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Table 1. Coefficients of the analytical expression used to reproduce refraction indices of alcohol and sugar syrup. Parameters are dimensionless. apparent imagine. It is drawn by superimposing its external segment (at the right side) to the corresponding blue ray (i.e. the blue ray starting from the same point). One can say that the object ‘O’ is mapped to the imagine ‘I’ in figure 7. One has also to account for two different light refractions at sections A, A’ of figure 7, in which light passes from air-to-glass-to-liquid and vice versa. The glass interface, though, is not thick enough to allow for a noticeable deviation and has not been included in our computation. The differential equation has been solved numerically with the Mathematica™ ‘NDSolve’ routine®.

Table 1 Coefficients of the analytical expression used to reproduce refraction indices of alcohol and sugar syrup. Parameters are dimensionless. apparent imagine. It is drawn by superimposing its external segment (at the right side) to the corresponding blue ray (i.e. the blue ray starting from the same point). One can say that the object ‘O’ is mapped to the imagine ‘I’ in figure 7. One has also to account for two different light refractions at sections A, A’ of figure 7, in which light passes from air-to-glass-to-liquid and vice versa. The glass interface, though, is not thick enough to allow for a noticeable deviation and has not been included in our computation. The differential equation has been solved numerically with the Mathematica™ ‘NDSolve’ routine®.

Related Figures (10)

Figure 1. Inferior mirage as seen from the point of view of an observer looking through the container.
Figure 1. Inferior mirage as seen from the point of view of an observer looking through the container.
Figure 2. Left: the sequential laser light ray paths which hit the bottle in several points represent the light rays coming from the object. Right: calculated ray paths using a numerical approach, as explained in the text. In order to have equally focused images in figure 1, the bottle was slightly tilted to compensate for the longer optical path length of the curved rays. In order to reproduce this same situation in the laboratory, we used a transparent glass container boiled, in order to avoid bubbles on the container’s walls. 4 The teacher may remark that a substance with a lower density can have a higher refractive index in order to distinguish neatly between the density of a substance and its optical density (or refractive index).
Figure 2. Left: the sequential laser light ray paths which hit the bottle in several points represent the light rays coming from the object. Right: calculated ray paths using a numerical approach, as explained in the text. In order to have equally focused images in figure 1, the bottle was slightly tilted to compensate for the longer optical path length of the curved rays. In order to reproduce this same situation in the laboratory, we used a transparent glass container boiled, in order to avoid bubbles on the container’s walls. 4 The teacher may remark that a substance with a lower density can have a higher refractive index in order to distinguish neatly between the density of a substance and its optical density (or refractive index).
Figure 4. Superior mirage as seen from the point of view of an observer looking through the container.
Figure 4. Superior mirage as seen from the point of view of an observer looking through the container.
T Lépez-Arias et al Figure 3. Alcohol—water refractive index gradient as a function of the height. The higher optical density on top leads to an inferior mirage.
T Lépez-Arias et al Figure 3. Alcohol—water refractive index gradient as a function of the height. The higher optical density on top leads to an inferior mirage.
Figure 6. Syrup—water refractive index gradient as a function of the height. The lower optical density on top leads to a superior mirage. Figure 5. Left: the sequential laser light ray paths which hit the bottle at several points represent the light rays coming from the object. Right: calculated ray paths using a numerical fit of the experimental refractive index gradient, as explained in the text. Note that the bottle stands with a different tilt with respect to the inferior mirage.
Figure 6. Syrup—water refractive index gradient as a function of the height. The lower optical density on top leads to a superior mirage. Figure 5. Left: the sequential laser light ray paths which hit the bottle at several points represent the light rays coming from the object. Right: calculated ray paths using a numerical fit of the experimental refractive index gradient, as explained in the text. Note that the bottle stands with a different tilt with respect to the inferior mirage.
Figure 8. Refracted (blue) and imaginary (red) paths to explain the (inferior) mirage on the bottle. Figure 7. Schematic view of the ray tracing elements. The blue ray is the ‘physical’ path (refracted). The red ray has the same initial conditions (Xp, Yo) as the blue one (right-hand side) but it follows a straight line path and so constitutes a geometric virtual source.
Figure 8. Refracted (blue) and imaginary (red) paths to explain the (inferior) mirage on the bottle. Figure 7. Schematic view of the ray tracing elements. The blue ray is the ‘physical’ path (refracted). The red ray has the same initial conditions (Xp, Yo) as the blue one (right-hand side) but it follows a straight line path and so constitutes a geometric virtual source.
Figure 9. Ray tracings in alcohol (inferior mirage) showing the construction of the non-inverted, squeezed image (top) and of the inverted image (bottom).
Figure 9. Ray tracings in alcohol (inferior mirage) showing the construction of the non-inverted, squeezed image (top) and of the inverted image (bottom).
Mirages in a bottle Figure 10. Inferior mirage main features (see figures 1 and 9). ‘A’ section: squeezed, non-inverted image (figure 9, top); “B’ section: inverted image (figure 9, bottom).
Mirages in a bottle Figure 10. Inferior mirage main features (see figures 1 and 9). ‘A’ section: squeezed, non-inverted image (figure 9, top); “B’ section: inverted image (figure 9, bottom).
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Physics EducationRay TracingCurriculum and Pedagogy
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