A new class of rainbows is created when a droplet is illuminated from the inside by a point light source. The position of the rainbow depends on both the index of refraction of the droplet and the position of the light source, and the rainbow vanishes when the point source is too close to the center of the droplet. Here we experimentally measure the position of the transmission and one-internal-reflection total-internal-reflection rainbows, and the standard (primary) rainbow, as a function of light-source position. (C) 2003 Optical Society of America

Adler, Charles L.

Ekelman, Diana

Keating, Brian

Lock, James A.

Mulholland, Jonathon

English

EngagedScholarship @ Cleveland State University

Experimental Observation of Total-Internal-Reflection Rainbows

Cleveland-Marshall College of Law

Cleveland State UniversityEngagedScholarship@CSUPhysics Faculty Publications Physics Department1-20-2003Experimental Observation of Total-Internal-Reflection RainbowsCharles L. AdlerJames A. LockCleveland State University, j.lock@csuohio.eduJonathon MulhollandBrian KeatingDiana EkelmanFollow this and additional works at: https://engagedscholarship.csuohio.edu/sciphysics_facpubPart of the Physics CommonsHow does access to this work benefit you? Let us know!Publisher's StatementThis paper was published in Applied Optics and is made available as an electronic reprint with thepermission of OSA. The paper can be found at the following URL on the OSA website:http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-42-3-406. Systematic or multiplereproduction or distribution to multiple locations via electronic or other means is prohibited and issubject to penalties under law.Repository CitationAdler, Charles L.; Lock, James A.; Mulholland, Jonathon; Keating, Brian; and Ekelman, Diana, "Experimental Observation of Total-Internal-ReflectionRainbows" (2003). Physics Faculty Publications. 43.https://engagedscholarship.csuohio.edu/sciphysics_facpub/43This Article is brought to you for free and open access by the Physics Department at EngagedScholarship@CSU. It has been accepted for inclusion inPhysics Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contactlibrary.es@csuohio.edu.Original CitationAdler, Charles L., James A. Lock, Jonathon Mulholland, Brian Keating, and Diana Ekelman. "Experimental Observation of Total-Internal-Reflection Rainbows." Applied Optics 42 (2003): 406-411.Experimental observation oftotal-internal-reflection rainbowsCharles L. Adler, James A. Lock, Jonathon Mulholland, Brian Keating, and Diana EkelmanA new class of rainbows is created when a droplet is illuminated from the inside by a point light source.The position of the rainbow depends on both the index of refraction of the droplet and the position of thelight source, and the rainbow vanishes when the point source is too close to the center of the droplet.Here we experimentally measure the position of the transmission and one-internal-reflection total-internal-reflection rainbows, and the standard primary rainbow, as a function of light-source position.© 2003 Optical Society of AmericaOCIS codes: 010.1290, 290.1310, 290.5850.1. IntroductionIn the standard geometrical optics theory of the rain-bow, the position of the rainbow is derived by assum-ing that a bundle of parallel light rays are refractedinto a water droplet, make a number of reflectionsinside the droplet, and refract out. The rainbow oc-curs at a relative minimum in the deflection angle asa function of the impact parameter of light rays en-tering the droplet. The assumption that the raysare parallel means that the light source illuminatingthe droplet is infinitely far away. Realistic modelsof the natural rainbow also must take into accountthe finite angular width of the Sun, but we will ignorethis complication.1 We can imagine taking a pointsource of light and moving it from a great distanceaway from the droplet, bringing it closer and closer,and finally inside the water droplet. Does the na-ture of the rainbow change when we bring the lightsource inside?It does, in two ways:a First, a fundamentally new class of rainbows iscreated total-internal-reflection TIR rainbowsTIRRs when the light source is moved inside thedroplet.b Second, the standard rainbows and TIRRs donot exist for all positions of the light source. Therainbow angles are a function of the position of thelight source and cease to exist when the light sourceis moved too close to the center of the droplet.2. TheoryA semiclassical treatment of the scattering of light byan internal light source in a spherical droplet has beenderived recently.2 In this paper we look only at whatgeometrical optics can tell us about this problem.Figure 1 shows a diagram of a droplet of radius a witha point source of light inside it at a distance r from thecenter. The index of refraction of the droplet is n,whereas the index of the surrounding medium is 1. Alight ray from the source is internally reflected p  1times before leaving the droplet. We define severalangles as follows:  is the angle that the light raymakes with respect to a diameter of the droplet thatruns through the light source,  is the angle that theray makes with the surface normal to the droplet as itexits the droplet,  is the angle between the ray and thesurface normal when the ray is refracted out of thedroplet, and 	 is the total deflection angle from 0°.From Snell’s law and the law of reflection, we canderive several relationships among these angles:rasin   sin , (1)n sin   sin , (2)	    p  1  2p  1  . (3)C. L. Adler cladler@smcm.edu, J. Mulholland, B. Keating, andD. Ekelman are with the Department of Physics, St. Mary’s Collegeof Maryland, St. Mary’s City, Maryland 20686. J. A. Lock is withthe Department of Physics, Cleveland State University, Cleveland,Ohio 44115.Received 23 January 2002; revised manuscript received 12 April2002.0003-693503030406-06$15.000© 2003 Optical Society of America406 APPLIED OPTICS  Vol. 42, No. 3  20 January 2003Because 	 can be larger than 180°, we define thescattering angle, , to be 	 reduced to the range0–180°. Because the light source is inside the drop-let, for certain values of ra and , light will be totallyinternally reflected inside the droplet. Figure 2shows  as a function of  for ra  0.8, n  43, andp  1 i.e., no internal reflections. For  betweenapproximately 80° and 110°, the light is totally inter-nally reflected. What is interesting is that for val-ues of  just beyond the TIR region,  goes through ashallow minimum for a scattering angle that is oth-erwise forbidden because of TIR. This minimum an-gle represents a scattering angle for which theintensity of light from geometrical optics diverges:in other words, a rainbow.By considering the conditions under which d	d 0, we can examine what range of values of   ra forwhich a TIRR can occur. From Eqs. 1–3,d	d 1  2p  1cos cos  ncos cos . (4)Setting the derivative to 0, and using Eqs. 1–3 tosubstitute for cos  and cos , we find1 nx1  n221  x212 2p  1x1  21  x212 0, (5)where x  cos  in this expression. Since x  1,  1n if there is to be a zero for this expression; thusthis TIRR occurs for values of ra between 0.75 and 1for n  43. For this index the rainbow angle 	varies between 131° for ra  0.75 and 158.5° forra  1.If light is reflected once before exiting the waterdroplet, potentially two different types of rainbowcan be created: the standard p  2 rainbow, and,because some of the rays from the point source willbe totally internally reflected, a p  2 TIRR. Fig-ure 3 is a graph of 	 versus  for the p  2 case forra  0.8 and n  43. The position of the p  2standard rainbow and the p  2 TIRR are indicatedon the graph. The dent in the graph near   66°is an artifact caused by 	 exceeding 180° at thatpoint. The behavior of the p  2 standard rain-bow as the position of the light source is varied isinteresting. For ra  0.6, the p  2 rainbow doesnot occur; we can show this by setting x  1 in Eq.5 and solving for . For ra  0.6 the rainbowangle is 180°, which is essentially rainbow-enhanced glory scattering.3,4 As ra approaches 1,the p  2 standard rainbow and the p  1 TIRRmerge; this is because a point source located nearthe edge of the droplet will have the TIR regionlocated almost directly behind it on the nearer edgeof the droplet.2 In principle, TIRRs can exist for allvalues of p, and the standard rainbow can exist forall p  2, but we do not consider them in this paper,because they become faint; experimentally, we didnot see any of the p  2 rainbows.Fig. 1. Ray diagram for light scattered by a point light sourceinside a spherical droplet.Fig. 2. Scattering angle 	 as a function of the internal impactparameter  for zero internal reflections p  1. For this dia-gram, ra  0.8 and n  1.33.Fig. 3. Scattering angle 	 as a function of internal impact pa-rameter  for one internal reflection p  2. For this diagram,ra  0.8 and n  1.33.20 January 2003  Vol. 42, No. 3  APPLIED OPTICS 4073. ExperimentFigure 4 shows a diagram of the experimental setupwe used to investigate TIRRs. We used a water-filled globe inner diameter, 16 cm; wall thickness,0.13 cm to simulate a water droplet and placed apoint source of light inside the droplet. Light exitingthe droplet was projected onto a white screen. Thepositions of the p  1 and p  2 TIRRs and the p 2 standard rainbow were measured on the screenwith a meter stick; in addition to this, we photo-graphed the rainbows, using a high-resolution digitalcamera. The point light source that we used was abulb from a mini-Maglite flashlight; to hold the bulbin place, we pushed wires through a double-barreledplastic coffee stirrer, looped them around the endprongs of the bulb, and pulled the end of the bulb intothe coffee stirrer. This gave us an electrical contactof such quality that we did not have to solder theleads to the wire. We then wrapped the end of thetube with waterproof tape to seal the bulb from wa-ter. The size of the filament is approximately 1 mmand is located 3 mm from the end of the bulb. Figure5 is a photo of the experimental setup; the bulb washeld on a mounting arm that was in turn supportedon a translation stage used to change the position ofthe light source. We used a low-voltage power sup-ply to illuminate the bulb; experimentally, we foundthat the bulb could be powered with a voltage of 2.5 Vwithout burning it out. The height of the bulb was 8cm above the bottom of the globe.Figure 6 shows photographs of the rainbows fromthe water-filled globe projected onto a white screen ata distance of L  67.5 cm behind the globe. The p 1 TIRR, the p  2 standard rainbows, and the p 2 TIRR are labeled on the figures. The sequence ofFig. 6 shows a monotonic decrease in ra from 0.96which is as close to the wall of the globe as we can getour light source down to ra  0.6, where the p  2standard rainbow ceases to exist. It is seen that atra  0.75, the p  1 and p  2 TIRRs vanish aspredicted. Note that they move in opposite direc-tions and that their colors are also oriented in theopposite manner. At ra  0.6, the p  2 standardrainbow scattering angle is 180°, and for values of raless than this, it is gone.Although it is not immediately obvious from thephotograph, there are in fact two p  2 standardrainbows, which we label  and . We see two p 2 standard rainbows because of the glass in the globe:Fig. 7 shows the two ray paths that contribute to the and  rainbows. Because the relative index dif-ference between glass and air is greater than forwater and glass, the  rainbow will be brighter thanthe . The angular separation between the  andthe  rainbows is proportional to the glass thicknessdivided by the radius of the globe.5,6 Because thereare two possible internal reflections for the p  2TIRR the internal reflection occuring at the water–glass or the glass–air interface, one might expect tosee 2 p  2 TIRRs; however, only one was seen. Weconjecture that the second is too faint to be seenagainst the background of scattered light from theglobe.Figure 8 shows a graph of the scattering angle as afunction of ra for the p  1 and p  2 TIRRs and thep  2 standard rainbow. Both experiment and the-ory are shown. For the theory, two corrections hadto be made to the rainbow angle: One was due to theglass thickness, and one was due to a geometricalcorrection resulting from the fact that all experimen-tal angles were measured with respect to the center ofthe glass globe and the scattering screen is not infi-nitely far away from the globe. The theory and ex-periment match well, except for the p  2 TIRR,where the theory and experiment are a few degreesapart but have the same functional dependence onra. We are not certain why the discrepancy is solarge for the p  2 TIRR but conjecture that it isbecause the globe is not a perfect sphere and thatdepartures from sphericity will have a larger effect onthe positions of the higher-order rainbows than onthe lower-order ones.7 Although we did not examinethe colors of these rainbows in detail, from experi-ment the angular width of the rainbows was of theorder of 1–2° for all four rainbows for ra 0.9. Onecan use Eqs. 1–5 and the dispersion of the refrac-tive index of water to predict the angular width; forra  0.9, the angular width of the rainbow is pre-Fig. 4. Experimental setup. In our experiment, L 67.5 cm anda  8 cm.Fig. 5. Photograph of droplet and assembly used to move the lightsource. The cross arm holding the light bulb was built on a lineartranslation stage, which was moved in 0.5-mm increments tochange the position of the bulb.408 APPLIED OPTICS  Vol. 42, No. 3  20 January 2003dicted to be 0.71° for the p  1 TIRR, 1.56° for the p 2 TIRR, and 0.46° for the p  2 standard rainbow.This is in decent agreement with observation, al-though it does not take into account the refractiveeffects of the glass or its dispersion.If we walk around the globe and look at the lightFig. 6. Photographs of the light scattered by the droplet projected onto a screen. a ra  0.96. 1. p  1 TIRR rainbow, 2. p  2 standardrainbow. b ra  0.8. 1. p  1 TIRR rainbow, 2. p  2 TIRR rainbow, 3. p  2 “standard” rainbow. c ra  0.75. At this light sourceposition, the p  1 TIRR vanishes, leaving only the p  2 TIRR and the standard rainbows. 1. p  2 TIRR rainbow, 2. p  2 standardrainbow. d ra  0.6. At this light source position, the p  2 standard rainbows converge at a scattering angle of 	  180°. Forsmaller values of ra, they cease to exist.Fig. 7. Ray diagram showing the origin of the  and  rainbows.Fig. 8. Position of the p  1 and p  2 TIRRs and the p  2standard  and  rainbows as a function of light source position.Points, experiment; solid curve, theoretical fit.20 January 2003  Vol. 42, No. 3  APPLIED OPTICS 409source, we will see some interesting things as wechange our viewpoint. Figure 9 shows a glare-spotsequence for ra  0.9. In this sequence we arelooking directly at the globe while changing the view-ing angle of the observer. The position of the lightsource is kept constant. We start out at a viewingangle corresponding to a scattering angle near 90°and increase the viewing angle through the sequenceto the point where the observer’s eye is in the regionto intercept rays from the p  1 TIRR. In Fig. 9aone image of the light source is visible, correspondingto a one-ray region. In Fig. 9b, when we increasethe viewer’s observing angle further, the light sourcecannot be seen because of TIR. In Fig. 9c two im-ages of the light source are seen, corresponding to thep  1 TIRR.4. ConclusionsBy illuminating a water droplet from the inside, wehave created and seen a fundamentally new class ofrainbows that to our knowledge have never been rec-ognized or closely examined before. One potentialfield where this research may be relevant is in theexcitation of morphology-dependant resonances andthe examination of single-molecule fluorescence inlevitated droplets, droplet streams, and solidparticles.8–11 There are intriguing data in the paperby Hill et al.11 in which they measured the angulardependence of the fluorescence of Coumarin dye mol-ecules in a micodroplet. For three-photon excitationof the dye there seems to be an excess of fluorescenceat angles near 150°. This may be significant, as therate of three-photon excitation is proportional to theproduct of the intensity of three laser fields inside thedroplet; because of this, the region where the proba-bility of three-photon excitation of the dye is high islocalized. Because the fluorescence is coming from alocalized region inside the droplet, the excess signal ispossibly an example of a p  1 TIRR.11 We arecurrently putting together an experiment to measurethe intensity of the rainbow as a function of scatter-ing angle, to allow further comparison with single-molecule fluorescence results.As far as examples from meteorological optics go,Young12 made an interesting observation during the2001 Meteorological Optics Conference: Considerreversing the direction of light rays so that we haverays from outside the sphere converging on a pointinside it. If we place an observer at this point, andconsider the bowl to be the observer’s sky, then theobserver will see multiple images of parallel rays thathit the sky at angles near the TIRR impact parame-ters. This is analagous to the refraction of light raysin a stratified atmosphere, which produces multiple,distorted images of the setting Sun.12We thank our two reviewers for a number of sug-gestions that strengthened the paper, especially thereviewer who called our attention to Refs. 9–11.This research was supported by National ScienceFoundation NSF grant PHY-9987862 and by NASAgrant NCC3-521.References1. R. L. Lee, Jr., “What are ‘all the colors of the rainbow’?” Appl.Opt. 30, 3401–3407 1991.2. J. A. Lock, “Semi-classical scattering of an electric dipolesource inside a spherical particle,” J. Opt. Soc. Am. A 18,3085–3097 2001.3. H. M. Nussenzveig, “Complex angular momentum theory ofFig. 9. Glare-spot sequence. ra  0.9. a One-ray region.Light source clearly visible; one image of it seen. b Zero-rayregion. Light source invisible because of total internal reflection.c Two-ray region. Two images of light source visible. The lightsource is in about the right position for the viewer to see the p 1 TIRR.410 APPLIED OPTICS  Vol. 42, No. 3  20 January 2003the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–10791979.4. D. S. Langley and M. J. Morrell, “Rainbow-enhanced forwardand backward glory scattering,” Appl. Opt. 30, 3459–34671991.5. J. A. Lock, J. M. Jamison, and C.-Y. Lin, “Rainbow scatteringby a coated sphere,” Appl. Opt. 33, 4677–4690 1994.6. C. L. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders,“Experimental observation of rainbow scattering by a coatedcylinder: twin primary rainbows and thin-film interference,”Appl. Opt. 40, 1548–1558 2001.7. C. L. Adler, J. A. Lock, and B. R. Stone, “Rainbow scattering bya cylinder with a nearly elliptical cross section,” Appl. Opt. 37,1540–1550 1998.8. M. Kerker and S. D. Druger, “Raman and fluorescent scat-tering by molecules embedded in spheres with radii up toseveral multiples of the wavelength,” Appl. Opt. 18, 1172–1179 1979.9. M. D. Barnes, K. C. Ng, W. B. Whitten, and J. M. Ramsey,“Detection of single Rhodamine 6G molecules in levitated mi-crodroplets,” Anal. Chem. 65, 2360 1993.10. M. D. Barnes, N. Lermer, C.-Y. Kung, W. B. Whitten, J. M.Ramsey, and S. C. Hill, “Real-time observation of single-molecule fluorescence in microdroplet streams,” Opt. Lett. 22,1265–1267 1997.11. S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J. P. Wolf, W. L. Pan,S. Holler, and R. K. Chang, “Enhanced backward-directedmultiphoton-excited fluorescence from dielectric microcavi-ties,” Phys. Rev. Lett. 85, 54–57 2000.12. A. Young, “Green flashes,” presented at the 7th TopicalMeeting on Meteorological Optics, Boulder, Colo., 6–8 June2001.20 January 2003  Vol. 42, No. 3  APPLIED OPTICS 411

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Experimental Observation of Total-Internal-Reflection Rainbows

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