Physics 1502: Lecture 34 Today’s Agenda Announcements: –Midterm 2: graded soon … –Homework 09: Friday December 4 Optics –Interference –Diffraction »Introduction to diffraction »Diffraction from narrow slits »Intensity of single-slit and two-slits diffraction patterns »The diffraction grating

Interference

A wave through two slits Screen   P=d sin  d In Phase, i.e. Maxima when  P = d sin  = n Out of Phase, i.e. Minima when  P = d sin  = (n+1/2)

A wave through two slits In Phase, i.e. Maxima when  P = d sin  = n Out of Phase, i.e. Minima when  P = d sin  = (n+1/2) + +

The Intensity What is the intensity at P? The only term with a t dependence is sin 2 ( ).That term averages to ½. If we had only had one slit, the intensity would have been, So we can rewrite the total intensity as, with

The Intensity We can rewrite intensity at point P in terms of distance y Using this relation, we can rewrite expression for the intensity at point P as function of y Constructive interference occurs at where m=+/-1, +/-2 …

Phasor Addition of Waves Consider a sinusoidal wave whose electric field component is Consider second sinusoidal wave The projection of sum of two phasors E P is equal to E0E0 E 1 (t) tt E 2 (t) E0E0  E P (t) ERER  /2 E0E0 tt E 1 (t)  t+  E0E0 E 2 (t)

Phasor Diagrams for Two Coherent Sources E R =2E 0 E0E0 E0E0 E0E0 E0E0 ERER 45 0 E0E0 E0E0 ERER 90 0 E R =0 E0E0 E0E0 E0E0 E0E0 ERER E R =2E 0 E0E0 E0E0

SUMMARY 2 slits interference pattern (Young’s experiment) How would pattern be changed if we add one or more slits ? (assuming the same slit separation ) 3 slits, 4 slits, 5 slits, etc.

Phasor: 1 vector represents 1 traveling wave single traveling wave 2 wave interference

N=2N=4N=3      N-slits Interference Patterns

Change of Phase Due to Reflection Lloyd’s mirror P2P2 P1P1 S I L Mirror The reflected ray (red) can be considered as an original from the image source at point I. Thus we can think of an arrangement S and I as a double-slit source separated by the distance between points S and I. An interference pattern for this experimental setting is really observed ….. but dark and bright fringes are reversed in order This mean that the sources S and I are different in phase by An electromagnetic wave undergoes a phase change by upon reflecting from the medium that has a higher index of refraction than that one in which the wave is traveling.

Change of Phase Due to Reflection phase change n1n1 n2n2 n 1 <n 2 n1n1 n2n2 no phase change n 1 >n 2

Interference in Thin Films Air Film t phase change no phase change A wave traveling from air toward film undergoes phase change upon reflection. The wavelength of light n in the medium with refraction index n is The ray 1 is out of phase with ray 2 which is equivalent to a path difference n /2. The ray 2 also travels extra distance 2t. Constructive interference Destructive interference

Chapter 34 – Act 1 Estimate minimum thickness of a soap-bubble film (n=1.33) that results in constructive interference in the reflected light if the film is Illuminated by light with =600nm. A) 113nm B) 250nm C) 339nm

Problem Consider the double-slit arrangement shown in Figure below, where the slit separation is d and the slit to screen distance is L. A sheet of transparent plastic having an index of refraction n and thickness t is placed over the upper slit. As a result, the central maximum of the interference pattern moves upward a distance y’. Find y’ where will the central maximum be now ?

Solution Corresponding path length difference: Phase difference for going though plastic sheet: Angle of central max is approx: Thus the distance y’ is: gives

Phase Change upon Reflection from a Surface/Interface Reflection from Optically Denser Medium (larger n) Reflection from Optically Lighter Medium (smaller n) by analogy to reflection of traveling wave in mechanics 180 o Phase ChangeNo Phase Change

Examples : constructive: 2t = (m +1/2) n destructive: 2t = m n constructive: 2t = m n destructive: 2t = (m +1/2) n

Application Reducing Reflection in Optical Instruments

Diffraction

Experimental Observations: (pattern produced by a single slit ?)

First Destructive Interference: (a/2) sin  = ± /2 sin  = ± /a m th Destructive Interference: (a/4) sin  = ± /2 sin  = ± 2  /a Second Destructive Interference: sin  = ± m /a m=±1, ±2, … How do we understand this pattern ? See Huygen’s Principle

So we can calculate where the minima will be ! sin  = ± m /a m=±1, ±2, … Why is the central maximum so much stronger than the others ? So, when the slit becomes smaller the central maximum becomes ?

Phasor Description of Diffraction Can we calculate the intensity anywhere on diffraction pattern ?  =  =  N   / 2  =  y sin (  ) /  = N  = N 2   y sin (  ) / = 2  a sin (  ) / Let’s define phase difference (  ) between first and last ray (phasor) 1st min. 2nd max. central max. (a/  sin  = 1: 1st min.

Yes, using Phasors ! Let take some arbitrary point on the diffraction pattern This point can be defined by angle  or by phase difference between first and last ray (phasor)  The arc length E o is given by : E o = R  sin (  /2) = E R / 2R The resultant electric field magnitude E R is given (from the figure) by : E R = 2R sin (  /2) = 2 (E o /  ) sin (  /2) = E o [ sin (  /2) / (  /2) ] I = I max [ sin (  /2) / (  /2) ] 2 So, the intensity anywhere on the pattern :  = 2  a sin (  ) /

Other Examples What type of an object would create a diffraction pattern shown on the left, when positioned midway between screen and light source ? A penny, … Note the bright spot at the center. Light from a small source passes by the edge of an opaque object and continues on to a screen. A diffraction pattern consisting of bright and dark fringes appears on the screen in the region above the edge of the object.

Fraunhofer Diffraction (or far-field) Incoming wave Lens  Screen

Fresnel Diffraction (or near-field) Incoming wave Lens Screen P (more complicated: not covered in this course)

Resolution (single-slit aperture) Rayleigh’s criterion: two images are just resolved WHEN: When central maximum of one image falls on the first minimum of another image sin  =  / a  min ~  / a

Diffraction patterns of two point sources for various angular separation of the sources Resolution (circular aperture)  min = 1.22 (  / a) Rayleigh’s criterion for circular aperture:

EXAMPLE A ruby laser beam ( = nm) is sent outwards from a 2.7- m diameter telescope to the moon, km away. What is the radius of the big red spot on the moon? a. 500 m b. 250 m c. 120 m d. 1.0 km e. 2.7 km  min = 1.22 (  / a) R / = 1.22 [ / 2.7 ] R = 120 m ! Earth Moon

Two-Slit Interference Pattern with a Finite Slit Size I diff = I max [ sin (  /2) / (  /2) ] 2 Diffraction (“envelope” function):  = 2  a sin (  ) / I tot = I inter. I diff Interference (interference fringes): I inter = I max [cos (  d sin  /  ] 2 smaller separation between slits => ? smaller slit size => ? The combined effects of two-slit and single-slit interference. This is the pattern produced when 650-nm light waves pass through two 3.0- mm slits that are 18 mm apart. Animation

Example The centers of two slits of width a are a distance d apart. Is it possible that the first minimum of the interference pattern occurs at the location of the first minimum of the diffraction pattern for light of wavelength  ? d a a 1st minimum interference: d sin  = /2 1st minimum diffraction: a sin  = The same place (same  ) : /2d = /a a /d =  No!

Application X-ray Diffraction by crystals Can we determine the atomic structure of the crystals, like proteins, by analyzing X-ray diffraction patters like one shown ? A Laue pattern of the enzyme Rubisco, produced with a wide-band x-ray spectrum. This enzyme is present in plants and takes part in the process of photosynthesis. Yes in principle: this is like the problem of determining the slit separation (d) and slit size (a) from the observed pattern, but much much more complicated !

Determining the atomic structure of crystals With X-ray Diffraction (basic principle) 2 d sin  = m  m = 1, 2,.. Crystalline structure of sodium chloride (NaCl). length of the cube edge is a = nm. Crystals are made of regular arrays of atoms that effectively scatter X-ray Bragg’s Law Scattering (or interference) of two X-rays from the crystal planes made-up of atoms