# General Physics I - Department of Physics and Astronomy Wave nature of light thin films, diffraction Physics 123, Spring 2006 03/01/20 Lecture VI 1 Intensity in Youngs experiment E=E1 +E2 E=E0 (sin(t)+sin(t+)) E 2 E0 sin t cos 2 2 IE 2 =0 =0: amplitude E(=0)=2E0 I(=0)=4E02 A B A B

sin A sin B 2 sin cos 2 2 03/01/20 Amplitude E()=2E0cos(/ 2) I()=4E02 cos2(/2) Lecture VI 2 Intensity in Youngs experiment I(=0)=4E02 I()=4E02 cos2(/2) 2d y L I ( ) 2 2 d cos cos I (0) 2

L Bright when cos=1, or -1 03/01/20 y d y m, m 0,1,2,3 L L y m d Lecture VI 3 Youngs experiment 03/01/20 Lecture VI r=700 nm

b=400 nm d=2000nm L=20cm First fringes (bright spots) yr, yb-? m=1: y=L /d yr=7cm yb=4cm Blue is closer to the center than red 4 Youngs experiment Two different - Distance between slits d Multiple slits (diffractive grating) same pattern, sharper lines 03/01/20 Interference pattern depends on Lecture VI

Maxima: d sin = m d sin = m 5 Coherence Why do not we observe an interference pattern between two different light bulbs? These two sources of light are incoherent: What does it mean for two sources to be coherent? Same (or close) frequency Constant shift in phase (not necessarily zero) 03/01/20 Lecture VI 6 Light in a medium (refraction) n1 n2

03/01/20 Lecture VI Huygens principle each point forces oscillations with frequency f f1=f2 v1=c/n1 v2=c/n2 n11=n22 E.g. go from air to medium n: /n n=/n 7 Light in medium 1=k1x=(2/)x=2600/400=3 + - =400nm x=600nm

Destructive interference n=2 + n=n400/2=200nm Extra phase =3 cos( / 2) cos(3 / 2) 0 2=k2x=(2/)x=2600/200=6 03/01/20 Lecture VI 8 E1 E2 0 Reflection of a transverse wave pulse Reflection from fixed end inverted pulse Reflection from loose end the pulse is not inverted. 03/01/20

Lecture VI 9 Reflection + + - + Reflect from medium with higher n2>n1 phase change by = Reflect from medium with lower n2

10 Soap film Soap film, air on both sides Thickness t n(soap)=1.42 n(soap)>n(air) Ray 1 at A n(air)

t=123nm Violet is thinner than red. 1 2 Relative shift -= t/n If m t/n=m or t(m=1)=n/2 Rays 1 and 2 are out of phase Destructive interference B If m t/n-=m or t(m=0)=n/4 Rays 1 and 2 are in phase Constructive interference 03/01/20 Lecture VI 11 Diffraction on a single slit z

l x Slit size D, z=-D/2 to D/2 dE Observe diffraction at angle Interference of wavescoming fromdz dz i ( kx t ( z )) E0 Im e D l z ( z ) 2 2 sin 03/01/20 Lecture VI 12

Diffraction on a single slit Integrate overdz z l x z ( z ) 2 sin E0 i ( kx t ) i ( z ) E0 i ( kx t ) E dE e e dz e e D D E0 E i 2D sin

03/01/20 sin i 2 z e i ( kx t ) e i 2 sin z dz D/2 D/2 Lecture VI E0 e i ( kx t ) sin( sin D / ) sin D /

13 Diffraction E E0 e i ( kx t ) sin( sin D / ) sin D / sin 2 ( sin D / ) I / I0 ( sin D / ) 2 Dark spot at sin D / m sin m D Except =0 must be bright spot: sin 2 ( sin D / ) I / I 0 lim 1 2 ( sin D / )

0 03/01/20 Lecture VI 14 Diffraction Single slit diffraction Angular half width of the first peak: 03/01/20 sin D Lecture VI 15