TB, BPN, UK, EJP/274478, 31/03/2008

IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 29 (2008) 1–15 UNCORRECTED PROOF

The problem of focusing and realimages

Pavlos Mihas

Department of Elementary Education, Demokritus University, Nea Chili, Alexandroupolis,Gr-68100, Greece

E-mail: pmichas@eled.duth.gr

Received 4 March 2008Published DD MMM 2008Online at stacks.iop.org/EJP/29/1

AbstractA historical discussion of the theories which deal with the formation of realimages in mirrors and lenses is presented in this paper. Speculations on mirrors See endnote 1appeared as early as Plato. Euclid’s, Hero’s and Ptolemy’s approaches to visualrays are described. The theory on burning mirrors starts with Diocles and laterwas continued by the Arabs. Al Haytham extensively studied the reflectionof light rays on concave mirrors. Huygens tried to find a shorter way to dothe calculations. With lenses Kepler gave a new way of finding the positionof images by using approximations. Huygens also gave a solution for theshape of a ‘perfect’ lens. Huygens’ principle on waves can be combined withFermat’s principle to explain the formation of images. These theories can beused in education to help students better understand the formation of images,the propagation of waves and the properties of lenses.

Introduction

The formation of real images in lenses and concave mirrors is one of the main subjects ofteaching in optics in secondary education and first-year physics in universities. Sometimesbooks confine their treatment in giving some ‘recipes’ of how to find the images (e.g. Antoniouet al 2000). Although it is emphasized that these models are valid only for paraxial rays andthin lenses, exercises are used to teach that all rays starting from one point of the objectconverge to one point of the image. The paraxial ray model is very useful in helping studentsmake predictions. Some drawbacks are that students get the impression of the main rays asnecessary for the formation of images; if they are asked to apply the laws of reflection orrefraction (for refraction qualitatively) to show their understanding of the functioning of anoptical instrument, usually they are perplexed and are unable to proceed.

In this paper these drawbacks are emphasized and some historical data on how people inthe past and, especially in the Middle Ages dealt with the problem of focusing are examined.

0143-0807/08/000001+15$30.00 c© 2008 IOP Publishing Ltd Printed in the UK 1

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Figure 1. Image X of A seen by B.

Finally, Huygens’ method of constructing a ‘perfect’ lens is used as an exercise so studentsthat can learn the constancy of the optical path from the source to the points of focusing.

We start with the images due to reflection.

The evolution of theories of reflection

Ancient Greeks

Plato in the dialogue Timaios (Plato 1993) presents a theory for the formation of images inplane and concave mirrors. He considers that in plane mirrors the left is reversed to right,while in concave mirrors this thing does not happen.

Euclid in his Catoptrics (Spandagos 2000) gives several propositions (theorems) for the See endnote 2

reflection of visual rays and the position of images.In proposition 6 it is stated that ‘if you place the eye between the centre (of curvature) and

the surface then the visual rays will sometimes meet and sometimes not’. This propositionsays that there are cases where the eye can be joined by more than one visual ray to the object,so there will be multiple images. In proposition 18 the position where it will be seen will beon the intersection of the visual ray starting from the eye with the line drawn from the objectand passing through the centre. Proposition 18 describes the method of finding the position ofthe image and it reminds a modern construction of real images, although it deals with visualrays. In modern terms it says that there are cases where the image is formed in front of themirror. In figure 1 the point where the image of A will be seen by the eye B is not on themirror but at point X.

Hero of Alexandria wrote a book on Catoptrics (Schmidt and Nix 1900). In this bookhe presents the principle of least path. For concave mirrors he presents the case of having animaginary image when the object is near the cylindrical mirror’s surface and a real image whenthe object is moved away from the surface. In the first case an inscription seems reversed,while in the second case it can be read normally. This can be implemented in a classroom. Thestudents were presented with a translation of a passage from Hero’s Catoptrics and asked tocomment on it, and the reaction of the students and in-service teachers was very enthusiasticwhen they were confronted with this simple experiment.

Ptolemy in his fourth book on optics deals with plane mirrors, convex and concave mirrors.According to Smith (1999), Ptolemy’s treatment of multiple images is more comprehensive

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Figure 2. Eye on H, three points which join source Z through reflection to H.

and more systematic than Euclid’s treatment. One important case that was examined byPtolemy covers those cases in which the eye and object lie upon a chord located between thecentre of curvature and the surface from which the reflection takes place. Ptolemy examinesthe case of three reflections from one arc while the eye lies at predetermined points. Heshows that no more than three reflections can occur, but there are cases where fewer than tworeflections (figure 2) occur. Al Haytham, in another context, took over the problem of multiplereflections, as will be discussed below. Ptolemy’s theory of visual rays predicted that imageswould be seen even when the method yielded no intersection, or the point of intersection wasbehind the eye.

Burning mirrors

The ancient Greeks studied burning mirrors to find ways to converge by reflection the solarrays on mirrors. This study differs from the study of images, which was done with visual rays(Bellosta 2002).

Concave spherical mirrors were explored by the ancient Greeks as to whether they werecapable of becoming burning mirrors.

Euclid, in his Catoptrics (which, according to Smith, is not really Euclid’s work), gavesome results for burning mirrors; in one of them he derives that the point where ignition canhappen will be the centre.

Diocles (Smith 1998) proved that it is impossible to have a real focus in a spherical mirror.This result is interesting and anticipates similar results derived by Al Haytham.

By using the properties of parabolas Diocles (Smith 1992, Bellosta 2002) proved that therays that are incident parallel to the optical axis are reflected in such a way that they passbetween the middle of the radius and the top of the mirror. Diocles (figure 3) proves that whena parallel ray meets the parabola at Q the ray is reflected towards D (focus of the parabola).Students can use an Excel file to observe that all the rays parallel to the y-axis are reflected tothe focus and that, starting at a certain level, their length (ABF in figure 4) is constant

Ibn Sahl: the elliptical mirror

Ibn Sahl studied the burning instruments: mirror and lenses. He developed the elliptical andthe parabolic mirror (Rashed 1992).

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Figure 3. Diocles proof for a parabolic mirror.

Figure 4. An Excel graph to show the constancy of the path.

Ibn Sahl proved that a ray emanating from one focus of the ellipse passes through theother focus upon reflection. If C and A are the two foci of the ellipse and I is a point onthe ellipse, then the perpendicular to the ellipse at I bisects the angle AIC. The bisector ofthe external angle AIT is tangent to the ellipse. Ibn Sahl proved it by reduction ad absurdum(figure 5).

Ibn Sahl’s proof gives a way to construct an ellipse point by point. With the centre Cwe draw a circle of radius CT = a. As we turn the radius around the centre C we draw the

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Figure 5. Ibn Sahl’s proof of reflection properties of the ellipse.

Figure 6. Reflection properties of ellipse in an Excel graph.

perpendicular on the middle of AT. The point where this perpendicular meets CT is a point ofthe ellipse. Since the ellipse is the locus of all points whose sum of distances from the two fociis constant, all the paths have the same length. The students can use an Excel file (figure 6)which permits them to observe this fact.

Al Haytham’s problem

Al Haytham proposed the problem: given a spherical reflecting surface and two fixed pointsnot on this surface to find the point on the surface at which light from one of the fixed pointswill be reflected on the other. He investigated the number of points which permit this tohappen (Mihas 2002). He found at most four rays. So the maximum number of points on thecircle that can connect through reflection two given points A and B is four.

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Figure 7. Al Haytham’s way of finding the number of points.

Al Haytham’s solution

Al Haytham used his vast knowledge on conics to solve his problem. He proposed six lemmasfor the solution. The most important is lemma 4 which states: let circle AB, with centre G, begiven, and let D, E be two given points; we wish to draw from E, D two lines, such as EA,DA, so that a line drawn tangentially to the circle, such as AH, will bisect the angle EAD.

His proof is quite involved and to follow it one must first study lemmas 1–3. The proofsare given in Sabra (1981) and are presented by software in http://utopia.duth.gr/∼pmichas/ See endnote 3

AlHaytham. By using lemma 4 and other lemmas we can get the solution: the number ofpoints is four, three or two. This number is given by the number of intersections of a hyperbolawith a circle constructed according to lemmas 1 and 3 (figure 7). If the number of cuts is four,then we have four points that join the source (E) with the eye (D). A rough numerical solutionusing an Excel file is presented in figure 8. Students can use this file to change the parametersand observe that the number of solutions is two or four. They also observe that the total lengthof light from D to E is either a local maximum or a local minimum (an extension of Hero’stheorem, figure 9).

Huygens’ solution to the Al Haytham problem

Huygens (Huygens 1669) gave a solution which again is based on the intersections ofhyperbolas, but this time he uses the properties of the dichotomy. His solution is based

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Figure 8. An Excel graph of a numerical solution for students.

Figure 9. Total length of D to E: An Excel application.

on the fact that if a line divides an angle of a triangle to two equal parts, then the ratio of thetwo sides is equal to the ratio of the segments of the base (figure 10).

As we can see although the ancient Greeks (Diocles) the Byzantines (Anthemios) and theArabs (Ibn Sahl, Al Haytham) knew the focusing of ellipses and parabolas, on consideringthe circle they were not ready to make approximations so as to develop a theory of imagingcomparable to what is taught in modern day’s school science.

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Figure 10. Huygens’ solution.

Images due to refraction

The law of refraction was studied by Ptolemy by a method of locating the image of an immersedtarget object. He also predicted that the refraction of the atmosphere causes the stars to beseen in a different position. Ibn Sahl, according to Rashed (1992), proposed Snell’s law sixcenturies before Snell proposed his law in 1626 (Sabra 1981). Ibn Sahl proved that if T is ona hyperbola (that is TA − TL = BM a given distance), then the bisector of ATL is tangent tothe hyperbola (figure 11).

If a point X is located inside a hyperbola consisting of a crystal, then the ray XT isrefracted and comes out of the crystal following the direction TA. If AT − TL = MB thenwe find point C on AT such that AC = MB; then TC = TL and the bisector of ATL cuts CLperpendicularly on its middle Z. The ratioTC/CB2 is constant (figure 12) and equal to 1/n(n the index of refraction). By using this law Ibn Sahl proposed the construction of a hyperboliclens, which has the property of perfect focusing. If the lens is plane convex, then a parallelbundle of rays, which is incident perpendicular to the plane side will be focused perfectly ata point. If the lens is biconvex then a bundle of rays emanating from one focus is perfectlyfocused on a second focus (figure 12).

Focusing 9

Figure 11.

Figure 12. Ibn Sahl’s proof that a parabola focuses on one point.

Sphere–rainbow

According to Rashed (1993), Al Haytham gave a rule for the refraction of rays in a sphere.This rule can be considered as giving a limit for rays which have a relatively good focusing See endnote 4

and which produce spherical aberration.Let a ray parallel to the axis OO′ be refracted for a second time on point G and meet the

continuation of OO′ at T (figure 13). It is impossible for a second ray parallel to the first tomeet the axis after its second refraction at point T, except this ray is refracted at a point of theperiphery that is described by G on the sphere. If the second ray is refracted at a point further See endnote 5

away than O′ in relation to G, then after its exit from the sphere nearer to the sphere than T.The rays depicted by S and S′ are limiting rays. The parallel rays that are incident at a distance

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Figure 13.

smaller than S and S′ from axis OO′, after the second refraction, meet the axis at points furtheraway than the rays S and S′. Al Farisi used these ‘critical’ rays (S and S′) for his theory of therainbow. These rays are the same that Descartes used for his theory of the rainbow.

Focusing: new ideas

Figure 13 is very similar to the figures that appear in the work of Della Porta De Refractione(Ronchi 1956). It seems that the problem of focusing had not yet got the form we have today.But the works of Maurolico and Kepler changed the whole scene. Ronchi (Ronchi 1956, p77) writes that for Kepler the image was something intentional.

According to Ronchi (1991, p 40), Kepler in 1604 published a Supplement to Witello(Ad Vitellionem paralipomena) that completely changed optics. Kepler got his cue from thewritings of Della Porta and found the key to the mechanism of vision. He explained that eachpoint of a body emits rectilinear rays in all directions, like a point star. If there is an eye in frontof this point star, all of the rays that enter the eye will form a cone having the star as the vertexand the pupil as the base. Kepler proposed that these rays, after refraction by the cornea andthe internal parts of the eye, go on to make a new cone whose base is again the pupil but whosevertex is a point on the retina. Kepler arrived at this fundamental conclusion by studyingrefraction in a sphere of water. In his study he introduced a very important innovation. Whena cone of rays is incident upon the sphere, the emergent rays cut the axis at many differentpoints. But if we observe the emerging group, we see a concentration of rays virtually passingthrough a point. Therefore, instead of allowing the whole sphere to function, we may limit itto a part by means of a very narrow diaphragm.

Kepler developed his theory of lenses and in January 1611 he published his Dioptrice(Dioptrics). Kepler’s approach to lenses made reasonable the observation by telescopes. Inthis book Kepler presents the idea of a lens’ focus as the point where a parallel beam gathersafter refraction through the lens (Kipnis 1993, p 82).

Kepler’s views influenced scientists in the way they regarded lenses (figure 14). Now allrays emanating from a point source either are converging to the real image or their extensionsto the back are converging at the imaginary image. For example, in Newton’s OPTICKS(fourth edition of 1730) figure 3 shows all the rays from 3 point sources lying on a plane toconverge all of them in their respective real images.

Focusing and wave theory

The wave theory of light introduces a model for the description of the phenomenon which isrelated to the nature of light. In most cases, rays can be considered perpendicular to the wave

Focusing 11

Figure 14. Kepler’s ideas: by adding a narrow diaphragm the focusing becomes sharp.

Figure 15. Focusing of waves when a diaphragm is used.

fronts (an exception occurs in the case of double refraction). The model of wave fronts andrays was, according to Shapiro (1973), introduced for the first time by Kepler who took thefirst significant step towards developing the concept of ray and the ‘luminous surface’ or wavefront.

In this paragraph we consider the application of wave theory to the formation of images.Wave theory is also presented in many books in an idealized form. After passing through

the lens the waves are depicted as converging towards the focus. A similar picture can be foundin Ronchi (1991, p 61). A simulation can show that the wave fronts really do not convergeto one point. An exception occurs on Ibn Sahl’s plane convex lens. For spherical mirrors thefocusing is again better if a diaphragm is inserted in front of the lens.

Students who find difficulties in understanding ray diagrams generally find that wavepictures that appear during the simulations are more understandable (figure 15).

Combination of Fermat’s principle and the Huygens method

Fermat (Sabra 1981) gave his least-time principle and explained Snell’s Law. Light movingfrom one point to another always chooses the path of shortest possible time. Some elementarydescriptions of optics use Fermat’s principle as an explanation of refraction phenomena (Hewit1995, Antoniou et al 2000).

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Figure 16. First step in applying Fermat’s principle.

Figure 17. Combination of Fermat’s and Huygens’ principle for finding the image with wavelets.

Fermat’s principle sometimes is questioned on the ground ‘how does light know how tochoose its path?’ Huygens’ construction can be helpful in answering this question.

This can be explained as follows: if a wave falls obliquely towards a surface, then Huygens’construction shows that if we consider some points on the surface, and the refraction is possible,then the wavelets cut each other (if enough time is allowed); if, on the other hand, there is nota refracted wave then the wavelets do not intersect.

If light is emitted from a point and part of it passes through a lens, then Fermat’s principlepermits us to find the points on the surface of the lens farther from the object point. We cancalculate the time needed for each path (figure 17). To continue the rays we need to know thedirection of the ray that comes out of the lens. For a lens we can consider all the points of thesurface of the lens where the light has reached. Since the time is not the same for all points,from each point we draw circles of radii c · (tfinal − tpoint). If neighbouring circles intersectthen we can find the ray. If the two circles do not intersect then it may be the case of totalreflection, and then the ray reaching the surface will not propagate.

One way to see the final result is to draw the rays coming out of the lens. It can be seen(figure 16) that they will not intersect at the same point. The results of this simulations can becompared with either the thin lens model or the thick lens model. It can be seen in figure 16that the rays give a caustic.

By graphing the wavelets for progressing time they appear as having a tangent curve thatsomehow converges towards the image point (figure 17).

(software ‘fakosfermat’ from http://utopia.duth.gr/∼pmichas).The student can at first seeFermat’s principle at different points. Then he can ask for a continuation of a ray and then hecan see what will be the continuation of a bundle.

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Figure 18. Student’s drawing of rays.

Figure 19. Huygens’ way of finding the shape of a ‘perfect’ lens.

(This figure is in colour only in the electronic version)

Huygens’ construction of the ‘perfect’ lens

As an incentive for the study of the lens with perfect focusing, an exercise is used wheredrawings of wave fronts passing through a spherical lens were distributed to the students(figure 18). The students were asked to draw some rays as perpendiculars to the wave fronts.The students were asked: (a) do all the emerging rays pass through the same point? (b) Countthe wave fronts from the source to the points where the most rays seem to pass, how manywave fronts can you count? The students who drew five or six different rays could easily findthat not all the rays seem to pass from the same point. For the second question all the studentswho correctly drew the rays answered that the same number of wave fronts were between thesource and the focus point.

After this Huygens’ (Huygens 1966) method for constructing a lens that will be capableof perfect focusing a certain point was presented. This method is the following.

There is a point X and a surface. Let us assume that the surface will be spherical. Thenif there is a point F where all the rays will converge as they pass through the lens, the totaltime the light wave spends on each ray will be the same. That means that the total number of

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Figure 20. A student’s application of Huygens’ solution.

wavelengths (in air and in the medium of the lens) will be constant. In the following figurethe number of periods of time is 25 from X to F.

This can be implemented in an activity in which the students are asked to draw the shapeof a ‘lens’ that has one face (towards the source) spherical, a given width in its centre and agiven point as the point where all the rays passing through the lens will pass. The teacher byusing special software distributed a diagram with the left side of figure 18 in which the rightside was missing. The student, by drawing the circles on the right side, had to find the cuts ofthe circles with the wave fronts inside the lens so as to make the optical path constant. Almostall the students succeeded in constructing a lens similar to figure 19 (cf figure 20)

Conclusion

History can guide us into considering several aspects of the theory of focusing. For the ancientGreeks the main obstacle was their theory of the nature of light. The procedure of findingthe ‘image’ started with Kepler who proposed using two rays for finding the image of a pointsource.

The ancient Greeks asked for exact constructions and this was continued by the Arabswho translated the Greek works. This was seen in Al Haytham’s approach. With Kepler’sconsideration of paraxial beams new vistas were opened for optics, astronomy and microscopy.Although the limitations of this approach are quoted in many textbooks, usually they are notconsidered. Only rarely do introductory textbooks include exercises for the student to see thelimitations (McDermott and Shaffer 1998).

Focusing 15

The wave theory of light is usually taught only in the context of physical optics, but thereare many exceptions (McDermott and Shaffer 1998, Halliday and Resnik 1976). Studentssometimes express their appreciation of wave theory for understanding refraction. The solutionof finding the shape of perfect focusing with lenses (Huygens) is similar to the solution ofperfect focusing in reflection (Ibn Sahl). Computers permit the use of simulations which canexplain the difficult concepts of paraxial rays, caustics and imaging.

References

Antoniou N, Valadakis A, Dimitriadis A, Papamichalis K and Papatsimpa L 2000 Physics for 2nd Year in Junior High See endnote 6School (Greek Ministry of Education and Religions)

Bellosta H 2002 Essay-review: burning instruments: from Diocles to Ibn Sahl Arabic Sci. Phil. 23 285–303Halliday and Resnick 1976 Physics parts A and B (Athens: Pnevmatikos) (Greek Transl.)Huygens C 1966 Traite de la Lumiere reprinting of the original 1690 edition by Kripssthoek, Netherlands)Huygens C 1669 Complete Works vol 20 pp 265–9Kipnis N 1993 Rediscovering Optics (Minneapolis: BENA Press)McDermott L and Shaffer P 1998 Tutorials in Introductory Physics (Englewood Cliffs, NJ: Prentice-Hall))Mihas P 2002 Some historical resources for teaching about curved mirrors Phys. Educ. 37 311–7Nazif M 1942 Al Hasan bin al Haitham buhuthuhu oua kushufughu al basariia, CairoNewton I 1952 Opticks (New York: Dover)Plato 1993 Timaios (Athens: Kaktos) (in Greek)Rashed R 1992 Optique et Mathematiques: Recherches sur L’Histoire de la pensee Scientifique En Arabe (Collected

Works) variorumRonchi V 1956 Histoire de La Lumiere (Paris: SEVPEN)Ronchi V 1991 Optics, the Science of Vision (New York: Dover)Sabra A I 1981 Theories of Light from Descartes to Newton (Cambridge: Cambridge University Press)Sabra A I 1981 Ibn al-Haytham’s lemmas for solving ‘Alhazen’s problem’ Arch. Exact Sci. 26 299–324Schmidt W and Nix L 1900 Heronis Alexandrini: Opera Quae Supersunt Omnia vol 2 (Leipzig: Teubner) Mechanica

et catoptricaShapiro A E 1973 Kinematic optics: a study of the wave theory of light in the 17th century Arch. History Exact Sci.

11 134–266Smith A M 1999 Ptolemy and the foundations of ancient mathematical optics: a source based guided study Trans.

Am. Phil. Soc. 89 1–172Smith A M 1996 Ptolemy’s theory of visual perception: an English translation of the optics with introduction and

commentary Trans. Am. Phil. Soc. 86 1–300Spandagos E 2000 Euclid’s Optics and Catoptrics (Athens: Aithra) (in Greek)Spandagos E 2000 Structure and analysis Int. J. Sci. Educ. 22 57–88

Endnotes

(1) Author: Please check whether the edits made in the sentence ’Speculations on mirrors . . . ’retain your intended sense.

(2) Author: Please confirm whether the reference citation ‘Spandagos (2000) should beSpandagos (2000a) or Spandagos (2000b) or both’.

(3) Author: Please confirm whether the reference citation ‘Sabra (1981)’ should be ‘Sabra(1981a)’ or ‘Sabra (1981b)’ or both here and elsewhere in the text.

(4) Author: Please check whether the edits made in the sentence ’This rule can beconsidered . . . ’ retain your intended sense.

(5) Author: The sense of the sentence ’If the second ray . . . ’ is unclear. Please check.(6) Author: Please provide the publisher’s location in reference ’Antoniou et al 2000’.(7) Author: Please check this paper carefully (so as to ensure that your meaning is not

affected) since more than the usual number of changes have been made.(8) Author: Please provide captions for figures 11 and 13.(9) Author: Please be aware that colour figure 19 in this article will only appear in colour

in the Web version. If you require colour in the printed journal and have not previouslyarranged it, please contact the Production Editor now.

(10) Author: Please check the discrepancy in figures 6, 11 and 12 between JPG files and sourcefile and reference PDF. We have used the one given in source file which also match withreference PDF.

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