Computation of the Disk of Least Confusion for Conic Mirrors

Cc

J

1

RiccnSfelcdrav

twrrptocl

c2eMP

2

6

omputation of the disk of least confusion foronic mirrors

orge Castro-Ramos, Oscar de Ita Prieto, and Gilberto Silva-Ortigoza

We use geometrical optics to compute, in an exact way and by using the third-order approximation, thedisk of least confusion �DLC� or the best image produced by a conic reflector when the point source islocated at any position on the optical axis. In the approximate case we obtain analytical formulas tocompute the DLC. Furthermore, we apply our equations to particular examples to compare the exactand approximate results. © 2004 Optical Society of America

OCIS codes: 080.0080, 080.1010, 080.1510.

waefcwpTsSwpaartsfllta

icw�mtttaie

b

. Introduction

otationally symmetric mirrors, and especially con-cs, have been the subject of considerable study be-ause of their practical applications in theonstruction of optical devices such as large astro-omical telescopes and microscope objectives.1,2

uch elements have many advantages over lenses;or example, it is well known that the behavior oflectromagnetic waves in glass depends on the wave-ength whereas mirrors have no such dependence be-ause the law of reflection is not wavelengthependent. Furthermore, by using only conic mir-ors, one can design an optical system free of some orll the monochromatic aberrations over a field ofiew.The geometric imaging properties of optical sys-

ems are closely related to the caustics associatedith the evolution of the light rays; for this and other

easons, there has been considerable and importantesearch on the subject.2–18 Whereas some papersresent purely theoretical results, others show howhe caustic surface is used to solve some technical orptical problems. Shealy and Burkhard3,4 haveomputed the caustic surface associated with theight rays reflected by an arbitrary curved surface

J. Castro-Ramos �[email protected]� is with the Instituto Na-ional de Astrofısica, Optica y Electronica, Apartado Postal 51 y16, Tonantzintla, Cholula, Puebla 72000, Mexico. O. de Ita Pri-to and G. Silva-Ortigoza are with the Facultad de Ciencias Fısicoatematicas de la Universidad Autonoma de Puebla, Apartadoostal 1152, Puebla, Puebla 72001, Mexico.Received 19 March 2004; revised manuscript received 13 July

004; accepted 14 July 2004.0003-6935�04�336080-10$15.00�0

c© 2004 Optical Society of America

080 APPLIED OPTICS � Vol. 43, No. 33 � 20 November 2004

hen the light source is located at any place in spaces the locus of singularities of the flux density of themanating radiation. By using the methods of dif-erential geometry of surfaces, Stavroudis and Fron-zek5 have computed the caustic surface associatedith the evolution of a wave front as the locus of itsrincipal centers of curvature. Schroader6 andheocaris7,8 used the caustic surface to analyze thehape of any optical surface to obtain its quality.healy and Burkhard9–11 and Theocaris and co-orkers12–15 have presented a thorough study of theroperties of the caustics obtained by illuminatingny conic reflector with a point light source lyinglong the principal axis of the reflector. The mainesult obtained by these researchers was a descrip-ion of the behavior of the properties of the causticurface depending on the shape of the particular re-ector, its aperture, and the relative position of the

ight source and the reflector. For a classification ofhe caustics as catastrophes, see Berry and Upstill16

nd the references cited therein.It has been shown by Conrady2 that one of the most

mportant applications of the caustic surface is theomputation of the best focus of any optical system,hich is commonly called the disk of least confusion

DLC�. By using the third- and fifth-order approxi-ations, Conrady showed that the DLC is located at

he intersection of the caustic surface defined by in-ermediate rays from one side of the aperture withhe marginal rays arriving from the other side of theperture. This important result has been general-zed to axially symmetric mirrors17,18 and has beenxplicitly applied only to the spherical case.By using a different procedure from that followed

y Shealy and Burkhard,3,4 in a previous paper18 we
omputed the caustic surface associated with the

lflteDpaetisatrstpfwDuppp

cflsStlFtoDmgctptgosdr

2

Btmart1pfpTmrt

asy

wa

wsd

TsWtfib

wtg

t

Fo

ight rays reflected by a rotationally symmetric re-ector when the point source is located at any posi-ion on the optical axis. This result was used toxplain how to obtain the condition that defines theLC for a rotationally symmetric mirror when theoint source is located at any position on the opticalxis. Only in the spherical case was this conditionxplicitly obtained. In the present paper we extendhese results. We first show that the procedure usedn Ref. 18 to compute the caustic for a rotationallyymmetric mirror with the point source on the opticalxis can be used to obtain the caustic associated withhe light rays reflected by an arbitrary smooth curvedeflector when the point source is located at any po-ition in space. Our result is equivalent to that ob-ained by Shealy and Burkhard3,4 by using a differentrocedure. Furthermore, we show that our generalormula reduces to that reported in Ref. 18. Second,e obtain explicitly the condition that determines theLC for any conic reflector in an exact way and bysing the third-order approximation. The presentaper is a logical continuation of the research re-orted in Ref. 18 and can be considered as its secondart.The paper is organized as follows. In Section 2 we

ompute the caustic associated with the rays re-ected by a smooth arbitrary surface when the pointource is located at any arbitrary place in space.ubsequently, we restrict our analysis to any reflec-or with axial symmetry when the point source isocated at any arbitrary point on the optical axis.inally, an equation is obtained to compute the caus-

ic surface for any conic mirror with the light sourcen the optical axis. In Section 3 we compute theLC in exact and approximate ways for any conicirror, developing first an expression for the mar-

inal surface associated with reflected rays for anyonic mirror with a point source located at any arbi-rary place on the optical axis. After that we com-ute equations that express the intersection amonghe marginal surface and that part of the causticiven by the surface of revolution with a singularityf the cusp type. Finally, in Section 4, obtained re-ults are applied to particular cases to discuss theifferences between the exact and the approximateesults.

. Computation of the Caustic Surface

ecause the DLC is defined by one of the intersec-ions between one of the sheets of the caustic and thearginal surface,2,18 we need to compute the caustic

ssociated with the light rays reflected by any coniceflector when the point light source is located alonghe optical axis. To this end, following Refs. 16 and8–21, we present the derivation of a formula to com-ute the caustic created from light rays emanatingrom an arbitrary smooth curved reflector when theoint light source is located at any position in space.he general formula is applied to a rotationally sym-etric reflector, and finally the caustic for the conic

eflector is obtained, in an exact way and by use of the
hird-order approximation, as a particular case. s
2

If the point light source is located at s � �s1, s2, s3�nd the surface of the reflector is given by z � z�x, y�,ee Fig. 1, then the light ray reflected at point r � �x,, z�x, y�� is given by

T � r � lR, (1)

here l is the distance along the reflected light raynd R is given by

R � I � 2�I � n�n, (2)

here n is the unit normal vector to the reflectingurface z � z�x, y�. Vector I gives the direction of theiverging ray from the point source and is given by

I �I

�I��

� x � s1, y � s2, z � s3�

�� x � s1�2 � � y � s2�

2 � � z � s3�2�1�2 . (3)

o obtain a vector field perpendicular to the reflectingurface, we define the function f �x, y, z� � z � z�x, y�.e observe that one level surface of this function is

he reflecting surface z � z�x, y�. Therefore a vectoreld perpendicular to the reflecting surface is giveny

n � ��zx, �zy, 1�, (4)

here zx � �� z��x��f�0 and zy � �� z��y��f�0. Finallyhe unit normal vector field to the reflecting surface isiven by

n � � ��zx, �zy, 1�

�1 � zx2 � zy

2�1�2��f�0

. (5)

By use of Eqs. �1�–�5�, a direct computation showshat, if s � �s1, s2, s3� is the position of the point

ig. 1. Arbitrary reflecting surface, the point source, the directionf the incident light ray, and the direction of the reflected light ray.

ource, then a light ray that is reflected by the arbi-

0 November 2004 � Vol. 43, No. 33 � APPLIED OPTICS 6081

tp

T

T

T

w

Botrwor

Eb��Tlcdmaac�fr1t

pbs

Bt

w

w

m

Tii

td

clscdTpfsadtJsTbl

tso

6

rary smooth curved reflector, given by z � z�x, y�, atoint r � �x, y, z�x, y�� is described by

1� x, y, z0� � x � � z0 � z� x, y��h1� x, y, s1, s2, s3�

h3� x, y, s1, s2, s3�� ,

2� x, y, z0� � y � � z0 � z� x, y��h2� x, y, s1, s2, s3�

h3� x, y, s1, s2, s3�� ,

3� x, y, z0� � z0, (6)

here

h1� x, y, s� � � x � s1��1 � zx2 � zy

2� � 2zx� zy� y � s2�

� s3 � z�,

h2� x, y, s� � � y � s2��1 � zx2 � zy

2� � 2zy� zx� x � s1�

� s3 � z�,

h3� x, y, s� � � z � s3��1 � zx2 � zy

2� � 2� zx� x � s1�

� zy� y � s2��. (7)

ecause we used the reflection law only one time tobtain Eqs. �6�, these equations describe the evolu-ion of the light rays that have experienced only oneeflection before leaving the system. In this papere assume that the parameters that characterize the

ptical system under consideration are such that thisestriction is satisfied by the reflected light rays.

Observe that, from a mathematical point of view,qs. �6�, with s fixed, represent a differentiable mapetween two three-dimensional subsets of �3, wherex, y, z0� are the coordinates of the domain space andT1, T2, T3� are the coordinates of the target space.o obtain the caustic associated with the reflected

ight rays, we introduce the definitions of critical andaustic sets of a differentiable map between three-imensional spaces. Let f:�3� be a differentiableap, with � and � differentiable manifolds. Then

ll the points in � such that f is not locally one to onere referred to as its critical set, and the image of theritical set is referred to as the caustic set of f.19–21 If

and � are three-dimensional differentiable mani-olds with local coordinates �x1, x2, x3� and �y1, y2, y3�,espectively, then f is given by yi � fi�xj�, where i, j �, 2, 3. Therefore the critical set is obtained fromhe condition

J� x1, x2, x3� �� y1, y2, y3�

�� x1, x2, x3�� 0. (8)

In accordance with the above definition, the set ofoints in the domain space such that the map giveny Eqs. �6� is not locally one to one �that is, the criticalet� is obtained from the condition

��T1, T2, T3�
J� x, y, z0� �� x, y, z0�
� 0. (9)p


y use of Eqs. �6� and �7�, a direct computation showshat Eq. �9� is equivalent to

H2�x, y��z0 � zh3

�2

� H1�x, y��z0 � zh3

�� H0�x, y� � 0,

(10)

here

H2 � h � ��h� x� � ��h

� y�� ,

H1 � h � �� r� x� � ��h

� y� � ��h� x� � � �r

� y�� ,

H0 � h � �� r� x� � � �r

� y�� , (11)

ith r � �x, y, z�x, y�� and h � �h1, h2, h3�.From Eq. �10� we find that the critical set of theap given by Eqs. �6� is given by

z0 � z0�� x, y� � z � h3��H1 � �H12 � 4H2 H0�

2H2�1�2

.

(12)

herefore the caustic set or simply the caustic, whichs obtained when we substitute Eq. �12� into Eqs. �6�,n vector form is given by

Tc�� x, y� � r � ��H1 � �H12 � 4H2 H0�

1�2

2H2�h. (13)

It is important to note that Eq. �13� is equivalent tohat obtained by Shealy and Burkhard3,4 by use of aifferent procedure.To understand the geometric meaning of the

austic associated with the evolution of reflectedight rays by any arbitrary surface, when the pointource is located at any position of the space, weompute dT1dT2dT3. By using Eqs. �6�, we obtainT1dT2dT3 � �J�x, y, z0��dxdydz0 because dT3 � dz0.hen dT1dT2 � �J�x, y, z0��dxdy. Now consider theencil of light rays reflected by the differential sur-ace dxdy of the reflector. When z0 � z, the cross-ectional area of this pencil of rays is exactly dxdy,nd as the light rays evolve this area is given byT1dT2 � �J�x, y, z0��dxdy. If �x, y, z0� belongs tohe critical set of the map given by Eqs. �6�, then�x, y, z0� � 0, and therefore in that case the cross-ectional area of the pencil of rays collapses to zero.his result shows that the caustic surface is definedy the focusing region associated with the reflectedight rays.

Now we show that the derived results reduce tohose obtained in Ref. 18 for any reflector with axialymmetry when the point source is located on theptical axis, which will be on the z axis. For this
articular case, s1 � 0 � s2, and z�x, y� � z�� where

r

w

Bf

Btb

S

� �x2 y2�1�2. By using these conditions, we caneduce Eqs. �7� to

h1 � �x��1 � z

2� � 2z�s3 � z��,

h2 � �y��1 � z

2� � 2z�s3 � z��,

h3 � � z � s3��1 � z2� � 2z, (14)

here z � � z��. By using Eqs. �6� and �14� we obtain

T1 �x � � z0 � z��1 � z

2� � 2z�s3 � z�

2z � �s3 � z��1 � z2�� ,

T2 �y � � z0 � z��1 � z

2� � 2z�s3 � z�

2z � �s3 � z��1 � z2�� ,

T3 � z0. (15) the equations for the caustic surface:

b1

c

wccg

3 3

y use of Eqs. �14�, a direct computation shows that,or this case, Eqs. �11� are reduced to

H0 � �1 � z2�� z � s3 � z�,

H1 � �1

�2�1 � z2��2z � �s3 � z�2z � 3z

� �s3 � z��1 � z2 � s3z � zz��,

H2 � �1

��1 � z2��2�s3 � z� z � ��1 � z

2��z

� z � zz2 � z

3 � 22z � 2s32z � 2z2z

� s3�1 � z2 � 4zz��. (16)

z0 � z0�� x, y� � � c2

1 � u� � ( c32 � �1�1 � u��1 �

c2 a1 � b1 u
z0 � z0� x, y� � �1 � u� � �c1 � d1 u� ,
2

y substituting Eqs. �16� into Eq. �13�, we find thathe critical set, for this axial symmetry case, is giveny18

z0 � z0� x, y� � z ��2z � � z � s3��1 � z

2��

2z�s3 � z� � �1 � z2�

,

(17)

z0 � z0�� x, y� � z �

�s3 � z � z�� z � s3��1 � z2 � z��

z�1 � z2� � 2s3

2z � 2z2z � ��z � z3 � 2z�

.

(18)

ubstituting Eqs. �17� and �18� into Eqs. �15�, we find

Thus we have shown that our general results giveny Eqs. �12� and �13� reduce to those obtained in Ref.8.For a conic reflector, we obtain the critical and the

austic sets by means of Eqs. �17�–�20� by taking22

z�� c2

1 � �1 � �1 � �c22�1�2 , (21)

here c is the paraxial curvature and is the coniconstant. In this case, by carrying out a directalculation one can show that the critical set isiven by

��s3�1 � �2 � �c22�� 1 � 2u�c2

s3 � c22� � 2��1 � �1 � �cs3�c22) , (22)

T1c� � 0,

T2c� � 0,

T3c� � z ��2z � � z � s3��1 � z

2��

2z�s3 � z� � �1 � z2�

, (19)

T1c �2x � 2z

3 � 3z � 2z2�s3 � z� � �s3 � z�2� z � z�

z�1 � z2� � 22z � �1 � z

2��s3 � z� � 2�s3 � z�2z� ,

T2c �2y � 2z

3 � 3z � 2z2�s3 � z� � �s3 � z�2� z � z�

z�1 � z2� � 22z � �1 � z

2��s3 � z� � 2�s3 � z�2z� ,

T3c ��22z

2 � z�3 � z2��s3 � z� � �1 � z

2��s3 � z�2

z �1 � z 2� � 22z � �1 � z 2��s � z� � 2�s � z�2z. (20)

� u2c

(23)


w

T

w

Wwticca

3tmpr

sov

i

Fm3

Fbs

6

here u, a1, b1, c1, and d1 are given by

u � �1 � �1 � k�c22�1�2,

a1 � c24 � c4 6 � c2�4 � c2�1 � �2� c22 � 5��s3

� 2�1 � c2�3 � 2 �2 � c4�1 � ��2 � �4�s32,

b1 � 2c24 � 4c2��1 � c2�1 � �2�s3 � �2 � �5

� 3 �c22 � �1 � ��2 � �c44�s32,

c1 � �2s3 � c(4s32 � c24��1 � 2c�1 � �s3�

� 2�3

� 2cs3�1 � 2 � �1 � �cs3��) ,

d1 � �2s3 � c�4s32 � 32�1 � c��1 � �s3�

� c24�1 � c�1 � �s3��. (24)

ig. 2. Part of the caustic given by Eqs. �25� for a hyperbolicirror with c � �1�2415��1�mm�, � �2, D � 1470 mm, and s3 �

0�c, 1.5�c, and 3.3�c.

he caustic set is given by � � 2�, with D the diameter of the conic mirror�. By

cacr

3c 0

here a2 and b2 have the following expressions:

a2 � c2�c2 2 � 3� � 2s3�1 � 2s3 c � 2c22s3��1

� 2 � � �c22 � s3 c��1 � ��,

b2 � c3 4��1 � cs3�1 � �� 2s3�1 � 2cs3�

� 3c2��1 � cs3�1 � ��. (27)

e observe that Eqs. �25� contribute to the causticith a straight-line segment, whereas Eqs. �26� give a

wo-dimensional surface of revolution with a singular-ty of the cusp type. In Fig. 2 we show the part of theaustic given by Eqs. �25� for a hyperbolic mirror with� �1�2415��1�mm�, � �2, diameter D � 1470 mm,
nd the point source is located at �0, 0, s3� with s3 �

0�c, 1.5�c, and 3.3�c, respectively. In Fig. 3 we showhe part of the caustic given by Eqs. �26� for the mirrorentioned above. Figure 3 shows clearly that the

art of the caustic given by Eqs. �26� is a surface ofevolution with a singularity of the cusp type.

To verify that Eqs. �26� describe a surface with aingularity of cusp type, we make a series expansionn them. To this end, we start with a change ofariables

x � cos �,

y � sin �, (28)

n Eqs. �26� �in these Eqs. �28�, 0 � � D�2 and 0 �

ig. 3. Part of the caustic surface given by Eqs. �26� for a hyper-olic mirror with c � �1�2415��1�mm�, � �2, D � 1470 mm, and3 � 30�c, 1.5�c, and 3.3�c.

arrying out a power series expansion of T1c, T2c,nd T3c around c � 0, we find that the part of theaustic given by Eqs. �26� to third order in c, iseduced to

T1c � 2c�1 � 2cs3 � �1 � ��cs3�2

s3�1 � 2cs3��3 cos �,

T2c � 2c�1 � 2cs3 � �1 � ��cs3�2

s3�1 � 2cs3��3 sin �,

T3c ��2cs3 � 1�s3 � 3c�1 � 2cs3 � �1 � ��cs3�

2�2

�1 � 2cs3�2 .

T1c� � 0,

T2c� � 0,

T3c� � z0�� x, y�, (25)

T1c� x, y� � 2xc2�1 � 2cs3 � �1 � �c2s32��1 � �c22 � 2�1 � u��

a2 � b2 u ,

T2c� x, y� � 2yc2�1 � 2cs3 � �1 � �c2s32��1 � �c22 � 2�1 � u��

a2 � b2 u ,

T � x, y� � z � x, y�, (26)

(29)

B

w

Fas

3

ADsofttbmrTot�rfiflst

w

ww

sf

wt

w

BE

wtatrs�t

ap�ocgct�ocsotd

y use of Eqs. �29�, a direct computation shows that

T1c2 � T2c

2

�2 � �2cs3 � 1��s3 � �2cs3 � 1�T3c�

3c�1 � 2cs3 � �1 � ��cs3�2� 3

,

(30)

here � is given by

� � 2c�1 � 2cs3 � �1 � ��cs3�2

s3�1 � 2cs3�� . (31)

or each fixed value of c, , and s3, Eq. �30� describestwo surface of revolution around the z axis with a

ingularity of cusp type.

. Calculation of the Disk of Least Confusion

s pointed out above, Conrady2 showed that theLC is located at the intersection of the caustic

urface defined by intermediate rays from one sidef the aperture with the marginal rays arrivingrom the other side of the aperture. In our casehis definition for the DLC is equivalent to our ob-aining the intersection of least area between theranch of the caustic described by Eqs. �26� and thearginal surface �which is constituted by the light

ays reflected by the rim of any conic mirror�.herefore to compute the DLC it is necessary tobtain an expression for the marginal surface. Tohis end, Eqs. �21� and �28� are substituted into Eqs.15�. After that, in the resultant equations iseplaced by m, � by �m, and z0 by z0m. Thus wend that the marginal surface linked with the re-ected rays by any conic mirror when the pointource is located at any arbitrary point on the op-ical axis is given by

T1m � �1 � �z0m �cm

2

1 � um��am

bm��m cos �m,

T2m � �1 � �z0m �cm

2

1 � um��am

bm��m sin �m,

T3m � z0m, (32)

here

m � � xm2 � ym

2�1�2,

um � �1 � �1 � �c2m2�1�2,

am � �1 � 2cs3 � c2m2��1 � um� � 2�cs3�1 � �

� 1�c2m2,

bm � �1 � 2um � c2m2�cm

2 � �1 � �2

� �c2m2�s3�1 � um�, (33)

ith zm � z�m� � z0m � �, 0 � �m � 2�, m � D�2
ith D being the diameter of the conic mirror. c
2

If c, , and s3 are given, the intersections amongurfaces yielded by Eqs. �26� and �32� are obtainedrom the following conditions:

T1c�, �� T1m�m, �m, z0m�,

T2c�, �� T2m�m, �m, z0m�,

T3c�, �� T3m�m, �m, z0m�, (34)

here � takes values in a 2� length interval. Equa-ions �34� are equivalent to

Tc�� Tm�m, z0m�, (35)

z0�� z0m, (36)

tan � � tan �m, (37)

here

Tc � �T1c2 � T2c

2�1�2, (38)

Tm � �T1m2 � T2m

2�1�2. (39)

y substituting Eq. �36� into Eq. �35� and by usingqs. �26� and �32�, we obtain

�2c3

�1 � 2cs3 � �1 � �c2s32��1 � �c22 � 2�1 � u��

a2 � b2 u �� m�1 � � c2

1 � u�

a1 � b1uc1 � d1u

�cm

2

1 � um��am

bm�� ,

(40)

tan � � tan �m, (41)

here 0 � �m � 2�. Equations �40� and �41� givehe conditions that bring about the intersectionsmong the part of the caustic given by Eqs. �26� andhe marginal surface given by Eqs. �32� to any coniceflector when the point source is located at any po-ition on the optical axis. By solving Eqs. �40� and41� for and � for each value of �m, we can computehe DLC in an exact way.

From a geometric point of view, Eqs. �40� and �41�ccept two nontrivial sets of solutions. A direct com-utation shows that one set of solutions is given by� � �m, � m� with 0 � �m � 2�. When this setf solutions is substituted into Eqs. �32�, we obtain aircle, which corresponds to the fact that the mar-inal surface given by Eqs. �32� and the part of theaustic given by Eqs. �26� are tangent to each other athis circle. The other set of solutions is given by ��

m �, � 1� with 0 � �m � 2�, where 1 isbtained in a numerical way from Eq. �40� with theondition 0 � 1 � m. When this second set ofolutions is substituted into Eqs. �32�, we obtain an-ther circle that is the DLC of any conic mirror whenhe point source is located on the optical axis. Airect computation shows that the radius and the
enter of the best image produced by the optical sys-

ti

w

tr

e�

t

Iioa

ac � �1�2415��1�mm�, � �2, and m � D�2 � 735 mm. ac � �1�2415��1�mm�, � �0.5, and m � D�2 � 735 mm.

6

em are given by R � Tc�1� and �0, 0, C � z0�1��,.e.,

here

u1 � u�1�, a1 � a1�1�, b1 � b1�1�,

c1 � c1�1�, d1 � d1�1�. (44)

Now we obtain the third-order approximations ofhe radius and the center of the DLC of any conic

R � 2c��1 � 2cs3 � �1 � �c2s

a

C � � c12

1 � u1� � � a1 � b1 u1

c1 � d1 u1� ,

eflector. To this end, we perform a power series t

ac � �1�2415��1�mm�, � �1, and m � D�2 � 735 mm.


xpansion around c � 0 and cm � 0 in Eqs. �40�,42�, and �43�. Under this approximation we find

hat the condition given in Eq. �40� is reduced to

�m�32 � m2�� 23�. (45)

t is important to emphasize that the condition givenn Eq. �45�, which provides the intersections, to third-rder approximation �among the marginal surfacend that part of the caustic with a singularity of cusp

�1 � �c212 � 2�1 � u1��

˜2 u1

�13, (42)

(43)

Table 1. Exact and Approximate Results for the Radius and the zCoordinate of the Center of the DLC as Functions of the Position of the

Point Source for a Hyperbolic Reflectora

s3 R Ra C Ca

1350 145.4730 111.0700 18,633.5854 16,625.52591523 73.8167 68.0696 7673.7130 7448.39221746 52.8709 51.1347 4795.2109 4732.26291969 43.7194 42.8802 3698.7549 3668.67032192 38.2524 37.6998 3120.5682 3102.21162415 34.4910 34.0406 2763.3994 2750.543522,832 11.2437 10.7856 1332.4752 1331.052343,249 10.1441 9.7050 1291.3869 1291.386963,666 9.7515 9.3203 1278.8722 1277.668384,083 9.5499 9.1230 1271.8873 1270.7118104,500 9.4272 9.0030 1267.6647 1266.5064� 8.9225 8.5101 1250.5300 1249.4400

ype� does not depend on either c or .


Point Source for a Parabolic Reflectora

s3 R Ra C Ca

1350 29.8221 30.4475 12,882.7061 12,861.11841523 26.7847 26.9889 6483.7222 6471.02021746 23.6782 23.5418 4300.5268 4291.32661969 21.2187 20.8756 3395.7344 3388.24932192 19.2228 18.7518 2900.6874 2894.28582415 17.5705 17.0203 2588.4026 2582.771722,832 1.9833 1.8003 1284.8220 1284.294443,249 1.0510 0.9504 1247.2779 1247.000063,666 0.7150 0.6456 1234.2765 1234.087984,083 0.5417 0.4888 1227.6805 1227.5378104,500 0.4361 0.3933 1223.6919 1223.5771� 0 0 1207.5000 1207.5000


Point Source for an Ellipsoidal Reflectora

s3 R Ra C Ca

1350 8.6216 9.8638 11,026.9923 10,978.91471523 6.1561 6.4485 5977.6633 5982.33421746 9.6886 9.7454 4070.9547 3248.03891969 10.0356 9.8733 3250.0240 3248.03892192 9.5730 9.2779 2792.8894 2790.32292415 8.8796 8.5102 2501.5549 2498.885922,832 3.0548 2.6924 1259.8901 1260.915643,249 3.9054 3.4269 1223.5014 1224.806663,666 4.2140 3.6917 1210.8917 1212.297884,083 4.3734 3.8282 1204.4928 1205.9507104,500 4.4707 3.9115 1200.6228 1202.1124� 4.8734 4.2550 1184.9100 1186.5300


Point Source for a Spherical Reflectora

s3 R Ra C Ca

1350 39.7208 50.1751 9561.7356 9096.71101523 12.9837 14.0918 5518.9932 5493.64821746 3.9867 4.0510 3107.7688 3850.39031969 1.1488 1.1290 3107.7688 3107.82842192 0.2039 0.1961 2686.3173 2686.35972415 0 0 2415 241522,832 8.4076 7.1851 1234.1295 1237.536543,249 9.1785 7.8042 1198.9058 1202.613163,666 9.4602 8.0291 1186.6913 1190.507584,083 9.6061 8.1453 1180.4913 1184.3637104,500 9.6953 8.2163 1176.7410 1180.6478� 10.0656 8.5101 1161.5100 1165.5600

32��

2 � b

ac � �1�2415��1�mm�, � 0, and m � D�2 � 735 mm.

s

Tcogmacet

Fc m

the center of the DLC for the same reflectors.

ac � �1�2415��1�mm�, � 2, and m � D�2 � 735 mm.

2

By solving Eq. �45� for , we find that the physicalolutions are given by

� m, �m

2. (46)

he first solution � m describes the fact that theaustic and the marginal surfaces are tangent to eachther at a circle, and the second solution � m�2ives the DLC. To obtain the third-order approxi-ations Ra and Ca of R and C, respectively, we obtainseries expansion to third-order around c � 0 and

m � 0 in Eqs. �42� and �43�. Then, in the resultingquations, we substitute � m�2 to obtain Ra and Cahat are given by

of the DLC as a function of the position of the point source when. �c� and �d� Similar results are presented for the z coordinate of

m3, (47)

�m2s3

2 � c��3m2 � 8s3

2�

2cs3�2 . (48)

ig. 4. �a� and �b� Exact results for five conic reflectors for the radius� �1�2415��1�mm�; � �2, �1, �0.5, 0, 2; and � D�2 � 735 mm


Point Source for an Oblate Spheroidal Reflectora

s3 R Ra C Ca

1350 124.7206 211.4203 5852.8671 1567.89611523 79.2989 96.2533 4037.3948 3538.90421746 56.8681 59.2366 3061.1533 2968.51761969 46.7303 45.1382 2567.2090 2546.98652192 41.3088 38.0920 2268.8732 2270.50832415 38.1156 34.0406 2069.1193 2079.456522,832 33.8826 25.1558 1120.9165 1144.020843,249 34.3675 25.3133 1090.4584 1113.839463,666 34.5535 25.3784 1079.8643 1103.346884,083 34.6516 25.4136 1074.4803 1098.0156104,500 34.7122 25.4356 1071.2216 1094.7892� 34.9679 25.5305 1057.9700 1081.6700

Ra � �1 � 2cs3 � c2�1 � �s32

4s3�1 � 2cs3��c

Ca ��4s3 � 6c2m

2s3 � 3c3�1 �

4�1 �


ccl1gapdRsl�pt1

4

NesemtFtwtai�sr 02

cttrssi

rs4nFteaimemi2ra

trg

5

IdDwwoatrttoctp�rctratw

btoo

tSny4kdtOrdd

R

6

Before presenting some particular examples, wearry out an analysis for well-known particularases �1� For the spherical mirror and point sourceocated at its center of curvature, i.e., � 0 and s3 ��c, a direct computation shows that the conditioniven in Eq. �40� is an identity, the radius is R � 0,nd C � 1�c. �2� For the parabolic mirror andoint source at infinity, i.e., � �1 and s3 � �, airect calculation shows that Eq. �40� is an identity,� 0, and C � 1�2c. �3� Ellipsoidal mirror with its

emimajor axis on the z axis and a point sourceocated at one of the foci. The foci are at �0, 0, �1 �

� ��1 �c��, where �1 � � 0. A direct com-utation shows that �a� if s3 � 1 �� 1 �c,hen R � 0 and C � 1 � �� 1 �c; and �b� if s3 �� 1 �c, then R � 0 and C � 1 �� 1 �c. In both cases Eq. �40� is an identity.

. Examples

ow we apply the obtained results to particularxamples, i.e., we calculate the radius and the po-ition �z coordinate� of the center of the DLC in anxact and approximate way for five specific conicirrors and we discuss the differences between

hem. The results were obtained as follows.irst a computer program in MATHEMATICA was writ-

en to solve Eq. �40� numerically �with s3 variedhen c, m � D�2, and are given� for � 1; after

hat their values were substituted into Eqs. �42�nd �43� to obtain R and C, respectively. Approx-mate results were obtained by use of Eqs. �47� and48�. In Tables 1–5 we present the numerical re-ults for R, C, Ra, and Ca for the following five coniceflectors: �1� hyperbolic, � �2; �2� parabolic,� �1; �3� ellipsoidal, � �0.5, �4� spherical, �

; and �5� oblate spheroidal, � 2; when c � �1�415��1�mm� and m � D�2 � 735 mm.From the results presented in Tables 1–5 we con-

lude that there are substantial differences be-ween the exact and the approximate calculations ofhe radius of the DLC for hyperbolic, oblate sphe-oidal, and spherical reflectors when the pointource tends to R�2 � 1207.5 mm and for the oblatepheroidal reflector when the point source tends tonfinity.

Finally, in Figs. 4�a� and 4�b� we show the exactesults for R as a function of the position of the pointource for the five conic reflectors. In Figs. 4�c� and�d� similar results are presented for the z coordi-ate of the center of the DLC for the same reflectors.rom Tables 1–5 and Figs. 4�a� and 4�b� we observe

hat when s3 � �1350 mm, 1414.16 mm� for thesexamples the conic reflector that gives the best im-ge as a function of the position of the point sources the ellipsoidal reflector. When s3 � �1414.14

m, 8245.33 mm� the best image is given by thellipsoidal and spherical reflectors. For 8245.33m � s3 � 20,000 mm we observe that the best

mage is given by the ellipsoidal reflector. For s3 �0,000 mm the best image is given by the paraboliceflector. Finally, from Tables 1–5 and Figs. 4�a�
nd 4�b� when 1350 mm � s3 � 1400 mm, we can see

hat the worst image is given by the hyperboliceflector, and for s3 � 1400 mm the worst image isiven by the oblate spheroidal reflector.

. Conclusions

n this paper we have described the necessary proce-ure to estimate the radius and the center of theLC, performed by use of an exact technique dealingith geometrical optics, applied to a conic reflectorhen the point source is placed at any position of theptical axis. In addition, by using the third-orderpproximation of geometrical optics, we have ob-ained the approximate analytic expression for theadius and the center of the DLC for any conic reflec-or when the point source is placed at any position onhe optical axis. The exact and approximate resultsbtained were applied to particular cases. We havealculated the radius and the z coordinate of the cen-er of the DLC as a function of the position of theoint source when c � �1�2415��1�mm�; � �2, �1,0.5, 0, 2; and m � D�2 � 735 mm. From the

esults obtained for these particular cases we con-lude that there are substantial differences betweenhe exact and the approximate calculations of theadius of the DLC for hyperbolic, oblate spheroidal,nd spherical reflectors when the point source tendso R�2 � 1207.5 mm and for the oblate sphericalhen the point source tends to infinity.Finally, it is important to emphasize that it could

e interesting to make a study of the deformation ofhe DLC for any conic reflector due to a displacementf the point source out of the optical axis as a functionf , c, and its diameter D.

The authors are deeply indebted to the referees ando Al Janis for their constructive comments. G.ilva-Ortigoza and O. de Ita Prieto acknowledge fi-ancial support from the Consejo Nacional de CienciaTecnologıa �Mexico� through grants 33725-E and

4515-F. G. Silva-Ortigoza and J. Castro-Ramos ac-nowledge financial support from Consejo Nacionale Ciencia y Technologia �Sistema Nacional de Inves-igadores �Mexico��. Finally, G. Silva-Ortigoza and. de Ita Prieto also acknowledge the partial support

eceived from Vicerrectoria de Investigacion y Estu-ios de Posgrado-Benemerita Universidad Autonomae Puebla through grant II-161-04�EXC�G.

eferences1. R. Kingslake, Lens Design Fundamentals �Academic, New

York, 1978�.2. A. E. Conrady, Applied Optics and Optical Design �Dover, New

York, 1985�, Pt. 1.3. D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irra-

diance for reflection and refraction from an ellipsoid, ellipticparaboloid, and elliptic core,” Appl. Opt. 12, 2955–2959 �1973�.

4. D. L. Shealy and D. G. Burkhard, “Flux density ray propaga-tion in discrete index media expressed in terms of the intrinsicgeometry of the deflecting surface,” Opt. Acta 20, 287–301�1973�.

5. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and thestructure of the geometrical image,” J. Opt. Soc. Am. 66, 795–800 �1976�.

6. I. H. Schroader, “The caustic test,” in Amateur Telescope Mak-

1

1

1

1

1

1

1

1

1

1

2

2

2

ing, A. G. Ingalls, ed. �Scientific American, New York, 1974�,Vol. 3, p. 429.

7. P. S. Theocaris, “Reflected shadow method for the study ofconstrained zones in cracked plates,” Appl. Opt. 10, 2240–2247�1971�.

8. P. S. Theocaris, “Multicusp caustics formed from reflections ofwarped surfaces,” Appl. Opt. 27, 780–789 �1988�.

9. D. L. Shealy and D. G. Burkhard, “Analytical illuminancecalculation in a multi-interface optical system,” Opt. Acta 22,485–501 �1975�.

0. D. L. Shealy, “Analytical illuminance and caustic surface calcu-lations in geometric optics,” Appl. Opt. 15, 2588–2596 �1976�.

1. D. G. Burkhard and D. L. Shealy, “Simplified formula for theilluminance in an optical system,” Appl. Opt. 20, 897–909�1981�.

2. P. S. Theocaris and E. E. Gdoutos, “Surface topography bycaustics,” Appl. Opt. 15, 1629–1638 �1976�.

3. P. S. Theocaris and T. P. Philippides, “Possibilities of reflectedcaustics due to an improved optical arrangement: some fur-ther aspects,” Appl. Opt. 23, 3667–3675 �1984�.

4. P. S. Theocaris, “Properties of caustics from conic reflectors.
1. Meridional rays,” Appl. Opt. 16, 1705–1716 �1977�.
2

5. P. S. Theocaris and J. G. Michopoulos, “Generalization of thetheory of far-field caustics by the catastrophe theory,” Appl.Opt. 21, 1080–1091 �1982�.

6. M. V. Berry and C. Upstill, Progress in Optics, E. Wolf, ed.�North-Holland, Amsterdam, 1980�, Vol. XVIII, p. 259.

7. A. Cordero-Davila and J. Castro-Ramos, “Exact calculation ofthe circle of least confusion of a rotationally symmetric mir-ror,” Appl. Opt. 37, 6774–6778 �1998�.

8. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Davila,“Exact calculation of the circle of least confusion of a rota-tionally symmetric mirror. II,” Appl. Opt. 40, 1021–1028�2001�.

9. V. I. Arnold, Catastrophe Theory �Springer-Verlag, Berlin,1986�.

0. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singu-larities of Differentiable Maps �Birkhauser, Boston, Mass.,1995�, Vol. 1.

1. V. I. Arnold, Mathematical Methods of Classical Mechanics�Springer-Verlag, Berlin, 1980�.

2. D. Malacara, Optical Shop Testing, 2nd ed. �Wiley-
Interscience, 1992�, Appl. 1, pp. 743–745.

Date post:	06-Oct-2016
Category:	Documents
Upload:	gilberto
View:	217 times
Download:	2 times

Computation of the Disk of Least Confusion for Conic Mirrors

Documents