Wave Phenomena Physics 15c

Transcription

1 Wve Phenomen Physics 15c Lecture Diffrction (H&L Chpter 11) Wht We Did Lst Time! Studied interference! or more wves overlp " Amplitudes dd up " Intensity = (mplitude) does not dd up! Thin-film interference! Reflectivity of thin film depends on thickness, plus hrd/soft-ness of the two boundries! Newton s rings! 1/4-wvelength nti-reflective cotings! Two-body interference! Intensity depends on ngle I = I! Young s experiment demonstrted wve-ness of light π d sin cos

2 Gols for Tody! Extend -body " n-body interference! Rdition becomes highly directionl! Diffrction grtings! Diffrction! Light through finite-size hole on the wll! Light does not follow the exct shpe of the hole! Fundmentl limit of the resolution of opticl system! Crystllogrphy! As n exmple of technology tht uses interference N-Body Interference! It s esy to extend -body interference to n bodies! Result is interesting! There re prcticl pplictions! Difference between pths from neighboring sources is d sin xm = x1 + ( m 1) dsin! Add up E from ll sources n n i( kxm t) i( kx1 t) ik ( m 1) d sin m= 1 m= 1 iknd sin i( kx1 ωt) e 1 = Ee ikd sin E = E e = E e e ω ω e 1 d x 1 d sin n m= 1 α ( m 1) x n n α 1 = α 1

3 N-Body Interference knd sin knd sin sin ( 1) sin i i iknd k n d ( 1 ) 1 i i kx ωt e i( kx1 ωt) e e ikd sin kd sin kd sin e 1 i i e e k( n 1) dsin i knd sin i( kx1 ωt) sin( ) = Ee e kd sin sin( ) E = E e = E e e! We re interested in E phse doesn t mtter knd sin sin( ) sin nx kd sin E = E = E kd sin x sin( ) sin x! How does this look like?! It s periodic function (period π) of x! Wht hppens t x =, π, π, where sinx =? Hlf the phse difference between neighbors N-Body Interference! Wherever sinx =, sinnx is lso sin nx nx +!! Use Tylor expnsion " lim = lim = n x sin x x x+! n = 5 n = 3 π π 3π x n = n = 4

4 N-Body Interference x! Clculte intensity reltive to = sin( ) sin nx E = E = E sin( ) sin x knd sin kd sin! Peks ner x =, π, π, gets nrrower s n increses! With lrge enough n, wves vnish except for very shrp peks t kd sin = = mπ rcsin m = d! First pek t sin = d I I I( ) sinnx = I() nsin x π π π Lrge n π 3π 3π x Diffrction Grting! Consider plte with fine stripes! e.g., trnsprent film with regulr scrtches light! Gps between scrtches trnsmit light! Like Young s experiment, with mny mny mny slits! Light is diffrcted t ngle only if sin =! Trivil solution: = for ny wvelength! First non-trivil solution: m = 1 " sin = d! Light goes to different depending on m d! Cn mesure light intensity s function of wvelength

5 Wide Slit! Now we mke wide slit on the wll! We know wht hppens: shdow shped just like the slit! We lso know tht we get spreding wves from nrrow slit! Observer behind the wll sees light with nrrow slit, but not with wide slit! Does this mke sense?! Wht hppens if the width of the slit is in between? Light Light Wide Slit! Huygens principle tells us wht to do! Imgine lot of wve sources in the slit! Wves spred from ech source! Add up the mplitude! This is n-body interference with big n! We know the nswer for ny n I( ) sinnx kd sin = x I() nsin x! Mke n " infinity while keeping nd = = width of the slit d

6 Wide Slit! Since nd =, kd sin k sin x = n k nx = sin! Tke the limit of n " infinity ksin ksin ksin sin sin sin lim = lim = n ksin ksin ksin nsin n n n n k sin π sin! Let s define α = I( ) sin α! Intensity is (this) = I() α sin nx sin = nsin x nsin = sinα α ksin ksin n Do you remember seeing this function before? Fourier Integrl! We sw function in Lecture #11 α! Fourier integrl of squre pulse ws ωt 1 ωt 1 sin F( ω) = sin = ωt πωt π! Is there connection?! In fct there is sinα T 1 T

7 Fourier Integrl! Represent the slit by the trnsmittivity s function of position y! It s squre function! Wves from y is E f y e f ( y) { ( sin ) ω } ( ) i k r y t! Integrte this to get the totl mplitude i{ k( r ysin ) ωt} i( kr ωt) ikysin ( ) ( ) i( kr ωt) = πee F( ksin ) E = E f y e dy = E e f y e dy y y r ysin r Fourier! Interprettion! Shpe of the slit nd the mplitude of the wves re relted by Fourier integrls f ( y ) Fk ( sin )! Width of f nd F re inversely proportionl to ech other! Remember the discussion of speed nd bndwidth? The wider the slit, the nrrower the spred of the direction of the wves! Wide slit " light continues stright! Nrrow slit " light spreds ll directions! If you remember the δ-function, you know tht perfectly stright bem of light must be infinitely wide

8 Frunhofer Diffrction! Let s look t I( ) sin α π sin = α I() α.1 = = 1 = = 5 = 1! First sinα = is sin = nrrow wide Fresnel Diffrction! So fr we considered diffrction t lrge distnce! Frunhofer diffrction s function of! At short distnce, things get much more complicted! Angle from ech source differs d! Pth difference not exctly dsin! Solution is Fresnel diffrction! Principle is unchnged, but the result cnnot be expressed in simple form " Must be clculted numericlly! See H&L Section 11.8 if you re curious x 1 d sin x n

9 Opticl Devices! Opticl devices use lenses to collect light from object! Telescopes, microscopes, cmers, humn eyes! Lenses hve finite perture! It s like pssing light through hole " Diffrction! Light spreds out ccording to! This blurs the imge sin = Opticl Resolution! Consider two points of the object d! Light from these points end up in two points! But they re spred out due to diffrction by sin d =! If the ngle between two points is < d, the points cnnot be distinguished (or resolved) in the imge! We cn consider d s the ultimte resolution! It s fundmentl due to the wve nture of light! It s determined by the perture of the device

10 Airy Disc! Frunhofer diffrction ws clculted for slit! Lenses re usully round! We need the diffrction formul for round hole! Clcultion tedious, but not fundmentlly different! Solution known s Airy Disc light sin d = 1. d Airy Disc! Actul shpe of light pttern looks like = = = 5! There should be fringes, but impossible to see

11 Diffrction Limit! Resolution of n opticl device with perture is sin > 1.! Clled the diffrction limit or Ryleigh criterion! It cn be worse, of course! Exmple: 1-inch telescope! Eyes good for 1-3 rd m 6 1. =.6 1 rd.54m! If mgnifiction > 4, you strt to see the blur! No point in mking >5 power 1-inch telescope! Common wisdom: 5x per inch Lrge Telescopes! The lrger, the better It s tht simple! Better resolution + light collection! Keck + KeckII (1 m) on Mun Ke, Hwii! Loction chosen to minimize the disturbnce due to ir density fluctution! Index refrction of ir limits ll telescopes on Erth

12 Crystllogrphy! Crystl = regulr rry of molecules! Ech molecule cn sctter light! Crystl mkes 3-D version of n-body interference! Consider it s light-reflecting plnes seprted by fixed distnce d d Wht is the condition for constructive interference? Brgg Condition! Constructive interference is chieved if dsin = n d d d sin d sin! Wves reflected by djcent lyers hve the sme phse! Brgg condition: dsin = n Bsis of crystllogrphy

13 Crystllogrphy dsin = n! There re mny different sets of plnes in crystl! Different ngle nd different plne spcing! Ech set reflect light when Brgg condition is met! Experiment cn control! Angle: rotte the crystl! Wvelength: use white light nd nlyze wvelength of the reflection using diffrction grting (spectrometer)! Crystlline structure cn be reconstructed from mesured reflection X-Ry Diffrction! Wvelength must be comprble to d " X-rys! To mke X-rys, ccelerte electrons nd hit wll! When the electrons stop, decelertion cretes rdition =.6975 Å on cubic SiC crystl -D diffrction pttern of lysozyme

14 Summry! Extended -body " n-body interference! Wves become more focused s n gets lrger! Diffrction grtings nlyze light spectrum! Discussed diffrction! Wide slits " Frunhofer diffrction! Slit nd diffrction connected by Fourier! The wider the slit, the nrrower the ngulr distribution! Diffrction limits resolution of opticl devices! Airy disc for round perture " 1.! Crystllogrphy " Brgg condition I( ) sin α = I() α! Next week: Optics with lenses nd mirrors I ( ) sinnx = I () nsin x kd sin x π sin α >1.