SPEC I A L F E A T U RE : C O L O U R

www.iop.org/journals/physed

Coloured rings produced ontransparent platesWilfried Suhr and H Joachim Schlichting

Institut fur Didaktik der Physik, Universitat Munster, 48149 Munster, Germany

E-mail: wilfried.suhr@uni-muenster.de

AbstractBeautiful coloured interference rings can be produced by using transparentplates such as window glass. A simple model explains this effect, which wasdescribed by Newton but has almost been forgotten.

Interference colours where they are notexpectedThe formation of interference colours is usuallyassociated with careful experimentation, special-ized illumination and small dimensions. Studentsdo not normally expect to see this phenomenonoutside the laboratory, as it is not commonlyknown that interference colours can be seen ontransparent material such as window panes whichare several millimetres thick. However, if theylook at a window in the usual way they will not seeany interference colours—a specific perspective isrequired to see the coloured rings.

Interference colours similar to those producedby diffraction gratings can only be seen using thickglass within a small solid angle whose centrelineis perpendicular to the plate. If the centrelinecombines a suitable light source with its reflectionon the plate it is possible to see coloured ringsconcentric around the reflection.

Some historical remarksIn 1704, Isaac Newton published the firstobservations of these kinds of interference coloursin his book ‘Opticks’ [1]. Towards the end ofthe 19th century, distinguished scientists such asThomas Young, John Herschel, George Stokesand Adolphe Quetelet (see [2]) investigated thismore thoroughly, and explained it on the basis of

the wave theory of light. Newton’s very detaileddescription of his experiments is an invitation forthem to be reproduced (see [1], p 289).

He used the Sun’s rays as a light sourceshining into a dark room through a small holein a shutter. He let the rays fall onto a concavespherical mirror with quicksilver on its convexreverse side. A piece of white cardboard witha small hole in it was fixed to the shutter. Byaligning the mirror so that its centre of curvaturecoincided with the hole in the cardboard it couldbe expected that all light passing through the holewould be reflected back to it by the mirror. Incontrast, he observed ‘upon the chart four or fiveconcentric irises or rings of colours, like rainbows,encompassing the hole . . .’ ([1], p 290).

This experiment can be reproduced using acommercial cosmetic mirror. When a modernshutter blind is nearly closed so that the gapsbetween the slats remain open and a white cardprovided with a circular hole of about 5 mm isfixed in front of it, we have the same setup asNewton. However, after adjusting the mirror sothat the light is focused back to the hole, nospecial effect can be observed. Obviously Newtondid not mention one necessary condition for theappearance of the rings: his mirror must have beencovered with impurities, causing part of the lightto be scattered. To achieve this scattering effectin our experiment, the front of the mirror must

566 P H Y S I C S E D U C A T I O N 42 (6) 0031-9120/07/060566+06$30.00 © 2007 IOP Publishing Ltd

Coloured rings produced on transparent plates

be covered by a thin layer of scattering particles.Domestic dust does very well, but you have to waittwo or three weeks for this! Instead, covering themirror with a thin film of salad oil instantly givesvery good results.

This idea of using a greasy film goes back tothe beginning of the 19th century, when AdolpheQuetelet and William Whewell were investigatingand writing about the phenomenon of colouredrings, and hence the coloured rings are also knownas Quetelet’s rings (see [3], p 158).

Coloured rings caused by a magic breathTo demonstrate the experiment in a classroomwe used a bright electric light source instead ofsunlight. Between the light and the mirror weplaced a 30 cm × 30 cm screen with a 5 mmcircular hole in its middle. The distance betweenthe screen and the mirror has to correspond to theradius of curvature of the concave mirror. This isthe case when the light spot reflected to the screenhas the same size as the circular hole. (If the spotspreads out due to the manufacturing tolerances ofthe mirror, a circular aperture covering the rim ofthe mirror will correct this.)

At the start of the demonstration the studentscan only see a very small band around the hole,receiving light from the mirror. This is just whatone would expect when the light is focused onthe hole, but by simply breathing against themirror colourful rings like rainbows are magicallycreated on the screen (see figure 2). However,the magic quickly fades away as the condensationevaporates, showing directly that this was thereason for the phenomenon.

To show experimentally which element ofthe mirror contributes to which of the observedrings, the illumination can be restricted to a smallsection of the mirror surface. This is best doneby directing the narrow beam of a laser pointerthrough the hole of the screen onto differentsections of the mirror. Regardless of whichelement of the mirror is hit by the laser beam,the rings appear in the same form and size as ifthe whole screen was illuminated. (Of course,when using monochromatic light the colorationdue to the spectral distribution is absent.) Fromthis observation it can be concluded that eachelement of the mirror produces a similar patternof rings (see figure 4). The bright ring systemproduced by the completely illuminated mirror can

be thought of as a superposition of the elementaryrings originating from each element of the mirrorsurface.

Virtual coloured rings ‘painted’ in oilThe projection of the coloured rings on a screenrequires a very bright light source. This hasthe advantage that many students can see thephenomenon at the same time. If a single observerlooks directly into the reflected light, only arelatively faint illumination is needed to make thecoloured rings visible. This alternative view hadalready been adopted by Newton when he glancedalong the optical axis towards the surface of theilluminated mirror. He recognized that the positionand the curvature of the visible sections of the ringsystems changed according to his visual angle.

In order to obtain sufficient luminosity fora real projection of the coloured rings, it isnecessary that all mirror elements contribute tothe superposed image. Obviously, our eyes aresensitive enough to manage with the relatively lowlight intensity scattered by particles on the smallsurface element of the mirror. Therefore, this typeof observation does not require the concentrationof the reflected light by means of a concave mirror,and a plane mirror will be adequate.

A simple hands-on experiment can help toillustrate this by using an ordinary mirror in adarkened room.

The mirror must be misted up (preferablywith small droplets of water) and the observershould be looking at it from a distance of about2.5 m. On holding a flashlight (whose reflector hasbeen removed) in front of the observer’s forehead,coloured stripes can be seen around the directreflection in the mirror. These stripes are sectionsof rings whose centre is displaced according to thedistance between the eye and the flashlight.

Even the flame of a candle is bright enoughto produce Quetelet’s rings. Quetelet used candleflames as a light source, and the small size of thecandle flame meant that it had the advantage thatit could be put between the observer’s eye and themirror without obscuring too much of the visualfield. A similar phenomenon based on a differentphysical principle was recently described by JamesBridge in this journal [4].

Ordinary glass panes can also be used forproducing interference rings. Quetelet’s ringscan be seen on windows which have not beencleaned for some time (see figure 1). Sometimes

November 2007 P H Y S I C S E D U C A T I O N 567

W Suhr and H Joachim Schlichting

Figure 1. Coloured rings can be seen on a dirtywindow pane around the reflection of the setting sun.

you can find by chance windows that look dirtyduring daylight but which display colourful stripesat night when illuminated by the headlights of acar (figure 5). This effect can be reproduced byapplying a thin coating of salad oil to a pane ofglass. This results in an impressive display ofcoloured rings around the reflection of the sun.They can be up 50 cm in diameter (figure 6).

A simple model explaining the colouredrings

The origin of the coloured rings will be explainedby a simple model. According to the setupof the experiment shown in figure 7, L denotesa point light source and O the observer’s eye.Their positions are described within a rectangularcoordinate system. The surface of the mirror islying in the xy-plane. In order to simplify thegeometry we presuppose that the thickness t of themirror glass is small compared to the distances lz

and oz . We consider a small element of the surfacewhere P denotes the position of a small particle.Figure 8 shows that the (almost parallel) rays l1

and l2 emitted by L impinge the surface: at pointA the ray l1 is refracted into the glass, reflectedat the silvered layer MM′ and afterwards hits theparticle P, at which point it is scattered. As the sizeof the particles is assumed to be of the magnitudeof some micrometres, the forwardly directed Miescattering prevails. In contrast, the ray l2 is atfirst scattered at P, and the scattered light is thenrefracted into the glass.

Figure 9 shows the geometric relationshipneeded to calculate the phase difference. Startingfrom the light source L the rays l1 and l2 arrive atthe points A and B with the same phase. Amongthe light rays scattered at the mirror element arethe rays o1 and o2, which travel to the eye O of theobserver. They have a constant phase shift whenleaving the points F and E. The optical retardation

Figure 2. Coloured rings produced with the setup shown in figure 3. On the left: concentric rings. On the right: ata slight distortion of the mirror, eccentric rings appear.

568 P H Y S I C S E D U C A T I O N November 2007

Coloured rings produced on transparent plates

Figure 4. Each element of the mirror produces a similarelementary ring system.

�s may then be calculated to give

�s = (AP′ + PF

)opt − (

BP + P′E)

opt . (1)

In this calculation of the optical path length1

we make use of the fact that the distance coveredby the light travelling from A via the mirror to the(real) particle P is the same as the distance to itsvirtual image point P′. According to this the lightpath between PE and P′E is the same, too.

The geometrical relations show that BP =AP · sin α and AC = AP · sin α′. Making use of thelaw of refraction and substituting sin α = n · sin α′we get: BP = n · AC and correspondingly FP =n · DE. Due to the identity of these optical pathlengths, the calculation of the optical retardation isreduced to

�s = (CP′ − P′D

)opt . (2)

Now we determine this retardation as afunction of the thickness t of the mirror glass andits index of refraction n. Crucial for the lengthscale is that (PP′)opt = 2nt . Regarding as wellthat � CP′P = α′ and � PP′D = β ′, the lengthsin equation (2) can be replaced by equivalenttrigonometric expressions to obtain

�s = 2nt (cos α′ − cos β ′). (3)1 The optical path length in a medium is the distance whichthe light would travel in the same time in the vacuum.

Figure 5. Coloured rings on a window pane producedby the front light of a car (photo: Eva Seidenfaden).

Figure 6. Virtual interference rings on a glass paneilluminated by the sun.

The relations between the trigonometricfunctions, the law of refraction and approximatingfor small angles permit us to transform in thefollowing way:

cos α′ =√

1 − sin2 α′ =√

1 − (sin2 α)/n2

≈ 1 − sin2 α

2n2.

Applying such transformations to equa-tion (3), we get

�s = 2nt

[(1 − sin2 α

2n2

)−

(1 − sin2 β

2n2

)].

(4)

November 2007 P H Y S I C S E D U C A T I O N 569

W Suhr and H Joachim Schlichting

Figure 7. Schematic representation of the experimentalsetup.

z

L

O

oz

βα

pxox

P

lz

x

Figure 8. Schematic representation of a surface elementof a mirror. The light hitting the particle is scattered.

´MM

A P

l1 l2

t

Figure 9. Selected path of the rays at the mirror element.

´MM

AP

l1 l2

t

F

t

o1 o2

α´

α

C

B

E

β´

β

P´

D

Figure 10. Concentric interference rings. Upper part:photograph of an impure mirror. Lower part:representation of the corresponding model calculation.

Figure 11. Eccentric interference rings. Upper part:photograph of an impure mirror. Lower part:representation of the corresponding model calculation.

Summarizing and cancelling in equation (4)yields

�s = t

n(sin2 β − sin2 α). (5)

If instead of the angles only the distancesdepicted in figure 7 are known, one may benefitfrom the approximation sin2 α ≈ tan2 α, which is

570 P H Y S I C S E D U C A T I O N November 2007

Coloured rings produced on transparent plates

valid only for small angles. Because px /lz = tan α

and (ox − px )/oz = − tan β , equation (5) cantherefore be expressed as

�s = t

n

[(ox − px

oz

)2

−(

px

lz

)2]

. (6)

Comparison between experimental resultsand a model calculationThe model described above has been implementedusing a computer program calculating the opticalretardation for all points of the mirror surface.According to the RGB colours of the monitor,this calculation was performed for the wavelengthsof 605, 545 and 460 nm. The strength ofeach corresponding R, G, and B component wasdetermined by values proportional to the intensityof the superposed light. The respective mirrorelement was then dyed by the combination colourmade up of these components.

The graphic results of these model calcula-tions were compared with photographs taken dur-ing experiments which were performed with thesame parameters (see figures 10 and 11). In bothcases, for the development of concentric and ec-centric Quetelet rings, we obtained good agree-ment for the diffraction orders next to the whitezeroth order. For higher diffraction orders thesequence of calculated and photographed colourrings corresponds, while their location diverges.This deviation is presumably due to our very sim-ple colour model. This argument is supported bythe fact that the photographs of the interference

rings obtained with monochromatic light are a sur-prisingly good match to the calculated rings.

The photographs show that the light intensityof the coloured rings decreases with the distancefrom the zeroth order of diffraction. This decreasehas been modelled by taking into account therestriction of the coherence length of only a fewmicrometres.

Received 13 March 2007, in final form 6 April 2007doi:10.1088/0031-9120/42/6/001

References[1] Newton I 2003 Opticks new edn (New York:

Prometheus Books) (originally published:London 1730)

[2] de Witte A J 1967 Am. J. Phys. 35 301[3] Stokes G 1856 Trans. Camb. Phil. Soc. 9 147–76[4] James Bridge N 2005 Phys. Educ. 40 359–64

Wilfried Suhr gained his PhD in 1992from the University of Oldenburg. He is alecturer at the Institute of Didactics ofPhysics at the University of Munster(Germany), working in the field ofPhysics Education.

H Joachim Schlichting is a Professor ofPhysics Education, and he directs theInstitute of Didactics of Physics at theUniversity of Munster (Germany). He hasa Diploma and PhD from the Universityof Hamburg and Habilitation from theUniversity of Osnabruck. He was head ofthe section of Didactics of Physics of theGerman Physical Society.

November 2007 P H Y S I C S E D U C A T I O N 571