Interference and Diffraction

Transcription

1 Topics: What we are going to learn Wave behaviour of EM wave (In the case of light) Interference an iffraction Huygens' principle Interference of light fro ouble slit Thin fil Interferoeter iffraction of light fro Single slit Circular apterture this class, you shoul be able to: escribe the wave-property of EM waves (the case of light) eterine relevant paraeters (intensity, wavelength, path ifference, phase ifference, separating with of ouble slit) fro interference of light Waves or Particles? eterine relevant paraeters (intensity, wavelength, path ifference, phase ifference, separating with of ouble slit) fro iffraction of light passing through circular aperture. 3 4

2 History: Waves or Particles History: Christian Huygens Huygens' principle (678): "all points on a given wave front are taken as point sources for the prouction of spherical seconary waves, calle wavelets, which propagate outwar with spees characteristic of waves in that eiu" Sir Isaac Newton - light is a bea of particles (8:3) Huygen's assution: a source of light eits a spherical wave. Christian Huygens (69-69) History: Huygens' principle Huygens' assution: a source of light eits a spherical wave. 6 History: Thoas Young Young's ouble-slit experient (8) Wave at the interface generate new wavelets propagate in eiu. The experient was to prove the wave-property of light by eonstrating that light unergoes interference as o water waves, soun waves, an all other type of waves. Incient wave vδt v cδt vδt v Thoas Young (773-89) Reflecte wave Inex of refraction n Plane wave Spherical wave c v Young's ouble slit experient: 7:39 in ( 7 8

3 Interference Interference: what we are going to learn Interference n cause by n Phase shift arising fro N Reflection n n (n n) ighter to enser (v l > v) enser to lighter (vl < v ) Reflection phase shift. No reflection phase shift sin()δ y sin( ) ( ) sin() + Interference: ouble-slit Maxia-bright Maxia-ark Nature for all types of wave n n 9 Conitions Superposition between two or ore wavelets Phase ifference (φ) Coherent wave (φ reain constant) N Path length ifference Interference: N N v n c n ifference inex of refraction n n φb φ A A+B y Wave B Wave A φb φa + π/ ' Δ path ifference phase ifference Wave A Wave B φb φa + π axia interference bright fringes phase shift Δ inia interference ark fringes phase shift Δ + ( )

4 Interference: ouble-slit Interference: ouble-slit - Intensity Maxia-bright Maxia-ark r r Siplify: >> y r // r ' ' r sin( )Δ r Maxia-bright fringes sin() Maxia-ark fringes sin() +,,,... ( ) At point P ' E E sin (ω t) E E sin (ω t +ϕ) Δ path ifference,,,... How can intensity of the interference wave be eterine? 3 Interference: ouble-slit - Intensity E E sin (ω t +ϕ) E E E sin (ω t) E Φ E Interference: ouble-slit - Intensity To cobine the E-fiel coponents At point P E 4 P E E sin (ω t)+ E sin ( ω t+ ϕ) Apply trigonoetric ientity E E (+ cos ϕ)4 E cos ωt I 4 I cos ( ϕ) ϕ Intensity at screen As intensity (I) is proportional to the aplitue β E E Φ E β ωt E E sin (ω t) E E sin (ω t +ϕ) E E + E E cos(β) I E I E I4 I cos where 4I ϕ ϕ I I The phase ifference (φ) ( π ) ( path ifference) π ϕ( ( sin ) ) π sin () π ϕ 4π 3π. π π.. π π 3π. 4π π φ for axia. for inia. Δ / 6

5 Interference: Thin-fil Interference: Exaple What is the istance on screen of the first axia near the centre of the interference pattern. The wavelength of light is 46 n, the slit separation is., an the slitscreen separation is c. (Assue that is sall enough) y ()(46 9 )( ) Interference: Thin-fil Interference: Phase shifts by reflection Conitions observer n n r n3 r i 8 Thin transparent fil Unifor thickness () with refractive inex n Soap bubble / oil fil n n3 Maxiu interference if r an r are in-phase at the eyes n n n n ~ path ifference ~ Unable to count phase change irectly as: ighter to enser (vl > v) Path ifference occurs in eiu Reflection ay inuce the phase ifference n < n 9 Reflection phase shift. enser to lighter (v < vl) n > n Reflection phase shift

6 Interference: Thin-fil observer Reflection Point a (n n): r. Point b (n n): no change Point c (refraction): no change on r c i b a MgF Reflection r : φ. r : φ. Glass Path ifference (for bright fil, inphase) ½ o n (in-phase) ½ even n (out-of-phase) where n / n r Path ifference to have r. r r n.38 n3. n3 r r For the centre of visible light, n. eterination of interference n n n Interference: Exaple (Coate ens) o nuber n Interference: Wrappe up Interference n cause by arising fro iffraction N n Phase shift Reflection Path length ifference n n N N v n c n ifference inex of refraction N iffraction: n n n n ( n n) Phenoena (flare) which occur when a wave encounters an obstacle or a slit ighter to enser (v l > v) enser to lighter (vl < v) Reflection phase shift. No reflection phase shift sin()δ y sin( ) ( ) sin() + 3 4

7 iffraction: Single Slit iffraction of light through a single slit of with a coparable to the wavelength ocating the Minia: two zones P r a/ b r P Δx a/ iffraction: Single Slit ocating the Minia: four zones Assuption light is escribe by spherical wavelets eitte fro a istribution of points locate in the slit. The light superposes we observe bright an ark fringes on screen. a/4 r P a/4 a/4 sin() sin() tan () y / a sin () a sin () sin() tan () y / P P Spiting up slit into ore zone, we can generalize equation to escribe ark fringes as: a sin () (,, 3,...) iffraction: Single Slit ocating the Minia: ulti-zones Δ x a sin () 4 Δx a/4 >> a approxiating r // r // r3 // r4 Δ x a sin () The ark fringe coul be fore by any two rays The first ark fringe: r4 The first ark fringe: >> a approxiating r // r r3 a/4 The path ifference for light for the top ege an that fro the ile Δx sin() a/ r P 6 iffraction: Single Slit Intensity Calculation of the intensity Single-slit ark fringes are prouce when the path length ifference between the top an the botto rays,, 3,... a sin () a/ (,, 3,...) P r (,, 3,...) If the screen is place at a large istance fro the slit ay If the screen is place at a large istance fro the slit a/ I I ax r b Δx P where α ( sin α α ) πa sin Position of the ark fringes on the screen >> a approxiating r // r // r3 // r4 sin() tan () y / y a (,, 3,...) 7 8

8 iffraction: Checkpoint Circular Aperture Two wavelengths, 6 an 43 n, are use separately in a single-slit iffraction experient. The results as graph of intensity (I) are shown on the left. I iffraction: Circular Aperture A B What colour will be seen in the cobine iffraction pattern at a) A 43 n b) B 6 n y sin. a First iniu 9 iffraction: Circular Aperture Resolvability iffraction: Checkpoint ens iages is the iffraction pattern an it is iportant when we want to istinguish two istant point objects. Accoring to Rayleigh's criterion two objects can be istinguishe as figure (c) when the central axiu of the iffraction pattern is centere on the first iniu of the iffraction pattern of the other The angular separation: a) too sall to be istinguishe b) clearly istinguishe c) arginally istinguishe (Rayleigh's criterion) 3 rsin ( Hint:. ).. a) iprove Since the angle is very sall r r Suppose that you can barely resolve two re ots because of iffraction by the pupil of your eye. If we increase the general illuination aroun you so that the pupil ecreases in iaeter, oes the resolvability of the ots iprove or iinish? b) iinish 3 3

9 iffraction: Exaple (Pointillis painting) iffraction: Circular Aperture Pointillis painting Observer Assue that the average centre-to-centre separation of the ots is.. Also, assue that the iaeter of the pupil of your eye is. an that the least angular separation between ots you can resolve is set by Rayleigh's criterion. What is the least viewing istance fro which you cannot istinguish any ots on the painting? r. For a sall angle tan. The Seine at Herblay by Maxiilien uce (89) 33 iffraction: Circular Aperture saller larger re ots ( 7 n) saller blue/violet ots ( 4 n) larger iffraction: ouble Slit ouble Slit Relative intensity Previously, in the ouble-slit interference, we assue that a << How Far Can egolas See? (. in): In practice, the conition a << is often not et for the visible light iffraction of a single-slit woul get involve Why are stars star shape? (3:8 in): 4&listUUUHW94eEFW7hkUMVaZz4eg (eg) Relative intensity (eg) Relative intensity 3 The intensity is the cobination of these interference an iffraction (eg) 36

10 iffraction: ouble Slit iffraction Grating When the iffraction effects is taken into account I ()I (cos β) ( sinαα ) πa sin where β an π α sin 37 iffraction: Gratings iffraction: Gratings ispersion Intensity We exten the iea of ouble-slit fro two slits to N slits sin 38 iffraction grating can be use to separate out ifferent wavelengths of light fro a spectru of wavelength for,,, (axia) sin ispersion () escribes the ability of a iffraction grating to sprea apart the various wavelength. The quantity is given by: >> Optical evices which is siilar ouble-slit but has uch greater nuber of N slit Δ / Δ The with of the lines is expresse in ters of the half-with of the central line (Δhw at ) Therefore Intensity Therefore Central line where Δ is the angular separation between two line with wavelength ifferent Δ N sin Δ hw Δ hw N for,,, (axia) or Δ hw >> Δhw N cos ine at o 39 sin ( ) ( ) ( ) ( ) 4

11 iffraction: Gratings Resolving power sin ( ) ( ) ( ) iffraction: Su-up ( ) iffraction Optical Coponents Because / sin Wave properties of EM wave Single-Slit sin cos Grating Therefore >> cos ouble-slit Mixe pattern Circular aperture,, 3,... Interference iffraction The ispersion increase as the istance between rulings gets saller 4 4