and intensitiesf,g,local smoothness ofuwas obtained by Guti′errez-Tournier[10]and Karakhanyan[15].

This paper is arranged as follows.In Section 2,we introduce a class of nonlinear optimisation problems,and give a general existenceresult for maximisers.Then in Sections 3.1 and 3.2,we discuss the refractor problems with a point source and with a parallel source,and prove Theorems 1.1 and 1.2,respectively.

2 Nonlinear optimisation

In this section,we introduce a class of nonlinear optimisation problems,which enclose theaboverefraction models.Let?,??betwo bounded domains in Rn(Notethat thefollowing argument works for general manifolds.For example,when applying to Theorem 1.1,?is a subset of Sn.).Assumethe constraint function

φ=φ(x,y,t,s):?×??×R×R→R

isC1and strictly increasing int,s.By the implicit function theorem there is aC1function?=?(x,y,s)strictly increasing inssuch that the constraintφ≤0 can be written as

We assume further that there exists a constantθ0>0 such that

and?satisf ies the condition.

(H1)For eachx0∈?,for any(p,t)∈Rn×R,there is at most one pair(y,s)∈Rn×R such that

(?x,?)(x0,y,s)=?(p,t),

namely,?x(x0,y,s)=?pand?(x0,y,s)=?t.And,for eachy0∈??,for any(q,s)∈Rn×R,thereis at most one pair(x,t)∈Rn×R such that

namely,(?y/?s)(x,y0,s)=?qand?(x,y0,s)=?t.

By straight computation,onecan verify that theconstraint functions(1.1)in thepoint light source and(1.15)in the parallel light source both satisfy the above conditions(2.2)and(H1)under the corresponding hypothesesof Theorems 1.1 and 1.2,respectively.

In general,we consider the nonlinear optimisation problem of maximising the functional

among all pairs(u,v)∈Kand the mappingT∈T,def ined in(1.3)and(1.8),respectively,where the functionsf,gsatisfy the balance condition(1.4)and the bounded condition(1.5).

Def inition 2.1.Wesay apair(u,v)∈K isadual pairwith respect toφif it satisf ies

Lemma 2.1.Assumethat theconstraint function?satisf ies(2.2)and(H1).For each dual pair(u,v)∈K,thereexists an associated mapping T(u,v):?→??that solves theequation

for any otherx′∈?.Note that since?isC1smooth,one can seeuis locally Lipschitz,and thus differentiable almost everywhere.Letx∈?be a differentiable point ofu,by differentiation we have

Therefore,for the f ixedx∈?,settingt=u(x)andp=Du(x)one can see that

?x(x,y,v)=?p, and?(x,y,v)=?t.

From the condition(H1),we then obtain the mappingy=T(u,v)(x)solving the equation(2.5).Sinceuis differentiable almost everywhere on?,the mappingT(u,v)is uniquely determined almost everywhere on?.

where the sequence?j→0 asj→∞.SinceTjis measure preserving,one has

From(2.9)and(1.5),the left hand side converges to

While by the construction of{hj},the right hand sideconverges to

To showTis measure preserving,for anyh∈C(??)one can see

for eachj,sinceTjis measure preserving.Taking thelimit by lettingj→∞,from(2.9)we then obtain

which implies thatTis measure preserving.One can similarly show thatT?1is measure preserving as well.Therefore,the mappingT∈T and the proof is done.

Lemma 2.3.Under thehypotheses of Theorem2.1,thereexists a dual maximiser(u,v)∈K and T∈Tsuch that I(u,v,T)=supK,TI.

Proof.Given any pair(u,v)∈K,we claim thatI(u,v,T)does not decrease ifvis replaced by

In fact,by the continuity ofφandu,for eachy∈??there is somex∈?such that

φ(x,y,u(x),v?(y))=0≥φ(x,y,u(x),v(y)),

where the last inequality follows from the assumptionφ(x,y,u,v)≤0 in?×??.By(2.1)–(2.2),v?≥v.Furthermore,φ(x,y,u(x),v?(y))≤0 for all(x,y)∈?×??.

Sincev?≥v,by(2.2)we have

I(u,v?,T)≥I(u,v,T).

Similarly,if we def ine

thenφ(x,y,u?(x),v?(y))≤0 in?×??,and

I(u?,v?,T)≥I(u,v?,T)≥I(u,v,T).

Thus we do not decreaseI(u,v,T)by replacing(u,v)by(u?,v?).The claim is proved.

C0+?(x,y,s)≤u(x)+?(x,y,s)≤0, for allx∈?.

Then by(2.2)again,thereexists a constantC1,such thats≤C1.This implies that

φ(x1,y1,u?(x1),v?(y1))=0,φ(x2,y2,u?(x2),v?(y2))=0.

Then by(2.1),we have

0=φ(x2,y2,u?(x2),v?(y2))?φ(x1,y2,u?(x1),v?(y2))

+φ(x1,y2,u?(x1),v?(y2))?φ(x1,y1,u?(x1),v?(y1))

=u?(x2)?u?(x1)??x(?x,y2,v?(y2))·(x2?x1)+φ(x1,y2,u?(x1),v?(y2)),

u?(x2)?u?(x1)≥?C2|x2?x1|,

where the constantC2=sup(|?x|+|?y|).

On the other hand,replacingφ(x1,y2,u?(x1),v?(y2))byφ(x2,y1,u?(x2),v?(y1))in the above calculation,we have

u?(x2)?u?(x1)≤C2|x2?x1|.

Therefore,the Lipschitz constant ofu?on?is controlled by

By switchingxandyin theabove argument,wecan obtain the Lipschitz continuity ofv?on??,

Last,lettingIT(u,v)=I(u,v,T),we show that whenC0<0 suff iciently small,

Def ine

Since the constraint function?isC1smooth insand by(2.1)–(2.2),except a setE??and a setE′???of measure|E|=|E′|=O(Nεn),

whereδ:=mini/=j{dist(xi,xj)}.Therefore,by(1.4)and the mean value theorem we have

Lemma 2.4.Let(u,v)∈K and T∈Tbe the maximiser obtained in Lemma2.3such that I(u,v,T)=supK,TI.Then,one has0=I(u,v,T)and T=T(u,v)is the associated mapping to thedual pair(u,v),given in Lemma2.1.

Proof.Leth∈C(??)and|?|<1 suff iciently small.Def ine

Then(u?,v?)∈Kand(u0,v0)=(u,v).

Since(u,v)satisf ies(2.4),by Lemma 2.1 for everyx∈?thesupremum(2.4)isattained at pointy0=T(u,v)x.We claim that at thesepoints we have

To prove(2.16),f irst we show thatLHS≤RHS.

0=u(x)+?(x,y0,v(y0))

=u(x)+?(x,y0,v?(y0))+?(x,y0,v(y0))??(x,y0,v?(y0))

≤u(x)?u?(x)+?s(x,y0,ˉv)(v(y0)?v?(y0))

≤u(x)?u?(x)???s(x,y0,ˉv)h(y0),

u?(x)?u(x)≤???s(x,T(u,v)x,v(T(u,v)x))h(T(u,v)x)+o(?).

0≥u(x)+?(x,y?,v(y?))

=u(x)+?(x,y?,v?(y?))+?(x,y?,v(y?))??(x,y?,v?(y?))

=u(x)?u?(x)+φs(x,y?,?v)(v(y?)?v?(y?)),

u?(x)?u(x)≥???s(x,T(u,v)x,v(T(u,v)x))h(T(u,v)x)+o(?).

This implies thatLHS≥RHS,and(2.16)follows.

Next,letI(?):=I(u?,v?,T),whereT∈T is the maximiser obtained in Lemma 2.3.Since the functionI(?)achieves maximum at?=0,we obtain

SinceT∈T,T?1is measure preserving,one has

Hence,for anyh∈C(??),we obtain

which implies thatT=T(u,v).Therefore,from(2.5)

The proof is f inished.

Corollary 2.1.Let(u,v)∈K be a maximiser pair of I,the associated mapping T(u,v)∈Tis measurepreserving.

In the following context,we shall simply writeT(u,v)asTwithout causing any confusion.As a consequence,we have the following

Corollary 2.2.Assumethecondition(1.4)holds.If theoptimal mapping T iscontinuousdifferentiable,then

Proof.From Corollary 2.1,one hasTis measure preserving in the sense of(1.7).WhenTisC1smooth,by the formula of change of coordinates,

for anyh∈C(??).Hence the Jacobian ofDTsatisf ies(2.17).

3 Formulation of ref lector problems

3.1 Ref lector with point source

In order to formulate the near f ield refraction problem with a point source to an optimisation problem,we need the notion ofCartesian oval,which has a special refraction property:all rays emitting from the originOare always refracted into a focus pointY.

In the polar coordinate system,such a Cartesian oval can be represented by

S={Xρo(X):X∈Sn}.

In the caseκ<1,by the Snell law of refraction one has

wherepis the focal parameter,and?(t):=(p?κ2t)2?(1?κ2)(p2?κ2|Y|2)fort∈R.For non-degenerate Cartesian ovals,there are some physical constraints for refraction

Note that such a Cartesian oval is uniquely determined byYandp.We refer the readers to[8]for more physical interpretationsand detailed calculations.

If we regardp=p(Y)as a focal function on??,we then have a family of Cartesian ovals.Represent the refraction surface R in polar coordinatesystem as

Rρ={Xρ(X):X∈?},

whereρis a positive function.Recall that[8],R is a near f ield refractor if at each pointXρ(X)∈R there exists a supporting Cartesian oval,i.e.,for someY∈??,ρ(X′)≤ρo(X′,Y,p(Y))for allX′∈?with equality holds atX′=X.Therefore,the radial functionρsatisf ies

From the energy conservation,for eachY∈??there is an ovalρ0(·,Y,p(Y))supporting to Rρ.We also have

Def inition 3.1.For any X∈?,def inetheset-valued mapping Tρ:?→??by

By settingη=1/p,the pair(ρ,η)satisf ies the dual relation

Similarly to[19,24]we now formulate the refractor problem with a point light source to a nonlinear optimisation in the form of(2.3).Letu=logρandv=logη.Set the functional

whereT∈T def ined in(1.8)and(ρ,η)∈K given by

with the constraint function

Letρbe a function def ined on?.The tangential gradient ofρis def ined by

where(gij)=(gij)?1.Note that?ρ(X)∈TXSn,the tangent space,so〈?ρ,X〉=0.Let Γρ={Xρ(X):X∈?}be the graph ofρover?.By computation,the unit normal ofΓρatXρ(X)is

The length of ref lected segment isd:=|Y?Xρ(X)|.One has

Observe that at differentiable pointXofρ,there exists a unique supporting Cartesian ovalρ0(X,Y,η)atXρ(X),with fociO,Yandη=1/p(Y)withpis the focal function,in(3.1).By(3.3)and the Snell law of refraction,one hasκ|Xρ(X)?Y|=p?ρ(X)and thus

Therefore,

On the other hand,by differentiating the constraint function in(3.8)and Lemma 2.1,with(ρ,η)being a dual maximising pair of(3.7)–(3.8),one has the associated mappingT:?→??.From the uniqueness of Lemma 2.1,in order to showT=Tρ,it suff ices to show thatTρsatisf ies the equation(2.5),where the constraint functionφis given in(3.8).In other words,by substituting〈X,Y〉in(3.13)into(3.8),we need to verify that atXandY=Tρ(X)thereholds

or equivalently

where?(X,Y,η)=(η?1?κ2〈X,Y〉)2?(1?κ2)(η?2?κ2|Y|2).

By direction computation,ρin(3.15)solves a quadratic equation

By completing the square,one has

κ|Xρ(X)?Y|=η?1?ρ,

which is exactly the Snell law of refraction,and thus holds forY=Tρ(X).This implies that the equality(3.14)holds forY=Tρ(X).Then by the uniqueness of Lemma 2.1,we obtainT(X)=Tρ(X)whenXis a differentiable point ofρ.

Proof of Theorem 1.1.Letu=logρ,v=logη.In order to apply Theorem 2.1,we need f irst to verify(2.2)for the constraint functionφin(3.8)(or equivalently(1.1)).Heuristically,from(3.3)onecan seethat when thefocal functionp(=η?1)increases,theradial functionρalso increases,which implies that for a dual pairρ,ηwith respect to(3.8),ifρincreases,the dual functionηwill decrease.Sinceφ(ρ,η)≤0 andφt>0,it is also true thatφs>0.

Rigorously,it is easier to verify this by checking(1.1).Letp=e?v.From the physical constraint(R1)in Theorem 1.1,p0.

Then by the dual relation(2.4)and(3.8),one can see thatρandηsatisfy(3.6).From Theorem 2.1,one knows thatρis Lipschitz continuous.Actually,sinceρhas supporting Cartesian oval at any point of its graph,it is semi-convex and twice differentiable almost everywhere[16].Hence,by Lemma 3.1Tρ=Ta.e.,whereTρis the mapping def ined in(3.5).Moreover,from Corollary 2.1,Tis measure preserving.Therefore,sinceTρ=Ta.e.,we see thatTρis measure preserving as well,namelyρis a solution of the refractor problem.

Remark 3.1.We remark that in the limit case when|Y|→∞,the Cartesian oval(3.1)converges to an ellipsoid which has the uniform refraction property in the far f ield case[7,8].To be specif ic,whenκ<1,lettingYf=Y/|Y|andp=κ|Y|+Cfor a positive constantC,then from(3.1)one has

as|Y|→∞.Whenκ>1,if|Y|→∞,the refracting oval also converges to the semihyperboloid in the far f ield case,see[8]for more detailed computation.

Remark 3.2.In the general case as in Theorem 2.1,by a further differentiation of the constraint functionφand from Corollary 2.2 we have the following Monge-Amp`ere type equation[19,20]:

In the refraction problem whenκ<1,recalling thatu=logρand from Lemma 3.1 one can derive the same equation forρas in[8],where??is contained in a hypersurface{Y∈Rn+1:ψ(Y)=0}:

and

3.2 Ref lector with parallel source

In the case of a parallel source,we shall consider a part of the special surface called“inverse”ellipsoid of revolution,which has the refraction property that all rays parallel toen+1from??{xn+1=0}being refracted into a single pointY∈???{yn+1=h}.It is the graph of the function

ry(x)=u(x),andry(x′)≥u(x′)for all x′∈?.

In thiscase,wedenotey∈Tu(x).

Similarly as before,we can now formulate the refractor problem with parallel source to the nonlinear optimisation,where the functionalIis given in(1.16)and the constraint functionφis in(1.15),that is

Lemma 3.2.Let(u,v)bea dual maximising pair of(1.16)and(3.18),and T betheassociated optimal mapping.Then T=Tu at any differentiablepoint of u,where Tu istheref lection mapping in(3.2).

Proof.The proof is similar to that of Lemma 3.1.In this case the refraction surface Ruis the graph of functionu,namely

By computation,the unit normal of Ruis given by

Assume the ray through(x,u(x))∈Ruis refracted into the point(y,h),wherey=Tu(x)∈??.The unit refraction direction is

The Snell law of refraction says that the vectorκen+1?Yris parallel to the normal vectorγ,namely

for somet∈R.Note that this property uniquely determines the pointy∈??.

On the other hand,from the constraint function(3.18)and Lemma 2.1,for each dual maximising pair(u,v)thereisan associated mappingT:?→??.In order to showT=Tu,it suff ices to verify thaty=T(x)satisf iesthe equality(3.19).By differentiating(3.17)inxi,one has

Combining(3.19)and(3.20)we have

Hence,for(3.19)holds,it suff ices to verify

which implies(3.19)holds.Therefore,T=Tuis verif ied.

Proof of Theorem 1.2.Thisproof followsthelinesof that of Theorem 1.1.Themonotonicity condition(2.2)and condition(H1)is easier to verify for the constraint functionφin(3.18).The energy conservation condition(1.4)is assumed.Hence by Theorem 2.1 we have a dual maximising pair(u,v).From Lemma 3.2,the associated mappingT=Tua.e.,whereTuis in Def inition 3.2.Therefore,from Corollary 2.1,Tuis measure preserving,and thusuis a solution of the refractor problem.

Acknowledgments

This work is supported by the Australian Research Council Grant No.DP170100929 and DP200101084.