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Physics 1202: Lecture 26 Today’s Agenda Announcements: –Midterm 2: Friday Nov. 6… –Chap. 18, 19, 20, and 21 No HW for this week (midterm)No HW for this week (midterm) Optics –Interference –Diffraction
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Interference
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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)
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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 … (destructive at m= ±1/2, ±3/2 …
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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 180 0 An electromagnetic wave undergoes a phase change by 180 0 upon reflecting from the medium that has a higher index of refraction than that one in which the wave is traveling.
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Change of Phase Due to Reflection 180 0 phase change n1n1 n2n2 n 1 <n 2 n1n1 n2n2 no phase change n 1 >n 2
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Interference in Thin Films Air Film t 1 2 180 0 phase change no phase change A wave traveling from air toward film undergoes 180 0 phase change upon reflection. The wavelength of light n in the medium with refraction index n is The ray 1 is 180 0 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
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Chapter 26 – 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
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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 ?
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Solution Corresponding path length difference: Phase difference for going though plastic sheet: Angle of central max is approx: Thus the distance y’ is: gives
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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
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Examples : constructive: 2t = (m +1/2) n destructive: 2t = m n constructive: 2t = m n destructive: 2t = (m +1/2) n
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Application Reducing Reflection in Optical Instruments
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Diffraction
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Experimental Observations: (pattern produced by a single slit ?)
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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
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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 ?
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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.
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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 ( ) /
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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.
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Fraunhofer Diffraction (or far-field) Incoming wave Lens Screen
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Fresnel Diffraction (or near-field) Incoming wave Lens Screen P (more complicated: not covered in this course)
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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
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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:
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EXAMPLE A ruby laser beam ( = 694.3 nm) is sent outwards from a 2.7- m diameter telescope to the moon, 384 000 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 / 3.84 10 8 = 1.22 [ 6.943 10 -7 / 2.7 ] R = 120 m ! Earth Moon
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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
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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!
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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 !
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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 = 0.562 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
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