By using the paraxial approximation of geometricaloptics, Berry, in an extraordinary work [1], studied,among other things, the image of an arbitrary one-dimensional object obtained by reflection on an arbi-trary smooth surface (of rippled water). He foundthat, under certain conditions, the object and its im-age do not have the same topology. To explain thisbeautiful phenomenon, he introduced the so-calledcaustic-touching theorem, which states that changesof image topology occur when the object touches thecaustic associated with the family of imaginary lightrays emitted by the observing eye.It is worthwhile describing, with little convenient
changes, the procedure followed by Berry to obtainthe image of an arbitrary object curve by reflectionon an arbitrary smooth surface. Without loss of thegenerality, we assume that: the object curve is lyingon a plane perpendicular to the z axis, the two-dimensional reflecting smooth surface is locally gi-ven by z ¼ f ðx; yÞ, and the position of the observingeye is given by ~S ¼ ðs1; s2; s3Þ, (see Fig. 1). At first in-
stance (sight), one could think that, to obtain the im-age of the one-dimensional object, one has to takeinto account the family of light rays associated witheach of its points, that is, a family of light rays char-acterized by three parameters (one of these para-meters provides the position of an arbitrary pointon the one-dimensional object and the other two givethe direction of the light ray emitted from that point).However, the only important light rays that give con-tribution to the image formation are those that reachthe observing eye and these, in accordance with thereciprocity principle [2], can be regarded as belong-ing to the single family emitted by the eye. In otherwords, the original problem of image formation isequivalent to a new problem where the observingeye is replaced by an imaginary point light source.From this new point of view, a point on the reflectingsurface belongs to the image associated with the one-dimensional object if it can be associated, via a re-flected light ray emitted by the imaginary pointsource, with a point of the one-dimensional object,as it is graphically described in Fig. 2. Since, in gen-eral, the curvature of the reflecting surface is notequal to zero, then the reflected light rays emittedby the imaginary point light source will focus at a
region in the space. This region is the caustic asso-ciated with the imaginary reflected light rays. Froma mathematical point of view, the evolution of the re-flected light rays is described by a map between twosubsets of R3 or equivalently by a one-parameterfamily of maps between two subsets of R2 (seeSection 2). The caustic associated with the reflectedlight rays is determined by looking for the points, onthe reflecting surface, where these maps are notlocally one to one. Therefore, if the one-dimensional object is placed outside the causticregion, there will be a one-to-one correspondence be-tween its points and the points of its associated im-age. That is, the observing eye will see only oneimage. In this case, one says that the object andits image have the same topology. Remember that,roughly speaking, two curves are said to be topologi-cally equivalent if one can be transformed into thesame shape as the other without connecting or dis-connecting any points. However, if the object is lo-
cated at the caustic region, the maps are not oneto one and its position with respect to the caustic willbe crucial to describe correctly its image. The caustic-touching theorem establishes that new image loopsappear when the object curve touches the caustic as-sociated with the reflected light rays emitted by theimaginary point light source. Such disruption may beelliptic (a loop born from an isolated point) or hyper-bolic (a loop pinched off from an already existingone). In these cases, the object and its image donot have the same topology and the observer maysee several images corresponding to a single object.As remarked by Berry, it is what happens in nature’soptics, whose elements may be the reflecting surfaceof rippled water. The first aim of the present work isto obtain, within the geometric optics approximation,an exact set of equations to study the change of topol-ogy of a one-dimensional object obtained by reflectionon an arbitrary smooth surface.
From the research developed by several authors[1,3–8], it is quite clear that the multiple image for-mation process in natural optical systems, whosecomponents may be the reflecting surface of rippledwater, refractive-index gradients in the atmosphere,or the gravitational field associated with a matterdistribution, can be explained within the geometricoptics limit by using the caustic-touching theorem.On the other hand, conventional optical systems,such as microscopes and telescopes, whose basic com-ponents are mirrors and lenses, are designed to pro-duce a single image that is, of course, as perfect aspossible. So, at first glance, one could infer that non-multiple images associated with a single object canbe seen by using these optical devices. However, ingeneral, it is not true. The second and main contribu-tion of this work is to remark that, even with a per-fect mirror, it is possible, under certain conditionsdetermined by the caustic-touching theorem, to ob-serve image disruption. In particular, we remarkthat the caustic-touching theorem allows us to de-scribe the pattern, referred to as the Ronchigramin the well-known Ronchi test, when the grating isplaced at the caustic region.
The Ronchi test [9], developed by Ronchi in the1920s, is one of the most simple and powerful meth-ods to extract information about the aberrations ofan optical system. For these reasons, it has beenthe subject of innumerable publications, from boththe physical and the geometric points of view (see[10] and the references cited therein). From the phy-sical point of view, the fringes are interpreted asbeing due to interference between several wavefrontsproduced by the ruling acting as a diffraction grating,while from the geometric point of view, the fringesare interpreted as shadows of the ruling slits. Whenthe frequency of the ruling is not very high, the twopoints of view predict the same result. Since, by ana-lyzing the fringes of the real and ideal Ronchigrams,one can determine the type and, in principle, themagnitude of the aberrations present at the exit
Fig. 1. Schematic of the object curve, which we assume is lying ona plane perpendicular to the z axis, the two-dimensional reflectingsmooth surface is locally given by z ¼ f ðx; yÞ, and the position of theobserving eye is given by ~S ¼ ðs1; s2; s3Þ.
Reflecting smooth surface
X
Y
ZImarinary point
light source
One dimentional object
z0
Imaginary emittedlight ray
Imaginary reflectedlight ray
Point belonging to the objetc’s image
Fig. 2. New geometric arrangement to compute the image of aone-dimensional object under reflection on the arbitrary smoothsurface z ¼ f ðx; yÞ. The observing eye has been replaced by an ima-ginary point light source. In this diagram, we show an imaginaryemitted light ray and the corresponding reflected light ray, whichwe assume arrives at a point of the one-dimensional object. There-fore, the point of the surface where the imaginary light ray is re-flected belongs to the image of the one-dimensional object.
pupil of the system, then it is fundamental to knowthe properties of the Ronchigrams.In this work we consider the geometric point of
view of the Ronchi test. Under this assumption,the essential features of the Ronchi test for a concavemirror when the point source is located on the opticalaxis may be described by reference to Fig. 3. The lightrays emitted by the point light source are reflected bythemirror under test and they focus to a region in thespace. As previously mentioned, this region is thecaustic associated with the reflected light rays.The grating, which is referred to as the Ronchi rul-ing, is located at different positions on the opticalaxis. The pattern observed through the grating onthe surface of the mirror is referred to as the realRonchigram. From the geometric point of view, thefringes of the real Ronchigram are interpreted asshadows of the ruling bands. By comparing these realfringes with the ideal ones obtained by simulation,one can deduce the defects of the mirror under test[11]. Therefore, one of the cornerstones of the Ronchimethod is the ideal Ronchigram that is obtained bysimulation. Thus, Sherwood [12] has calculated theRonchi pattern of a paraboloidal mirror when testednear the center of curvature; Malacara [11] devel-oped an algorithm to predict the geometric Ronchi-gram of any spherical or aspherical mirror whentested at any point along the optical axis. These re-sults have been generalized by Cordero et al. [13] toan arbitrary mirror when the point source is locatedat any position; in particular, their equations can beused to simulate Ronchigrams for the cases of cen-tered and off-axis conic sections with the point lightsource at any location. Cordero et al. [14] have pre-sented a simple algorithm that allows the simulationof Ronchigrams for any optical system in which it ispossible to make an exact ray tracing. In accordancewith the results obtained by these authors, it is clearthat the form and structure of the ideal Ronchigram
associated with a given perfect mirror depend on thelocation of both the point source and the Ronchi rul-ing. Furthermore, it is well known that when theRonchi ruling is placed at the caustic region, it iscommon to see closed loop fringes in the Ronchigram.However, to our knowledge, an explanation of theclosed loop fringes in the Ronchigram using geo-metric optics has not been presented. Actually, inpractice, for an easy interpretation of the Ronchi-gram, it is common to avoid any closed loop fringe,which is possible only when the grating is locatedoutside the caustic region associated with the re-flected light rays [10]. In this work, we remark thatthe explanation is given by the caustic-touching the-orem and, therefore, the relative position of the grat-ing with respect to the caustic associated with thereflected light rays emitted by the point light sourceis crucial to describe correctly the structure of the as-sociated Ronchigram. Our justification for these as-sertions is as follows. From Figs. 2 and 3, it is obviousthat the procedure followed by Berry to compute theimage of an arbitrary object under reflection on anarbitrary smooth surface when the observer is lo-cated at an arbitrary position in the space is exactlythat used by several authors to simulate the Ronchi-gram associated with an arbitrary smooth reflectorby using a point light source located at different po-sitions in the space. Remember that, in Berry’s pro-cedure, the observer is replaced by an imaginarypoint light source. If, in addition, we replace theone-dimensional object by a grating, in particular aRonchi ruling, then we get the desired configurationto simulate the corresponding Ronchigram. Eventhough simulation of the Ronchigrams has been im-plemented for many years, to our knowledge, no onehas remarked on the fundamental role played by thecaustic associated with the reflected light rays to de-scribe the structure of the Ronchigrams when theRonchi ruling is located at the caustic region. Actu-ally, as we will show in Section 2, the equations thatwe obtain to compute the image associated with anarbitrary object obtained by reflection on an arbi-trary smooth surface are exactly those obtained byCordero et al. [13] in the context of the Ronchi andHartmann tests. However, these authors did notstudy the properties of the Ronchigrams when theRonchi ruling is located at the caustic place.
An important observation is that, in the Ronchitest, the point light source is not an imaginaryone, as in Berry’s procedure, but it is a real oneand the pattern, the image (in Berry’s problem), isobserved through the grating on the surface of themirror. For practical purposes, it is more convenientto plot the simulated Ronchigram in a plane close tothe reflecting surface. Therefore, in this work we plotthe corresponding images (in Berry’s problem) or pat-terns (in Ronchi’s test) in the plane z ¼ 0.
The aim of the present work is twofold: first, withinthe geometric optics limit, we obtain an exact set ofequations to study the changes of image topology un-der reflection on an arbitrary smooth surface and,
T2
T1
X
YSurface under
test
Point lightsource
Emittedlightray
Reflected lightray
Ronchiruling
One ronchigramfringe
Fig. 3. Schematic of the Ronchi test arrangement. In this dia-gram, we have the surface under test locally given by z ¼ f ðx; yÞ,a real point light source located on the optical system, and aRonchi ruling. The pattern observed through the grating on thesurface of the mirror is referred to as the real Ronchigram.
second, we apply our results to two particular casesto show that, even with a perfect spherical mirror,one can observe the change of topology. In particular,we remark that the structure of the Ronchigramswhen the Ronchi ruling is located at the caustic re-gion can be described by using the caustic-touchingtheorem.The organization of the present work is as follows.
In Section 2, we use geometric optics to obtain anexact set of equations to compute the image of aone-dimensional object produced by reflection onan arbitrary smooth surface. We find that this setof equations is exactly that previously obtained inthe context of the Ronchi and Hartmann tests in[13]. In Section 3, we compute the caustic surface as-sociated with the light rays reflected by an arbitrarysmooth surface when a point light source is located atan arbitrary position in the space. In Section 4, wecollect the two sets of equations to study in an exactway the changes of image topology. Finally, in Sec-tion 5, we present some examples to illustrate thechange of image topology when the reflecting surfaceis a spherical mirror and the objects are circles or linesegments. In particular, we describe the structure ofthe Ronchigram associated with a spherical mirrorwhen the point light source is located on the opticalaxis and the Ronchi ruling is placed at the causticregion.
2. Computation of the Image of a One-DimensionalObject Obtained by Reflection on an Arbitrary SmoothSurface
In this section, following Berry’s procedure, we ob-tain the exact set of equations to compute the imageof an arbitrary one-dimensional object obtained un-der reflection on an arbitrary smooth surface. Re-member that the original problem is changed by anew one, where the observer is replaced by an ima-ginary point light source as previously explained andshown in Fig. 2. Since this new problem is exactlythat of simulating Ronchigrams in the Ronchi testand there the point light source is a real one, fromnow on, we will use the term point light source to re-fer to both the real and the imaginary point lightsources, understanding that in the Ronchi test it isreal and in the Berry’s procedure it is an imaginaryone. Thus, assuming that in R3 we have a smootharbitrary surface and a point light source with arbi-trary position, we first obtain a parametric represen-tation of the maps that describe the evolution of thereflected light rays by the arbitrary smooth surface,and then we obtain the set of equations that allowsus to compute the image of a one-dimensional objectobtained by reflection.If the point light source is located at ~S ¼ ðs1; s2; s3Þ
and the smooth arbitrary surface is locally given byz ¼ f ðx; yÞ (see Fig. 4) then the light ray reflected atthe point ~r ¼ ðx; y; f ðx; yÞÞ is described by
~T ¼~rþ lR; ð1Þ
where l is the distance along the reflected light ray, Ris given by
R ¼ I − 2ðI · NÞN; ð2ÞN is the unit normal vector to the reflecting surfacez ¼ f ðx; yÞ, and I gives the direction of the divergingray from the point light source. From Fig. 4 we have
I ¼~I
j~I j¼ ðx − s1; y − s2; f − s3Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx − s1Þ2 þ ðy − s2Þ2 þ ðf − s3Þ2p : ð3Þ
To obtain a vector field perpendicular to the reflect-ing surface, we define the function Gðx; y; zÞ ¼z − f ðx; yÞ. Observe that one level surface of this func-tion is the reflecting surface z ¼ f ðx; yÞ. Therefore, avector field perpendicular to the reflecting surface isgiven by
~N ¼ ð−f x;−f y; 1Þ; ð4Þwhere f x ¼ ð∂f =∂xÞjG¼0 and f y ¼ ð∂f =∂yÞjG¼0. Finally,the unit normal vector field to the reflecting surfaceis given by
N ¼ ð−f x;−f y; 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ f 2x þ f 2yÞ
q : ð5Þ
By using Eqs. (1)–(5), a direct computation showsthat, if ~S ¼ ðs1; s2; s3Þ is the position of the point lightsource, then a light ray that is emitted in the direc-tion I and reflected by the arbitrary smooth curvedreflector at the point~r ¼ ðx; y; f ðx; yÞÞ is described by
T1 ¼ xþ lh1
α ; ð6Þ
T2 ¼ yþ lh2
α ; ð7Þ
Reflecting smoothsurfaceX
Y
Z
Point lightsource
Object curve lying on a perpendicularplane to z axis
z0
S=(s , s , s )1 2 3
z=f (x,y)r = (x, y, f(x,y))
R
Î
N
Fig. 4. Schematic of the optical system and the vectors used tocompute the image of a one-dimensional object obtained by reflec-tion on an arbitrary smooth surface locally given by z ¼ f ðx; yÞ. Inthis diagram, S ¼ ðs1; s2; s3Þ denotes the position of the point lightsource, I is the direction of an emitted light ray,~r ¼ ðx; y; f ðx; yÞÞ isthe point on the smooth surface where the emitted light ray is re-flected in the direction R, and N is the normal vector to the smoothsurface at the point of reflection.
h1 ¼ ðx − s1Þð1 − f 2x þ f 2yÞ − 2f x½f yðy − s2Þ þ s3 − f �;h2 ¼ ðy − s2Þð1þ f 2x − f 2yÞ − 2f y½f xðx − s1Þ þ s3 − f �;h3 ¼ ðf − s3Þð−1þ f 2x þ f 2yÞ þ 2½f xðx − s1Þ þ f yðy − s2Þ�;
α ¼ ð1þ f 2x þ f 2yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs1 − xÞ2 þ ðs2 − yÞ2 þ ðs3 − f Þ2
q;
ð9Þ
and l, which gives the position of an arbitrary pointalong the reflected light ray, is such that f ðx; yÞ ≤l < ∞. Furthermore, we assume that xmin ≤ x ≤ xmaxand ymin ≤ y ≤ ymax. The values of xmin, xmax, ymin,and ymax are determined by the dimensions of the re-flecting surface, which we assume are known.Since we are interested in the intersection of the
reflected light rays with an arbitrary planez ¼ constant, then it is convenient to take T3 ¼ z0and, thus, from Eq. (8), we have that
l ¼ α�T3 − f ðx; yÞ
h3
�¼ α
�z0 − f ðx; yÞ
h3
�; ð10Þ
so that Eqs. (6)–(8) can be rewritten in the followingform:
where f ðx; yÞ ≤ z0 < ∞.Before continuing, it is important to explain the
geometric meaning of the parametric map given byEqs. (11). For this end, we assume that the positionof the point light source is fixed. Then for fixed valuesof x and y, that is, for a point on the reflecting surface,z ¼ f ðx; yÞ, Eqs. (11) describe a line segment, whichstarts at the point ðx; y; f ðx; yÞÞ and goes to infinityas z0 does. Therefore, as x and y take all their allowedvalues, this map describes a family of line segmentsthat start at the points of the reflecting surface andgo to infinity. This family of line segments are thereflected light rays. On the other hand, from a math-ematical point of view, Eqs. (11) describe the para-metric form of a map between two subsets of R3,where ðx; y; z0Þ are local coordinates of the domainspace and ðT1;T2;T3Þ are local coordinates of the tar-get space. Equivalently, Eqs. (11), can be seen as aone-parameter family of maps between subsets ofR2, where each map is characterized by a particularvalue of z0. This family is explicitly given by
Each member of the family, characterized by a spe-cific value of z0, maps points of the reflecting surfaceto points on the plane z ¼ z0. It is important to re-mark that in Eqs. (12), z0 is not a coordinate as itis in Eqs. (11), but it is a parameter characterizinga particular member of the family of maps, and issuch that f ðx; yÞ ≤ z0 < ∞.
The sets of Eqs. (11) and (12) are those maps wewere referring to in the introduction, which describethe evolution of the reflected light rays by the arbi-trary smooth surface after being emitted by the pointlight source. We remark that, because we have usedthe reflection law only one time to obtain Eqs. (11)and, therefore, Eqs. (12), these equations describethe evolution of the light rays that have experiencedonly one reflection before leaving the smooth surface.In this work, we assume that the parameters thatcharacterize the optical system under study are suchthat this condition is fulfilled by the reflected lightrays.
Assuming that, in the plane z ¼ ~z0, a coordinatesystem ðTx;TyÞ is introduced with origin at ð0; 0;~z0Þsuch that Tx and Ty are parallel to the axes x and y,respectively (see Fig. 5), a one-dimensional object ly-ing on this plane can be described in a parametricway by
Tx ¼ ΓðσÞ; Ty ¼ ΣðσÞ; ð13Þwhere σ is a parameter that labels the points on theobject. If we eliminate the parameter σ, we have thatthe object could be described by
Reflecting smoothsurface
X
Y
ZPoint light
sourceObject curve
Tx
Tyz0
z
z0=S=(s , s , s )1 2 3
Point belonging to theobject´s image
Fig. 5. Schematic of the two sets of coordinate systems used tocompute the image of a one-dimensional object obtained by reflec-tion on an arbitrary smooth surface locally given by z ¼ f ðx; yÞ. Wealso have included an emitted light ray such that its associatedreflected light ray connects a point of the smooth surface with apoint of the one-dimensional object. The point on the smooth sur-face where the emitted light ray is reflected belongs to the image ofthe one-dimensional object.
Therefore, the image of this object that an observer,with position ~S ¼ ðs1; s2; s3Þ, may see on the surfaceof reflection is given by all the points of the formðx; y; f ðx; yÞÞ such that x and y are solutions to
In other words, a point on the reflecting surface be-longs to the image of the object in Eqs. (13), if it canbe connected, via a reflected light ray, with a point ofthat object (see Fig. 5). In the explicit examples thatwe present later on, we will not plot the image on thereflecting surface, but on the plane z ¼ 0. This is be-cause when one takes a photograph of an object itsimage is normally printed on a plane. Therefore, inour examples, the image will be given by all thepoints ðx; y; 0Þ such that x and y satisfy Eqs. (15).Equations (15) together with the caustic associated
with the reflected light rays will allow us to study thechange in the topology of the image of an arbitraryone-dimensional object obtained under reflectionon an arbitrary smooth surface.
3. Computation of the Caustic
Since to study the change of image topology of an ar-bitrary one-dimensional object obtained by reflectionit is required to compute the caustic associated withthe reflected light rays described by Eqs. (11) or,equivalently, by Eqs. (12), in this section, followingour previous work [15,16], we review the computa-tion of the caustic by using Eqs. (11) and, further-more, we point out its relationship to the causticassociated with Eqs. (12). To this end, we introducethe following:Definition: Let h : M → N be a differentiable
map, with M and N differentiable manifolds. Theset of points in M where h is not locally one to oneare referred to as its critical set, and the image of thecritical set is referred to as the caustic set of h[17–19]. If M and N are differentiable submanifoldsofRn with local coordinates (xi) and (yj), respectively,then locally h is given by
yi ¼ hiðxjÞ; where i; j ¼ 1; ;n: ð16Þ
Therefore, in this case, the critical set is obtainedfrom the condition
J ≡ det�∂yi∂xj
�¼ 0: ð17Þ
In general, this condition can be written in the fol-lowing way:
Fðx1;…; xnÞ ¼ 0; ð18Þ
If this equation can be solved, for example, for xn,then locally the critical set of h is given by
xn ¼ gðx1;…; xn−1Þ; ð19Þ
which, in the general case, is a set of surfaces of di-mension n − 1 in the domain space, which we are as-suming has dimension n. Therefore, the caustic set,which is the image of the critical set, is obtained bysubstituting Eq. (19) into Eqs. (16). That is, the caus-tic set associated with the map (16) is locally given by
In accordance with the above definition, the criti-cal set of the map given by Eqs. (11), that is, the set ofpoints in the domain space with coordinates ðx; y; z0Þsuch that the map is not locally one to one, is ob-tained from the following condition:
Jðx; y; z0Þ ¼ det�∂ðT1;T2;T3Þ∂ðx; y; z0Þ
�
¼�∂T1
∂x
��∂T2
∂y
�−
�∂T1
∂y
��∂T2
∂x
�¼ 0:
ð21Þ
By using Eqs. (11), a direct computation shows thatthis condition is equivalent to
Jðx; y; z0Þ ¼ H2ðx; yÞ�z0 − fh3
�2þH1ðx; yÞ
�z0 − fh3
�þH0ðx; yÞ ¼ 0; ð22Þ
where
H2ðx; yÞ ¼ ~h ·��
∂~h∂x
�×�∂~h∂y
��;
H1ðx; yÞ ¼ ~h ·��
∂~r∂x
�×�∂~h∂y
�þ�∂~h∂x
�×�∂~r∂y
��;
H0ðx; yÞ ¼ ~h ·��
∂~r∂x
�×�∂~r∂y
��; ð23Þ
with
~r ¼ ðx; y; f ðx; yÞÞ; ~h ¼ ðh1;h2;h3Þ: ð24Þ
From Eq. (22) we find that the critical set associatedwith the map given by Eqs. (11) is given by
Therefore, the caustic set associated with the mapgiven by Eqs. (11), which, by definition, is obtainedby substituting Eq. (25) into Eqs. (11), can be writtenin the following form:
It is important to remark that Eq. (26) is equivalentto that obtained by Shealy and Burkhard [20,21] byusing a different procedure. From Eq. (26), it is clearthat the caustic set or, simply, the caustic associatedwith the reflected light rays described by Eqs. (11), ingeneral, is composed by two branches, which, for veryparticular forms of the reflecting surface and parti-cular positions of the point light source, reduce toa single point. For example, this happens when thereflecting surface is part of a perfect spherical mirrorand the point light source is located at its center ofcurvature. In the general case, the two branches ofthe caustic are two-dimensional surfaces, which,when they are stable under small deformations ofthe reflecting surface and the position of the pointlight source, locally have singularities of well-knowntypes: the swallowtail, the pyramid or elliptic umbi-lic, and the purse or hyperbolic umbilic [17–19]. Thefact that the caustic might not be stable under smalldeformations of the reflecting surface and the posi-tion of the point light source is related to the symme-tries of the system formed by the reflecting surfaceand the point light source.Since we are assuming that xmin ≤ x ≤ xmax and
ymin ≤ y ≤ ymax, then, in our case, the caustics willbe located within a finite region of the space. In par-ticular, there exist real numbers zcmin� and zcmax�such that zcmin� ≤ T3c� ≤ zcmax�.Now we compute the critical and caustic sets asso-
ciated with the one-parameter family of maps givenby Eqs. (12) and we point out their relationship tothose associated with the map given by Eq. (11).To this end, we select the map given by Eqs. (12)when z0 ¼ ~z0 ¼ constant. That is,
where ~z0 is a real number such that f ðx; yÞ ≤ ~z0 ≤ ∞.Remember that this map sends points from the re-flecting surface to points on the plane z0 ¼~z0 ¼ constant. The critical set associated with thismap is obtained from the following condition:
~Jðx; yÞ ¼ det
∂ðT1;T2Þ∂ðx; yÞ
!
¼�∂T1
∂x
��∂T2
∂y
�−
�∂T1
∂y
��∂T2
∂x
�¼ 0; ð28Þ
or equivalently from
~Jðx; yÞ ¼ H2ðx; yÞ�~z0 − fh3
�2þH1ðx; yÞ
�~z0 − fh3
�þH0ðx; yÞ ¼ 0: ð29Þ
In Eq. (29), ~z0 is a number; it is not a variable as z0 isin Eq. (22). Observe that the relationship between~Jðx; yÞ and Jðx; y; z0Þ is given by ~Jðx; yÞ ¼ Jðx; y;~z0Þ.This means that only when
zcmin� ≤ ~z0 ≤ zcmax�; ð30Þ
the map given by Eqs. (27) will be not locally one toone. Therefore, if ~z0 satisfies the above condition,from Eq. (29) we have that, locally, the critical set as-sociated with the map given by Eqs. (27) may be writ-ten in the form
y ¼ Ψðx;~z0Þ: ð31Þ
In Eq. (31), we have included ~z0, which is a number,only to show that it is the critical set associated withthemap given by Eqs. (27). The caustic set associatedwith this particular map, which is obtained by sub-stituting Eq. (31) into Eqs. (27), is given by
In the general case, it is a curve or family of curves inthe plane z0 ¼ ~z0 ¼ constant. For very particularcases it reduces to a point. When the caustic is stableunder small deformations of the optical systemformed by the reflecting surface and the point lightsource, then, in accordance with theWhitney’s singu-larity theory [17–19], it locally has singularities offold or cusp type.
From the computations presented in this section. itis clear that the caustic associated with the map ofEq. (27) given by Eq. (32) is equal to the intersectionof the caustic associated with the map of Eq. (11), gi-ven by Eq. (26), with the plane z0 ¼ ~z0 ¼ constant.
Before closing this section, we explain the geo-metric meaning of the caustic associated with theevolution of the reflected light rays by the smooth ar-bitrary surface when the point source is located atany position of the space. To this end, consider thepencil of light rays reflected by the differential sur-face dxdy of the reflector. From Eqs. (27), we havethat, when ~z0 ¼ f , the cross-sectional area of this
pencil of rays is exactly dxdy, and as the light raysevolve, this area is given by dT1dT2 ¼ j~Jðx; yÞjdxdy ¼j~Jðx; y;~z0Þjdxdy. If ðx; yÞ belongs to the critical setof the map given by Eqs. (27), then ~Jðx; yÞ ¼Jðx; y;~z0Þ ¼ 0 and, in that case, the cross-sectionalarea of the pencil of rays collapses to zero. This resultshows that the caustic is defined by the focusing re-gion associated with the reflected light rays and,therefore, also can be defined as the singularitiesof the flux density [20,21]. Finally, we remark that,in the literature, one can find explicit expressionsand plots of the caustic associated with particular re-flectors and certain positions of the point light source[22–24].
4. Equations to Study the Change of Image Topology
In this section, we summarize the two sets of equa-tions that allow us to study, in an exact way, thechanges in the topology of the image of an arbitraryobject obtained by reflections on an arbitrary smoothsurface:
• The image of an arbitrary one-dimensionalobject, locally given by Eqs. (13) and located on theplane z ¼ ~z0 ¼ constant, obtained by reflection onthe smooth reflecting surface locally given by z ¼f ðx; yÞ, is computed by solving Eqs. (15); that is,
and the plane z ¼ ~z0 ¼ constant, or equivalently gi-ven by Eq. (32).
Remember that the map given by Eqs. (27) mapspoints on the reflecting surface to points on the planez ¼ ~z0 ¼ constant. If the plane z ¼ ~z0 ¼ constant,which contains the object, is outside the caustic re-gion; that is, there is not intersection between thisplane and the caustic given by Eq. (34), then themap given by Eqs. (27) is locally one to one. Thismeans that, for each point ðTxðσÞ;TyðσÞ;~z0Þ on the ob-ject, there is a unique solution ðx; yÞ to Eqs. (33) and,therefore, under this condition, the observer will reg-ister a single image. In other words, the object and itsassociated image, in this case, have the same topol-ogy. Now we assume that the plane z ¼ ~z0 ¼ constantis located at the caustic region given by Eq. (34). In
this second case, the map given by Eqs. (27) is notlocally one to one at those points that belong tothe intersection of the caustic and the plane z ¼~z0 ¼ constant. Therefore, if the object lying on thisplane is outside the caustic, then the observer willregister a single image, but if the object reachesthe caustic in such a way that they become tangentto each other; that is, there is a touch between them,then the object and its associated image do not havethe same topology and, therefore, the observer cansee multiple images corresponding to a single object(see the examples in Section 5).
We close this section with the following obser-vations:
• The set of Eqs. (33) have been reported in theliterature by Cordero et al. [13] in the context of theRonchi and Hartmann tests. These authors showedthat Eqs. (33) allow describing the main featuresof both tests. In particular, they found that theRonchigram associated with an arbitrary reflectingsurface, when the point source is located at an arbi-trary position and the Ronchi ruling is at the planez ¼ ~z0 ¼ constant with its rulings parallel to the xaxis, is given by the level curves of TyðσÞ.
• The caustic, Eq. (34), has also been reported in[16,20,21] and, in particular, it was used to computethe circle of least confusion associated with a rota-tionally symmetric mirror when the point lightsource is located on the optical axis [15,16,24,25].
• Though the two sets of Eqs. (33) and (34) havebeen reported, to our knowledge, they have not beenused to explain the structure of the Ronchrigramswhen the Ronchi ruling is located at the caustic re-gion. The main contribution of this work is to realizethat the procedure implemented by Berry is equiva-lent to that used in the Ronchi test and, therefore,that the closed loop fringes observed in the Ronchipattern when the Ronchi ruling is located at thecaustic region can be explained by using the caustic-touching theorem. (The present work can be consid-ered as a logical continuation of research reported in[13,15,16].)
5. Examples: Spherical Mirror with the Point LightSource on the Optical Axis
To illustrate everything we have presented in theprevious sections, we assume that the reflectingsmooth surface is a part of a spherical mirror withradius r and diameter D given by
z ¼ f ðx; yÞ ¼ r −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − x2 − y2
q; ð35Þ
where −D=2 ≤ x ≤ D=2 and −D=2 ≤ y ≤ D=2. Equiva-lently, this equation can be written in the followingform:
z ¼ f ðρÞ ¼ r −ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
and the corresponding branches of the caustic set by
T1c−ðx; yÞ ¼ 0; T2c−ðx; yÞ ¼ 0; T3c−ðx; yÞ ¼ z0−;
ð40Þ
T1cþðx; yÞ ¼2xðr − sÞ2ρ2
r2�3r2 − 4rsþ 2s2 þ 3ðs − rÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
p � ;
T1cþðx; yÞ ¼2yðr − sÞ2ρ2
r2�3r2 − 4rsþ 2s2 þ 3ðs − rÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
p � ;T3cþðx; yÞ ¼ z0þ: ð41Þ
Equations (40) describe a line segment on the opticalaxis, that is, on the z axis. This part of the caustic
corresponds to the fact that the rays that leave thepoints in a circle on the mirror with its center onthe optical axis converge at a point. All the conver-gent points of all the rays that leave all the possiblecircles on the mirror with centers on the optical axisform this part of the caustic (the line segment). Bycontrast, Eqs. (41), in general, describe a surface ofrevolution with a degenerated singularity of cusptype. If we intersect this surface with a plane con-taining the z axis then we obtain a curve with a sin-gularity of cusp type. However, if we intersect it witha plane perpendicular to the z axis, then we obtain a
circle of radiusffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2
1cþ þ T22cþ
q. It is an example where
one of the branches of the caustic is not a two-dimensional surface; this fact is related to the axialsymmetry of the optical system under study. Evenmore, as it is clear from Eqs. (39)–(41), for the veryparticular case s ¼ r, that is, when the point lightsource is located at the center of the spherical mirror,
200100 0100200y
7000
8000
9000
10 000
11 000
z
200100
0100200
x
Fig. 6. Intersection of the caustic given by Eqs. (40) and (41) withthe plane y ¼ 0, for the special case r ¼ 2415mm,D ¼ 1470mm ¼ 2ρmax, z ¼ z0 ¼ 11; 000mm, and the point lightsource is located at (0, 0, 1350mm).
the two branches of the caustic reduce to an iso-lated point.By using Eqs. (39), a direct computation shows
that when s ≥ r, then
T3c−ðρmaxÞ ≤ T3c−ðρÞ ≤rs
2s − r;
T3cþðρmaxÞ ≤ T3cþðρÞ ≤rs
2s − r: ð42Þ
Therefore, there is an interval on the z axis whereT3c−ðρÞ ¼ T3cþðρÞ (see Fig. 6). Thismeans that, in thiscase, the intersection of the caustic with a plane z ¼z0 ¼ constant such that z0 ¼ T3c−ðρÞ ¼ T3cþðρÞ, is acircle and an isolated point (see Fig. 7). The circleis obtained from the intersection between the planeand that part of the caustic that is a surface of revo-lution, while the point is obtained from the intersec-tion between the plane and the line segmentcorresponding to the other branch of the caustic.For this case, the one-parameter family of maps be-
tween points on the spherical mirror and points of anarbitrary plane z ¼ z0 ¼ constant, is explicitly givenby
T1ðx; yÞ ¼ x
�1þ
�z0 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q �GðρÞ
�;
T2ðx; yÞ ¼ y
�1þ
�z0 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q �GðρÞ
�:
ð43Þ
From the previous discussion we have, if z0 is suchthat z0 ¼ T3c−ðρÞ ¼ T3cþðρÞ, then the caustic asso-ciated with this map is a circle and a point that co-incides with the center of the circle.
If, in the plane z ¼ z0 ¼ constant, we have a one-dimensional object given by Eqs. (13), then its imageis obtained by solving, for x and y, the following set ofequations:
TxðσÞ ¼ x
�1þ
�z0 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q �GðρÞ
�;
TyðσÞ ¼ y
�1þ
�z0 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q �GðρÞ
�: ð44Þ
Given TxðσÞ, TyðσÞ, z0, and r, the image on the sphe-rical mirror is given by all the points (x, y,r −
Eqs. (44). In this work, instead of plotting the imageon the smooth reflecting surface, we will plot it on thez ¼ 0 plane.
For the explicit computations we take r ¼2415mm, D ¼ 1470mm ¼ 2ρmax, the point lightsource at (0, 0, 1350mm), and z ¼ z0 ¼ 11; 000mmas the plane where the object is located. Using thesedata, in Fig. 6, we present the intersection of the twobranches of the caustic given by Eqs. (40) and (41)with the plane y ¼ 0. In Fig. 7 we present the inter-section of these two branches of the caustic withthe plane z ¼ 11; 000, that is, with the plane wherethe one-dimensional object is located. The intersec-tion is a circle of radius Rc ¼ 4:1643mm and itscenter. (In the plots presented in Subsection 5.A
-4-2
02
4x
-4
-2
0
2
4
y1100011000110001100011000
4-2
02
4x
z
Fig. 7. Intersection of the two branches of the caustic with theplane z ¼ 11; 000.
Fig. 8. (Color online) (a) Object space and (b) image space for k ¼ −7.
corresponding to the object space, the planez ¼ 11; 000, we show only that part of the caustic thatcorresponds to the circle; we are not including thepart corresponding to the isolated point.)
A. First Example: Circular Object
In this example we assume that the object is a circleof radius 1 given by
ðTx − kÞ2 þ T2y ¼ 1; ð45Þ
where k is a real constant. That is, we are interestedin obtaining the image of the object that is a circlewith radius equal to 1 with center at ðk; 0Þ, whichis located in the plane z ¼ 11; 000. From Eqs. (44)and (45), we obtain that, for a fixed value of k, the
image of the circle of Eq. (45) in the plane z ¼ 0 isgiven by all the values of x and y such that
�x
�1þ
�11000 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q �GðρÞ
�− k
�2
þ y2�1þ
�11000 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q �GðρÞ
�2¼ 1: ð46Þ
We have written a computer program inMathema-tica to study the change of topology of the image forthis case and those cases presented later on. That is,we give a particular value to k and, in the objectspace (Tx, Ty, 11,000) we plot the object (circle of ra-dius 1) and the caustic (remember that we will plotonly the circle of radius Rc). After that, the set of
Fig. 9. (Color online) (a) Object space and (b) image space for k ¼ −Rc.
Fig. 10. (Color online) (a) Object space and (b) image space for k ¼ −1:5.
Eqs. (46) are solved for x and y under the conditions−ρmax ¼ −735 ≤ x ≤ ρmax ¼ 735 and −ρmax ¼ −735 ≤
y ≤ ρmax ¼ 735. Finally, these values obtained for xand y are plotted in the plane z ¼ 0. In what follows,we present some plots where the change of topologyis clearly observed.In Fig. 8 we show the object and image for k ¼ −7.
In this case, we get only one image because the mapthat sends points from the spherical mirror to pointsof the object is one to one; that is, the object is out ofthe caustic region. In accordance with the caustic-touching theorem, we will observe a change of imagetopology when the object and the caustic touch each
other, that is, when they become tangent to eachother. If we assign values to k from −7 to 7, thenthe first change of topology is obtained when k ¼−Rc − 1 (remember that Rc ¼ 4:1643). For this valueof k a new image appears from an isolated point. InFigs. 9(a) and 9(b), we present the case correspondingto k ¼ −Rc; here it is clearly observed the second im-age. There is another change of topology whenk ¼ −Rc þ 1; for this value of k a third image appears.In Figs. 10(a) and 10(b), we present the case k ¼ −1:5,where the third image is clearly observed. Anotherchange of topology is obtained for k ¼ −1 becausethe caustic, in the plane z ¼ 11; 000, is constituted
Fig. 11. (Color online) (a) Object space and (b) image space for k ¼ 0.
Fig. 12. (Color online) Set of circles with their centers on Tx.
Fig. 15. (Color online) Image of the set of concentric circles with their centers on Tx.
Fig. 16. (Color online) (a) Object space: set of line segments parallel to the Ty axis with Ty ∈ ½−6; 6� and the caustic, which is a circle ofradius Rc ¼ 4:1643mm and its center. (b) Image space: the corresponding images. In the Ronchi test, the set of lines (a) is the grating orRonchi ruling and its image (b) is referred to as the associated Ronchigram.
by the circle of radius Rc and its center. For this par-ticular value of k the object is tangent to that part ofthe caustic that corresponds to the isolated point. Fork ¼ 0, the images are three concentric circles (seeFig. 11). Now it is clear that there will be otherchanges of image topology when k ¼ 1, Rc − 1, andRc þ 1.
Finally, in Fig. 12, we show a set of circles withtheir centers on the Tx axis in the object space andin Fig. 13 we present the corresponding images. InFig. 14 we show a family of objects correspondingto concentric circles with different positions with re-spect to the caustic and in Fig. 15 we present theirassociated images. It is important to remark that
Fig. 17. (Color online) (a) Object space and (b) image space for k ¼ −6.
Fig. 18. (Color online) (a) Object space and (b) image space for k ¼ −4.
Murty and Shoemarker [26] have presented the the-ory of a method to test optical systems that is similarto that of Ronchi, but instead of using straight linesfor the grating, they used concentric circles. Theyhave presented characteristic patterns for the usualaberrations of optical systems. In some of these pat-terns it is possible to see the change of image topol-
ogy. However, this fact was not pointed out by theseauthors.
B. Second Example: Linear Object
In this second example, which we believe is the mostrelevant due to its connection to the famous Ronchitest used to test in particular conic mirrors, we
Fig. 19. (Color online) (a) Object space and (b) image space for k ¼ −2.
Fig. 20. (Color online) (a)–(d) Object space and (e)–(h) image space for k ¼ 0, 2, 4, and 6, respectively.
assume that the object is a line segment parallel tothe Ty axis. Actually, in the Ronchi test, the grating isformed by ruling bands; for our purposes we can as-sume that the line segments that we consider coin-cide with the center of the ruling bands of theRonchi ruling. That is, we assume that the objectis given by
Tx ¼ k; ð47Þ
where k is a real constant. In other words, we are in-terested in obtaining the level curves associated withthe function T1, given in Eqs. (43). From Eqs. (44)and (47) we have that, for k fixed, the image in the
plane z ¼ 0 is given by all the points of the formðx; y; 0Þ such that x and y satisfy the following equa-tion:
xh1þ
�11000 − rþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − ρ2
q GðρÞ
i¼ k: ð48Þ
In exactly the same way as in the previous case, tostudy the change in the topology of the image, wegive different values to k in the interval [−6, 6]. InFig. 16(a) we show the caustic and the object for dif-ferent values of k. In Fig. 16(b), the correspondingimages can be observed. In the Ronchi test,Fig. 16(a) corresponds to the Ronchi ruling andFig. 16(b) corresponds to what is referred to as theideal Ronchigram associated with a spherical mirror.The Ronchigram, Fig. 16(b), has been reported in theliterature, but its closed loop fringes never had beenexplained by using the caustic-touching theorem, aswe are doing in this work.
We start with the analysis of the change of the im-age topology: in Fig. 17(a) we show the object (linesegment given by Eq. (48) with k ¼ −6) and the caus-tic, which are in the plane z ¼ z0 ¼ 11; 000. InFig. 17(b) we show its image, which is not a segmentof a line but is a distortion of it. For this value of k,that is for k ¼ −6, we obtain only one image, becauseany point of the line segment can be reached by onlyone reflected light ray. The first change in the imagetopology is obtained when the line segment is tan-gent to that part of the caustic corresponding tothe circle of radius Rc; that is, when k ¼ −Rc, at thisvalue of k, a new image is born as an isolated point.The second image that is born is visible for k ¼ −4, ascan be seen in Figs. 18(a) and 18(b). In Figs. 19(a) and19(b), we have presented the case k ¼ −2. The follow-ing change of topology occurs when k ¼ 0; it is
�300 �200 �100 0 100 200 300�300
�200
�100
0
100
200
300
x
y
Fig. 21. (Color online) Pattern obtained by Ronchi.
Fig. 22. (Color online) Simulation of the Ronchi ruling and the caustic curve.
because the center of the circle with radius Rc is alsopart of the caustic. In Figs. 20(a)–20(h), we show theobject space and the image space for k ¼ 0, 2, 4, and 6,respectively. Now it is clear that, for this example, if
k ∈ ð−Rc; 0Þ, the object and its image have the sametopology. The second image that was born when k ¼−Rc transforms into a circle when k ¼ 0. Fork ∈ ð0;RcÞ, we have two images. Finally, the last
Fig. 23. Image of the Ronchi rulings or Ronchigrams.
change in the image topology is obtained whenk ¼ Rc. For this value of k, one of the images reducesto an isolated point, so that for k > 0, we have only asingle image.Remember that, when k ¼ −Rc in the image space,
there appears a new image from an isolated point,which, as k goes to zero, transforms into a circle.Something similar happens for 0 < k < Rc but, inthis second case, the circle reduces to an isolatedpoint when k ¼ Rc. In Fig. 21 we have isolated someof these loop images for some values of k. It is impor-tant to remark that this pattern was obtained byRonchi [9] when he was studying a lens with a re-markable aberration. He also obtained the patterncorresponding to k ¼ 0.Finally, in the set of Figs. 22 and 23, we show the
object space and the image space associated with aset of objects emulating the Ronchi ruling and theirassociated Ronchigrams.
6. Conclusions
We have obtained an exact set of equations to studythe change of image topology of an arbitrary objectobtained by reflection on an arbitrary smoothsurface. We have pointed out that the equations thatallow computing the image of an arbitrary one-dimensional object have been reported in the litera-ture [13] in the context of the Ronchi and Hartmanntests. However, to our knowledge, no one has re-marked that the closed loop fringes observed inthe Ronchigram when the Ronchi ruling is locatedat the caustic place are due a disruption of imagesor fringes. We believe that the main contributionof this work is to realize that the procedure developedby Berry to study the changes of image topology ofthe Sun disk is equivalent to that used in the simula-tion of ideal Ronchigrams. From this observation, itis clear that the caustic plays a major role in describ-ing correctly this kind of pattern. We claim that theresults established in this work could provide a de-scription of the patterns obtained in other tests thatuse gratings.Our general results were illustrated by several ex-
amples when the reflecting surface is a spherical mir-ror and the point light source (in the Ronchi test) orobserving eye (in Berry’s procedure) is located on theoptical axis. We believe it could be worthwhile tostudy the change of image topology when the pointlight source is out of the optical axis and the reflect-ing surface is a conic reflector.We remark that analogous results can be obtained
for lenses. In a future paper, we will report theseresults.
The authors thank M. Berry and an unknownreferee for helpful comments on the manuscript.E. Román-Hernández was supported by a ConsejoNacional de Ciencia y Tecnología (CONACyT) scho-larship and G. Silva-Ortigoza acknowledgesfinancial support from Sistema Nacional de Investi-gadores (SNI) (México) and CONACyT.
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