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5, , FERMAT’S PRINCIPLE, APLANATIC AND, CARDINAL POINTS, , agation of light in straight lines thy,, , , , Oug,, , Geometrical opti Is with the prop: k, rical optics deals wi Pp egarding the nature of light. The rex,, , a medium, without making any assumptions 1 fn, is that the order of wavelength of light is very small (= 10°’ m) and the dimension,, , the apertures, lenses etc. are large in comparison to the wavelength of light,, geometrical optics, the path of rays are traced from objects down lo their ima., from the experimental knowledge of the behaviour of rays of light at the inter,, between the different optical media. The well-known laws of geometrical optics tal, , (3) Rectilinear propagation, , (ii) Laws of reflection, , (iii) Laws of refraction., , Our aim is now to seek for 2 single princip', This principle is the Fermat’s principle., 5-1. Fermat’s Principle of Extremum Path, , Fermat, in 1658, gave a general principle which is stated as follows :, , ‘A ray of light in passing from one point to another by any number of reflection:, refractions chooses a path along which the time taken is the least or minimum,’, , This principle is called the principle of least time. One should note thatt, principle of least time holds for reflection and refraction at plane surfaces onl}., spherical surfaces, the actual path of light will be that for which the time taket, either maximum or minimum, but no other value i.e., the light ray follows a path:, , which the time taken is the extremum. Thus a general statement of Fermat’s print’, is made as follows : j, , ‘A ray of light in passing from one point to the other through any numb, reflections or refractions follows a path for which the optical path* is either minim, maximum i.e., extremum (or in other words, the optical ‘, path is stationary)., , In Fig. 5-1, let a ray of light passes from a point P to a, point Q through the actual path PAQ. There are number of, other paths between the points P and Q. Consider one such, path PBQ for which the lateral difference AB = ds, According to Fermat’s law. if this difference ds is very small, of the first order, then dt difference of times taken along, PAQ and PBOQ is very small of the second order ie., Fig. 5°1- Fermal, , le which can explain all the three jy, , , , sel, , * Optical path between two points is directly proportional to the tim, distance lin a medium of refactive index /, the optical path in the medium we ‘, path is the distance which light would have travelled in air (or vacuum) during 8, , Scanned with CamScanner
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Fermat's Principle, Aplanatic and Cardinal Points | 533, , a. w(5.1), , ds, A given path is said to be the actual i if z i i, g M c path if and only if all the neighbouring paths, , of the given path involve time difference i i, Iteral differences of path. ifferences which are small of an order higher than the, , Now we shall try to prove t, optics, V1Z+» (i) rectilinear propagation, (ii) reflection,, jn Fermat’s principle. ,, (1) Rectilinear propagation of light, in a homogeneous medium a straight, he path of least time. Thus light trave!, , s of geometrical, , hat all the three fundamental law:, are embodied, , (iii) refraction,, , __ Between two points line 1s the path of least, distance and hence it ist Js in straight lines in, homogeneous medium, (2) Reflection of light from a plane surface, , cordance with the, , A beam of light is reflected from an optical surface in acc, following two laws :, , . (i) The reflected ray lies in the plane of a, incidence /.e., 1n the plane which contains the, , 1 at the point of, , , , incident ray and norma, incidence., (ii) The angle of incidence is equal to the x Y, P M P’, x—<_— eo -x, , ———|, 2. Reflection of light from a, , plane surface, , angle of reflection., In Fig 5.2, let a ray of light reaches from a, , point A toa point A’ after reflection from a, plane surface XY following the path AMA’., The normals from the points A and A’ on the, surface XY are respectively AP and A’P’. Let AP = 4 A'P! =b, PP’ =¢ and, PM =x, hence MP’ =c -x., , ath AMA’ in air, hence optical path travelled by, , Since light travels the entire p:, d A‘ 1s, , the light in between Aan, 5 = AMA’ =AM+ MA’, = VAPE+ PM? + VA'P! + MP?, aVate t+ vb? +c =x), he reflecting surface is v,, , Fig. 5°, , f light in the medium above t the time, , Live +x + Ver+-2)), , 5, t=-=vv, , If the speed 0, taken in travelling the distance 5 18, , nciple, for the actual path a =0, , , , , , By Fermat’s pri, # 1[ty—* gil uc-x(-1) | _ 9, mo ovi|2 Vere 2 Vb2 + (c -x), ° x = cox or PM _ MP", = Tia Ve+e-9 4M MA’, or sini = sinr or! =7 (5.2), = angle of reflection., , of incidence =, , n of light., , a plane surface, , dium L (refractive index #4) reaches, through a surface XY, , or angle, This is the law of reflectio, (3) Refraction of light from, , In Fig 5.3, a light ray from a point A in me, a point A’ in medium 2 (refractive index #42) after refraction, , Scanned with CamScanner
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Year (second Paper) ", , _<-gecond at Aand Ate, 534 | Unified ee normals from the poin n the SUrFacg %, through the path ae Let AP=4,A’P'=b, pp, respectively AP an and MP’ =c-x “Py, , Now if the velocities of j; he, tand 2 be 1 and v9 respectively the Mey, , taken by light to travel from A 1, 4,1, Y AM , MA’ Wily, , = +, Vy vo, +, Vea? RN, vy SY, , By Fermat’s principle, fo, 4,, ' dt Cw, Fig. 5-3. Refraction of light from paths =0, , a plane surface, , Hy, , , , dt 1 1B] 3p SSI., ay |2Vae+x v2 [2 Ve2+(-x2 | ~°, 1 x 1 _©=*, or wVetx Y2Vb?+(¢-2), 4,.£0 Ly ME or A gint=+ sine, oF ¥, ° AM 2 MA! vy v2, sini Yl, or sinr V2 wa of, , where #2 is the refractive index of second medium with respect to th:, , medium, , Thus the sine of the angle of incidence bears a constant ratio with the sined!:, angle of refraction. This is the law of refraction of light., , Note that in the above two cases (i.e., reflection and refraction at the p, , dt 1g 5 :, surfaces) we can see that Fl Somes out to be positive which means that in these, , the actual path is the path of least time. Now we shall see that this is not alva’, case, but the actual path is cither maximum or minimum. In some cases it wilt, path of maximum time, while in other cases it will be a path of minimum time.lt, of reflection at spherical surfaces, we can show that for reflection at the, surface, the actual path is the longest path while for reflection at convex sutla:, actual path is the least path which will prove that Fermat principle is esset!*, peace of extremum path and not of least path,, , flection from a concave surf: io. M, , Fi 5 ‘ace —, reflection at concave mirror LMN, Pot i eae shows ihe, path of ray reaching from a point A to the poi is tie eh, at M from the curved surface Cc O the point B after reflection /’, through M with A and B as tee -onsidering the ellipse passing, , AM +MB= AM’ +R a’ nave, (by the property of ellipse), , At, For any other nei 6.4) Bh i, ghbour te Fb nace, , the actual ray AMB and the ray ANN the path difference for trom, , , , , , 4 = ANB~ AMR, , iis (AN NB) (AM + MB), Scanned with CamScanner
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wn 55, t to the fir, , he sine of t, , at the pitt, , , , , , Fermat’s Principle, Aplanatic and Cardinal Points | 535, , From eqn. (5.4),, A, , (AM' — NM! + NB) — (AM’ + M'B), — NM' - M'B+NB, ' , = — (NM’ + M’B- NB) (5.5), , id a and M'Bare the two sides of a triangle NM'B whose third side is NB., SNR e et hand’ side of eqn. (5.5) is always negative, which means, A j or the actual path AMB is the maximum path, Any other neighbouring, path is smaller than the actual path., , Reflection froma convex surface — Fig. 5.5 shows the reflection at convex mirror, LMN. | Light reaches at point B from a point A after, reflection from the curved surface at M, so that AMB is the, actual path of the ray., , Now consider an ellipse passing through M and having, A and B as its focii. Rotating the ellipse about AB axis, we, get an ellipsoid., , By the property of ellipse (i.e., sum of the distances of, a point on the circumference of an ellipse from its focii is Fig, 5-5, Reflection from, , , , constant), a convex surface, AM + MB = constant for all points on the surface of ellipse, ie, AM+MB =AM'+M’B (5.6), , For any other neighbouring ray ANB, the path difference for the actual ray, AMB and the ray ANB is, A =ANB-—AMB, = (AN + NB) — (AM + MB), (AM’ + M'N + NB)—(AM + MB), , ", , From eqn. (5.6),, , Ul, , (AM + M’N + NB)—(AM’ + M’B), , = (M’N + NB - M’B). (5.7), nd NB are the two sides of a triangle M’NB whose third side is M'B,, of eqn. (5.7) is positive, which means ANB > AMB or, hn and any other neighbouring path is longer than the, , But M'Na, therefore the right hand side, the actual path is the /east pat, actual path. :, , 5-2. Aplanatic Points, The rays of light incident close to the principal axis of a spherical reflecting or, refracting surface are called the paraxial rays. While the rays of light incident far from, the principal axis are called the marginal rays. For a point object on the principal axis,, the paraxial rays after reflection or refraction form a point image, but due to marginal, rays same aberrations are produced in the image which we shall study in detail in, chapter 6. However, there are some positions of the object point with respect to the, curved surface for which sharp point image is obtained from all the rays (both, paraxial and marginal). This property of the surface by virtue of which all the rays, starting from a particular point object on its axis, after reflection or refraction at the, surface, converge or appear to diverge from a single point image, is called aplanatism, and the particular object and image points are called the aplanatic points while the, surface is said to be the aplanatic surface with respect to those two points. ;, Aplanatic points of a mirror— Fig 5.6 shows an ellipse of which O and O ae, the focii. For any position of a point P on the ellipse, PO and PO’ will make equa, , Qo:, Faery Vorg, , Scanned with CamScanner
