The “rainbow” in the dropGiovanni Casini and Antonio Covello Citation: Am. J. Phys. 80, 1027 (2012); doi: 10.1119/1.4732530 View online: http://dx.doi.org/10.1119/1.4732530 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v80/i11 Published by the American Association of Physics Teachers Related ArticlesReflection of a polarized light cone Am. J. Phys. 81, 24 (2013) Energy and momentum entanglement in parametric downconversion Am. J. Phys. 81, 28 (2013) Novel cases of diffraction of light from a grating: Theory and experiment Am. J. Phys. 80, 972 (2012) A simple experiment that demonstrates the “green flash” Am. J. Phys. 80, 955 (2012) The Optics of Life: A Biologist’s Guide to Light in Nature. Am. J. Phys. 80, 937 (2012) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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APPARATUS AND DEMONSTRATION NOTESThe downloaded PDF for any Note in this section contains all the Notes in this section.

Frank L. H. Wolfs, EditorDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14627

This department welcomes brief communications reporting new demonstrations, laboratory equip-ment, techniques, or materials of interest to teachers of physics. Notes on new applications of olderapparatus, measurements supplementing data supplied by manufacturers, information which, while notnew, is not generally known, procurement information, and news about apparatus under developmentmay be suitable for publication in this section. Neither the American Journal of Physics nor the Editorsassume responsibility for the correctness of the information presented.

Manuscripts should be submitted using the web-based system that can be accessed via the AmericanJournal of Physics home page, http://ajp.dickinson.edu and will be forwarded to the ADN editor forconsideration.

The “rainbow” in the drop

Giovanni CasiniDipartimento di Fisica, Universita di Roma “Tor Vergata,” Roma 00133, Italia

Antonio CovelloLiceo Vittoria Colona di Roma, Roma 00186, Italia

(Received 17 March 2008; accepted 18 June 2012)

We describe an apparatus that can visualize the creation of rainbows using a cylinder of acrylic

glass. The apparatus allows one to observe rainbows up to the sixth order. A brief theoretical

introduction and a method to quantitatively analyze the observation are discussed. A simple

lighting system is described and its divergence is computed. The effect of light divergence on

rainbow formation is analyzed. VC 2012 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4732530]

I. INTRODUCTION

A demonstration that creates a rainbow in the laboratorycan capture the attention of our students.1 Although mostdemonstrations of this type allow us to measure the anglebetween the white incoming rays and the emerging coloredrays, they do not allow us to see the reflections and refractionsthat occur inside the drop. The apparatus described in this pa-per allows us to visualize the paths of the rays outside andinside the “drop.”2 We can see that the light emerging fromthe “drop” forms rainbows up to sixth order. Furthermore, wecan quantitatively compare the measured and calculated exitangles. When an intense light source and a good camera areused, it is possible to use the apparatus as a classroomdemonstration.

II. A SHORT ESSAY ON THE THEORY OF THE

RAINBOW

The rainbow is part of the light that comes back to oureyes from raindrops illuminated by the sun.3 Consider abeam of parallel rays incident on a spherical drop of radiusR. Fig. 1 shows a particular ray, incident with an impact pa-rameter p and an angle of incidence hi. Part of the incidentray is reflected from the surface and part is refracted into thedrop. The refracted ray will meet the water–air surface whereit will again be partially refracted into the air and partiallyreflected back into the interior of the drop. The ray will con-tinue to undergo multiple reflections inside the drop, losingintensity at each encounter with the water–air surface.

The exit angle h can be calculated by adding the devia-tions of the ray each time it meets the water–air interface.The deviation of the incident ray when it refracts into thedrop is hi � hr. A ray refracting from the interior of the dropinto the air experiences the same deviation hi � hr. The raythat emerges from the drop after two refractions has a devia-tion of 2ðhi � hrÞ. The deviation for a reflected ray isp� 2hr . The exit angle for a ray that undergoes k reflectionsinside the drop before being refracted into the air is

hkðhi; kÞ ¼ 2ðhi � hrÞ þ kðp� 2hrÞ¼ 2hi þ kp� 2hrðk þ 1Þ: (1)

Since the angle hr is a function of the incident angle hi andthe index of refraction nðkÞ, the exit angle is a function of hi,k, and k. Light reflected once inside the drop, k¼ 1, is cus-tomary referred to as “light of first order”; “light of secondorder,” k¼ 2, is reflected twice inside the drop. A plot of hk

as function of hi for the first four orders is shown in Fig. 2.Each order has a minimum exit angle, customary called theCartesian angle hCk. In order to find the incident angle thatproduces the Cartesian angle, the angle hr must be elimi-nated from Eq. (1) using Snell’s law. The correspondingincident angle hi;Ck is given by

hi;Ck ¼ arccos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½nðkÞ�2 � 1

kðk þ 2Þ

s0@

1A: (2)

1027 Am. J. Phys. 80 (11), November 2012 http://aapt.org/ajp VC 2012 American Association of Physics Teachers 1027

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The presence of a distinct exit angle h for different wavelengthsseems sufficient to explain the origin of rainbows. However,this difference in exit angles occurs for any angle, while the cir-cular shape of rainbows can be explained only if the colors areseparated around a specific angle, about 41� � 42�.

In order to understand color separation, we examine Fig. 3.Figure 3 shows the exit angle of first-order light, obtainedfrom Eq. (1), for k ¼ 633 nm (red) and k ¼ 404 nm (violet).For a given exit angle h1 and wavelength k, the correspondingangles of incidence can be determined. Drawing a horizontalline at the exit angle to be studied, we can look at the intersec-

tions of this line with the curves h1ðhi; kÞ. If the line crossesthe curves associated with red and violet light, it must crossall curves corresponding to wavelengths between 404 nm and633 nm. The scattered light at that particular exit angle thuscontains all wavelengths of the visible spectrum. If the rela-tive intensities of the scattered wavelengths are not very dif-ferent, the light that emerges will be white. For example, forh ¼ 150 �, the line crosses the red and violet curves in two dif-ferent regions of hi. Since the relative intensities of the differ-ent wavelengths are similar,4,5 the light that emerges ath ¼ 150� will be white.

Outgoing colored light will emerge from a drop only ifthere are values of h for which not all wavelengths are pres-ent or if their relative intensities are very different. The firstcondition occurs at values of h for which the red and violetcurves are not intersected by the same horizontal line.Figure 3 shows that there are two areas where this happens:on the far right of the graph (88� < hi < 90� and165� < h1 < 167�), where the incident rays are almost tan-gential to the drop, and in the region around the minima ofthe curves (50� < hi < 70� and 137� < h1 < 139�). The in-tensity of the colored light for 88� < hi < 90� is very low fortwo reasons. First, the ratio of the intensities of reflected andrefracted light is an increasing function of hi; for88� < hi < 90�, almost all incident light is reflected and verylittle is refracted into the drop. Second, the range of impact

Fig. 1. Path of a light ray incident on a drop and subsequent reflections and

refractions. Only the first two internal reflections are shown.

Fig. 3. (Color online) The exit angle h1ðhi; kÞ, obtained using Eq. (1) for

k ¼ 632 nm (red light, long dashed) and k ¼ 404 nm (violet light, long

dashed-point-point), as function of the angle of incidence hi. In the region

around the minimum, graphs for k¼ 589 nm (yellow, continuous line),

k¼ 546 nm (green, dashed), k¼ 486 nm (cyan, long dashed-point),

k¼ 436 nm (blue, dotted) are also shown, assuming indices of refraction of

water at 20 �C equal to nred ¼ 1:33211, nyellow ¼ 1:33335, ngreen ¼ 1:33483,

ncyan ¼ 1; 33749, nblue ¼ 1:34059, and nviolet ¼ 1:34312.

Fig. 2. (Color online) The exit angle h as function of the angle of incidence

hi obtained from Eq. (1) for red and violet lights and k¼ 1, 2, 3, and 4.

The index of refraction of water was assumed to be nR ¼ 1:33211 and nV

¼ 1:34312 for red (633 nm) and violet (404 nm) light, respectively.

1028 Am. J. Phys., Vol. 80, No. 11, November 2012 Apparatus and Demonstration Notes 1028

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parameters associated with the interval 88� < hi < 90� issmall compared to Dhi ¼ 2� intervals for non-tangential rays(see Fig. 1). As a result, the scattered white component hasan intensity that is at least 3 orders of magnitude larger thanthe colored component, and no colors are visible.11

Descartes was the first one to address the formation of rain-bows by drops of water.7 He calculated the exit angle formany incident rays and noticed that rays were strongly con-centrated around the minimum exit angle hC.8 He concludedthat the intensity of the emerging rays should have a maxi-mum around hC and that this angle should be the rainbowangle. His hypothesis was confirmed by measurements eventhough his ideas about the origin of the individual colors wereinadequate and a complete explanation of the rainbow phe-nomenon was achieved only with Newton’s contribution.9

Consider what happens in the area around the minimumexit angle, shown in Fig. 3 and in more detail in Fig. 4. Sincered light has the lowest minimum exit angle, at hC1ðkredÞonly red light is visible. For other colors, this does not hap-pen. At the minimum exit angle for yellow light, red light ispresent too, and the yellow light is superimposed on the redlight. For the same reason, green light is superimposed onyellow and red light, and so on. At the minimum exit anglehCkðkÞ, each color has a maximum intensity. Since the mini-mum exit angle hCkðkÞ depends on wavelength, each colorhas a maximum intensity at a slightly different exit angleand a single wavelength will dominate at each exit angle. Inthe geometrical optics approximation, the angular width ofthe intensity distributions shown in Fig. 4 is similar to thedivergence of solar rays,19 and soft color separation results.Equation (1) for k > 1 provides a similar behavior, as shownin Fig. 2.

In order to understand several key features of a rainbow, itis useful to visualize the information contained in Fig. 2 fork¼ 1, 2 using a graphical representation of the rays emergingfrom the drop, as shown in Fig. 5. First-order light emergesfrom the drop within a cone of opening angleaR ¼ p� hC1ðkredÞ þ d=2, where d is the divergence of the

solar rays. Second-order light emerges from the dropoutside a cone of opening angle bR ¼ hC2ðkredÞ � p� d=2.Colored rays are scattered at angles between aV ¼ p�hC1ðkvioletÞ � d=2 and aR for the first-order rainbow and atangles between bR and bV ¼ hC2ðkvioletÞ � pþ d=2 for thesecond-order rainbow. The colors for the first- and thesecond-order rainbow appear in reverse order. No light offirst or second order emerges in the angular range betweenaR and bR.

The role played by raindrops based on their position in thesky is illustrated in Fig. 6. The axis of the rainbow passesthrough the eye of the observer and is parallel to the direc-tion of the sun rays. Since the angle c between the axis andthe line of sight of a raindrop is equal to the scattering angle,we can subdivide the sky in five zones, delimited by four

Fig. 5. Schematic of the light scattered by a raindrop illuminated by sun

rays. The direction of the incident rays is specified by the arrow shown on

the left. First-order light is scattered at angles less than aR while second-

order light is scattered at angles greater than bR. First-order rainbows form

between aV and aR; second-order rainbows form between bR and bV .

Fig. 4. (Color online) Details of Fig. 3, expanded in the region around the minimum exit angle. The bar on the right schematically shows the color of the light

leaving the drop at the corresponding angle. The distributions on the right-hand side qualitatively show the intensity distributions of the scattered light as func-

tion of the exit angle h for various wavelengths in arbitrary units. The peak shape is consistent with the divergence of solar light.

1029 Am. J. Phys., Vol. 80, No. 11, November 2012 Apparatus and Demonstration Notes 1029

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conical surfaces with opening angles of aV , aR, bR, and bV .Zone W1, within a cone of opening angle aR, returns first-order white light to the observer. Zone R1, between aV andaR, corresponds to the primary rainbow. Zone D, between aR

and bR, is called the Alexander’s10 dark zone and cannotreturn light of first and second orders to the observer. Itappears remarkably darker. Zone R2, between bR and bV ,corresponds to the secondary rainbow. Compared to the pri-mary rainbow, the secondary rainbow appears larger,weaker, and with colors in reverse order. Zone W2, outsidethe cone of angular opening bV , returns white light of secondorder to the observer.

For third and fourth orders, the minimum scattering anglesare greater than p=2 and rainbows of third and fourth ordercannot be observed because they are overwhelmed by thesun itself.

Since the angles of incidence associated with rainbows arenear the Brewster angle, the light forming the rainbows isalmost completely polarized in a direction parallel to theplane of incidence.12

The picture we have shown is an approximation becausethe geometrical optics theory has two significant limitations.First, the angular dependence of the intensity of the diffusedlight is not affected by the radius of the drop, which entersthe equation only as an overall factor. This suggests that arainbow forms each time we illuminate drops, independentof their dimension. But light diffused by the clouds whichcontain drops with diameters between 0.1 lm and 100 lm iswhite.13 Light diffused by fog, which contain drops withdiameters smaller than 60 lm, may show a white bow.14

Observations show that rainbows only form in the presenceof relatively big drops, such as rain, with diameters between200 lm and 2–3 mm, and that the bigger the drops thebrighter the colors.6,15 Second, the geometrical optics theorycannot explain the presence of the supernumeraryrainbows.4–6,13,16,17 These are little weak rainbows, placedjust inside the primary rainbow and external to the secondaryrainbow. Both these failures can be overcome with a wave-based theory that takes into consideration the interferencebetween different wave fronts emerging from the drop withthe same direction of propagation but with different phases.Other theories have been developed to explain otherdiscrepancies.16,17

III. THE EXPERIMENTAL APPARATUS

The experimental apparatus we developed relies on the cy-lindrical symmetry of the process that generates the rainbow.Since the reflection and refraction occur in a plane, we canuse a cylinder instead of a sphere in our apparatus. A draw-ing of the apparatus is shown in Fig. 7. It is made of a flatand smooth surface disk (2), 200 mm in diameter. A pivot(4) is attached to the center of the disk, protruding from oneside. On the same side, a plastic ring (3), about 40 mm wideand with the same outer diameter as the disk, is glued to theedge of the disk. On the other side of the disk, a circularsheet of paper (5) is glued and at its center, a cylinder ofacrylic glass18 (1) is fixed with transparent double sided ad-hesive tape. The cylinder is 20 mm thick and has a diameterof 60 mm. A strip of white thin cardboard (6) is glued to theedge of the disk, on the external surface of the plastic ring,protruding about 25 mm on the acrylic cylinder side. On itsprotruding side, a 10 mm wide slit (8) is cut for the incidentlight. The inner face of this white strip is the screen wherethe scattered light is observed. In order to shield this screenfrom ambient light, it is necessary to place a second biggerscreen around the entire apparatus (7). This screen, made ofblack thin cardboard, must have a corresponding slit. Piecesof black cardboard and black adhesive tape are attached tothe inner screen to absorb light rays reflected from the sur-face and light rays that are refracted without an internalreflection. A support with a suitable hole for the pivot allowsour apparatus to rotate freely.

The slit width is adjusted using a piece of black adhesivetape. To determine the proper width, the information con-tained in Table I can be used. Table I shows all expected val-ues of the Cartesian angles for two wavelengths, 632.8 nm(red) and 435.8 nm (blue), and refraction indices ofn(632.8)¼ 1.489 and n(435.8)¼ 1.5025.20 To first order, thedifference between the Cartesian incident angles for red andblue light is Dhi;C ¼ 470, corresponding to a slit width ofabout 0.27 mm on the external screen. For higher orders, thisvalue is smaller. We used a slit width of about 1.5 mm tocover an adequate range of hi around hi;C for any order.

The incident beam of light used with our apparatus musthave a divergence similar to those of solar rays. According

Fig. 7. (Color online) A sectional drawing of the experimental apparatus: (1)

acrylic glass cylinder, (2) disk, (3) plastic ring, (4) pivot, (5) white paper, (6)

white thin cardboard screen, (7) shielding black cardboard, and (8) slit. The

light entering the slit is scattered by the acrylic glass cylinder and is visible on

the paper glued on the disk surface and on the white thin cardboard screen.

Fig. 6. (Color online) Role played by raindrops according to their position

in the sky: observer eye OE, rainbow axis RA, a raindrop RD with its own

scattering graph, first-order white zone W1, primary rainbow zone R1,

Alexander’s dark zone D, secondary rainbow zone R2, and second-order

white zone W2. The limit angles of each zone are indicated.

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to geometrical optics, each point on the surface of the illumi-nated object is reached by rays emitted from each point onthe source surface, as shown in Fig. 8 for a spherical sourceand object. The maximum divergence of the beam comingfrom the source depends on the distance between the sourceand the object and on their sizes. The total divergence willdepend on the divergence associated with the dimension ofthe source and the divergence associated with the dimensionof the object. The divergence associated with the size of thesource is equal to

dS ¼ 2 arctanS

2D

� �’ S

D; (3)

where S is the diameter of the source and D the distancebetween the source and the object. The divergence associ-ated with the size of the object is equal to

dO ¼ 2 arctanO

2D

� �’ O

D; (4)

where O is the diameter of the object. The total divergenceof the beam is equal to

d ¼ dS þ dO ’S

Dþ O

D: (5)

In order to understand the effect of divergence on our experi-ment, we consider two rays entering the drop at the samepoint with different incident angles, as shown in Fig. 9. RayA (solid line) represents the usual ray without divergence,while ray B (broken line) represents a ray with a divergenced. In reference frame Oxy, ray A has an incident angle hi. Theexit angle can be calculated using Eq. (1): hA ¼ hkðhi; kÞ.Equation (1) cannot be used to determine the exit angle ofray B since the incident ray is not parallel to the x axis. How-ever, in reference frame Ox0y0 , rotated by an angle d withrespect to Oxy, we can use Eq. (1) with an angle of incidence

of hi þ d. The exit angle of ray B in reference system Oxy isequal to

hBðhi; k; dÞ ¼ hkðhi þ d; kÞ � d: (6)

For a point source, the first term in Eq. (5) is zero and no tworays enter the drop at the same point with different angles ofincidence. For a raindrop of radius r at a distance D from thepoint source, the divergence is equal to

dðhiÞ ¼ arctanr sin hi

D� r cos hi

� �: (7)

Figure 10 shows the exit angle obtained by inserting Eq. (7)in Eq. (6). Figure 10 also shows the exit angle in the absenceof divergence. The effect of divergence due to object sizeresults in a reduction of both the minimum exit angle and theincident angle at which the minimum exit angle occurs. Theleft shift is due to the use of the angle hi instead of the inci-dent angle hi þ d. The down shift is a result of the fact thatour observations are made in the Oxy reference frame. Sincein our experiment we measure the exit angle hCk with respectto the incident angle hi þ d, we can neglect this kind ofdivergence.

For a point object, we have the superposition of incomingrays, distributed across an angular range of 6d=2 for eachhi. For solar light, S in Eq. (5) is the diameter of the sun andthe divergence is about 320.19 Figure 11 shows the effect ofsource size on the scattering process. The dotted curveshows the relation between exit and incident angles for apoint source. The solid and dashed curves showhBðhi; k; d=2Þ and hBðhi; k;�d=2Þ, assuming a divergenced ¼ 640, twice as large as the solar divergence. The effectof the divergence depends on the value of the incident anglehi. For a fixed wavelength, the spread of the incoming raysin the 6d range causes a broadening of the exit angularrange where the intensity of the scattered light peaks. Thistype of divergence thus increases the overlap of contiguous

Table I. Incident angle, spread of the incident angle, rainbow angle, colors spread calculated for the wavelengths kR ¼ 632:8 nm and kB ¼ 435:8 nm. The cor-

responding indices of refraction for polymethyl methacrylate (PMMA) are nR ¼ 1:489 and nB ¼ 1:5025, respectively. The measured angles of the red side of

the rainbow and the differences with expected values are reported in the last two columns.

hi;C Dhi;C hC DhC Measured values

Order k kR kB iCR � iCB kR kB hi;CB � hi;CR hM6DhM hM � hC

First 1 50�260 49�390 470 156�090 157�230 1�140 156�3006400 þ0�210

Second 2 67�030 66�390 240 264�530 267�180 2�250 264�5906400 þ0�060

Third 3 73�270 73�100 170 366�190 369�460 3�270 366�5006800 þ0�310

Fourth 4 76�590 76�460 130 465�160 469�430 4�270 466�06061000 þ0�500

Fifth 5 79�150 79�040 110 563�050 568�290 5�240 564�36061200 þ1�310

Sixth 6 80�500 80�410 90 660�150 666�360 6�210 659�17061200 �0�580

Fig. 8. (Color online) The calculation of the beam divergence relies on geometrical optics and the assumption that each point on the surface of the object is

reached by rays emitted by each point on the surface of the source.

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wavelengths and washes out the colors. It is thus importantto keep the divergence of the source in our experiments ator below the solar divergence.

To generate the proper beam for our experiment, we triedvarious approaches. In this paper, we describe two simple,cheap, and easily reproducible solutions. The first approachuses a tungsten light bulb, placed in a black box with a holepositioned such that the light cone comes almost exclusivelyfrom the filament. The bulb is a 65W Osram Halostar Ecowhose filament is a coil, 1.95 mm in diameter and 4.3 mm inlength. For this lamp, the divergence can be estimated usingEq. (5), where S is taken to be the diameter of the filamentcoil, assuming that the lamp is positioned with the axis of thefilament coil parallel to the slit of the apparatus and O is thewidth of the slit. In order to have d ¼ 320, the lamp must beplaced about 42 cm from the axis of the apparatus. Using thislight source, we can observe rainbows up to third order.

In order to obtain a more intense light source, we use thesame tungsten lamp in the box and add a lens and an iris.Good results are obtained with a f¼ 150 mm lens. For thissystem, the beam divergence can be calculated using Eq. (5).In all configurations useful to focus the light with the appro-priate divergence, S is the lens diameter or the iris aperture,D is the lens-slit distance and O is the slit width. In ourmeasurements, we set the filament-lens distance to a valueslightly less than the focal length. An iris with an aperture of29 mm was placed in front of the lens, while the apparatus

was placed 2 m beyond the lens. With this method, weobtained the required divergence and a light intensity suffi-cient to see rainbows up to fourth order. Using photographswe can observe fifth-order rainbows, even though it isdirected toward the slit of the apparatus and only a weaktrace is visible near the acrylic cylinder.

Using sunlight and sufficient shielding, we succeeded inphotographing even a weak sixth-order rainbow.

IV. RESULTS

We illuminated the apparatus slit as previously describedand turned its support in the horizontal plane by about 3� 300

so that the light grazed the paper on the disk such that thepath of the rays was clearly visible. Rotating the apparatuson its pivot, we let the beam enter the acrylic cylinder atwell-defined angles of incidence. This measurement can beused both as a starting point to explain why rainbows appearand to demonstrate the previously described theory of colorsseparation near the minimum exit angle. It can be used toexplain many rainbow features: the form, the angle, the posi-tion of the first- and second-order rainbows, and the darkband.

Our results are shown in Figs. 12–15. Figure 12 shows apicture of the apparatus illuminated by a tungsten lamp, setup to observe the primary rainbow. Figure 13 shows a pictureof the apparatus set up to observe two and three reflections,corresponding to the conditions required for second- andthird-order rainbows. Figure 14 shows a picture of the appa-ratus illuminated by a tungsten lamp and configured to showrainbows of third and fourth order. Finally, Fig. 15 shows apicture of the apparatus illuminated by solar rays wherefourth, fifth, and a pale sixth-order rainbows are visible. Thephotograph was taken outside with a thick black cover allaround the apparatus and the camera. Due to the very weakintensity of those orders, the picture has been taken usinglight with its polarization parallel to the plane of incidence.This technique does not alter the intensity of the rainbow butreduces the intensity of the light diffused for k¼ 0. To takethe photos, long exposure times are required and a tripodmust be used. To use the experiment as a classroom demon-stration, we have used a commercial camera at its maximumsensitivity.

Fig. 10. (Color online) Exit angle as function of incident angle for 632.8 nm

light from a point source incident on a finite drop with (dashed curve) and

without (solid curve) divergence. The incident angle hi is defined in Fig. 9.

The divergence is calculated using Eq. (7) for r¼ 30 mm and d¼ 425 mm,

an actual case studied with our setup using direct lighting.

Fig. 11. (Color online) Effect of divergence due to source size on drop scat-

tering for 632.8 nm light. The dotted line represents scattering without diver-

gence, the solid line correspond to �d=2 divergence and the dashed line

correspond to þd=2 divergence, where d ¼ 640, twice the solar divergence.

On each point of the drop surface, the incoming rays are distributed over a

6d=2 range, and the curves should be replaced by a band.

Fig. 9. (Color online) Impact of beam divergence on the outgoing direction.

The scattering of the usual ray A is compared with that of ray B whose direc-

tion is changed by an angle d due to divergence. The scattering of ray B is

evaluated in the rotated reference system Ox0y0 .

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In order to make measurements, we have to take photo-graphs with the camera lens aligned with the center of the“drop.” The photographs must include the entire disk and theinner screen of the apparatus. The geometrical informationon the photo, i.e., the angles between the path of the rays,can be easily extracted by making measurements with rulerand a protractor on the photo print. Alternatively, one canuse the software package GIMP 2, which is available for freeon internet.21

GIMP 2 has a tool to measure distances (inpixels) and angles (in degrees). With GIMP 2, one can find theangular position of the "software ruler" with high resolution.We made test measurements with GIMP 2 on photos of sheetswith lines drawn at known angles and determined that themeasurements are repeatable within an error of about 50.22

Similar errors were obtained for photos similar to thoseshown in Figs. 12–15 if the border of the light path is sharp.Otherwise, larger uncertainties must be assumed. The sharp-ness of the light beam depends primarily on rainbow orderand we made a set of measurements for each order. Weassumed an error of three standard deviations for each set.23

We used GIMP 2 to measure the divergence of solar rays.Rotating the apparatus in a way that light does not enter theacrylic cylinder, the path of the beam is clearly visible andits divergence can be measured. The values obtained variedbetween 2806100 and 3606100, in agreement with theaccepted value of 320. The measured divergence of the beam

Fig. 14. (Color online) A photo of the apparatus illuminated by a tungsten lamp

in the condition of the third- and fourth-order rainbows. The labels on the photo

indicate slit S, reflected rays R, and each number refers to the order of the light.

Fig. 15. (Color online) A photo of the apparatus illuminated by the sun. The

labels on the photo indicate slit S, reflected rays R, and each number refers to

the order of the light. We can see the fifth-order rainbow emerging just after

the first-order light on the left side of the cylinder, going toward the slit. A

pale sixth-order rainbow can be seen between the second order and the

reflected rays. Brightness and contrast of the picture have been increased for

best visual quality.

Fig. 12. (Color online) A photo of the apparatus illuminated by a tungsten

lamp in the condition of the first-order rainbow. The inset shows the zone

delimited by the dashed rectangle photographed with a reduced exposure.

The labels on the photo indicate slit S, reflected rays R, and each number

refers to the order of the light.

Fig. 13. (Color online) A photo of the apparatus illuminated by a tungsten

lamp in the condition of the second-order and third-order rainbows. The

labels on the photo indicate slit S, reflected rays R, and each number refers

to the order of the light.

1033 Am. J. Phys., Vol. 80, No. 11, November 2012 Apparatus and Demonstration Notes 1033

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obtained with the tungsten lamp was lower by about 50 com-pared to the expected values obtained with Eq. (5). Wenoticed that the measured divergence depends on the expo-sure. By increasing exposure, part of the twilight zonebecomes visible and the measured divergence increases.

During our measurements of the angles between incidentand diffused rays, we noticed that the red side of the rainbowhad a sharp border but the violet side did not. When weincreased the exposure time, the red border was quite stablebut the violet border shifted. As a consequence, we decidedto make measurements only for the red side of the rainbow,and these are reported in Table I, starting from the first to thesixth order.

We calculated the error introduced by a rotation in thehorizontal plane of the apparatus by about 3� � 4� andfound that it to be very small, no more than 0.2%. The errordue to the imperfect roundness of the cylinder, which ismade from an extruded bar, depends on the point of inci-dence of the beam, but we found that rotating the cylinderwith respect to the beam or cylinder substitution did notchange the results obtained. A cause of systematic error inthe measurement of the first-order angle can be the caus-tic:24 the red border follows the caustic profile and isslightly curved. This profile is more curved near the dropand approaches the Cartesian ray far from it. Due to the slitwidth used in our measurements, the caustic does not formcompletely because it lacks a portion of the beam that isnecessary for its formation. Unfortunately, a reflection gen-erates an identical weaker complete caustic. It can be easilyrecognized and should be ignored in our measurements byusing sections of the rays furthest away from the dropwhere the caustic is almost straight and close to the direc-tion of the Cartesian ray. The most significant contributionto the error is the loss of intensity of the scattered lightalong its path. It causes an increasing shift of the red borderof the light path going away from the cylinder. The loss ofintensity produces a systematic error that increases themeasured exit angle.

If one observes Figs. 14 and 15 online, one will see manythin colored rays in the fan of light that emerges from thecylinder for k¼ 0 and k¼ 1. These rays are produced by su-perficial imperfections on the acrylic extruded rod fromwhich the cylinder is made. They cause little differences inthe optical path that increase with the incidence angle. Theresulting interference produces thin fringes with differentspacing for each color.

ACKNOWLEDGMENTS

The authors thank Professor Mauro Casalboni for encour-aging them several times to write this article. They thank allthe professors of the Gruppo Arcobaleno Lauree Scientifichefor their precious hints for teaching. They also thank

Dr. Roberta Forte, Elisabetta Forte, and Professor Maria Co-rona for helping them in the English translation.

1We recall two experiments of this kind: Frank S. Crawford, “Rainbow

dust,” Am. J. Phys. 56 1006–1009 (1988); Harold A. Daw, “A 360� rain-

bow demonstration,” Am. J. Phys. 58, 593–595 (1990).2Our method can be considered as an evolution of that given by R. D. Rus-

sel “A rainbow for the classroom,” Phys. Teach. 27, 262–263 (1989).3An introductory, geometrical approach to the rainbow may be found in R.

J. Whitaker, “Physics of the rainbow,” Phys. Teach. 12, 283–286 (1974).4A very good paper on the rainbow is J. D. Walker “Multiple rainbows

from drop of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).5W. J. Humphreys, Physics of the Air, 3rd ed. (MacGraw-Hill, New York,

1940). Reprint (Dover Publications, New York, 1964), pp. 476–500.6M. Minnaert, The Nature of Light & Color in the Open Air, translated by H.

M. Kremer-Priest, revised by K. E. Brian Jay (G. Bell & Sons, Ltd., London,

1938). Reprint (Dover Publications, New York, 1954), pp. 167–189.7R. Descartes, Les meteores, Discours huitieme, “de l’arc-en-ciel,” A recent

and complete translation of this may be found in R. Descartes, The Worldand Other Writings, translated and edited by S. Gaukroger (Cambridge

U.P., New York, 1998), pp. 85–86.8From the point of view of Descartes, h0 was the maximum deviation

because, if compared to our mode, he measured the angle of deviation of

the rays from the other side, that is, the complement to 360�.9Newton’s letter to Oldenburg, secretary of the Royal Society, containing his

new theory about light and colors, published in the Philos. Trans. 80,

3075–3087 (February 19, 1672). Available online at <http://rstl.royalsociety

publishing.org/content/6/69-80/3075.full.pdf>. The complete Newton’s

theory of the rainbow is in his Opticks, 4th ed. (William Innis, London,

1730), first book, Part II, Prop. IX, Prob. IV, pp. 147–156, freely available

online at <http://books.google.com>.10Alexander of Afrodisia, a philosopher lived between the and the 3rd cen-

tury BC11See Ref. 4, pp. 424–425 and Ref. 17, pp. 246–251.12This is true for the natural rainbows and for ours experiment in the range

of interest. Other specific situation with different refraction index or higher

orders must be appropriately calculated.13See Ref. 6, pp.178–179 and David K. Lynch and William Livingston, Color

and Light in Nature, 2nd ed. (Cambridge U.P., New York, 2001), p. 120.14F. Palmer, “Unusual rainbow,” Am. J. Phys. 13, 203–204 (1945).15J. D. Walker observed the vanishing of the color for drops under 300 lm

of diameter, see Ref. 4, p. 432.16For a popular review on the rainbow theory see H. Moyses Nussenzveig,

“The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).17For a complete review of the rainbow’s theory see J. A. Adam, “The mathe-

matical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).18With “acrylic glass,” we mean polymethyl methacrylate (PMMA).19Divergence of solar ray is equal to apparent diameter of the sun that vary

with the earth–sun distance between aphelion and perihelion. For our pur-

pose, we considered the apparent diameter of the sun at the mean distance,

that is, about 32’. See Encyclopedia of the Solar System (Academic Press,

San Diego, CA, 2007).20We have found accurate refractive index data of PMMA at <http://refractive

index.info/?group¼PLASTICS&material¼PMMA>.21See the <http://www.gimp.org> web site.22On photos 4752� 3168 displayed at 1:1 zoom, the position of the elec-

tronic ruler can be set with a such accuracy.23An exception is the 6th order that we were able to observe with sufficient

intensity to perform a measure only in two photo.24J. E. McDonald, “Caustic of the primary rainbow,” Am. J. Phys. 31,

282–284 (1963).

1034 Am. J. Phys., Vol. 80, No. 11, November 2012 Apparatus and Demonstration Notes 1034

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