Fermat's Principle in Optics

DEEN DAYAL UPADHYAYA GORAKHPUR UNIVERSITY, GORAKHPUR GORAKHPUR Presentation on : Presentation on : Fermat Fermat s Principle PRESENTED BY : DR. PRESENTED BY : DR. PRASHANT PRASHANT SHAHI SHAHI

CONTENTS CONTENTS 1 1. . Ferma Fermat t s s Principle Principle 2. 2. Examples to verify that the optical path is Examples to verify that the optical path is extremum extremum 3 3. . Deduction Deduction of laws of reflection from of laws of reflection from Ferma Fermat t s s principle principle 4 4. Deduction of . Deduction of Snel Snell l s s law of refraction from law of refraction from Ferma Fermat t s s principle principle 5. 5. Aplanatic Aplanatic points of a spherically refracting surface points of a spherically refracting surface

FERMAT S PRINCIPLE The The A A light light ray of of media media by a a path path along minimum minimum or Ferma Fermat t s s ray in in passing passing from by any any number number of along which which time or a a maximum maximum or principle principle from one of reflections reflections or time taken taken by or stationary stationary . . can can point to be be other through through a a set or refractions refractions choses by the the light light ray ray is is either stated stated as as- - set one point to other choses either a a

Or, ds/c= a minimum or a maximum or a stationary. Since speed of light c in vaccum is always constant . So we must have, ds= a minimum or a maximum or a stationary. Here, ds = optical path = a minimum or a maximum or a stationary. Examples to Examples to verify extremum extremum verify that the optical path is that the optical path is Let PAQ be the actual path having terminals P and Q. A light ray starting reaches point Q after being reflected at the point A of the mirror AB. Draw an ellipse passing through A such that P and Q are the foci of the ellipse. from P

By the property of an ellipse, PA + AQ = constant. Here A is the variable point on the surface of the ellipse. Let PBQ be the neighbouring path to the actual path PAQ. From first figure of ellipse, the difference between the actual and varies paths is PBQ PAQ = ( PR + RB + BQ ) - ( PA + AQ ) PBQ - PAQ = ( PR + RB + BQ ) - ( PR + RQ ) PBQ PAQ = RB + BQ RQ = Positive Quantity. Hence PBQ > PAQ. This means that any neighbouring path is longer than the actual path. In other words the actual path is shortest. From second figure of ellipse, PBQ PAQ = ( PR RB + BQ ) ( PA + AQ ) PBQ PAQ = ( PR RB + BQ ) ( PR + RQ ) PBQ PAQ = -RB + BQ RQ PBQ PAQ = BQ ( RB + RQ ) = Negative Quantity. Hence PBQ < PAQ. This means that any neighbouring path is shorter than the actual path. In other words the actual path is maximum. BB

DEDUCTION OF LAWS OF REFLECTION FROM FERMAT S PRINCIPLE First law First law - - From figure, we see that 1. PO > PO and 2. QO' > QO Hence the time taken by the light ray along path PO Q will be greater than that along the POQ , which is contrary to Fermat s Principle. So the points O and O' must coincide. Thus the incident ray , reflected ray and normal to the mirror lie in one plane.

Second law - Let MM' = d, MO= x,OM' = d-x As the light ray travels the entire path in air, the length of the optical path b/w P and Q, I = POQ = PO+OQ I = (a2+x2)+ {b2+(d-x)2} According to Fermat s principle, the length of the optical path is minimum or maximum or stationary. This is possible only if dI /dx = 0 (1) Diff. ( Diff. (1 1) w. r. to x, we get, ) w. r. to x, we get, dI dI / /dx dx = [ x / = [ x / (a2+x2) ] [ (d-x) / {b2+(d-x)2} ] But according to Fermat principle, dI /dx = 0 [ x / [ x / (a2+x2) ] [ (d-x) / {b2+(d-x)2} ] = 0 Or, [ x / [ x / (a2+x2) ] = [ (d-x) / {b2+(d-x)2} ] (2)

From figure, [ x / [ x / (a2+x2) ] = MO/PO = sini [ (d-x) / {b2+(d-x)2} ] = OM/OQ = sinr From eq(2), we have sini = sinr i.e., angle of incidence = angle of reflection This is second law of reflection. DEDUCTION OF SNELL'S LAW OF REFRACTION FROM FERMAT'S PRINCIPLE Let MM' = d, MO= x,OM' = d-x the length of the optical path b/w P and Q, I = POQ = n1.PO + n2.OQ I= n1 (a2+x2)+n2 {b2+(d-x)2}.. (1) According to Fermat s principle, the length of the optical path is minimum or maximum or stationary. This is possible only if dI /dx = 0

Diff. ( Diff. (1 1) w. r. to x, we get, ) w. r. to x, we get, dI dI / /dx dx = = n1[ x / [ x / (a2+x2) ] n2[ (d-x) / {b2+(d-x)2} ] But according to Fermat principle, dI /dx = 0 n1 [ x / [ x / (a2+x2) ] n2[ (d-x) / {b2+(d-x)2} ] = 0 Or, n1[ x / [ x / (a2+x2) ] = n2[ (d-x) / {b2+(d-x)2} ] (2) From figure, [ x / [ x / (a2+x2) ] = MO/PO = sini [ (d-x) / {b2+(d-x)2} ] = OM/OQ = sinr From eq(2), we have n1sini = n2sinr or, sini / sinr = n2/ n1 This is Snell s law of refraction.

APLANATIC APLANATIC POINTS POINTS Defenition - A Surface is aplanatic with respect to two conjugate points if all the rays originating from one of the two points, after refraction or reflection at the surface, converge or appear to diverge from another point. This property of the surface is called Aplanatism. Such a pair of points are termed as Aplanatic points.

Proof In LOC , sini /sin = OC /LC But CL = R , OC = n2/ n1R Therefore sini /sin = n2/ n1 According to Snell's law, sini / sinr = n2/ n1 therefore sini /sin = sini / sinr sin = sin r or = r In ILO, exterior / LOC = + / ILO Or = + (r-i) {sin ILO = r-i} r = + r i so = i In ILC , sin /sinr = CL/IC sini / sinr = R/ IC CI = R sinr/sini = R n1/ n2 This means that the image formed by the lenses are just opposite to the object situated at any point.