Data Provided: A formula sheet and table of physical constants are attached to this paper.

Transcription

1 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Spring semester Optics 2 hours Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer THREE re mrked questions. out All of questions ten. The rebrekdown mrked out of on twenty. the right-hnd The brekdown side onof the the right-hnd pper side is ment of the pper s guide is ment to sthe guide mrks to the tht mrks cn tht be obtined cn be obtined from from ech ech prt. prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. PHY227 TURN OVER 1

2 1. () Explin why glss is visible in ir, but my be nerly invisible in wter. Justify your nswer using the Fresnel formule for norml light incidence. [3] (b) Collimted light is incident from ir on surfce of dielectric titnium dioxide. The light contins two different colours experiencing different refrctive indices in this mteril: red t λ r = 700 nm with n r = 2.3, nd violet t λ v = 400 nm with n v = The reflected light is observed through liner polriser while the titnium dioxide surfce is slowly tilted, so tht the ngle of incidence nd reflection re grdully chnged. (i) How should the trnsmission xis of the polriser be orientted so tht t certin ngles the reflected light observed through it contins single colour only? Justify your nswer. (ii) Find the ngles t which the reflected light observed through the polriser, orientted to stisfy the condition in prt (i), contins single colour. [3] (c) Light propgtes through mteril with the dielectric constnt ε 1. It is incident t n ngle θ i on the plnr surfce of second dielectric with the dielectric constnt ε 2 (ε 1 < ε 2 ). For the light reflected off this surfce, consider the electric field component directed out of the plne of incidence. Derive its phse with respect to tht of the out-of-plne electric field component of the incident light. Use the reltion describing the out-of-plne electric field mplitudes for the incident, A, nd reflected, R, light: [3] R = n 1cosθ i n 2 cosθ t n 1 cosθ i + n 2 cosθ t A Here n 1 nd n 2 re the refrctive indices of the first nd second dielectric. Hint: sin(α β) = sinαcosβ cosαsinβ. [4] (d) An elecromgnetic wve hs n electric field given by E(z, t) = e x E 0 sin(ωt kz), where e x is the unit vector long xis x, E 0 is the electric field mplitude, ω is the ngulr frequency, k is the wvenumber. (i) Find the mgnetic induction B, when the wve propgtes in vcuum. Your nswer should include the speed of light c. [4] (ii) How does the expression you found in nswer to prt (i) need to be modified when the wve propgtes in medium with dielectric constnt ε? [3] PHY227 CONTINUED 2

3 2. () Describe how circulrly polrised light cn be produced using qurter-wve plte nd piece of glss hving flt side. Describe the limittions of this method. [4] (b) A bem of unpolrised light with intensity I 0 is incident on liner polriser (LP1) hving its trnsmission xis long the verticl direction. Another liner polriser (LP2) with its trnsmission xis oriented horizontlly is plced fter the first one. A hlf-wve plte (HWP) nd third liner polriser (LP3) re lso provided. Either the HWP or LP3 re inserted between the LP1 nd LP2 nd re rotted to chieve the mximum intensity for light pssing through ll three opticl elements. (i) Find the mximum intensity trnsmitted through the three optic elements in the combintion LP1-HWP-LP2. In your nswer stte the orienttion of the min xes of the HWP with respect to the trnsmission xes of LP1 nd LP2 polrisers. [4] (ii) Find the mximum intensity trnsmitted through the three optic elements in the combintion LP1-LP3-LP2. In your nswer stte the orienttion of the trnsmission xis of LP3 with respect to the trnsmission xes of polrisers LP1 nd LP2. [4] (c) Linerly polrised light is incident on two qurter-wve pltes plced one fter nother. The incident light is polrised verticlly. (i) Describe how the fst nd slow xes of the qurter-wve pltes should be orientted so tht the light still remins linerly polrised fter pssing through both wve-pltes. Justify your nswer. [4] (ii) Describe how the fst nd slow xes of the qurter-wve pltes should be orientted so tht the light tht hs pssed through both wve-pltes hs liner polristion rotted by 20 degrees with respect to the originl direction. [4] PHY227 TURN OVER 3

4 3. () A diffrction grting hs 8500 slits ruled cross width of 4 cm. Wht is the wvelength of the light whose two fifth order mxim re seprted by n ngle of 90 degrees? [3] (b) A diffrction grting hs 1200 rulings uniformly spced over 10 mm. It is illuminted t norml incidence by yellow light from sodium vpor lmp. This light contins two closely spced lines of wvelengths nm nd nm. (i) In wht diffrction order will the two lines be spectrlly resolved if only qurter of the rulings re illuminted? [3] (ii) Wht should be the width of the ruling so tht this diffrction order is clerly observed? [3] (iii) How mny rulings need to be illuminted so tht the doublet is spectrlly resolved in the first order diffrction mximum? [2] (iv) In the cse described in prt (iii), wht re the ngles t which the diffrction mxim corresponding to the two wvelengths re observed? [2] (c) Sketch schemtic digrm of grting spectrometer nd explin the principle of its opertion bsed on your drwing. Explin how spectrum is mesured. [7] PHY227 CONTINUED 4

5 4. () Consider Fbry-Perot interferometer. Monochromtic light is incident on the interferometer nd the trnsmitted light intensity is mesured. Clculte the reflectivity of the mirrors if the minimum trnsmitted intensity obtined by tuning the wvelength of the incident light corresponds to 50% of the mximum trnsmitted intensity. [3] (b) Light with wvelength 400 nm is incident t n ngle of 60 degrees on Fbry-Perot interferometer consisting of two mirrors with n ir gp between them. The size of the gp is t = 0.1 cm. A mximum trnsmitted light intensity is observed by detector plced behind the interferometer. By grdully turning the interferometer, so tht the ngle of incidence is chnged, minimum in intensity is obtined. Find the new ngle of incidence. [3] (c) Light hving mixture of two wvelengths nm nd nm is incident normlly on Fbry-Perot interferometer consisting of two mirrors hving reflectivity R nd seprted by n ir gp. The thickness of the gp is t = 0.1 cm. Find the vlue of R for which the two wvelengths re resolved using the interferometer. [4] (d) Light is incident from ir on thin film with refrctive index n=1.5 t n ngle φ = The light is linerly polrised normlly to the plne of incidence. Find n expression for thicknesses of the film for which light with wvelength λ = 500 nm will be reflected most strongly. [5] (e) Wht is the min dvntge of Fbry-Perot interferometer compred with grting spectrometer in opticl spectroscopy pplictions? Justify your nswer by providing relevnt formule nd numericl estimtes. [5] PHY227 TURN OVER 5

6 5. () Sketch grph of the intensity of light trnsmitted through silic fibre s function of the wvelength of light. Wht ttenution mechnisms re responsible for the min fetures on your grph? Indicte the trnsprency windows nd stte their pproximte wvelengths. [5] (b) The core of wveguide is mde from mteril with refrctive index n w. The wveguide is cld on both sides with the mteril hving refrctive index n c = 1.5. The wveguide is designed to work t wvelength 1.55 µm. The wveguide core hs thickness of 4 µm. Find the mximum n w for which the wveguide only supports propgtion of single mode. [4] (c) A stellite is positioned in n orbit 250 km bove the surfce of the Erth. A cmer instlled on the stellite is designed to imge objects situted 3 m prt on the Erth surfce. The sensitive element of the cmer hs 300 pixels per mm. Find the dimeter nd focl length of the cmer lens, for which such imging would be possible. [4] (d) Sketch Michelson stellr interferometer nd explin how it cn be used to mesure n ngulr size of str. Justify your nswer with relevnt mthemticl expressions describing light interference. [7] END OF EXAMINATION PAPER 6

7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

8 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x x 2 2 +x 2 1 x x x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

9 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

10 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk