PY3101 Optics. Course overview. Revision. M.P. Vaughan. Wave Optics. Electromagnetic Waves. Geometrical Optics. Crystal Optics

Transcription

1 Revisio M.P. Vaugha Course overview Wave Optics Electromagetic Waves Geometrical Optics Crystal Optics

2 Wave Optics Geeral physics of waves with applicatio to optics Huyges-Fresel Priciple Derivatio of Laws of Optical Propagatio Rectiliear motio Reflectio Refractio Diffractio Diffractio gratigs (use i spectroscopy) Electromagetic Waves The wave equatio The electric susceptibility tesor Light propagatio i isotropic media Refractive idex ad dispersio Optical loss Polarisatio Polarisig optical elemets (liear, retardatio plates)

3 Geometrical Optics Fermat s Priciple (least time) Derivatio of Laws of Optical Propagatio Imagig with leses ad mirrors Perfect imagig Spherical leses ad mirrors Paraxial approximatio Aberratios Systems of leses ad mirrors Crystal Optics Crystal Optics light propagatio i aisotropic media The idex ellipsoid Birefrigece 3

4 Huyges-Fresel Priciple Huyges Priciple You should be able to Write dow the form of a spherical wave Defie a wavefrot State Huyges Priciple Usig Huyges Priciple Derive the Law of Rectiliear Propagatio Derive the Law of Reflectio Derive the Law of Refractio 4

5 Spherical waves Spherical waves Sice the itesity of a EM wave is proportioal to the squared modulus of the amplitude, by the coservatio of eergy, the amplitude must vary as /r. Moreover, the requiremet that the amplitude be fiite at r = 0 meas that the spherical wave must be of the form E E r r iωt ( r, t) = e si kr. 5

6 Geometric wavefrot A geometric wavefrotis the surface i space cotaiig all poits i a optical field that have the same phase. A rayis a path through space that is everywhere perpedicular to the wavefrot. Geometric wavefrot - spherical Wavefrots cotours of costat phase 6

7 Geometric wavefrot - spherical Wavefrots cotours of costat phase Rays everywhere perpedicular to wavefrots Geometric wavefrot - plae Wavefrots cotours of costat phase 7

8 Geometric wavefrot - plae Wavefrots cotours of costat phase Rays everywhere perpedicular to wavefrots Huyges Priciple Each poit o a wavefrot acts as a source of secodary, spherical wavelets. At a later time, t, a ew wavefrot is costructed from the sum of these wavelets. 8

9 Huyges Priciple rectiliear propagatio All poits o the wavefrot act as sources of spherical wavelets costat phase over surface of sphere 0 z Huyges Priciple rectiliear propagatio Sice all poits o the spheres must have the same phase, the taget to the leadig edge of all the spheres must also be at a costat phase. 0 z 9

10 Huyges Priciple - reflectio Huyges Priciple - reflectio 0

11 Huyges Priciple - reflectio Huyges Priciple - reflectio

12 Huyges Priciple - reflectio Huyges Priciple - refractio

13 Huyges Priciple - refractio Huyges Priciple - refractio 3

14 Huyges Priciple - refractio Huyges Priciple - refractio 4

15 The Huyges-Fresel Priciple Huyges-Fresel Priciple You should be able to Explai what is meat by coherece ad iterferece State the Huyges-Fresel Priciple Explai how this is differet to Huyges Priciple 5

16 Coherece If two beams of light are coheretwith each other, the there is a fixed relatio betwee their phases Iterferece Two coheretbeams may add together via the Priciple of Liear Superpositio to obtai iterferece 6

17 The Huyges-Fresel Priciple For light of a give frequecy, every poit o a wavefrot acts as a secodary source of spherical wavelets with the same frequecy ad the same iitial phase. The wavefrot at a later time ad positio is the the liear superpositio of all of these wavelets. Diffractio 7

18 Huyges-Fresel Priciple You should be able to Describe the diffractio regimes (ear-field ad far field) Derive the sigle-slit diffractio patter Geeralise to the multiple-slit case Explai the Rayleigh criterio Aalyse diffractio gratigs Discuss the applicatios of diffractio Sketch ad explai basic moochromator desigs What is diffractio? Diffractio is the bedig of waves aroud objects or through apertures It is a iterferece effect 8

19 Light passig through a arrow aperture Huyges-Fresel Priciple Light passig through a arrow aperture Maximum possible path differece max = AP BP = AB = D. 9

20 Limitig cases: λ >> D max always less tha λ wavelets add costructively i all directios. Emerget field looks like poit source. Limitig cases: λ << D Wavelets add costructively i this regio Both costructive ad destructive iterferece outside shaded regio 0

21 Fresel ad Frauhofer diffractio Near field (Fresel diffractio) Far field (Frauhofer diffractio) Sigle slit diffractio E L is the field stregth per uit legth E P is the total field a the poit P

22 Sigle slit diffractio Field at x de = E L dx. Cotributio to field E P due to de EL dep = siωt kr( x) dx r x Total field E P E ( ) / = D P D / E r L ( x) [ ]. ( x) si[ ωt kr ] dx. Sigle slit diffractio r(x) is give by the cosie rule r r ( x) R + x Rx cos( π θ) = or x R x R ( x) = R + siθ. x / To fid a closed form solutio, we must approximate this expressio.

23 Taylor series expasio of r(x) The Taylor series expasio for a fuctio ( + ξ) / is Hece, ad r ξ ξ / ( + ξ) = + + K x x ( x) = R siθ+ cos θ+ K kr R 8 R ( x) = kr kxsiθ+ cos θ+k. kx R The Frauhofer coditio The third term i the expressio for kr(x) takes its maximum whe x ± D/ ad θ = 0. That is kx R cos kd θ = 8R πd 4λR The coditio that this term makes a egligible cotributio to the phase is πd 4λR << π.. 3

24 The Frauhofer coditio Neglectig the factor of 4 i the deomiator of the coditio just foud, it may be re-writte as D R << λ. D This is the Frauhofer coditio for far field diffractio. Far field approximatios Assumig that the Frauhofer coditio is valid, the third term i the expressio for kr(x) may be eglected ad we have kr ( x) kr kx siθ. The /r(x) factor appearig i the itegral for E P is less sesitive to chages i r(x) tha the phase ad we may simply put. r x R ( ) 4

25 Itegratig over x Usig these approximatios, the expressio for the total field E P becomes E To perform this itegral, we ote that si P = EL R D / [ ωt kr kxsiθ] dx. si + D / Im{ }. i( ωt kr+ kxsiθ) [ ωt kr+ kxsiθ] = e The total field E P Itegratig over the x-depedet part where D / D / e ikxsiθ ikxsiθ e dx= ik siθ β = Hece, the total field E P is E P D / kd siθ. D / ELD siβ = si( ωt kr). R β siβ = D, β 5

26 Itesity profile for a sigle slit Averagig E P over time gives E P = ELD siβ. R β The squared modulus of this will be proportioal to the itesity, i.e. siβ β ( θ) I( 0). I = Itesity profile for a sigle slit 6

27 Itesity profile for a sigle slit The zeros of the peaks occur at values of β = kd siθ = mπ, where m is a iteger. Hece, the first zeros aroud the cetral peak are give by siθ = λ. D Note that this result is oly valid for λ < D. I other cases, there are o zeros from π to π. Itesity profile for a circular aperture 7

28 The Airy disc Airy disc Airy patter by Sakurambo. A computer-geerated image of a Airy disk. URL: The Airy disc Laser Iterferece by Petrov Victor. Diffractio of red laser beam by a circular aperture. URL: 8

29 The Rayleigh criterio The first zero of the itesity profile for diffractio from a circular aperture occurs at λ siθ.. D This represets the miimum agular separatio that two poits ca be so that they may be separately resolved. Usig the small agle approximatio, this becomes λ θ.. D This is kow as the Rayleigh criterio. Diffractio limited imagig Itesity profiles for two resolvable distat poit sources. Merged itesity profiles for uresolvable distat poit sources. 9

30 Itesity profile for multiple slits I si Nα N siα siβ β ( θ) = I( 0). Itesity profile for multiple slits 30

31 Diffractio coditio Note that the coditio for costructive iterferece is a siθ = mλ. Diffractio coditio We ca re-write this as But this is just ka siθ =πm. α =πm, which gives the coditio for the local maxima of the itesity I si Nα N siα siβ β ( θ) = I( 0). 3

32 Diffractio wavelegth depedece Red (loger wavelegth) light is diffracted to a greater extet tha blue (shorter wavelegth). a siθ = mλ. (Yellow arrow is icidet light ad specular reflectio) Diffractio coditio k= kẑ, Icidet ad diffracted wavevectors: k ' = k( xˆ siθ+ zˆ cosθ). 3

33 Diffractio coditio: off-axis icidece AO = a siθ, OB= a siθ. i m The gratig equatio For off-axis trasmissio, the diffractio coditio is ow a ( siθ + siθ ) mλ. m i = This reduces to the case of ormal icidece whe θ i = 0. This result may be further geeralised by takig the icidet agle aroud to the frot of the gratig i.e. makig the gratig ito a reflectio gratig. 33

34 Reflectio gratigs Reflectio gratigs Light strikes the reflectio gratig at a agle θ i. For certai agles θ m, the diffractio coditio will be met: The path legths of rays from the icidet wavefrot via the successive ruligs of the gratig ad leavig at the same agle must differ oly be itegral multiples of the wavelegth λ. 34

35 Derivatio of the reflectio gratig equatio Icidet wavefrot AC Path from A to wavefrot BD AB= a siθ m. Path from C to wavefrot BD CD= a siθ i. Path differece AB CD= a( siθ siθ ). m i The reflectio gratig equatio The diffractio coditio for a reflectio gratig may the be expressed mathematically as mλ = a( siθ siθ ). m i This is kow as the reflectio gratig equatio. 35

36 Dispersio The dispersio of a gratig is defied as D θ = dθm. dλ Differetiatig the gratig equatios, we have dλ m = a cosθ m. dθ m Number of orders Recall that a ( siθ ± siθ ) mλ. m i = So the highest order m is the largest itegral value of a ( siθm± siθi). λ Hece a m max <. λ 36

37 Resolvig power The resolvig power of a gratig is defied as R= λ, λ where, via Rayleigh s criterio, λ is the miimum resolvable wavelegth betwee the peaks of two wavelegths with midpoit λ. Commo example of a gratig 37

38 Diffractio aroud a razor blade X-ray diffractio (o-optical) (See PY305) 38

39 Maxwell s equatios Maxwell s equatios You should be able to Defie the electric susceptibility tesor Derive the wave equatio for a isotropic medium Write dow plae-wave solutios of the wave equatio Explai the pheomea of dispersio Describe the mechaism of optical loss Startig with a complex wave vector, derive the light itesity with distace ad the absorptio coefficiet 39

40 Maxwell s equatios D= ρ, B= 0, B E=, t D H= j f +, t D= ε 0E+ P ad H= M. µ f 0 The electric susceptibility tesor The electrical polarisatio P of a medium is give by P=ε 0χ EE, where χ E is the electric susceptibility tesor. χ E characterises the frequecy respose of the medium to a applied electric field E. 40

41 Liear, isotropic ad homogeeous media I a liear, isotropic ad homogeeous (LIH) medium χ E χ0 = χ χ 0 Wave equatio i a LIH medium I a dielectric the free charge desity ad free curret are zero, so, from Maxwell s equatios ad D D = 0 D H=. t ( I+ χ ) E ε ε, = ε 0 E = 0 E where ε is the relative permittivity. 4

42 Wave equatio i a LIH medium Similarly H= B, µ µ where µ is the relative permeability. Hece ad D= ε 0 ε E= 0 H= B= µ µ 0 E B= ε 0εµ 0µ. t 0 D t E = ε 0ε, t Wave equatio i a LIH medium From Maxwell s equatios B E E = = ε 0εµ 0µ. t t Usig the vector idetity ad E= E ( E) E = 0, 4

43 Wave equatio i a LIH medium we have The wave speed is where is the refractive idex. E E = ε 0εµ 0µ. t c v = 0 0 = ( ε εµ µ ) /, = ( εµ ) / Plae-wave solutios We look for solutios of the form i( k r ωt ( t) = E e ), E r, 0 where E 0 is a Joes vector cotaiig iformatio about the polarisatio. Now t k ω. = k, 43

44 Plae-wave solutios So E E = ε 0εµ 0µ k E= ω ε 0εµ 0µ E, t where k 0 is the free space wave- which gives We may put vector, so k k 0, ω v=. k ω v=. k Frequecy depedece A simple aalysis yields ε χ 0 E q mi =, + iω τ i i ( ω ω ) i i showig that χ E is frequecy depedet. This gives rise to the pheomeo of dispersio (differet frequecies of light travellig at differet speeds i a optical medium). 44

45 Frequecy depedece The relative permittivity is This meas ad ε = + χ = ε. iε E = i k = k ik, where k = k 0. Relative permittivity 45

46 Optical loss Usig E, 0 z = v ( z t) E exp iω t, Substitutig for v usig the complex refractive idex, E 0 z z = c c ( z, t) E exp iω t exp. The absorptio coefficiet Now the itesity of the radiatio I(z) is proportioal to the squared modulus of the field So where I ω c ( z) E = E exp. I 0 αz ( z) = I( 0) e, α = is the absorptio coefficiet. ω c z 46

47 Polarisatio Polarisatio You should be able to Defie the polarisatio of light Describe ad aalyse plae-polarisatio circular polarisatio elliptical polarisatio Describe liear polarisers retardatio (wave) plates Apply the Joes calculus to states of polarisatio ad optical elemets 47

48 Liear polarisatio A plae-wave may be writte E= E i 0e ω ( t k r ), where E 0 is a Joes vector, cotaiig iformatio about the polarisatio. Liear polarisatio For liear polarisatio at a agle θ to the x-axis E 0 is give by cosθ E0 = E0. si θ 48

49 Liear polarisatio Liear polarisatio special cases θ = 0, 0 = E0, 0 E x x-liearly polarised π θ =, 0 0 = E0, E y y-liearly polarised 49

50 y-liearly polarised light y-liearly polarised (x-liearly polarised aliged with x-axis) The dichroic sheet 50

51 The liear polariser Elliptical polarisatio The most geeral form of polarisatio is elliptical polarisatio the electric field spirals aroud the propagatio axis tracig out a ellipse. This may be uderstood by resolvig the electric field ito orthogoal compoets. So log as these compoets remai i phase, the polarisatio will be liear. 5

52 Elliptical polarisatio If, however, a phase shift is itroduced o to oe of the compoets, the polarisatio will become elliptical. This is illustrated i the ext slide... Elliptically polarised light 5

53 Retardatio This phase shift is kow as the betwee orthogoal compoets is kow as the retardatio Γ. For the polarisatio This is E= E 0 e e iφ x iφ y cosθ e siθ Γ=φy φ x. i ( ωt k r ), Retardatio We may re-write the polarisatio as E = cosθ Γ si e i e θ e iφ x i E ω 0 cosθ E0 = E0. iγ si e θ ( t k r ). Sice the x-phase factor is arbitrary, 53

54 Circular polarisatio I the case ad we have e = e ± i Γ i π / =± i π θ =, 4 E0 E0 =. ± i Circular polarisatio x compoet 54

55 Circular polarisatio Γ= π/ I the case we have Re [ E] e E= = iγ The real part is the = e iπ / E i e i = i, ( ωt k ). 0 r ( ωt k r) E0 cos si( ). ωt k r Circular polarisatio y compoet E = E e y x iπ / 55

56 Circular polarisatio left polarised E = E y x e iπ / Circular polarisatio Γ= π/ I the case we have Re e i Γ E= The real part is the [ E] = i / = e π E i e i = i, ( ωt k ). 0 r ( ωt k r) E0 cos si( ). ωt k r 56

57 Circular polarisatio y compoet E y = E x e iπ / Circular polarisatio right polarised E y = E x e iπ / 57

58 Elliptical polarisatio: geeral case E E y 0 y E + E x 0x E E y 0 y E E x 0x cosγ= si Γ. Elliptical polarisatio: geeral case E y Γis arbitrary, E x E0 0 y. E α E x 58

59 Wave plates A retardatio or wave plate is a optical elemet that produces some retardatio betwee the orthogoal compoets of the wave. The physical origi of this retardatio is due to the pheomeo of birefrigece Wave plates Birefrigece is a pheomeo i which the compoets of the wave see a differet refractive idex depedig o the orietatio of the polarisatio withi some aisotropic material This leads to light with differet polarisatio directios havig differet phase velocities 59

60 Fast ad slow axes The idex ellipsoid The speed of the waves is determied by the refractive idex that it sees. This is determied by a costructio kow as the idex ellipsoid 60

61 Types of wave plate Quarter wave plate Itroduces a phase shift of Produces circularly polarised light from liearly polarised light Half wave plate Itroduces a phase shift of Reverses sig of y-compoet ±π / ±π Hece, for liearly polarised light at a agle θ to the wave plate axis, the light is rotated by θ. The aalysis of polarised light Polarised light may be aalysed by passig it through a liear polariser I the case of iitially liearly polarised light, the emerget itesity follows Malus Law Here, we cosider the geeral case 6

62 The aalysis of polarised light The power itesity through a aalyser is give by I ( θ) = I ( cos θ+ r si θ+ r cosθ siθ cos ), 0 Γ where r= E E 0 y 0 x. The aalysis of liearly polarised light For x-liearly polarised light, we have which gives E0 y = 0, ( θ) I cos 0 θ. I = This is kow as Malus Law. 6

63 The aalysis of circularly polarised light For circularly polarised light, we have which gives r= ad π Γ=±. ( θ) = ( θ+ si θ). I cos = I0 I other words, the time-averaged itesity is costat. Joes vectors Liear polarisatio Geeral case cosθ E0 = E0. si θ x-liearly polarised 0 = E0, 0 E x y-liearly polarised 0 0 = E0, E y 63

64 Joes vectors Elliptical polarisatio Geeral case left-circularly polarised right-circularly polarised E + cosθ E0 = E0. iγ si e θ E0 =. i E E0 =. i Joes matrices Polarisers Geeral case x-liear polariser P x P θ cos θ = cosθ siθ 0 =. 0 0 cosθ siθ. si θ y-liear polariser P y 0 0 =. 0 64

65 Joes matrices Retardatio plate Geeral case Quarter wave-plate Half wave-plate M± π M Γ 0 / =. 0 ± i 0 0. i e = Γ M π 0 =. 0 Joes matrices The effect of a series of optical elemets may be modelled by multiplyig the correspodig Joes matrices together to form a combied elemet. 65

66 Fermat s Priciple Fermat s Priciple You should be able to Defie the optical path legth State Fermat s Priciple Use Fermat s Priciple to derive the Law of Reflectio derive the Law of Refractio Demostrate perfect imagig 66

67 Geometrical wavefrot These poits are all i phase with oe aother ad costitute a geometric wavefrot. The optical path legth Puttig T = t - t 0, we may the multiply T by c to express the propagatio time i dimesios of space Λ ( r) = c( t ). The quatity Λ(r) is kow as the optical path legth ad is a fuctio of distace. t 0 67

68 The optical path legth Now, if the ds dt = v= c ( r) = cdt ds, d Λ =, so Λ = 0 r ( r) ( x, y, z) ds. Homogeeous medium I a homogeeous medium ( x, y, z) = 0, that is: the refractive idex is the same everywhere. 68

69 Isotropic medium I a isotropic medium that is: ( x y, z), =, 0 the refractive idex is the same i all directios Optical path legth LIH medium We also make the usual assumptio that the respose of the medium is liearly proportioal to the applied field. For a liear, isotropic homogeous (LIH) medium becomes Λ r ( r) ( x, y, z) Λ = 0 = 0 r ( r) ds. ds 69

70 Fermat s Priciple The path take betwee two poits by a ray of light is the path that ca traversed i the least time or, equivaletly i terms of optical path legth, Light traverses the route betwee two poits for which the optical path legth is a miimum. Rectiliear propagatio Usig the calculus of variatios, we ca use Fermat s Priciple to derive Law of Rectiliear Propagatio I a LIH medium, light propagates i straight lies. 70

71 Reflectio We may also apply Fermat s Priciple uder costrait. For istace, we may impose the costrait that light travellig betwee A ad B i a medium of refractive idex must touch some poit o the iterface betwee this medium ad aother of refractive idex. This is the required costrait for reflectio. Reflectio 7

72 Reflectio Λ= S + S S S A B = = A B ( x + y ) / ([ x x] + y ) /. B A Applyig Fermat s Priciple, we have., B Λ x S A SB = + = 0. x x Reflectio S x A S x B = = ( x + y ) / B ([ x x] + y ) B x A x x = B x S A /, = x x S B B. But x S A xb x = siθ i ad = siθr. S B 7

73 Reflectio So Fermat s Priciple implies Λ x = ( siθ siθ ) 0. i r = This is satisfied whe or si θi = siθ r θ i = θ r Reflectio Thus, Fermat s Priciple reproduces the Law of Reflectio I a LIH medium, the agle of reflectio equals the agle of icidece. 73

74 Refractio Refractio Λ= S + S S A B = = S A B ( x + y ) / ([ x x] + y ) /. B A Applyig Fermat s Priciple, we have,. B Λ x = S x S x A B + = 0. 74

75 Refractio S x A S x B = = ( x + y ) / B ([ x x] + y ) B x A x x = B x S A /, = x x S B B. But x S A xb x = siθ i ad = siθt. S B Refractio So Fermat s Priciple implies Λ x = siθi siθt = 0. This is satisfied whe i.e. by Sell s Law. siθ i = siθt, 75

76 Refractio Thus, Fermat s Priciple reproduces the Law of Refractio I a LIH medium, the Law of Refractio is give by Sell s Law siθ i = siθt, Perfect imagig hyperbolic les 76

77 Perfect imagig elliptical les Perfect imagig elliptical mirror 77

78 Perfect imagig elliptical mirror Perfect imagig parabolic mirror 78

79 Spherical leses ad mirrors Spherical leses ad mirrors You should be able to State the paraxial approximatio Defie the focal legth Recall the thi les equatio the les-maker s equatio the Gaussia les formula the expressio for a series of thi leses i close combiatio Recall ad apply the rules for image costructio Calculate trasverse magificatio Defie the optical power of a les 79

80 Imagig by a spherical les Caot obtai perfect imagig Reasoable approximatio possible Les sig covetios 80

81 Les sig covetios Light is always take to propagate from left to right. If A is to the left of B, the s o is take to be positive (ad vice versa). If C is to the right of B, the s i is take to be positive (ad vice versa). If the cetre of the sphere is to the right of B, R is take to be positive. This is a covex les. If the cetre of the sphere is to the left of B, R is take to be egative. This is a cocave les. Imagig by a spherical les Requiremet for perfect imagig is that all the rays have equal optical pathlegth. That is, we require Λ to be a costat 8

82 Imagig by a spherical les The optical path legth lo + l i R= s l i i s l o o. However l l o i ( φ, s ) = lo o, = l( φ, s ), i i so o closed form solutios. 8

83 The paraxial approximatio Small agle approximatio siφ φ, cosφ. So l o s o ad l i s i. The paraxial approximatio With these approximatios, we obtai s s ( ). R + = o i 83

84 The focal legth I the case s s0, ( ). R = i From this, we may defie the focal legth withi the les f i s = R i f i. The focal legth Similarly, whe s i, we may defie the focal legth outside the les f i s = R o f o. 84

85 A thick les A thick les 85

86 86 A thick les For the surface with radius of curvature R, ( ). R s s i o = + ( ). R s s o i = + For the surface with radius of curvature R, A thick les Combiig these results ( ) + = + o i o i s s R R s s Isertig i, o s d s =

87 87 The thi les equatio ( ) ( ). i i o i s d s d R R s s = +. = + R R s s o i Takig the limit ad puttig 0, d,, o o i i s s s s = = This is the thi les equatio. The les maker s formula. = R R f If either or i s s o. or = = R R s R R s i o But the term o the RHS is a costat, which we defie to be the focal legth f This is the les maker s formula.

88 The Gaussia les formula We must also have s i + s o = f. This is the Gaussia les formula. Thi leses i close combiatio I geeral f = i. fi 88

89 Mirror sig covetios Mirror sig covetios Light is always take to propagate from left to right. The object distace s o is positive whe it is to the left of the mirror surface. The image distace s i is positive whe it is to the left of the mirror surface (real image). The image distace s i is egative whe it is to the right of the mirror surface (virtual image). The radius R is positive if the mirror surface is to the right of the cetre of the sphere (covex mirror) The radius R is egative if the mirror surface is to the left of the cetre of the sphere (cocave mirror) 89

90 Spherical mirrors From the Law of Reflectio From ispectio of the figure θ = π φ θ = α+ φ. ( π β) = β φ. Hece φ = β α. Spherical mirrors Hece s o + s i =. R Takig limits as before ad defiig the focal legth f, we the have s o + s which is the same expressio as the Gaussia les formula. i = f, 90

91 Image costructio for covex leses Sketch a ray from the tip of the object parallel to the horizotal (the priciple axis) to the cetre lie of the les. From there, sketch aother ray passig through the focus associated with the left-had les surface. Sketch a ray from the tip of the object directly through the cetre of the les without deviatio. Sketch a ray from the tip of the object passig through the focus associated with the right-had les surface to the cetre lie of the les. From there sketch a lie parallel to the priciple axis towards the image. Covex les 9

92 Covex les The image at i is real ad iverted. The magificatio of the image M is give by M = y y i o. Sice y i is egative, so is M (iverted image). Image costructio for cocave leses Sketch a ray parallel to the optical axis of the les. Sice f < 0, this must pass through f o the left of the les (as a virtual ray). Sketch a ray passig through the cetre of the les without deviatio Sketch a ray followig the lie through f o the right of the les (this extesio is virtual o the right) ad emergig parallel to the optical axis. The parallel lie is the exteded to the left as a virtual ray 9

93 Cocave les Cocave les I this case, x o is the distace betwee the object ad f o the right of the les. x i is the distace betwee the image ad f o the left of the les. 93

94 Other types of leses plao-covex plao-covex doublet Optical power The optical power P of a les is a measure of the degree to which it coverges or diverges light. P is defied as the reciprocal of the focal legth. P = f. The SI uit of P is called the dioptre (m - ). 94

95 Optical istrumets Optical istrumets You should be able to Discuss the aatomy ad fuctio of the huma eye Describe commo visual impairmets Derive the agular magificatio for a magifyig glass Describe differet types of refractig telescope ad their desigs Describe the desig of reflectig telescopes 95

96 The aatomy of the eye The les The curvature of the les may be chaged by muscle cotractios i the eye. 96

97 Near ad far poits The ear poit is the closest distace for which the les ca focus light o the retia Typically at age 0, this is about 8 cm It icreases with age, ~ 5 cm for a adult The far poit of the eye represets the largest distace for which the les of the relaxed eye ca focus light o the retia Normal visio has a far poit of ifiity Farsightedess hyperopia Distat objects may be focussed but ot earby objects. 97

98 Correctig farsightedess Nearsightedess myopia axial myopia Les too far from retia refractive myopia Les-corea system too powerful to focus properly oto the retia 98

99 Correctig earsightedess Agular size of uaided image The object at o subteds a agle of α u at the viewig poit. 99

100 Magifyig glass - aided image The image i of a object placed at o withi the focal legth of a covex mirror subteds a agle of α a at the viewig poit. Agular magificatio The agular magificatio M α is defied as the ratio of the aided ad uaided viewig agles M = α a α α u. We shall employ the paraxial approximatio to obtai a expressio for this. 00

101 Agular magificatio From the diagrams α u ho =, α a D = h d o o = hi d i. So a M α = u = α α D d o. where D is the distace to the object i the uaided case. Agular magificatio From the diagram h f o = d i hi + f. From the expressio for α a, h o = h i d d o i, Leadig to f + = d i d o. 0

102 Agular magificatio From We the have M = α D d o, M α = D f + d i. Thus the shorter the focal legth, the greater the agular magificatio. Types of refractig telescope Terrestrial Produces upright image Employs a cocave les for the eyepiece Astroomical Iverts the image Employs a covex les for the eyepiece 0

103 Galilea (terrestrial) telescope Gives upright image. Note the focal poits coicide. Kepleria (astroomical) telescope Gives iverted image. Note the focal poits coicide. 03

104 Agular magificatio Galilea telescope Keplaria telescope From the system matrix of each telescope, both are foud to have a agular magificatio of M α = f f o e. Newtoia telescope 04

105 Newtoia telescope Essetially, we have the same magificatio system as for the astroomical telescope. Hece, the agular magificatio is M α = f f o e. Aberratios 05

106 Aberratios You should be able to Discuss what is meat by third order aberratio List ad describe commo forms of moochromatic aberratio Explai the origi of chromatic aberratio ad strategies to correct it Third order correctio Previously, we employed the paraxial approximatio ad I reality siθ θ cosθ. 3 θ siθ = θ 3! 5 θ + 5! +L The secod term is referred to as the third order correctio. The secod term i the expasio of cos is also used i correctios to this order. 06

107 07 Third order correctio Third order correctio = + i i o o i o s R s s R s R R s s φ

108 Third order correctio Note that the Rφ term gives a measure of the displacemet of the itersectio of the ray with the les from the optical axis. Thus, i the third-order treatmet, the ew term icreases i proportio with the square of the agular displacemet. Third order correctios Spherical Aberratio Coma Astigmatism Field Curvature Distortio 08

109 Spherical Aberratio Due spherical curvature of les Logitudial ad trasverse spherical aberratio The logitudial spherical aberratio L SA is defied as the distace betwee the itersectio of a ray with the optical axis ad the paraxial focus. L SA = s o s '. o The trasverse spherical aberratio L SA is defied as the perpedicular distace above (or below) the paraxial focus that a ray actually passes. 09

110 Spherical Aberratio Due spherical curvature of les Soft image focusig 0

111 Coma Due to off-axis object poits Trasverse magificatio is a fuctio of ray height Patter looks like a comet Coma i a les

112 Coma i a parabolic mirror Coma Corrective leses for Newtoia telescopes with f umbers less tha f/6 have bee desiged These employ a dual les system of a plao-covex ad a plao-cocave les fitted ito a eyepiece adaptor A example of a correctio strategy for coma is Baader Rowe Coma Correctio

113 Baader Rowe Coma Correctio Compariso of the coma i a ucorrected f/3.9 Newtoia telescope vs the affects of coma with the Baader Rowe Coma Corrector. Astigmatism Vertical plae is the tagetial plae Horizotal plae is the sagittal plae Astigmatism results i differet focal legth i each plae 3

114 Field Curvature A thi les images a spherical surface oto a spherical surface Image is distorted i the image plae Importat i les desig for close objects Distortio All poits i the object plae are imaged to poits i image plae Distortio arises whe the magificatio of off-axis image is a fuctio of the distace to the les ceter 4

115 Barrel distortio magificatio decreases with distace from the optical axis Picushio distortio magificatio icreases with distace from the optical axis 5

116 Moustache distortio Moustache distortio, i which iitially the magificatio decreases with distace from the optical axis, whilst at further distaces, the magificatio icreases with distace. Chromatic Aberratio Blue refracts more tha red (greater refractive idex for ormal dispersio 6

117 Achromatic doublet A achromatic doublet (achromat) is ofte used to compesate for the chromatic aberratio. Achromatic doublet We require f R = f B = f R f B Thus, we eed to choose parameters such that ( ) + ( ) B R B R = 0 R R R R3 7

118 Achromatic doublet Example of the use of a achromatic doublet, usig a doublet as the objective. The idex ellipsoid 8

119 The idex ellipsoid You should be able to Describe the modes of vibratio of the light Explai the modes of vibratio i terms of the idex ellipsoid ad the k-vector directio Explai how the optic axes of the crystal are determied. Hece, describe the differet optical classes Derive the expressio for the idex ellipsoid from the eergy desity Explai birefrigece ad apply to problems i uiaxial crystals Describe the use of birefrigece i wave plates Explai double refractio i aisotropic crystals The wave equatio x i i ( E) E = µ ε µε. i 0 0 i E t We look for solutios of the form E = E e i i ( k r ωt ). 9

120 0 The wave equatio We obtai a eigevalue problem with the characteristic equatio. ω κ i i ck = where we have defied 0, = z z z y z x z y y y y x z x y x x x κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ The wave equatio I the geeral case ad ( ), z z y y x x a a a a + + = ( ) ( ) ( ) x z y y z x z y x a a a b + + =. z y x c =

121 The idex ellipsoid The solutios for correspod to two modes of vibratio Differet compoets of the polarisatio see differet refractive idices These modes of vibratio may be visualised by meas of the idex ellipsoid The idex ellipsoid

122 The idex ellipsoid Cosider some arbitrary wavevector k Takig the itersectio of the plae perpedicular to k with the idex ellipsoid defies a ellipse The semi-axes of this ellipse give refractive idices ad, which correspod to the two modes of vibratio D ad D Optic axes I geeral, a ellipsoid has two circular cross-sectios I the case of just two distict semiaxes, we have a spheroid ad there is just oe circular cross-sectio The ormals to these cross-sectios are kow as the optic axes of the crystal

123 Optic axes Optic axes optical classes Biaxial crystals two optic axes (these are show as N ad N i the previous Fig.) Uiaxial crystals oly oe optic axis (take, covetio, to be alog the z-axis) Isotropic crystals No optic axis refractive idex the same i all directios 3

124 Optic axes For certai special k directios, the quadratic will have repeated roots. I these cases, the optical field will oly see oe refractive idex (the cross-sectio with the idex ellipsoid is circular) These directios are therefore the optic axes of the crystal ad determied by the coditio b 4ac= 0. Uiaxial crystals 4

125 Uiaxial crystals I a uiaxial crystal, we have (by covetio) x = y = o ad z = e. These are kow as the ordiary ad extraordiary refractive idices respectively. The coefficiets of the quadratic equatio for are the a= b= [ + ( ) a ], o e [ ( a ) + ( a )] 0 0 z e z ad c= 4 o e. o z Optic axes The discrimiat is b 4ac= 4 o ( a ) ( ). z o e For o e, the discrimiat is zero whe a z =, i.e. whe k is parallel with the z-axis. Thus, this is the optic axis of the crystal 5

126 Uiaxial crystals The solutios for are the e = e + az o ad = 0. Uiaxial crystals k 6

127 Uiaxial crystals Puttig a z = cosθ, we may obtai the explicit agular depedece ( ) e θ = + cos θ. e o Idex ellipsoid (easier derivatio) The eergy desity due to the electric field is give by u = D E= ( D E + D E + D E ). E x x y y z z This may be re-writte u E = Dx ε 0ε x + D 0 y ε ε y D z +. ε 0ε z 7

128 Idex ellipsoid (easier derivatio) Makig the chage of variable (associated with a scalig) x = i D i u 0 Eε, we the have x x + y y + z z =. This is the equatio of the idex ellipsoid. Uiaxial crystal For a uiaxial crystal, we have x o + y o + z e =. Takig x = 0 with o loss of geerality, from the figure, x= 0, y= z= ( θ) cos ( θ) siθ. θ, 8

129 Uiaxial crystal Substitutig these expressios ito the idex ellipsoid ( θ) cos θ ( θ) Rearragig this, we obtai as foud earlier. + o e si θ =. ( ) e θ = + cos θ, e o Birefrigece i a uiaxial crystal The birefrigece is defied as = ( θ). For a wave with extraordiary ad ordiary compoets, E e ad E o, propagatig i a directio r, we may write ad E e E o 0 ( θ) = E 0 exp iω t c r 0 exp = or E iω t. c 9

130 Birefrigece i a uiaxial crystal The secod of these equatios may be re-writte E o ( θ) 0 exp r ω = E iω t exp i o r c c [ ( θ) ]. Hece, after a distace r, the ordiary wave acquires a retardatio Γ ( r) ω θ = c ( ) r. This provides the physical basis for retardatio plates (see Polarisatio). Uiaxial crystal Further defiitio accordig to the relative sizes of e ad o. A egative uiaxial crystal has e < o E.g. calcite CaCO 3 ad ruby Al O 3. For e > o, we have a positive uiaxial crystal E.g, quartz SiO 30

131 Uiaxial crystal wave plates I egative uiaxial crystal, the extraordiary axis is aliged with the fast axis of the plate, sice c/ e > c/ o. For a positive uiaxial crystal, we have the opposite case ad the extraordiary axis is aliged with the slow axis. Birefrigece - double refractio 3

132 Double refractio example: calcite 3