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15C Sp16 Fourier Intro Outline 1. Ray Optics Review 1. Image formation (slide 2) 2. Special cases of plane wave or point source (slide 3) 3. Pinhole Camera (slide 4) 2. Wave Optics Review 1. Definitions and plane wave example (slides 5,6) 2. Point sources and spherical waves (slide 7,8) 3. Introduction to Huygen’s wavelets 1. Plane wave example (slide 9) 4. Transmission of a plane wave through a single slit 1. Narrow slits and Huygen’s wavelets (slides 10,11) 2. Wider slits up to the geometric optics limit (slide 12) 3. Time average intensity patterns from single slits (slide 13) 5. Transmission of a plane wave through two slits 1. Narrow slits and Huygen’s wavelets (slides 14-17) 2. Calculating the time averaged intensity pattern observed on a screen (slides 20-22) 6. Wave view of lenses (slides 24,25) 7. 4f optical system and Fourier Transforms 1. Introduction (slide 26) and reminder that the laser produces a plane wave in the z direction, so the kx and ky components of the propagation vector are 0 2. Demonstration that the field amplitude pattern g(x,y) produced by the LCD results in diffraction that creates non-zero k vector components in the x and y directions such that the amplitude of the efield in the fourier transform plane corresponds to the kx ky components of the fourier transform of g(x,y) Lec 2 :1 Ray Optics Intro Assumption: Light travels in straight lines Ray optics diagrams show the straight lines along which the light propagates Rules for a converging lens with focal length f, with the optic axis defined as the line that goes through the center of the lens perpendicular to the plane containing the lens: 1. Rays passing through the center of the lens are not deflected 2. Rays parallel to the optic axis are deflected so they pass through the focal point, which is the point on the optic axis one focal length in front of the lens Rules for finding an image using ray optics diagrams 1. Draw a line from an edge of the object that passes undeflected through the center of the lens 2. Draw a line from an edge of the object that travels parallel to the optic axis, this ray will deflect and pass through the focal point 3. The image is the point where the two rays intersect Focal Point Image Optic Position Axis Object Focal Length Ray Optics Examples Object at infinity=Plane wave, focuses at the focal point Point Image Focal Length A point source at the focal point, produces a propagating plane wave Point Source Focal Length Pinhole Camera By en:User:DrBob (original); en:User:Pbroks13 (redraw) - Lec 2 :4 Wave Optics Light is a wave whose propagation is governed by a wave equation. At a given time, at each position in space the wave can be described by specifying the amplitude and the phase. Linearly polarized plane wave propagating in the z direction is characterized by single k vector in the z direction with magnitude 2 A wave crest moves a distance during T,one oscillation period of the wave. Since the speed of light is c, T=c. T=2 and k= so 2 =2/(kc) ‐> w=kc E(x,y,z)= Eo Cos(kz – t) ŷ)= Eo Cos(k( z –c t )) ŷ Complex Notation E(x,y,z)= Eo Exp[i(kz – t)] ŷ)= Eo Exp[ik (z –c t )] ŷ k Wavelength= 2/k The wave moves at velocity v. It moves a distance in one oscillation period T= Propagating Plane Wave Waves Instantaneous Intensity Time Average Intensity Re[Exp[ i (kz-wt)] 2 Cos [kz-wt] T 2 Cos[kz-wt] 1/T 0 ∫ Cos [kz-wt] dt =1/2 E E*= 1/2 1.0 1.0 1.0 0.5 0.5 0.5 Out[419]= Out[420]= Out[421]= -2 -1 1 2 -2 -1 1 2 -2 -1 1 2 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 Out[749]= Out[172]= Out[285]= 1 1 1 Out[750]= Out[173]= Out[286]= 4 4 4 Out[751]= Out[174]= Out[287]= 1 1 1 Out[752]= Out[175]= Out[288]= 2 2 2 Out[753]= Out[176]= Out[289]= 3 3 3 Out[754]= Out[177]= Out[290]= 4 4 4 Out[755]= Out[178]= Out[291]= Out[756]= 1 Out[179]= 1 Out[292]= 1 Out[757]= Out[180]= Out[293]= Lec 2 :6 Wave propagation from a point source Efield amplitude= In spherical Re[Exp[ i (kr-wt)]/r coordinates, for =Cos[kr-wt)/r a point source at the origin, the wave vector always points in the r direction Lec 2 :7 Two In Phase Sources Very Close (gives resolution limit of microscopes) Looks like 1 Source Key Point: Can’t resolve objects much closer than a wavelength Field due to 2 Sources Intensity due to 2 Sources Lec 2 :8 Huygen’s Wavelets Example • He proposed that the forward propagation of a wave can be reproduced by choose a wavefront (surface of constant phase) and in phase point sources at every position on that wavefront. In practice Using sources separated by the wavelength is good enough. • Below a plane of sources creates plane waves Lec 2 :9 Plane Wave Transmission through a Small Slit slit < almost no wave gets through Real Picture of a Water Wave Going Through a Slit slit ~ wave reaches everywhere behind the slit slit > wave reaches everywhere behind the slit, but is less intense at large http://www.fas.harvard.edu/~scidemos/Oscillati angles onsWaves/RippleTank/RippleTank004.jpg Lec 2 :10 Huygen’s Wavelets Example • For a plane wave going through a slit, choose a phase front at the slit • Replace the plane wave by spherical waves all along the open slit. • For a narrow slit all of the point sources for Huygen’s wavelets are very close to each other -> propagating wave looks like a spherical wave Lec 2 :11 Transmission through slits with increasing d/ Lec 2 :12 Ray Optics are Limit of Wave Optics when << system size Particles follow ray optics Plane wave traveling through a single slit Wave Picture Time Average Intensity Picture Fields due to 2 separate sources Lec 2 :14 Increasing the Spacing Between the Sources Increases the Number of Nodes http://webphysics.davidson.edu/Applets/ripple4Lec 2 :15 / What you will measure is the Resulting Time Averaged Intensity at a Screen http://www.youtube.com/watch?v=DfPeprQ7oGcLec 2 :16 Electron Double Slit ( both wave and particle clearly visible) 100 Electrons 3000 Electrons 70000 Electrons http://www.physics.brocku.ca/courses/1p22/images/electron_two_slit.jpgLec 2 :17 Calculating Intensity at Different Positions on the Detection Screen At the top of the screen, the phase difference is pi (180 degrees). The waves interfere destructively and the intensity is zero 3/4 of the way up the screen, the phase difference is pi/2 (90 degrees). The intensity is 2. At the center of the screen (half way up), the phase difference is 0. The waves interfere constructively and the intensity 4. ¼ of the way up the screen, the phase difference is -pi/2 (-90 degrees). The intensity 4. At the bottom of the screen, the phase difference is - pi (-180 degrees). The waves interfere destructively and the intensity is zero stc: 18 Top view of the total wave and Graph of the Intensity at the screen as a function of position • With both sources the probability of reaching a given point on the screen is NOT the sum of the probabilities for the individual sources. At a given point, the probability can be higher or lower than the sum of the individual probabilities Lec 2 :19 How can you find the pattern without a Computer? • Draw Pictures showing the wave fronts – each red line represents a peak in the displacement – peaks are separated by lambda • For two sources draw pictures for each source separately – the TOTAL intensity is a maximum where the wave fronts coincide indicating peaks for BOTH waves Lec 2 :20 Line intersection -> Constructive interference Half separation -> Destructive interference Lec 2 :21 2 Slits using Huygen’s Wavelets Just After Sources Start spherical waves can be seen coming from each source At a later time the two waves are interfering everywhere Note the maxima, where the wave fronts from both waves coincide and minima where the wavef ronts from the two sources are as far separated as possible http://www.phy.ntnu.edu.twLec 2 :22 Atoms are Waves NIST Prentiss GroupLec 2 :23 Linking Lensing in Geometric Optics with Lensing in Wave Optics Point source at the focal point creates a plane wave Point Source Focal Plane Length Wave Point Source Lec 2 :24 Wave Picture of a Lens Converting a diverging Spherical Wave from the focus point to a Plane wave ( convert green sphere phase front to green plane phase front) Lens thickness d(x,y)= do( 1 - ½ (x2+y2)/f) Phase Lag = L x R=f K is in the z direction L=Sqrt(R2+x2) Phase Lag = (x) + k L = k R -> (x) = k (R –L) ~ k R ( 1- (1+1/2 x2/R2)) 2 = ½ k x /R. If the phase shift at the center of the lens is k nglass do, then the phase shift everywhere else should be smaller by ½ k x2/R. 2 so (x) = k nglass do( 1 - ½ x /R), so it decreases quadratically with x. Lens thickness d(x) given by do( 1 - ½ x2/f), Lec 2 :25 Huygen’s Wavelets and Fourier Optics Goal: Use a “4f” optical system to explore two dimensional Fourier transforms.
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