Physics.Ed-Ph] 10 Feb 2005 Iie E.I H Ne Frfato Sthen Is Refraction of Index the Ex- Sec.II

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arXiv:physics/0502054v1 [physics.ed-ph] 10 Feb 2005 iie e.I h ne frfato sthen is refraction of index The ex- Sec.II. is deflection hibited photon momentum with and associated energy laws the conservation from Eq.(1) of derivation The ..tebnigo ih asi rvttoa field. gravitational a in lens, rays gravitational light particle of the the bending Sec.V of the in application i.e. consider an As we geometric general viewpoint, Sec.IV. more in a velocity limit from photon and optics the Sec.III and in discussed refraction is of index relationship The the between momentum. and energy photon of terms em fteidcso ercin nl’ a frefractio of law that Snell’s 3] refraction, asserts[2, of indices the of terms ray. a light of the along position moves and it as momentum photon the respectively, denote, I.1 ih a oe rmamdu ihidxo re- of index with medium a from moves ray light fraction A 1: FIG. a edrvdfo h atce(..poo)ve of Hamil- view refraction photon photon) the of how laws show (i.e. we tonian the particular, particle how In the show rays. pur- from light to The derived is be light. note may of this theory of depend wave derivations[1] pose electromagnetic the the wherein on courses physics tary r osdrdt emd po htn ihmomenta with photons of up made be to considered are and h ercino ih a ssoni i..In Fig.1. in shown is ray light a of refraction The nl’ a frfato suulydsusdi elemen- in discussed usually is refraction of law Snell’s p 2 respectively. H n 1 ( n p noamdu ihidxo refraction of index with medium a into 1 , emter flgtry.Ti atrpril hsc view physics 42.15.-i,42.15.Dp,41.85.-p particle numbers: PACS latter This refraction. of rays. laws light the understanding Snell’ of theory. wave theory electromagnetic beam Maxwell the of terms in r sin utb eie.I hswork, this In derived. be must ) nl’ a flgtdflcinbtenmdawt ieeti different with media between deflection light of law Snell’s .INTRODUCTION I. θ 1 = n 2 nl’ a rma lmnayPril Viewpoint Particle Elementary an from Law Snell’s sin θ hsc eatet otesenUiest,Bso A02 MA Boston University, Northeastern Department, Physics 2 (Snell ′ Law) s .DodffadA Widom A. and Drosdoff D. . defined n p 2 Rays . and (1) p in n r 1 q.2-4 imply Eqs.(2)-(4) foedoes one if eoiyo h htn.Tevlct fapoo na in photon a of velocity by The determined is photons. ray the of velocity qiaett h oeuuldefinitions usual more the to equivalent htin that sa hscldmnin feeg n momentum. and energy the of have dimensions not physical do usual “momentum” and “energy” so-called the of htn ihmomenta with photons hc osiue ipeytrgru eiainof energy formal derivation rigorous yet simple law. Snell’s a constitutes which htnmmnu opnnsprle otepaeare plane i.e. the the ; media, to conserved direc- two parallel the in components separating momentum invariance plane photon translational the to is parallel there tions are Since energies tively. photon the that assume nt of units nryadmmnu.Frexample, For momentum. and energy nie frfato r endby defined are refraction of indices h nriso h htn r locnevd i.e. conserved; also are photons the of energies The ntrsof terms In h entoso h nie frfato nE.4 are Eq.(4) in refraction of indices the of definitions The easm httetory nFg1aemd pof up made are Fig.1 in rays two the that assume We lhuhsmlrt te ramnswihdsusa discuss which treatments other to similar Although Joule-sec/meter a a lob eie rmaphoton a from derived be also may law s our dcso ercini sal discussed usually is refraction of ndices Joules ramn eol ee othe to refer only we treatment sb a h otsml n for one simple most the far by is I OSRAINLAWS CONSERVATION II. physical o ngeneral in not I.POO VELOCITY PHOTON III. n n a and n n momentum and 1 1 = n = p nohrmr omltreatments, formal more other In . 1 1 cp E sin htneeg n oetm the momentum, and energy photon u sin c omlmomentum formal 1 1 1 v E θ p θ 1 1 1 = 1 and and 115 = = = and identify ∂E ∂ n p E p 2 2 2 n n sin p p sin . . 2 2 2 a ovninlunits conventional has = = θ θ epciey ealso We respectively. 2 2 u . u cp E E , E c ihtephysical the with 2 2 1 2 a conventional has 4 ] estress we 5], [4, . physical and . E 2 photon respec- (7) (4) (6) (5) (2) (3) 2

In general for light rays moving through continuous media[6]

E ∂E u 6= |v| since 6= . (8) p ∂p

In the Maxwell electromagnetic wave theory[7, 8, 9] of light, u is the phase velocity while v is the group velocity. Let us consider this in more detail from a purely particle physics viewpoint. FIG. 2: An incident photon denoted by γi with impact parameter b is refracted from a spherical gravitational lens. The refraction index is n(r)=1+(Rs/r). The deﬂection angle Θ IV. GEOMETRICAL HAMILTONIAN OPTICS describes the ﬁnal photon denoted by γf .

In inhomogeneous continuous media, the index of refraction in general depends on the photon energy as well in a gravitational ﬁeld. Newton’s formulation of gravity as position. Eq.(4) implies the particle energy restriction attributes the weight of an object to a gravitational ﬁeld g grad c|p| = − Φ. (15) E = . (9) n(r, E) The weight of an object of mass m is

In principle, the implicit Eq.(9) can be solved for the w = mg. (16) energy in the form The gravitational ﬁeld in a region of space is weak if for E = H(p, r). (10) all points within the region the gravitational potential obeys The particle of light, i.e. the photon in the ray, obeys 2 Hamilton’s equations[10, 11, 12, 13]; They are |Φ(r)| ≪ c . (17)

∂H(p, r) ∂H(p, r) For a region of space with a weak gravitational field, r˙ = and p˙ = − . (11) ∂p ∂r light rays bend in a manner described by the index of refraction, The velocity of the photon v = r˙ obeys[14] 2Φ(r) nu Ep cp n(r)=1 − 2 + ... , (18) v = wherein u = = . (12) c [n + E(∂n/∂E)] p2 np independently of the energy E of the photon. For ex- The force on the photon f = p˙ obeys ample, a spherical astrophysical object will induce in the neighboring space a potential Φ = −(GM/r) and thereby E gradn f = . (13) a spherical lens with index of refraction [n + E(∂n/∂E)] 2GM n(r)=1+ + ... , (19) For a discontinuity in the index of refraction, such as pic- c2r tured in Fig.1, the impulsive photon force is normal to the surface of the discontinuity. That there is no force par- To a sufficient degree of accuracy, Eqs.(9), (17) and (19) allel to the surface of the discontinuity leads directly to imply Snell’s law as in Sec.II. We note in passing that Hamil- 2 2 E 4GM ton’s Eq.(11) follow from minimizing the optical path |p| ≈ 1+ . (20) c c2r length[15, 16, 17] The identities L[Path, E]= n(r, E)|dr| (14) Path 2 2 2 2 Z r |p| = |r × p| + |r · p| , 2 2 2 2 2 over sufficiently small sections of the path of the ray. r |p| = J + r pr, (21) and V. PHOTON DEFLECTION DUE TO GRAVITY Eb J = (angular momentum), c Astrophysical gravitational lenses are naturally formed 2GM Rs = (gravitational radius) (22) and can be understood on the basis that light rays bend c2 3 together with Eq.(20) imply a radial photon momentum geometrical optics refraction cross section

2 2 E 2Rs b dσ bdb Rs pr = ± 1+ − , (23) = = 4 . (27) c s r r dΩ sinΘdΘ 4 sin (Θ/2)

and its associated scattering angular deflection ∞ bdr Θ= π − 2 2 2 (24) VI. CONCLUSIONS rmin r 1+(2Rs/r) − (b/r) Z p as shown in Fig.(2). The impact parameter thereby obeys The laws of refraction have been shown to follow most easily by employing a classical Hamiltonian, E = b = −Rs cot(Θ/2). (25) H(p, r), for a photon moving through transparent continuous media. Snell’s law of refraction at the boundary For small angles and large impact parameters[11, 18], of two different media is then a simple consequence of 2Rs 4GM the conservation laws inherent in the Hamiltonian de- Θ= − = − 2 (b ≫ Rs), (26) scription of the photon. The use of ray optics in the b c b design of lens systems is well known. We have here illus- which was used by Einstein to predict the bending of trated the use of the photon Hamiltonian for the case of light around the sun[19, 20]. We note in passing, the gravitational lens effects in astronomy.

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