2252 J. Opt. Soc. Am. A/Vol. 27, No. 10 /October 2010 Avendaño-Alejo et al.
Properties of caustics produced by a positive lens:meridional rays
Maximino Avendaño-Alejo,1,* Luis Castañeda,2 and Ivan Moreno3
1Universidad Nacional Autónoma de México, Centro de Ciencias Aplicadas y Desarrollo Tecnológico,Apdo. Postal 70-186, C. P. 04510, Distrito Federal, México
2Benemérita Universidad Autónoma de Puebla, Instituto de Física, Apdo. Postal J-48, C. P. 72570, Puebla, México3Universidad Autónoma de Zacatecas, Facultad de Física, C. P. 98060, Zacatecas, México
. INTRODUCTIONhe optics of caustics has been a topic of research forany years. Many works have been recently published
oncerning caustics by refraction, also known as diacaus-ic [1,2], and caustics by reflection, also known as cata-austic [3–5].
The caustic can be defined as the envelope of a systemf orthotomic rays and also as [6] the locus of the princi-al centers of curvature of a wavefront. A normal congru-ncy of rays incident on either a refracting or a reflectingurface gives rise to an aggregate of refracted or reflectedays, respectively, which also constitutes a normal con-ruence.
Alternatively, rays that originate from a point source inn isotropic medium form a normal congruence. In otherords, if equal optical paths are measured along each ray
rom the source, the surface constructed by the end pointsill be normal to all of the rays in the congruence. These
urfaces are the phase fronts of the wave system, forhich the rays are the geometrical optics approximation
7].Stavroudis and collaborators presented a general solu-
ion to the eikonal equation for propagation in a homoge-eous medium [8–10]. A general solution was presented
n Cartesian coordinates and expressed in terms of an ar-itrary function, which was denominated as the-function. As noted by Stavroudis [10], the k-functionontains all information about the aberrations introducedo the wavefront by the refracting (reflecting) surface. Aolution of the eikonal equation gives the equations forhe wavefront and caustic surfaces either after reflectionr after refraction. Recently, in [11] an explicit evaluationf the k-function was presented for both cases of refrac-ion and reflection of a plane wave from a rotationallyymmetric surface, showing an agreement with previousesults in the literature. The authors have also evaluatednd analyzed the caustic surfaces and wavefront pro-
uced by the first plane–aspheric lens of a Keplerian laseream shaper. On the other hand, by using the concept ofifferentiable map between two three-dimensional sub-ets of R3 or, in other words, between the coordinates ofhe object space and the coordinates of the image space, in12–14] the authors have obtained a formula for the caus-ic associated with the reflected light rays and have gen-ralized the study to rotationally symmetric mirrors with
source placed at an arbitrary location on the opticalxis. Applying these results, they have provided a formulaor the circle of least confusion (CLC) for a spherical re-ector. Alternatively, a formula for illuminance (flux den-ity) in [15] was derived in a vectorial manner with theormal use of differential geometry. Since the caustic sur-ace can also be defined as the locus of points for whichhe flux density is infinite or, in other words, as the locusf singularities of the flux density of the emanating radia-ion, the authors have also obtained a formula for theaustic surface produced by refraction. It is important toomment that in all these papers [6–15] the caustic sur-ace is considered as a three-dimensional problem. From aifferent point of view, with a theory based on the proper-ies of the Legendre transformation, it is possible to con-ect the caustic surface with the transverse aberration ofhe optical system in the meridional plane as was re-orted in [16].Although the caustic either by reflection or refraction is
well known subject, the contribution in this work will beo study some properties of the caustic surface and its re-ationship with the Seidel coefficients caused by refrac-ion on a spherical refracting surface (plano-convex andonvex-plane lenses) exclusively in a meridional plane.dditionally, a formula for the principal surface is pro-ided. Furthermore, we apply this formula to evaluate theLC by a positive lens following a procedure similar to
17], where a formula for the CLC for a concave sphericalirror is obtained. Recently, by using the caustic’s equa-
010 Optical Society of America
tsc
2Wwwctsspcawstf1TcfWpboa
Tirg
Tdm(ew
wv(t
TmwwtaF−e=w
Fl
Avendaño-Alejo et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. A 2253
ion it has been possible to design Hartmann’s nullcreens to evaluate quantitatively the properties of plano-onvex spherical lenses under test [18].
. PRELIMINARIESe define that the Z axis is parallel to the optical axis ande assume that the Y–Z-plane is the plane of incidence,hich is a cross section of a spherical refractor of radius R
entered at C, and the origin of the system O is placed athe vertex of the plano-convex lens. There is rotationalymmetry about the Z axis. Without loss of generality weuppose that light rays enter from the left and that theoint source is placed at infinity, and a plane wave is in-ident on the lens along the optical axis. We consider ide-lly a bundle of rays crossing the plane face of the lensithout being deflected, and transported to the spherical
urface. P1 is an image point formed by the intersection ofhe refracted rays A and B, and P2 is the image pointormed by the refracted rays B and C as is shown in Fig.. In this way, every refracted ray cuts the one next to it.he locus of their points of intersection is the requiredaustic by refraction, or diacaustic. We consider the re-racted ray P0P2 as representative of all refracted rays.
ithout loss of generality we have assumed that P0 is aoint at the spherical surface whose coordinates are giveny P0= �t−R�1−cos �i� ,R sin �i�, where t is the thicknessf the lens. �i and �a are the incident and refractionngles, respectively. Therefore, P0P2 can be written as
y cos��a − �i� + z sin��a − �i� = �t − R�sin��a − �i� + R sin �a.
�1�
his is a parametric family of refracted rays as is shownn Fig. 2. The diacaustic is the envelope of this family ofays. As we know, �a is related through Snell’s law to �iiven by
A
B
C
na
ni na
t
R
C
O
Y
h
qi1
qi
qi2
P0
Spheric
Opti
IncidentRays
qa1
qa
qa2
ig. 1. Process of refraction produced by a plano-convex lensocated at infinity.
�a = arcsin� ni
nasin �i�,
��a
��i=
ni cos �i
�na2 − ni
2 sin2 �i
. �2�
his means that an incident ray is traveling from a me-ium with index of refraction ni (commonly an isotropicedium) to a different medium with index of refraction na
which is usually the index of refraction of the air). Differ-ntiating Eq. (1) with respect to �i and reducing further,e have
− y sin��a − �i� + z cos��a − �i�
= �t − R�cos��a − �i� +���a/��i�R cos �a
��a/��i − 1, �3�
here ��a /��i is given in Eq. (2). In order to obtain the en-elope of the family of refracted rays [19], we solve Eqs.3) and (1) for �z ,y� where we have substituted its deriva-ive by using Eq. (2); in this way finally we get
z��i� = t − R + � niR
na2�ni
2 − na2���nina
2 cos3 �i
+ �na2 − ni
2 sin2 �i�3/2,
y��i� =Rni
2 sin3 �i
na2 . �4�
his is the diacaustic for a plano-convex lens produced byeridional rays as a function of the angle of incidence,hen the point source is placed at infinity, or—in othersords—Eq. (4) gives the coordinates of the locus of points
hat parametrically represent the diacaustic produced bypositive lens in a meridional plane as a function of �i.
or �i→0, Eq. (4) is reduced simply to z�0�= t+ �Rna� / �nina�= f, where f is a singular point, and is related to theffective focal length in the following way: EFL= f− t�Rna� / �ni−na� according to our frame of reference,here we have considered that R�0.
Z
EFL
P1
P2
face
is
Paraxial Plane
Refracted Rays
associated parameters by considering that the point source is
al Sur
cal Ax
f
and its
avtdh=
Tartf�ttpan=af
wf
wp
pid
wpc=pahatt�
f=a=dsta
3PIbgdktlpcw
Ft
2254 J. Opt. Soc. Am. A/Vol. 27, No. 10 /October 2010 Avendaño-Alejo et al.
Throughout this paper we consider h as a height vari-ble of each incoming ray and H as a predeterminatealue for one incident ray on the optical surface. Alterna-ively, we can write the diacaustic equation in a slightlyifferent way by considering the following relationship:=R sin �i; therefore, by substituting into Eq. (4) for �iarcsin�h /R� and reducing further, this becomes
z�h� = t − R +ni�nina
2�R2 − h2�3/2 + �R2na2 − h2ni
2�3/2
na2�ni
2 − na2�R2
,
y�h� =h3ni
2
R2na2 . �5�
his is the equation for the diacaustic as a function of anrbitrary height for each one of the incident rays in a me-idional plane. We also consider h as the entrance aper-ure. From Eq. (5) the first radical for z�h� leads to theact that h� R; the second radical demands that hni
Rna or, in other words, if hni= Rna, there are rayshat undergo total internal reflection. Thus, for an arbi-rary lens the condition for total internal reflection im-oses that sin��c�=na /ni=h /R, where �c is the criticalngle according to Snell’s law, and where we consider thati�na. When the height for a marginal ray satisfies hRna /ni=Hca, where the subscript ca means criticalngle, then the caustic surface touches the spherical sur-ace as is shown in Fig. 2.
Considering that h� R, physically this means thate are regarding the paraxial region. Expanding Eq. (5)
or z�h� in a Taylor’s series to first order in h2 yields
zp�h� � t +naR
ni − na�1 −
3
2� hni
Rna�2� ,
yp�h� =Rna
ni� hni
Rna�3
, �6�
here the subscript p means paraxial. Therefore, Eq. (6)rovides a parametric equation for the caustic in the
Rays that undergo
Total Internal Ref
h
H
HM
(ZM ,YM
ig. 2. (Color online) Caustic produced by a plano-convex lens who obtain the CLC.
araxial region. Alternatively, from Eq. (6) upon eliminat-ng h we obtain a nonparametric form for the paraxialiacaustic, given by
Yp = Kpc1/2Zp
3/2, �7�
here Kpc=8�ni−na�3 / �27nani2R� is a constant that de-
ends on the parameters of the lens. Additionally we havehosen a change of coordinates given by Zp= f−zp and Ypyp. Therefore, as is well known, the diacaustic in thearaxial region is a semi-cubic parabola with origin at fccording to [20–23] as is shown in Fig. 3(a), where weave placed the effective focal length for each one lensest a common paraxial plane. The constant Kpc modulateshe shape of the diacaustic as is shown in Fig. 3(b). Fur-hermore, this equation is a good approximation for F / #1, but this approximation fails for F / # �1.For the ray tracing we have used three lenses with the
ollowing parameters: R1=38.76, R2=77.52, and R3116.28 mm, whose thicknesses are t1=32.68, t2=11.45,nd t3=8.0 mm, respectively, considering na=1 and ni1.517 for �=589 nm; additionally, we have also used aiameter of D=75 mm and −25 mm�h�25 mm as ishown in Fig. 3(a). An important fact is that by reducinghe entrance aperture both paraxial and exact causticsre very close even for F / # �1 as we can see in Fig. 3(b).
. CIRCLE OF LEAST CONFUSION FOR ALANO-CONVEX LENS
t is well known that there is a plane where the focusedundle of rays has a minimum diameter; this plane wouldeometrically be regarded as the best focus [24–26]. Theisk of light that is formed at this best focus so defined isnown as the CLC. The way to obtain the CLC is rela-ively simple [17]. If we consider a marginal ray from theower part of the lens as is shown in Fig. 2, the CLC islaced at the intersection between a marginal ray and theaustic surface of the incident rays from the upper parthose coordinates we defined as �Z ,Y �.
Caustic Surface
le of Least Confusion
Paraxial Focus
Marginal Ray
from below
(Zi ,Yi)
point source is located at infinity. Also shown is the process used
es
lection
Circ
)
en the
i i
th=�ast
T(stpdn
Etf
Sb
Z
wmh=Ffitssp
a
Wt
FF oom ex
Avendaño-Alejo et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. A 2255
In order to obtain the CLC, we need to provide an equa-ion for a marginal ray; in this way we choose H as theeight for the marginal ray, regarding that �aarcsin��niH� / �naR��, �i=arcsin�H /R�, and where
zM ,yM�= �t−R+�R2−H2 ,−H� provides the coordinates forpoint on the spherical surface from below of the lens. By
ubstituting these values into Eq. (1), we obtain the equa-ion for the marginal ray which can be written as
y =H�ni
2 − na2��z − ��R2 − H2 + t − R��
na2�R2 − H2 + ni�na
2R2 − ni2H2
− H. �8�
he next step it is to solve for �z ,y� by using Eqs. (5) and8). Analytically it is very difficult, because this requiresolving an equation of 12th degree for either z or y. Fur-hermore, if we try to solve this using the paraxial ap-roximation from Eqs. (6) and (8), we need to solve a sixthegree equation for either z or y. In both cases there areumerical methods to solve these equations.An alternative method to obtain the CLC is by using
q. (8) and to meet it with an arbitrary ray arriving fromhe upper of the lens whose height h is unknown. There-
MarginalRays
F/#=1F/#=2F/#=3
h
(a)
ig. 3. (Color online) (a) Comparison between exact and paraxia/#. The aperture for all the cases is −25 mm�h�25 mm. (b) Z
ore, we can write Eq. (1) as follows: s
fi
F
y =− h�ni
2 − na2��z − ��R2 − h2 + t − R��
na2�R2 − h2 + ni�na
2R2 − ni2h2
+ h. �9�
olving for �z ,y� from Eqs. (8) and (9) gives a point whereoth rays meet as a function of h, given by
i�h� = t − R
+ni�h�na
2U + niV��niu + v� + H�na2u + niv��niU + V�
�ni2 − na
2��na2�Hu + hU� + ni�Hv + hV��
,
Yi�h� =hHni�v − V + ni�u − U��
na2�Hu + hU� + ni�Hv + hV�
, �10�
here the subscript i means the intersection between thearginal ray and a refracted ray with an arbitraryeight. Additionally, we have defined U=�R2−H2, u�R2−h2, and also V=�na
2R2−ni2H2 and v=�na
2R2−ni2h2.
rom Eq. (10) h is the only variable quantity and we cannd the value for which it creates a maximum. Differen-iating Eq. (10) with respect to h for either Zi or Yi andolving in order to provide their critical values, as wasuggested in [20,21], this method is shown in Fig. 4 for aarticular plano-convex spherical lens. Alternatively,
l
Exact ParaxialF/#=1
F/#=2
F/#=3
Yp
Zp
(b)
tics produced by plano-convex lenses considering three numbersclusively of the caustics.
olving Yi�h�=y�h� from Eqs. (5) and (10) gives
h2ni
HR2na2 =
�na2R2 − ni
2h2 − �na2R2 − ni
2H2 + ni��R2 − h2 − �R2 − H2�
na2�H�R2 − h2 + h�R2 − H2� + ni�H�na
2R2 − ni2h2 + h�na
2R2 − ni2H2�
,
nd after a bit of algebra we get
h2ni�na2�Hu + hU� + ni�Hv + hV�
− HR2na2�v − V + ni�u − U� = 0. �11�
ithout loss of generality we can consider that h�R;hus, from Eq. (11) by expanding in a Taylor’s series to the
rst order in h2 and reducing further, we have
2naR��na + ni�R − �ni�R2 − H2 + �na2R2 − ni
2H2��
− 3ni�na + ni�h2 = 0. �12�
inally, solving for h gives the height for the incident ray
CausticSurfaces
ParaxiaPlane
l caus
wb
h
Ishdbl�sfpouft
Tyb
4LAfFat=lw
w=ww
alP
wm
Toptd
Fb s a fun
2256 J. Opt. Soc. Am. A/Vol. 27, No. 10 /October 2010 Avendaño-Alejo et al.
hich provides the CLC’s coordinates, whose solution cane written as
CLC =�2naR��na + ni�R − �ni�R2 − H2 + �na2R2 − ni
2H2��
3ni�na + ni�.
�13�
f we consider that H�R, again by expanding in a Tayloreries to first order in H2 and reducing further, we have
CLCpar�H�ni /3; but, as was explained above, ni is the in-
ex of refraction of the lens, which traditionally rangesetween 1.5�ni�1.7 for incident rays whose wave-engths are in the visible range. Then, 0.7071H�hCLCpar
0.7527H, which is in good agreement with [20,21]. Sub-tituting Eq. (13) into either Eq. (5) or Eq. (10) providesor z�hCLC�=Zi the length along the Z-axis where the CLClane lies; for the Y-axis, y�hCLC�=Yi provides the radiusf the CLC. In the particular case when the marginal rayndergoes total internal reflection, the maximum heightor the incident ray is given by Hca=Rna /ni; thus substi-uting this value into Eq. (13) yields
hca =�2na
3ni�1 −�ni − na
ni + na�R. �14�
herefore, introducing Eq. (14) into Eq. (5) yields that�hca� gives the radius of the CLC for a marginal ray justefore undergoing total internal reflection.
. DIACAUSTIC FOR A CONVEX-PLANEENSs was explained above we need only a representative re-
racted ray to obtain the caustic surface as is shown inig. 5. An incident ray impinging on the spherical surfacet the point Pa whose coordinates as functions ofhe angle of incidence are provided by PaR�1−cos �a ,sin �a� is refracted inside the lens along the
ine PaPi and reaches the plane face of the lens. In thisay P P can be written simply by
-20 -10
-20
2
4
ym
zm
Yi(h)
Zi(h)
ig. 4. (Color online) Graphical method to obtain the maximum vy using ni=1.517, na=1, R=38.76 mm for a plano-convex lens a
a i
y − R sin �a = − tan��a − �i��z − R�1 − cos �a�,
here �i is related to �a through Snell’s law as �iarcsin��na /ni�sin �a. Pi gives the coordinates of a pointhere the incident ray meets with the plane Z= t; in thisay we have
Pi = �t,R sin �a − tan��a − �i��t − R�1 − cos �a��, �15�
nd, finally, this ray is refracted out of the lens along theine PiP1 as is shown in Fig. 5. Therefore, using Eq. (15)iP1 can be written as
y = R sin �a − tan �t�t − R�1 − cos �a� − tan �r�z − t,
here �t=�a−�i and �r=arcsin��ni /na��t; reducing a bitore we get
y cos �r + z sin �r = R� cos �r sin �i
cos �t� + t sin �r
+ �R − t�cos �r tan �t. �16�
his is a parametric family of refracted rays as a functionf �a as is shown in Fig. 6. The diacaustic as was ex-lained above is the envelope of this family of rays and, inhis way, differentiating Eq. (16) with respect to �a and re-ucing further we have
− y sin �r + z cos �r
= R� cos �r cos �i
cos �t� ��i/��a
��r/��a−
R sin �r�sin �i + sin �t�
cos �t
+t cos��r − �t�
cos �t+ �R − t + R sin �i sin �t�
�� cos �r
cos2 �� ��t/��a
�� /��. �17�
10 20h
Solutions
hich gives the radius of the CLC and the plane where it is placedction of the height.
0
0
alue w
t r a
Fr
w
Ff
Fs
Avendaño-Alejo et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. A 2257
inally, to obtain the envelope of the family of refractedays, we solve Eqs. (17) and (16) for �z ,y�, obtaining
z��a� =Q cos2 �r
cos �t+ t,
H
h
Paraxial Princi
( )Z Yi , i
ig. 6. (Color online) Caustic produced by a convex-plane lensormed by it.
na
ni
t
R
O
Y
h
Incident
Ray qa
Pa
Pi
PP
qi q
t
ig. 5. (Color online) Process of refraction produced by a conveource is located at infinity.
y��a� = −Q sin �r cos �r
cos �t+ �R − t�tan �t +
R sin �i
cos �t,
�18�
here we have defined
Caustic Surface
Paraxial focus
Marginal Ray
from below
ircle of least confusion
cipal Surface
ane
(Zi ,Yi)
the point source is located at infinity and the principal surface
ZP11
Optical Axis
Paraxial Plane
Refracted Ray
BFL
qr
EFL
F
qr
e lens and its associated parameters considering that the point
C
Prin
pal Pl
when
C
na
x-plan
ao
�zpt−acp
Wcl
AErff
Aw
w
FF
2258 J. Opt. Soc. Am. A/Vol. 27, No. 10 /October 2010 Avendaño-Alejo et al.
Q = R cos �i� ��i/��a
��r/��a� + � R − t
cos �t+ R tan �t sin �i�
�� ��t/��a
��r/��a� ,
nd for brevity we have omitted to write at length each
ne of the derivatives. It is important to say that for f
awrweFttgpIwhtt
cocglppc
a→0 from Eq. (18) it provides the paraxial focus given by�0�=naR / �ni−na�+ �ni−na�t /ni=F, where F is a singularoint and is related to the back focal length �BFL�hrough the following relationship: BFL=F− t=naR / �nina�−nat /ni=EFL−nat /ni, where EFL has been definedbove. By substituting �a=arcsin�h /R� into Eq. (16) andonsidering that y=h we obtain the coordinates for therincipal surface �PS� as is shown in Fig. 6. In parametric
orm its surface can be written simply as
PS = �t +�R − t − �R2 − h2��na
2R4 − h2��ni2R2 − na
2h2 − na�R2 − h2�2
nah2 + �ni2R2 − na
2h2�R2 − h2,h� . �19�
hen h→0 we recover the position for the paraxial prin-ipal plane �PPP� which according our frame of referenceeads to
PPP = �ni − na�t/ni.
gain, considering also that h� R, and expanding fromq. (18) for both coordinates z�h� and y�h� in a Taylor’s se-
ies to the first order, we obtain the paraxial caustic sur-ace for a convex-plane spherical lens that in a parametricorm can be written as
zp�h�
� F −3h2�ni
2R�ni3 − 2na�ni
2 − na2� − �ni − na�4�na + ni�t
2na�ni − na�ni3R2
,
yp�h� �h3�ni
2R�ni3 − 2na�ni
2 − na2� − �ni − na�4�na + ni�t
na2ni
3R3.
�20�
lso, as was done above, from Eq. (20) on eliminating he obtain a nonparametric form for the paraxial caustic,
Yp = Kcp1/2Zp
3/2, �21�
here Zp=F−zp, Yp=yp, and where we have defined
MR
F/#=F/#=2F/#=3
h
(a)
ig. 7. (Color online) (a) Comparison between exact and paraxia/# for an aperture of −25 mm�h�25 mm. (b) Zoom exclusively
Kcp =8�ni − na�3ni
3
27na�ni2�ni
3 − 2na�ni2 − na
2�R − �ni − na�4�na + ni�t�,
nd the subscript cp means convex-plane lens. Thus, asas pointed out in [20–22] the diacaustic in the paraxial
egion is a semi-cubic parabola with origin at F; also, asas done above, we have placed the back focal length forach lens at the paraxial plane as is shown in Fig. 7(a).rom Eq. (21) we can see that the constant Kcp modulateshe shape of the diacaustic as is shown in Fig. 7(b), andhe values used for the ray tracing through the lenses areiven above. Furthermore, from Eq. (21) this is a good ap-roximation for lenses with F / # �1, but fails for F / # �1.t is important to note that from Eq. (20) h� �−� ,��,hile from Eq. (18) h� �−niR /na ,niR /na. On the otherand, this means that the lens does not undergo total in-ernal reflection unless na�ni. Another important fact ishat Kcp depends on the thickness of the lens.
Mathematically Eqs. (7) and (21) are equivalent but theonstants Kpc and Kcp are directly related to the amountf spherical aberration produced by the lens, and as wean see from Figs. 3(b) and 7(b) the area under the curveives the amount of spherical aberration produced by theenses for 0�h�H. Therefore, as is well known, a convex-lane lens produces less spherical aberration than alano-convex lens when they are being illuminated with aollimated beam, and we conclude that Kpc�Kcp.
sticaces
ParaxialPlane
Exact ParaxialF/#=1
F/#=2
F/#=3
Yp
Zp
6
4
2
-2
-4
-6
-25 -20 -15 -10 -5
(b)
tics produced by convex-plane lenses considering three numberscaustics.
arginalays
CauSurf
1
l causof the
5At
wraartxawepG(twaasrcfi+cpt
bppbted
onkfcmiireascwai
was(
Esip
Iocrd
FT
Avendaño-Alejo et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. A 2259
. SPHERICAL ABERRATIONs far as we know the Seidel aberration [22] can be writ-
here b1 to b5 are commonly called the third-order aber-ations; x ,y are the exit (or entrance) pupil coordinates;nd , are the coordinates of the image point with origint O�. We have assumed that both the wavefront and theeference sphere are chosen to pass through the center ofhe pupil so that W must be zero at the origin O on the,y coordinate system as is shown in Fig. 8. It is custom-ry in the Hamiltonian theory to postulate that in theavefront aberration expansion the reference sphere forach field angle has its center at the Gaussian imageoint, i.e., the aberrations are measured taking theaussian image as the ideal image point [22]. From Eq.
22) the first term b1�x2+y2�2 is called spherical aberra-ion, and the second term b2y�x2+y2� is named coma,hich increases linearly with the field angle or distancend quadratically with the pupil radius. The third Seidelberration b3y22 is called astigmatism; in the meridionalection there is an increment of curvature since the aber-ation depends quadratically on y, and the increment inurvature in the y-section depends on the square of theeld angle. The fourth primary aberration term b42�x2
y2� is called field curvature; the astigmatism and fieldurvature are often grouped together since they both de-end on the square of the field angle. The last aberrationerm b5y3 is known as distortion.
As was explained above, we consider exclusively aeam of rays emanating from an on-axis object pointlaced at infinity in meridional Y–Z-section (a wavelane coming to the lens along the optical axis). Thiseam of rays and its caustic are symmetric with respect tohe optical axis. There are two on-axis monochromatic ab-rrations of the optical system: spherical aberration andefocus. The defocus is a translation of caustic along the
X
Z
Y
O
Reference Sphere
Aberrated Wavefront
Entrance Pupil
Coordinates C
ig. 8. (Color online) Systems of references: x, y, and z are the ehe aberrations are measured taking the Gaussian image as the
ptical axis away from the image plane. The defocus doesot influence the form of the caustic [16]. Note thatnowledge of the on-axis meridional caustic is sufficientor the construction of the whole caustic surface in thease of an optical system with rotational symmetry. Foreridional rays, the curvatures of the wavefront still lie
n and perpendicular to the meridional plane. The caustics somewhat more complicated for skew rays than for me-idional rays since the principal directions will not in gen-ral lie in the plane of incidence. The procedure for evalu-ting the caustic surfaces for skew rays refracted by apherical lens is well explained in [15]. In this way, weonsider exclusively spherical aberration because it isell known that the spherical aberration is symmetricalbout the principal ray. Following all the steps explainedn great detail in [22] the caustic surface can be written as
= −3/2
3R2�− n
3b1, �23�
here R is the radius of the reference sphere �R=EFL�nd n is the refractive index in the image space. By sub-tituting n=na, R=naR / �ni−na�, =Yp, and =Zp into Eq.23) we have
Yp = −Zp
3/2
3 �ni − na
naR �2� na
− 3b1. �24�
quating Eqs. (7) and (24), and solving for b1 which is thepherical aberration coefficient at third order, we obtaints value for a plano-convex lens PCL as a function of thearameters of the lens given by
b1pc=
ni2�ni − na�
8na2R3
. �25�
t is important to note that if we know a priori the valuef b1, for example, by using an interferometric test, wean get the values for either the index of refraction or theadius of curvature for an arbitrary plano-convex lens un-er test.
x
h´yp
z´Zp
L
Coordinates
at the Ideal
Point Plane
O’
y
il coordinates; �, , and are the coordinates of the image point.point.
R=EF
hief Ra
xit pupideal
sCl
Aidftt
6Wdcituocltfhttmbbql
ATAsdC8Gc
R
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2260 J. Opt. Soc. Am. A/Vol. 27, No. 10 /October 2010 Avendaño-Alejo et al.
In the same way by equating Eqs. (21) and (24), andolving for b1 we obtain its value for a convex-plane lensPL as a function of the parameters of the lens which
eads to
b1cp= �ni − na�
��ni2�ni
3 + 2na�na2 − ni
2�R + �na − ni�4�na + ni�t
8na2ni
3R4 � .
�26�
s we are considering exclusively third-order aberration,t is possible to add algebraically these aberrations, in or-er to calculate the aberration contributions at each sur-ace by using the Petzval contribution [27,28], obtaininghe same results given by Eqs. (25) and (26) for b1, respec-ively.
. CONCLUSIONSe have obtained simple formulas for the caustic pro-
uced by positive convex-plane and plano-convex spheri-al lenses when a point source of light is located at infin-ty. By using these equations a paraxial approximation forhe caustics is provided in both configurations. Also, bysing these equations it is possible to obtain the third-rder coefficient of spherical aberration. The shape of theaustic can be modified by changing the parameters of theens. A simple formula for the CLC has been provided forhe particular case when the source is placed at infinity. Aormula for the principal surface as a function of theeight is also given. We believe that the method to obtainhe caustic that we have reported is straightforward, ob-aining a relationship between caustics, wavefronts, andeasures of the spherical aberrations. Further work will
e done in order to design new Hartmann null screensased on the knowledge of the caustic by refraction to testualitatively either plano-convex spherical or asphericalenses.
CKNOWLEDGMENTShis work has been partially supported by Programa depoyo a Proyectos de Innovación Tecnológica, Univer-idad Nacional Autónoma de México (PAPIIT-UNAM) un-er project number IN113510 and Consejo Nacional deiencia y Tecnología (CONACyT) under project number2829. The corresponding author is grateful to I. Goméz-arcía and the referees for their valuable assistance and
omments.
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