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Everyday lens phenomena frequently deviate from the paraxial approximation. (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, 8 and 11.
Source publication
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 too...
Contexts in source publication
Context 1
... symmetry aligns with the two 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. ...
Context 2
... 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. ...
Context 3
... 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. ...
Context 4
... object located at this position will blur. Figure 12. Both the eye caustic and the light caustic stem from the same geometrical construction. ...
Context 5
... 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´>f´> R). The eye is positioned at point A on the optical axis, denoted by the z-axis, directed to the left. ...
Context 6
... 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 j between r 1 and the x-axis (see figure 15). ...
Context 7
... the unit vector e a = (x a , z a ) of the direction of view from A to E 1 is obtained by normalizing the vector from a to 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 5 and 8, the illuminance diminishes. ...
Context 8
... 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 n sin sin 1 2 a b = 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): Figure 14. The figure illustrates the function z a (a) for a ball lens with R = 7.5 cm and n = 1.54. ...
Context 10
... 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. ...
Context 11
... 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. ...
Context 12
... 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 t is 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. ...
Context 13
... determine the shape and quantity of optical images, we introduce an object plane positioned behind the sphere at a distance g and 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. ...
Context 14
... determine the shape and quantity of optical images, we introduce an object plane positioned behind the sphere at a distance g and 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. ...
Context 15
... a given j, the equation for one or more image points corresponding to the object point (20) is as follows (see appendix): Figure 18 presents a set of curves (T, T i ) obtained from equations (20) and (21), where g is a variable and a is fixed at 15 cm. These curves demonstrate certain characteristics influenced by the relationship between g and the focus of the eye caustic at z a . ...
Context 16
... curves demonstrate certain characteristics influenced by the relationship between g and 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 a can represent either an observer or a point light source. ...
Context 17
... diagram illustrates the process of calculating the images of a circle in the object plane Σ at 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. ...
Context 18
... α is 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 ρ and the angle θ ( figure 17). The angle θ represents the rotation of the observation plane, while ρ indicates the distance from the object point (T 1 , T 2 ) to the z-axis. ...
Context 19
... the object plane, we need to include π or 2π depending 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 = 0 fixed, T b increases incrementally. ...
Context 20
... 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. ...
Context 21
... law of refraction in its vectorial form states ( ) ( ) n n e e e e n n 1 1 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: A.2. Calculation of equations (12) and (13) The normal vector ( ) e n 2 is related to the vectors = -= -where d is the distance between the incidence points E 1 and E 2 (see figure 15) (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. ...
