The Eye Caustic of a Ball Lens

Abstract

Lens phenomena, such as caustics, image distortions, and the formation of multiple images, are commonly observed in various refracting geometries, including raindrops, drinking glasses, and transparent vases. In this study, we investigate the ball lens as a representative example to showcase the capabilities of Berry’s eye caustic as an optical tool. Unlike the conventional paraxial approximation, the eye caustic enables a comprehensive understanding of image transformations throughout the entire optical space. Through experimental exploration, we establish the relationship between the eye caustic and traditional light caustics. Furthermore, we provide mathematical expressions to describe both the caustic and the image transformations that occur when viewing objects through the ball lens. This approach could be of interest for optics education, as it addresses two fundamental challenges in image formation: overcoming the limitations of the paraxial approximation and recognizing the essential role of the observer in comprehending lens phenomena.

Full text

The eye caustic of a ball lens
Thomas Quick
*
and Johannes Grebe-Ellis
University of Wuppertal, Faculty of Mathematics and Natural Sciences, Gaußstraße 20,
D-42119 Wuppertal, Germany
E-mail: quick@uni-wuppertal.de and grebe-ellis@uni-wuppertal.de
Received 15 August 2023, revised 19 March 2024
Accepted for publication 15 April 2024
Published 10 May 2024
Abstract
Lens phenomena, such as caustics, image distortions, and the formation of
multiple images, are commonly observed in various refracting geometries,
including raindrops, drinking glasses, and transparent vases. In this study, we
investigate the ball lens as a representative example to showcase the capabilities
of Berry’seye caustic as an optical tool. Unlike the conventional paraxial
approximation, the eye caustic enables a comprehensive understanding of image
transformations throughout the entire optical space. Through experimental
exploration, we establish the relationship between the eye caustic and traditional
light caustics. Furthermore, we provide mathematical expressions to describe
both the caustic and the image transformations that occur when viewing objects
through the ball lens. This approach could be of interest for optics education, as
it addresses two fundamental challenges in image formation: overcoming the
limitations of the paraxial approximation and recognizing the essential role of
the observer in comprehending lens phenomena.
Keywords: light caustic, eye caustic, ball lens, paraxial approximation,
embedded perspective, detached perspective, image transformation
1. Introduction
Raindrops, drinking glasses, and transparent vases showcase an appealing array of lens
phenomena (figure 1). When observing objects through water-filled glasses, a multitude of
image changes become apparent, including magnification, reduction, sharpness, blurriness,
European Journal of Physics
Eur. J. Phys. 45 (2024)045301 (22pp)https://doi.org/10.1088/1361-6404/ad3eef
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Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the
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© 2024 The Author(s). Published on behalf of the European Physical Society by IOP Publishing Ltd 1

and, in particular, the occurrence of multiple images of objects positioned behind differently
shaped glasses. These image alterations depend on the properties of the refracting object as
well as the relative positions of the observed object and the observing eye, leading to sig-
nificant variations in the observed images, including changes in their topology (figures 1(a),
(c)) [1,2]. If, on the other hand, the same refractive object is exposed to a light source (e.g.
the Sun, a candle), fascinating light patterns, known as caustics, emerge (figures 1(b),(c))
[3,4]. The shape of the caustic body can be influenced by the movement of the refracting
object or the screen, while its structure remains independent of the position of the observer.
In this article, we explore the fundamental yet intriguing symmetry of vision and lighting
within the context of ball lens imaging, delving into Berry’s concept of the eye caustic or
imaginary caustic [5,6]. The traditional notion of a light caustic typically refers to the
envelope of a family of rays, signifying the boundary between regions of space with varying
ray densities. However, by substituting the luminous object with an observing eye and
considering the family of rays reaching that eye, we arrive at the concept of the eye caustic,
which offers a novel perspective in understanding image transformations caused by lens
phenomena. The eye caustic addresses the crucial role of the observer’s eye and allows for a
more comprehensive analysis of extended objects by focusing solely on the rays reaching the
eye from the object. The symmetry between light caustic and eye caustic is striking. It
reminds us of Kepler’s distinction between the optical imaging in space (pictura)and the
image actually seen (imago)[7]. While the light caustic is related to the luminous object, the
caustic of the eye is associated with the observing eye. This symmetry aligns with the two
Figure 1. Everyday lens phenomena frequently deviate from the paraxial approx-
imation. (a)A solitary rain drop hanging from a branch exhibits a compressed and
inverted image of the surrounding scene, decorated with further images at the periphery
of the sphere (Photo: Wikimedia Commons).(b)Dewdrops in the Sun create small
caustics, potentially leading to leaf burn (reprinted with kind permission of H.-J.
Schlichting).(c)Seeing and illuminating at the same time: a sunlit water-filled wine
glass creates complex light patterns on the wall and shows three images of a ruler.
Picture (c)was taken with a Sony alpha 7 III digital camera and the 28–75 mm F/2.8
Di III RXD objective from Tamron, as were the pictures in figures 3,4,8and 11.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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perspectives on lens phenomena mentioned above: the detached perspective on imaging
luminous objects onto a screen (object →lens →screen), and the embedded perspective,
which refers to observing an object through the lens (eye →lens →object). In general, both
perspectives are possible and can occur simultaneously in optical lenses (figure 1(c)). This
conceptualization may have methodological potential and could be of interest to optics
education for at least two reasons.
First, many of the everyday optical phenomena, while fascinating, do not comply with the
assumptions of the paraxial approximation in geometric optics, which assume small angles
and distances. To simplify the ray-tracing process for thin lenses, apertures are often used in
optical instruments, so the paraxial approximation is valid. Whenever we move away from the
paraxial approximation and discuss aberrations, we also have to take into account the effects
of caustics, which are deviations from ideal point-to-point imaging [8]. However, they are
usually neglected or just briefly mentioned in optics education.
Second, students often struggle to relate their own observations to the ray model of light,
particularly when the eye’s involvement in image formation is more pronounced [9]. This
challenging aspect has been extensively studied in the context of thin lenses [10–13]. Lens
phenomena always refer to either the detached or to the embedded perspective. How these are
connected and can be related to each other often remains conceptually unclear. We argue that
the concept of the caustic of the eye addresses both difficulties (paraxial approximation, role
of the observer’s eye)and thus may bridge the gap to the traditional treatment of thin
lenses [14].
The article is organized as follows: section 2introduces visualization techniques for the
eye caustic and examines its relationship with light caustics for a ball lens. In addition to the
known paraxial imaging equation, we formulate a corresponding equation for the observing
eye and identify familiar focal points with a novel interpretation. Section 3provides a
mathematical calculation of the caustic of a ball lens, applicable to both the eye and the light
caustic. From this calculation, we derive specific image transformations for the ball lens,
which illuminate the underlying principles governing lens phenomena.
2. Exploring the eye caustic
2.1. Observing and counting images: the eye caustic
To visualize the eye caustic, we study the image transformations of a candle flame and a ruler
when viewed through a ball lens (figures 2,3and 4). Since the principal planes coincide in the
ball lens as a special case of spherical lenses, the object and image distance can be measured
from the center. The ball lens we use has a radius of R=7.5 cm and is composed of crown
glass (crystal glass), whose optical properties are not known in detail. Its focal length was
determined by imaging a distant small light source onto a movable screen whose distance to
the ball lens was measured with a millimeter scale. Repeated adjustments of the screen
yielded a result of f=10.7 ±0.1 cm. Inserting this into the known equation for the effective
focal length of the ball lens [3,15]:
() ()fnR
n21 1=-
and rewriting the equation as n=2f/(2f−R), we obtain n=1.54 ±0.01 for the refractive
index of the glass, which is a reasonable value for typical crown glass.
For the following investigation, the eye is fixed in the observation space at a distance of
a=15.0 cm from the center of the ball lens (figure 2). To keep its position stable a pinhole
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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aperture with diameter Ø =5 mm is used. The candle is positioned at the object distance
g=8.5 cm and then stepwise displaced away from the optical axis. As when viewing an
object within the focal length of an ordinary magnifying glass, an enlarged virtual image of
the flame can be seen in the center (figure 3(a)). In the case of the ball lens, i.e. beyond the
paraxial approximation, a further image element is added: a bright ring near the periphery of
the sphere. Moving the candle perpendicular to the optical axis, the ring splits into two
smaller light arcs. On closer examination, these turn out to be two further images of the flame:
a total of three distinct images of the flame become discernible (b–d). As the candle is further
displaced, the two flame images on the left side merge, gradually fade away (e–g), leaving
only a single upside-down image of the flame on the opposite side of the sphere (h).
To reliably distinguish and count the images of an object viewed through a ball lens with
g<f, we insert a short intermediate consideration and compare the view of an upright ruler
through a water-filled glass cylinder and a ball lens (figure 4). In the case of the cylinder, the
central image is accompanied by two symmetrical lateral images (a). When switching to
spherical geometry, these merge to form a ring (b), which due to rotational symmetry in
principle consists of an infinite number of lateral images. To keep it simple, we consider the
ring in the symmetric case as a single image and therefore count a total of two images.
Moving the ruler to the left causes the central image and the left lateral image to move
towards each other, merging on the sides facing each other (figures 4(c),(d)), but remaining
distinguishable by their outer edges until they disappear (not shown in figure 4). This
observation validates the counting in figures 3(e)–(g), because the bright spot on the left side
of the sphere is effectively comprised of two distinct image elements.
We return to exploring the images of the candle flame without changing the position of the
eye and now enlarge the movement space of the candle (or another small test object)to the
whole object space and thereby identify positions at which images of the flame disappear or
appear for the observing eye, leading to a change in the number of images (from three to one
or vice versa). Thus, we investigate how an object behind the lens is transformed into multiple
images depending on its position. The track of marked positions at the boundaries, where the
number of images changes, is depicted in figure 5. Its shape approximates a caustic line,
which divides the plane of the object space into two distinct regions. When the candle is
positioned within this caustic, the observer typically perceives three images (a, two in the
symmetric case from the optical axis), while placing it outside the caustic reveals only one
Figure 2. Schematic configuration for observation from the embedded perspective:
view through the ball lens onto images of a candle flame (figure 3).
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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image (b). There are also places where no image can be seen, forming a blind spot where an
object can be hidden behind the transparent sphere (c). In summary, the structure of the object
space, derived from the analysis of images with fixed eye, appears to be similar to the more
common light caustic. It is therefore referred to as the eye caustic or the imaginary caus-
tic [5,6].
2.2. The light caustic—‘image’of the eye caustic
In the field of optics, caustics refer to distinctive patterns of light rays that are concentrated or
dispersed due to reflection or refraction [16,17]. A simple modification of the experimental
setup described above reveals the fundamental relationship between the eye caustic and the
light caustic. If we switch to the detached perspective by replacing the observing eye with a
small bright light source (figures 6and 7), an illuminance distribution behind the sphere
emerges, known as light caustic (figure 8). It exhibits the same geometric characteristics as the
eye caustic, including its size and structured regions. The inner region (a)appears sig-
nificantly brighter than the outer region (b), and there are also unilluminated areas behind the
sphere (c).
In figure 9, the construction of the light caustic is illustrated for the given scenario. A point
light source Lsituated at the object distance gin the object space emits light in all directions.
As it traverses through the ball lens, it undergoes refraction twice, once at the front surface
and once at the back surface, following the law of refraction. This results in a field of light
rays that generates a light caustic in the image space as envelope of the rays. The paraxial
focus distance z
g
at the cusp of the light caustic is determined by the object distance g. Using
the paraxial imaging equation
Figure 3. Transformation of a candle flame’s image, seen through a ball lens with
R=7.5 cm and g=8.5 cm, as the candle is displaced away from the optical axis, while
maintaining a fixed eye position at a=15.0 cm. The distance of the candle wick from
the optical axis is indicated below in each case. In the symmetric position (a), the
central flame image is surrounded by a bright ring. Moving the candle laterally splits
this ring into two lateral images. Two of the three images (b)merge and disappear in a
red glow (g), one remains (h).
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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()
zgf
111
,2
g
+=
z
g
is given with (1)by
() ()
z
gnR
gn nR21 .3
g
=--
Substituting the provided values of R=7.5 cm, g=15 cm, and n=1.54 into (3), we derive
z
g
=b=37.3 cm, in reasonable accordance with the measured value (b=36.8 ±0.5 cm).
In terms of spherical aberration, the caustic represents a distorted image of the light source
L. With respect to a screen at z
g
, the extension of marginal rays forms rings of confusion
where they intersect the image plane. Closer examination would have to address the extre-
mities of the caustic image body, which is formed by the totality of the rays passing through
the imaging system (circle of least confusion, density of light rays, focal points, [8,18]).
To relate the light caustic to the previous eye caustic study, we note that in the light
caustic, the structure of the eye caustic becomes visible as an illumination phenomenon, i.e.
the relationship between light caustics and eye caustics is based on the fact that the state of
Figure 4. View through a water-filled cylinder (a)and a ball lens of crown glass (b)–(d),
both with diameter Ø =10.0 cm, onto a ruler with object distance g=6.0 cm and
camera distance a≈25 cm.
Figure 5. Mapping the positions of a small test object behind the ball lens, at which two
images (optical axis, red), three images (a, green), one image (b, blue)or no image (c,
gray)is visible to the eye at a distance of a=15.0 cm. The dashed lines are intended to
visualize the boundaries at which the number of object images changes. In particular,
the green caustic lines connect positions where the number of object images changes by
two, leading to image annihilation.
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illumination at a given position in the light caustic depends on how many images of the light
source can be seen from there.
To illustrate this entanglement of detached and embedded perspective, we have extended
the setup of figure 7with an auxiliary screen and drilled a small hole in the original projection
screen within the three-dimensional caustic (figure 10). This pinhole projects a pinhole image
of the lighting situation onto the second semi-transparent screen. The pinhole image then
displays what is visible from the respective pinhole position while moving the first screen
through the caustic body [19]. Within the strongly illuminated region (figure 8(a)), three
images are perceived (figure 11, positions 1–5), while outside this region only one image is
visible (position 6). Dark areas (figure 8(c)) situated behind the sphere do not produce
discernible images. The topology of the images changes exactly when the pinhole is on the
fold of the caustic [5]. Therefore, the illuminance distribution of the light caustic indicates the
number of observable images.
2.3. The symmetry between eye caustic and light caustic
When examining with an eye at a distance ain the observation space, only the light paths that
actually reach the eye are taken into account for image formation (figure 12). We will refer to
these selected light paths as object rays, since they originate from the object in the object
space. We denote the caustic focus of the eye caustic as Z
a
. When a point-like object is
positioned within the eye caustic, any location where three object rays pass through results in
Figure 6. Relationship of imaging parameters between embedded and detached
perspective.
Figure 7. Schematic configuration for imaging with the ball lens (detached
perspective): viewing sections through the image body of the caustic (see figure 8).
The image distance bis defined here as the distance at which the light caustic shows its
strongest constriction, and the image of the light source on a screen perpendicular to the
optical axis appears sharpest.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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three distinct visible images of the object. The eye caustic proves to be particularly influential
when dealing with extended objects. In such cases, the eye caustic predicts which part of the
object is imaged either once or three times, especially when the object is intersected by the
caustic.
As gof the light source changes, the related light caustic also changes. For each position in
the observation space, there is a corresponding eye caustic, and its orientation and shape
depend on the position of the observer at a. The two caustics, the eye caustic and the light
caustic, are connected and influence each other (figure 13). When the eye passes through the
Figure 8. By exchanging the eye for a light source, in this case, a luminous filament at
g=15.0 cm, a light caustic is generated which shows the totality of the image locations
examined in figure 5. From region (a), three images of the light source are generally
visible (two from the optical axis); from region (b), exactly one image is seen; and from
region (c), no images are visible. However, region (c)is limited upwards and
downwards by the area that is directly illuminated passing the sphere, see the analogous
case in figure 5. The focus of the caustic shows the image of the luminous filament at
an image distance of b=36.8 ±0.5 cm. The calculated value is b=37.3 cm.
Figure 9. The ball lens refracts light rays twice, enveloping a light caustic (dashed line)
with focal point Z
g
that varies with distance g. Each point within the caustic is traversed
by three light paths, an observer will therefore see three images of the light source Lat
that point. Beyond the caustic, only one image will be visible.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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fold of the light caustic, the object also passes through the fold of the eye caustic. The
appearence/disappearance of images marks the boundary line, such as a fold or cusp, of both
caustics. This symmetry has another implication: the eye caustic for the ball lens, and con-
sequently for any refracting geometry, can be visualized by employing a point light source
and examining its corresponding light caustic. Substituting the point light source with an
observing eye does not alter the structure of the eye caustic. This facilitates quick predictions
of the image transformations that will arise for specific eye positions.
2.4. Focal points and imaging conditions
By replacing the object distance gwith the eye distance a, an equation analogous to the focus
of the light source can be obtained from (3)for the focus distance z
a
of the eye caustic:
Figure 10. Experiment to illustrate the entanglement of detached and embedded
perspective. While the projection screen with the pinhole (Ø=1.3 mm, thus beyond
diffraction effects)is moved through the caustic, the respective pinhole image on the
auxiliary screen shows what can be seen from the location of the pinhole: multiple
images of the luminous object (here a filament). The pinhole image distance
is
b10.0 0.1 cm¢= 
.
Figure 11. Pinhole images of a luminous filament at g=15.0 cm, imaged with the
pinhole moving transverse through the light caustic of the ball lens at a distance of
b=30.0 cm (figure 10). Two images disappear when the pinhole lies on the fold of the
light caustic (from position 5–6). This image series corresponds to the series in figure 3.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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() ()
z
anR
an nR21 .4
a
=--
Analogously, the caustic focus of the eye caustic converges to the focal point Ffor
increasing awith focal length f.Asadecreases, the focus of the eye caustic shifts away from
the sphere. At
a
f=
¢
the caustic boundary deviates from the optical axis, transitioning from
convergence to divergence. When the eye caustic focus lies at infinity, the position acor-
responds to the second focal point F
¢
on the frontside side of the ball lens. We obtain the
second focal length from (4)by setting the denominator to zero and solving for a, which leads
to equation (1)again.
Analogously to (2), we obtain an imaging equation
() ()
za
n
nR f
112 1 1 5
a
+= -=
which leads to a formulation of the formula for lenses, specifically tailored to the observing
eye. How should we interpret the lens equation (4)for the observing eye? In figure 14, the
graph of equation (4)as a function of z
a
(a)illustrates distinct regions in the a−g-plane where
either one or three images of an object are observed. When the object is precisely positioned
at the focus of the eye caustic, it will appear highly blurred. With respect to the embedded
perspective, we refer to this location as blur point. Consequently, we refer to the
corresponding graph of z
a
(a)as the blur line or blur curve.
We summarize the relevant conditions:
(i)a→∞: blur point of the eye caustic converges to focal point Fat distance f,
(ii)∞>a>f′: blur point is between Fand infinity,
(iii)a=f′: blur point is located at infinity and the eye is at the second focal point F
¢
,
(iv)f′>a>0: the caustic diverges, no blur point exist.
The ball lens we used has a radius of R=7.5 cm and a refractive index of n=1.54. When
we place an eye at a distance of a=15 cm from the lens, equation (4)show that the blur point
is z
a
=37.3 cm away from the lens. An object located at this position will blur.
Figure 12. Both the eye caustic and the light caustic stem from the same geometrical
construction. In the eye caustic, object rays envelope the caustic with paraxial focus
point Z
a
, determined by the distance a(embedded perspective). The eye caustic
organizes the object space based on observed images. An object within the caustic is
perceived three times, while outside the caustic, it appears only once.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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3. Modeling the eye caustic and image transformations
We present a mathematical model of the eye caustic for a ball lens. Previous research has been
conducted on imaginary caustics and image transformations for various reflecting and
refracting surfaces [20–25]. However, to the best of our knowledge, the didactically inter-
esting special case of the ball lens has not been studied before. Our derivation in this study
was inspired by the approach presented in [20]. During our mathematical considerations, the
distinction between the eye Aand the light point Lis not necessary. By simply replacing a
with g, we can easily obtain light caustics. All computations are carried out using
Mathematica.
3.1. Parametrization of the object ray field
The center of the ball lens, with radius R, is located at the origin of an cartesian coordinate
system. Since the problem can be solved in two dimensions, we will limit the calculation to
the x−z-plane in the following. Figure 15 illustrates a situation where the focal point of the
ball lens is between the lens and the eye (a>f´>R). The eye is positioned at point Aon the
optical axis, denoted by the z-axis, directed to the left. The angle εdescribes the apparent size
of the lens as observed from Aand is calculated as (
)
Ra2arctan
e
=. All rays within ε
contribute to the formation of the caustic line in the object space after passing through the ball
lens. First, we aim to describe the object ray field behind the sphere, which is parameterized
by the angle jbetween r
1
and the x-axis (see figure 15).
The vector a=(0, a)denotes the position of the eye in the x–z-coordinate system. At the
first rare-to-dense transition between air and ball lens at point E
1
,
()R
r
cos , sin
1
jj=
defines
the position vector of the point of incidence E
1
, and
() ( )
()
Rer1cos,sin
n
11
jj==
is the
normal vector at E
1
perpendicular to the sphere. Then, the unit vector e
a
=(x
a
,z
a
)of the
direction of view from Ato E
1
is obtained by normalizing the vector from ato r
1
, that is,
e
a
=(r
1
−a)/|r
1
−a|. This yields the following expressions for the two components of e
a
:
Figure 13. As the eye moves from position A
1
to A
2
, the latter outside region of the light
caustic but within region (b)of figures 5and 8, the illuminance diminishes.
Simultaneously, the number of observed images for the observer changes from two
(optical axis), to three (inside the light caustic), to one. This relationship is
understandable since the corresponding eye caustic also encounters a crossing of its
boundary line relative to the light source.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
11

()
() ()xR
aR aR
cos
2sin 6
a22 12
j
j
=+-
()
() ()
z
Ra
aR aR
sin
2sin .7
a22 12
j
j
=-
+-
To determine the direction of the refracted ray
()
() () ()
xze,
ooo
111
=
for the first transition, the
vectorial law of refraction establishes a connection between e
a
,()
e
n
1, and
()
e
o
1
such that the sine
relation
n
nsin sin
12
ab=
of the law of refraction is satisfied. For
()
e
o
1
we obtain with n
1
=1
(nearly vacuum)and n
2
=n(see appendix):
⎡
⎣
⎢⎛
⎝⎞
⎠
⎤
⎦
⎥
()
() ()
··()
() () () ()
nn n
eeeee ee
11 111.8
oananan
111
2
12
=- -- -
Evaluating (8), we get with the abbreviation
() ( )uaRa2sin
22 12
jj=+-
() () ()
()
xa
nu
a
nu
cos sin cos 1 cos 9
o
122
22
jj
jjj
j
=--
() () ()
()
z
a
nu
a
nu
cos sin 1 cos .10
o
1222
22
j
jjj
j
=- - -
Figure 14. The figure illustrates the function z
a
(a)for a ball lens with R=7.5 cm and
n=1.54. As the eye distance aapproaches infinity, the function converges towards the
focal point F. When the focus distance z
a
tends to infinity, we get F
¢
. An object placed
at the focus of the eye caustic appears significantly blurred, which is why the graph of
z
a
(a)is referred to as the blur line. For distance a=15.0 cm, the object appears blurry
at the calculated focus distance of g=z
a
=37.3 cm (yellow dot at P
2
). Whenever an
object crosses the focus, a topological transition of the observed images occurs (object
at P
1
is imaged once, object at P
3
is imaged three times).
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
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To determine the direction of the refracted object ray
()
e
o
2
after the second transition,we
use the incident vector
()
e
o
1
. The normal vector ()
e
n
2on the second boundary surface is cal-
culated as
()
()
Rer1
n
22
=
, where the vector r
2
points to the second point of incidence E
2
. For
()
e
o
2
we can write (see appendix)
⎡
⎣
⎢⎤
⎦
⎥
()
() ()
··()
() () () () () () ()
nn neeeee ee11 . 11
oonon on
21212 212
2
=- -- -
To trace the vectors
()
e
o
1
and ()
e
n
2in the second transition of (11)back to the vectors at the
first transition, and thus to the angle j, we use (8)for
()
e
o
1
. For ()
e
n
2we find (see appendix):
()
() () ()
d
R
eee 12
non
211
=+
with
() ()
d
Ra
nu
21 cos ,13
22
22
j
j
=-
where dis the distance between E
1
and E
2
. By using (11)with (8)and (12)we obtain long,
but analytical expressions for the components
()
x
o
2
and
()
z
o
2
of the vector
()
e
o
2
.
For the field of object rays behind the sphere, we write
() ·
()
ssor e,
o
22
j=+
, where r
2
is given by
·
()
d
r
er
o
211
=+
and sis the ray length (measured from the second surface at E
2
).
To express the components of o(s,j),wefinally have:
() · · ()
() ()
o
sdxR sx, cos 14
xoo
12
jj=+ +
() · · ()
() ()
o
sdzR sz,sin. 15
zoo
12
jj=+ +
Figure 15. The diagram illustrates the process of deriving the eye caustic. The diagram
shows an object ray originating from the observer’s eye at point A. This ray is refracted
twice at points E
1
and E
2
on the ball lens. The object ray field is paramterized by the
angle jbetween the x-axis and vector r
1
, which forms the eye caustic behind the ball
lens. The diagram also includes the focal point F
¢
and the focus of the caustic Z
a
.
Further details about the other quantities can be found in the accompanying text. Σ
denotes a plane positioned at distance g, perpendicular to the optical axis, aiding in
interpreting the subsequent figure 17.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
13

3.2. Calculation of the caustic
The components (14)and (15)of the object ray field behind the sphere can be interpreted as a
mapping from the parameter space (s,j)to the coordinate (or ‘control’)space (o
x
,o
z
). The
caustic can be found from the condition that the mapping (s,j)→(o
x
,o
z
)becomes singular.
By calculating the Jacobian and its determinant, the caustic can then be derived from the
condition of stationarity Jdet 0=[20]:
⎜⎟
⎛
⎝⎞
⎠·· ()Joso
oso
o
s
oo
s
o
det det 0. 16
xx
zz
xzz
x
j
jjj
=¶¶¶¶
¶¶¶¶ =¶
¶
¶
¶-¶
¶
¶
¶=
To simplify the problem, we adopt a more sophisticated approach to facilitate its treatment.
To achieve this, we concentrate on a specific, yet arbitrary, value of ()
¯
os z,
zj=and sub-
sequently transform the second component (15)into an equation for s:
()
¯·
()
()
()
s
zdz R
z
sin
.17
o
o
1
2
j
=
-+
Substituting (17)into the first component (14)yields the result
()
(¯)· ¯·()
() ()
()
()
o
zdx R zdz R x
z
, cos sin . 18
xoo
o
o
11
2
2
jj j=+ +-+
By reformulating the caustic condition as
(¯)oz,0
x
jj
¶
¶=
, we obtain from (18)an
equation that determines ¯
z
, which is dissolve after ¯
z
and produces a certain ¯¯
()
*
z
zj=, which
we replace again in (18). The equation
() ( (
¯)¯)()
**
o
oz z,19
cx
j=
than gives the eye caustic. (19)yields intricate yet analytical mathematical expressions. For
the particular case of R=7.5 cm and n=1.54, figure 16 shows the caustic curves for different
values of the eye distance a. When interpreting the caustic as a light caustic, we substitute a
with object distance g. If we set j=π/2, (19)gives the caustic focus coordinate as (5).
3.3. Mathematical description of the images in the sphere
The eye perceives the image of an object point somewhere along the image path
v(t,j)=a+t·e
a
of incident light rays entering the eye (see figure 15), where tis measured
from the first surface at E
1
. The exact location of the real or virtual image in the image space
is governed by the light caustic, i.e. where the image ray is tangential to the real or virtual
branches of the light caustic. For topological analysis, the precise position of the image is not
significant, enabling us to project all the image points onto a plane. This implies that each
object point, denoted T, can be associated with one or more image points, denoted T
i
. The eye
caustic, in turn, causes the images of extended objects to fragment, precisely at the points
where the objects touch the eye caustic.
To determine the shape and quantity of optical images, we introduce an object plane
positioned behind the sphere at a distance gand perpendicular to the optical axis (figure 17).
The object is placed in this plane (in figure 17 it is a circle)and mathematically described by
the coordinates T=(T
1
,T
2
)within the plane. Initially, we assume T
2
=0. The set of object
rays that form the eye caustic behind the sphere has already been derived, as shown in (14)
and (15). Suppose for a specific but arbitrary jan object point is encountered by an object ray
at
¯
zg
=
. From (18)we obtain the coordinate
() ( ¯)Tozg,
x1
jj==
as
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
14

()
() · · ( )
() ()
()
()
TdxR gdzR
x
z
cos sin 20
oo
o
o
111
2
2
jj j=+ +-+
For a given j, the equation for one or more image points corresponding to the object point
(20)is as follows (see appendix):
() ()TgaR
Ra
cos
sin .21
i
1
j
j
=-
-
Figure 18 presents a set of curves (T,T
i
)obtained from equations (20)and (21), where gis
a variable and ais fixed at 15 cm. These curves demonstrate certain characteristics influenced
by the relationship between gand the focus of the eye caustic at z
a
. With R=7.5 cm and
n=1.54 the value of z
a
is determined as 37.3 cm by solving (4), what we have to take
Figure 16. Caustic patterns are calculated for different positions of a, where acan
represent either an observer or a point light source. In this particular scenario, we have
a ball lens with a radius of R=7.5 cm and a refractive index of n=1.54, resulting in a
focal length of f=10.7 cm. (a)When the observer’s eye point is at infinity, the blur
point of the caustic pattern converges towards the focal point. (b)For values of a
greater than the focal length (a=15 cm, a >f´), a blur point exists. (c)If the observer’s
eye moves to the position where ais equal to the focal length (a=f´), the focus of the
caustic pattern touches the z-axis at infinity. (d)As the observer’s eye position moves
further away from the focal point (a<f´), the caustic pattern becomes divergent,
resulting in a virtual branch. The distance ais here set to 9 cm.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
15

negatively in our coordinate system (15). When the absolute value of gis greater than the
absolute value of z
a
, the curves T(T
i
)exhibit strict monotonicity. This indicates that each
object point corresponds to a single image point, and (T,g)is located outside the caustic. On
the other hand, when the absolute value of gis less than the absolute value of z
a
, each object
point can be associated with either one image point (when Tis outside the caustic)or three
image points (when Tis inside the caustic). The width ΔTof the caustic region for a given g
can be determined from this diagram. Experimental curves of (T,T
i
)can be found in [2], for
further details see [3,6,26].
Figure 17. The diagram illustrates the process of calculating the images of a circle in
the object plane Σat a distance g. The diagram does not include the ball lens (see
figure 15 for reference). Object points (T
1
,T
2
)that fall within the eye caustic are
imaged three times, while object points outside the caustic are imaged only once. To
address the three-dimensional nature of the problem, the determination of the image
points
(
)
TT,
ii
12
is carried out individually for each angle θ. In the T
1
−T
2
-system, we can
represent a circle with center (T
a
,T
b
)and radius rby using the parameters θand ρ,
where θrepresents the angle and ρrepresents the length.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
16

3.4. Example: images of a circular object
Now we aim to explicitly model the images of an object, a circle with the center (T
a
,T
b
)and
radius r. We can represent the circle using equation
(
)( )TT T T r
ab12222
-+- =
, where
·Tr Tcos a1a=+
and ·Tr Tsin b2a=+
(where αis the angle of rotation in the object
circle). To take advantage of the rotational symmetry of the ball lens, we can parameterize the
points (T
1
,T
2
)on the object circle in the object plane using the length ρand the angle θ
(figure 17). The angle θrepresents the rotation of the observation plane, while ρindicates the
distance from the object point (T
1
,T
2
)to the z-axis. For the length ρand the angle θ, we have
TT
1
2
2
2
r
=+
and (
)
TTarctan 21
q
=.
Using this parameterization, we transform the problem into a one-dimensional scenario for
each angle θ, and the solutions are given by equations (20)and (21). Therefore, according to
(20), we can solve the equation:
()
·· ()
() ()
()
()
dx R g dz R x
z
cos sin 22
oo
o
o
11
2
2
r
jj=+ +-+
for jin the range ε/2j180°−εnumerically and calculate according to equation (21)
the corresponding image points.
Figure 18. Characteristic curves (T,T
i
)are shown in the blue (a), green (b), and red (c)
graphs. The fixed value of ais set at 15 cm, which corresponds to z
a
being −37.3 cm.
The value of gvaries in each graph. In the blue graph, where gis −15 cm (i.e. |g|<
|z
a
|), all Tvalues within the interval ΔThave three corresponding images T
i
. On the
other hand, in the green graph, where gis −60 cm (i.e. |g|>|z
a
|), each Tvalue has
only one corresponding T
i
. The red graph represents the scenario where gis equal to z
a
,
resulting in a change in the topology of the images. The width of the eye caustic is
indicated by ΔT.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
17

In the object plane, we need to include πor 2πdepending on the quadrant. Figure 19
illustrates the images corresponding to the observation situation for the ball lens (R=7.5 cm,
n=1.54)shown in figure 3. The object circle at g=–8.5 cm has a radius of r=1 cm, and the
eye is positioned at a=15 cm. While keeping T
a
=0fixed, T
b
increases incrementally. Our
observations reveal a good correspondence between the experimental results (figures 3and
11)and the theoretical model.
The current model does not take into account dispersion, which may raise the question
whether dispersion effects can be incorporated into the concept of the caustic eye. In figure 3,
we have examined a persistent red glow just before the disappearance of the two images,
which is missing in figure 19. Dispersion can be considered by adjusting the refractive index
based on the wavelength. Since we do not have precise information on the glass composition
of the ball lens used, we assume the borosilicate crown optical glass BK7 (Schott)with the
following specifications: n
A
=1.5116 for line A (O
2
)with λ
A
=759.370 nm (red)and
n
L
=1.5334 for line L (Fe)with λ
L
=382.044 nm (blue). For varying refractive index, two
slightly displaced eye caustics appear for each respective wavelength (figure 20). As an bright
object in dark surrounding moves from the outer to the inner region of the caustic, it will first
exhibit a red border. The phenomenon is related to the green flash phenomenon [27].
4. Summary
In this paper, our primary objective is to showcase the versatility of the eye caustic as both a
pedagogical and a technical tool to explore and predict image transformations, using the ball
lens as an illustrative example. We emphasize the fascinating geometric similarity and
symmetry between the eye caustic and the light caustic that result from exchanging a point-
light source and an observing eye. We illustrate how these caustics can be associated with two
perspectives: the eye caustic plays a vital role in organizing the observed images when
looking through the ball lens (embedded perspective), whereas the light caustic characterizes
Figure 19. Image transformation of a circle at g=–8.5 cm with radius r=1cm
displaced perpendicular to the optical axis. The eye is positioned at a=15 cm. The
series corresponds to the photographed sequences from figures 3and 11.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
18

the imaging conditions on a screen (detached perspective). We have developed a novel
interpretation of the imaging equation for ball lenses, introducing the concept of the blur
point. In section 3, we present a model to describe image transformations of the ball lens
using geometric optics. We provide analytical solutions for light caustics and the eye caustic,
as well as a formalism to calculate the fragmentation and transformation of images.
By identifying and defining organizational categories for lens phenomena, we support the
careful and orderly description of the phenomena before relating them to the ray model. The
relationship between eye caustic and light caustic can be used in optics lessons as an example
of how the linking of embedded and detached perspective enables the exploration of optical
phenomena (for further examples see [19,28–31]). We show how the topological structure of
the eye caustic behind a ball lens can be explored with the help of a test object. This task may
enable even secondary school students to practise intersubjectively accurate and reproducible
observations that are not limited to paraxial space. The mathematical modeling of image
transformations on the ball lens presented in section 3certainly exceeds the level of high
school. The objective here is to show how the phenomenological exploration can be con-
tinued and deepened through mathematical modeling. Modeling phenomena and the question
of the fit between model and observation concern central properties of acquiring physical
knowledge, which graduate students can comprehend using the example shown.
Acknowledgments
The authors thank all anonymous referees for improving the clarity and presentation of this
document.
Data availability statement
All data that support the findings of this study are included within the article (and any
supplementary files).
Figure 20. Eye caustic for n
A
=1.5116 (red)and n
L
=1.5334 (blue)(a=15 cm,
R=7.5 cm). As an object moves from the outer to the inner zone of the caustics, it will
first take on a reddish hue.
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
19

Appendix
A.1. Calculation of equations (8)and (11):
The law of refraction in its vectorial form states ()( )
n
nee e e
nn11 2 2
´= ´indicating that the
unit vectors e
1
and e
2
in the direction of the incident and refracted rays, respectively, as well
as e
n
in the direction of the normal to the refracting surface at the point of incidence, are
coplanar. If we multiply the vectorial refraction law from the left with the cross product of e
n
,
the resulting equation can be simplified using the BAC-CAB rule:
(( )) (( )) ()
n
neee eee A1
nnn n11 22
´´ = ´ ´
()( )
() ( ) ()
n
nee e e e e e e e e .A2
nnn nnn11
2122
22
-=-
This equation needs to be solved for e
2
. For that, we need to transform the expression e
n
e
2
.
Because of the definition of the dot product and utilizing the trigonometric Pythagorean
theorem sin cos
1
22
bb+=
, we have initially:
ee cos 1 sin ,
n22
bb==-
where βrepresents the angle of refraction. Using the law of refraction in the form of
() ()
n
nsin sin
12
ab=
we get
⎜⎟
⎛
⎝⎞
⎠()
n
n
ee 1sin, A3
n2
1
2
2
2
a=-
where αrepresents the angle of incidence. Applying the trigonometric Pythagorean theorem
and the dot product in the form
ee cos
n1
a=
once again yields:
⎜⎟
⎛
⎝⎞
⎠
()
() ()
n
n
ee ee11 . A4
nn2
1
2
2
12
=- -
Substituting (A3)into (A2), considering that
e
1
n
2
=
,finally results in the expression for e
2
:
⎜⎟
⎛
⎝
⎜⎛
⎝⎞
⎠
⎞
⎠
⎟
()
() ()
n
n
n
n
n
n
eeeee ee11 .
nn n2
1
2
1
1
2
1
1
2
2
12
=- -- -
In the terminology developed in section 3.1, equation (8)yields for n
1
=1, n
2
=n,
()
ee
nn
1
=
,e
1
=e
a
and ()
ee
o
21
=
. Equation (11)yields for n
1
=n,n
2
=1,
()
ee
nn
2
=
,
()
ee
o
11
=
and ()
ee
o
22
=
.
A.2. Calculation of equations (12)and (13)
The normal vector ()
e
n
2is related to the vectors
()
e
o
1
and ()
e
n
1by
()
·()
() () ()
d
Rerr ee,A5
onn
121 21
=-= -
where dis the distance between the incidence points E
1
and E
2
(see figure 15). Thus, we get
(12)as
() () ()
d
R
eee.
non
211
=+
Eur. J. Phys. 45 (2024)045301 T Quick and J Grebe-Ellis
20

To determine dwe seek the intersection point of the line
·
()
d
x
re
o
11
=+
with the circle
x
2
=R
2
. Evaluating for d,wefind
–
()
d
re2
o
11
=
, and thus:
()
() ( )
() ()
d
Rx z2 cos sin A6
oo
11
jjj=- +
and with (9)and (10)we get (13)as
() ()
d
Ra
nu
21 cos .
22
22
jj
j
=-
A.3. Calculation of equation (21)
When an object point T
1
is on the object ray, we need to determine the corresponding image
ray and bring it to intersect with the object plane at g. This process yields the image points.
The image ray is given by v(t,j)=a+t·e
a
. From the component v
z
(t)=g, we can derive
with (7)the following equation:
·()
(())
gat Ra
aR aR
sin
2sin
22 12
j
j
=+ -
+-
and thus an equation for t
()( ())
() ()
t
gaa R aR
Ra
2sin
sin .A7
22 12
j
j
=-+-
-
By reintroducing (A7)into the first component
()vt T
x
i
1
=
we get (21).
ORCID iDs
Thomas Quick https://orcid.org/0000-0002-9201-6231
Johannes Grebe-Ellis https://orcid.org/0000-0003-0400-0780
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