Bubble optics: Leonardo’s cross revisited—Part 1, numerical methods

Markus Selmke; James A. Lock; James A. Lock

doi:10.1364/AO.426424

Applied Optics
Vol. 60,
Issue 21,
pp. 6213-6225
(2021)
•https://doi.org/10.1364/AO.426424

Bubble optics: Leonardo’s cross revisited—Part 1, numerical methods

Markus Selmke and James A. Lock

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Markus Selmke and James A. Lock, "Bubble optics: Leonardo’s cross revisited—Part 1, numerical methods," Appl. Opt. 60, 6213-6225 (2021)

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Abstract

Around 1508, Leonardo DaVinci observed a curious image of the sun formed by a floating bubble. Now, some 500 years later, we give a full picture of the surprisingly rich phenomenology of the light pattern formed below surface bubbles. Under shallow illumination, and for shallow depths of water, the caustic deviates from the symmetrical astroid shape obtained for larger depths and/or near-vertical illumination. Whereas the astroid caustic involves refraction through the outer meniscus only, more complex ray paths explain the asymmetric, multicomponent (composite) light pattern.

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Supplementary Material (1)

Name	Description
Dataset 1	Ray tracing output for floating bubbles

Data Availability

Data underlying the results presented in Section 3 of this paper are available as Dataset 1, Ref. [28].

28. M. Selmke, “Ray tracing output for floating bubbles,” figshare, 2021, https://doi.org/10.6084/m9.figshare.14762550.v1.

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Preceding	# $T_{i / o}$ in $[\dots]$ Even	# $T_{i / o}$ in $[\dots]$ Odd
$[\dots] T_{o} \to$	$◻_{i}$	$◻_{o} \lor ◻_{c}$
$[\dots] R_{o} \to$	$◻_{o} \lor ◻_{c}$	$◻_{i}$
$[\dots] T_{i} \to$	$◻_{o}$	$◻_{i} \lor ◻_{c}$
$[\dots] R_{i} \to$	$◻_{i} \lor ◻_{c}$	$◻_{o}$
	# $T_{i / c}$ in $[\dots]$ Even	# $T_{i / c}$ in $[\dots]$ Odd
$[\dots] T_{c} \to$	$◻_{i} \lor ◻_{c}$	$◻_{o}$
$[\dots] R_{c} \to$	$◻_{o}$	$◻_{i} \lor ◻_{c}$

$d$	Conceivable Ray Paths
1	$T_{o}$
2	$T_{c} T_{i}$ , $R_{c} T_{o}$ , $R_{o} T_{o}$ , $T_{o} R_{i}$
3	$T_{o} R_{i} R_{o}$ , $T_{o} T_{i} T_{i}$ , $R_{o} T_{o} R_{i}$ , $R_{o} R_{o} T_{o}$ , $R_{o} T_{c} T_{i}$ , $R_{o} R_{c} T_{o}$ , $T_{c} R_{i} T_{i}$ , $T_{c} T_{i} R_{o}$ , $T_{c} T_{c} T_{o}$ , $T_{c} R_{c} T_{i}$ , $R_{c} T_{o} R_{i}$ , $R_{c} R_{o} T_{o}$
4	$T_{o} R_{i} R_{o} R_{i}$ , $T_{o} R_{i} T_{o} T_{o}$ , $T_{o} T_{i} T_{i} R_{o}$ , $T_{o} T_{i} R_{i} T_{i}$ , $T_{o} T_{i} T_{c} T_{o}$ , $T_{o} T_{i} R_{c} T_{i}$ , $R_{o} T_{o} R_{i} R_{o}$ , $R_{o} T_{o} T_{i} T_{i}$ , $R_{o} R_{o} T_{o} R_{i}$ , $R_{o} R_{o} R_{o} T_{o}$ , $R_{o} R_{o} T_{c} T_{i}$ , $R_{o} R_{o} R_{c} T_{o}$ , $R_{o} T_{c} R_{i} T_{i}$ , $R_{o} T_{c} T_{i} R_{o}$ , $R_{o} T_{c} T_{c} T_{o}$ , $R_{o} T_{c} R_{c} T_{i}$ , $R_{o} R_{c} T_{o} R_{i}$ , $R_{o} R_{c} R_{o} T_{o}$ , $T_{c} R_{i} R_{i} T_{i}$ , $T_{c} R_{i} T_{i} R_{o}$ , $T_{c} R_{i} T_{c} T_{o}$ , $T_{c} R_{i} R_{c} T_{i}$ , $T_{c} T_{i} R_{o} R_{i}$ , $T_{c} T_{i} T_{o} T_{o}$ , $T_{c} T_{c} T_{o} R_{i}$ , $T_{c} T_{c} R_{o} T_{o}$ , $T_{c} R_{c} T_{i} R_{o}$ , $T_{c} R_{c} R_{i} T_{i}$ , $T_{c} R_{c} T_{c} T_{o}$ , $T_{c} R_{c} R_{c} T_{i}$ , $R_{c} T_{o} R_{i} R_{o}$ , $R_{c} T_{o} T_{i} T_{i}$ , $R_{c} R_{o} T_{o} R_{i}$ , $R_{c} R_{o} R_{o} T_{o}$ , $R_{c} R_{o} T_{c} T_{i}$ , $R_{c} R_{o} R_{c} T_{o}$

Ray Path	Corresponding Feature in Leonardo’s Cross
$T_{o}$	(i), (partial) astroid: $\pm x_{0}$ , $\pm y_{0}$ cusps [16,21]
$T_{c} T_{c} T_{o}$	(v), cap shadow ( $+$ parts of partial astroid)
$T_{c} T_{i} R_{o}$	(ii), (outer) $- x_{0}$ cusp [10]
$T_{c} T_{i} T_{o} T_{o}$	(iii), (inner) $- x_{0}$ cusp
$T_{c} T_{i}$ , $T_{o} T_{i} T_{i}$	(iv), sharp curved boundary (aperture effect)
$T_{c} R_{c} T_{i}$ , $T_{c} R_{i} T_{i}$	(vi), $+ x_{0}$ cusp/(outer, flanking) folds at $\pm y_{0}$
$T_{c} R_{c} T_{c} T_{o}$	(vii), $+ x_{0}$ cusp/(inner, flanking) folds at $\pm y_{0}$

cs	Category	Ray Paths
R	Right-rim interactions	$T_{c} T_{i} T_{o} [R_{o}]_{\geq 0} T_{o}$ , $T_{c} T_{i} R_{o} [R_{i} R_{o}]_{\geq 0}$ , $T_{c} T_{i} [R_{o} R_{i}]_{\geq 1}$
L	Left-rim interactions	$T_{o} R_{i} [R_{o} R_{i}]_{\geq 0}$ , $T_{o} [R_{i} R_{o}]_{\geq 1}$
T	Traverses	$T_{o} T_{i} T_{i}$ , $T_{o} T_{i} R_{i} T_{i}$ , $T_{o} T_{i} T_{i} R_{o}$ , $T_{o} T_{i} R_{i} T_{i} R_{o}$
C	Cap-related	$T_{c} T_{c} T_{o}$ , $T_{c} [R_{c}]_{\geq 1} T_{i}$ , $T_{c} [R_{c}]_{\geq 1} T_{c} T_{o}, \dots$

Preceding	# $T_{i / o}$ in $[\dots]$ Even	# $T_{i / o}$ in $[\dots]$ Odd
$[\dots] T_{o} \to$	$◻_{i}$	$◻_{o} \lor ◻_{c}$
$[\dots] R_{o} \to$	$◻_{o} \lor ◻_{c}$	$◻_{i}$
$[\dots] T_{i} \to$	$◻_{o}$	$◻_{i} \lor ◻_{c}$
$[\dots] R_{i} \to$	$◻_{i} \lor ◻_{c}$	$◻_{o}$
	# $T_{i / c}$ in $[\dots]$ Even	# $T_{i / c}$ in $[\dots]$ Odd
$[\dots] T_{c} \to$	$◻_{i} \lor ◻_{c}$	$◻_{o}$
$[\dots] R_{c} \to$	$◻_{o}$	$◻_{i} \lor ◻_{c}$

Abstract

References

Supplementary Material (1)

Data Availability

Cited By

Figures (11)

Tables (4)

Equations (1)

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