Polarization and visibility of higher-order rainbows

Gunther P. Können

doi:10.1364/AO.54.000B35

Applied Optics
Vol. 54,
Issue 4,
pp. B35-B40
(2015)
•https://doi.org/10.1364/AO.54.000B35

Polarization and visibility of higher-order rainbows

Gunther P. Können

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Gunther P. Können, "Polarization and visibility of higher-order rainbows," Appl. Opt. 54, B35-B40 (2015)

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Abstract

The degree of polarization of rainbows of order $k$ with $k \geq 3$ is bounded in the interval [75%, 78%], where 75% is the limit for $k \to \infty$ . A polarization filter can improve the signal-to-background ratio of the third and fourth rainbows by a factor of 2, which may lift their visibilities in natural circumstances above the threshold of human visual perception. Under optimal circumstances, the latter may be true for the recently photographed green fingerprint of the fifth rainbow, even without the aid of a polarization filter. The prospects for observing the sixth rainbow are unclear. There exists a possibility that the signal of the natural seventh rainbow (appearing at 64° from the Sun) may be separated from its background if photographed under perfect conditions through a polarization filter.

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Rainbow Order	Descartes Rainbow Angle (°)	Degree of Polarization (%)	Dominating Light Path of Background	Background Polarization (%)	Improvement Factor [Eq. (8)]
3	41.6	78.1	$p = 1$	$- 8.7$	1.95
4	43.8	77.0	$p = 1$	$- 9.5$	1.96
5^b	127.5	76.2	$p = 3$	56.5	1.13
5^c	129.4	76.8	$p = 0$	31.2	1.35
6	147.6	76.1	$p = 2$	26.5	1.39
7	63.8	75.9	$p = 1$	$- 8.2$	1.92
Infinity	—	75.3	—	—	—

Rainbow Order (Angle^b)	$B_{drop} (unpol)$ ^c	$S / B_{drop} (unpol)$	$S / B_{drop} (pol 1)$	Improvement Factor [Eq. (8)]
3, (40.8°)	1	0.045	0.086	1.89
4, (44.9°)	0.74	0.022	0.041	1.87
5^d, (128.8°)	0.11	0.047	0.045	0.95
5^e, (130.5°)	0.011	0.77	0.98	1.28
6, (146.2°)	0.15	0.032	0.041	1.28
7, (62.3°)	0.15	0.012	0.021	1.78

	$V i s (unpol) with B = B_{drop} +$			$V i s (pol 1) with B = B_{drop} +$
Rainbow Order	$+ 1 \times B_{L L}$	$+ 3 \times B_{L L}$	$+ 10 \times B_{L L}$	$+ 1 \times B_{L L}$	$+ 3 \times B_{L L}$	$+ 10 \times B_{L L}$
3	1	0.95	0.81	1.9	1.8	1.5
4	0.37	0.35	0.28	0.70	0.65	0.52
5, obscured by second^b	0.56	0.41	0.21	0.58	0.47	0.28
5, in Alex band^c	3.3	1.4	0.46	5.1	2.2	0.75
6	0.34	0.26	0.14	0.45	0.37	0.22
7	0.11	0.08	0.05	0.19	0.14	0.08

	Observer^a	Date (UTC)	Sun Elevation (°)	Visually Observed?	Discernable in Unprocessed Pictures?
Third rainbow	Grossman^b [1]	15 May 2011	8	Yes	Yes [14]
	Theusner [2]	11 June 2011	11	No	No [15]
	Antipov [12,16]	22 June 2013	15	Yes	Yes
	Edens [3]	13 July 2013	9	Yes	Yes
	Jones [17]	8 Sept 2013	10	No	No
	Edens [18]	22 Sept 2013	18	No	Yes
Fourth rainbow	Theusner^b [2]	11 June 2011	11	No	No [15]
	Antipov [12,16]	22 June 2013	15	No	No
	Edens [3]	13 July 2013	9	No	No
	Jones [19]	8 Sept 2013	10	No	No
	Edens [18]	22 Sept 2013	18	No	No
Fifth rainbow^c	Edens [18]	28 July 2009	17	—	Yes
	Edens [4]	4 Sept 2009	8	—	Yes
	Edens [18]	8 July 2010	34	—	Yes
	Edens [18]	29 Aug 2010	4	—	No
	Edens [18]	9 Sept 2010	1	—	No
	Edens^b [3,4]	8 Aug 2012	26	Not tried	Yes
	Edens [18]	8 Mar 2013	16	No	No
	Edens [3]	13 July 2013	9	No	No
	Edens [18]	16 Sept 2013	9	No	Yes
	Edens [18]	22 Sept 2013	18	No	No

Rainbow Order	Descartes Rainbow Angle (°)	Degree of Polarization (%)	Dominating Light Path of Background	Background Polarization (%)	Improvement Factor [Eq. (8)]
3	41.6	78.1	$p = 1$	$- 8.7$	1.95
4	43.8	77.0	$p = 1$	$- 9.5$	1.96
5^b	127.5	76.2	$p = 3$	56.5	1.13
5^c	129.4	76.8	$p = 0$	31.2	1.35
6	147.6	76.1	$p = 2$	26.5	1.39
7	63.8	75.9	$p = 1$	$- 8.2$	1.92
Infinity	—	75.3	—	—	—

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