FabryPerot Interferometer Bright rings 2 n Lcosq ml
Fabry-Perot Interferometer Bright rings 2 n. Lcosq = ml; m = 1, 2, 3. . .
Oblique Incidence on a Fabry-Perot Cavity Assume external reflection within the cavity. The point P on wave A propagates to the right reflector at Q where it is reflected. At Q, the wave experiences a phase change p. Then it propagates to R, and gets reflected again and experiences another p phase change. Right after reflection, the phase at R must be the same as that at the start point P; R and P are on the same wavefront. Df = Phase difference from P to Q to R = m(2 p)
Diffraction, Chapter 36 What is Diffraction: the process by which a beam of light or other system of waves is spread out as a result of passing through a narrow aperture or across an edge, typically accompanied by interference between the wave forms produced. A picture helps Diffraction of water waves
Diffraction from a single slit (Physical Optics) • In the Figure below, the prediction of geometric optics in (a) does not occur. Instead, a diffraction pattern is produced, as in (b). Geometric optics Physical Optics • The narrower the slit, the broader the diffraction pattern. • Light is a wave (EM waves) and hence diffracts. • Diffraction phenomena also occur for EM waves, sound, electrons (QM)
Diffraction What physical parameters control whether single slit diffraction occurs ? Ans: λ (wavelength), a (slit spacing). But this is not the full story. Ans: 2 nd part, distance to the screen (near field, Fresnel diffraction) and (far field, Fraunhofer diffraction)
Diffraction • According to geometric optics, a light source shining on an object in front of a screen should cast a sharp shadow. Surprisingly, this does not occur because of diffraction.
Diffraction and Huygen’s Principle • What is Huygens’s principle ? The Huygens-Fresnel principle states that every point on a wavefront is a source of wavelets. These wavelets spread out in the forward direction, at the same speed as the source wave. The new wavefront is a line tangent to all of the wavelets. So what is interfering in single slit diffraction ?
Fresnel and Fraunhofer diffraction by a single slit • The figure below shows Fresnel (near-field) and Frauenhofer (far-field) diffraction for a single slit.
Locating the dark fringes • Review the single-slit diffraction discussion in the text. • The figure below shows the geometry for single slit Fraunhofer diffraction. (Imagine the aperture is made of many tiny slits). What is the path difference between the two strips ? Ans: path difference =a/2 sin(θ) If path difference to P is λ/2, what do we see at Point P ? Dark fringe
Locating the dark fringes in single slit diffraction In general, at what angles do we find the dark fringes ? Ans: If path difference to P is mλ/2, we will find dark fringes Dark fringes
Locating the dark fringes in the single slit diffraction pattern
Examples of single-slit diffraction • The figure on the left is a photograph of a Fraunhofer pattern of a single horizontal slit. What features are especially notable ? • Example 36. 1: You pass 633 -nm light through a narrow slit and observe the diffraction pattern on a screen 6. 0 m away. The distance at the screen between the center and the first minima on either side is 32 mm long. How wide is the slit?
Quantitative Intensity in the single-slit diffraction pattern Intensity of a single slit diffraction pattern
Width of the single-slit pattern • The width of the single-slit diffraction pattern depends on the ratio of the slit width a to the wavelength λ.
Single-slit Diffraction So far in the multiple-slit interference problems we have assumed that each slit is a point source. Point sources radiate equally in all directions. Real slits have a non-zero extent – - a “slit width” a. The transmission pattern depends on the ratio of a to l. In general, the smaller the slit width, the more the wave will diffract. Small slit: Diffraction profile I 1 screen Large slit: Diffraction profile I 1 screen
Single-Slit Diffraction Slit of width a. Where are the minima? Use Huygens’ principle: treat each point across the opening of the slit as a wave source. The first minimum is at an angle such that the light from the top and the middle of the slit destructively interfere. This works, because for every point in the top half, there is a corresponding point in the bottom half that cancels it. a/2 P Incident Wave (wavelength l) a y d L The second minimum is at an angle such that the light from the top and a point at a/4 destructively interfere: q d Location of nth-minimum:
ACT 1 a 1 cm = W 2 m Which of the following would broaden the diffraction peak? a. reduce the laser wavelength b. reduce the slit width c. move the screen further away d. a. and b. e. b. and c.
ACT 1 - Solution a 1 cm = W 2 m Which of the following would broaden the diffraction peak? a. reduce the laser wavelength b. reduce the slit width c. move the screen further away d. a. and b. e. b. and c.
Multiple Slit Interference What changes if we increase the number of slits, e. g. , N = 3, 4, 1000, . . . (for now we’ll go back to very small slits, so neglect diffraction from each of them) S 3 d The positions of the principal interference maxima are the same for any number of slits! d sinq = ml S 2 d 2 d For equally spaced slits: If slit 1 and 2 are in phase with each other, than slit 3 will also be in phase, etc. We will almost always consider equally spaced slits. y we can First look at the principal maxima. P S 1 L
N-Slit Interference The Intensity for N equally spaced slits is given by: * y Derivation (using phasors) is in the supplementary slides. As usual, to determine the pattern at the screen, we need to relate f to q or y = L tanq: q d f is the phase difference between adjacent slits. You will not be able to use the small angle approximations unless d >> l. * Your calculator can probably graph this. Give it a try. L
Multiple Slits (1) Double slit pattern The fringes of the double slit pattern fade away from centre and disappear at the single slit minimum. Three-slit pattern There is a subsidiary maximum between the double slit maxima. The fringes become narrower and sharper. http: //www. matter. org. uk/schools/Content/Interference/gratings. html
Multiple Slits (2) • The fringes become sharper as the number of slits is increased. • The subsidiary maxima become less and less significant as the number of slits is increased. http: //www. matter. org. uk/schools/Content/Interference/grating. Explored. html
Diffraction Grating • A large number of equally spaced parallel slits is called a diffraction grating. • A diffraction grating can be thought of as an optical component that has tiny grooves cut into it. The grooves are cut so small that their measurements approach the wave length of light.
Diffraction Gratings • A diffraction grating splits a plane wave into a number of subsidiary waves which can be brought together to form an interference pattern.
Action of Diffraction Grating X θ θ Y d θ Path difference = d sin θ • If d is the slit spacing then the path difference between the light rays X and Y = d sin θ. • For principal maxima, d sin θ = nλ. • The closer the slits, the more widely spaced are the diffracted beams. • The longer the wavelength of light used, the more widely spaced are the diffracted beams.
Number of Diffraction beams • Since sin θ 1, n=2 θ 1 θ 2 n=1 n=0 n=1 The highest order number is given by the value of d/λ rounded down to the nearest whole number.
Using a diffraction grating to measure the wavelength of light • A spectrometer is a device to measure wavelengths of light accurately using diffraction grating to separate. Turntable Collimator C Diffraction grating θ Light source Achromatic lenses Telescope T Eyepiece Cross-wire Eye
View through Diffraction Grating • Diffraction grating placed in front of a methane air flame § Spectrum of a star - Procyon
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