LECTURE 14 PHASORS & GRATINGS. Instructor: Kazumi Tolich

Transcription

1 LECTURE 14 PHASORS & GRATINGS Instructor: Kazumi Tolich

2 Lecture 14 2 Reading chapter 33-5 & 33-8 Phasors n Addition of two harmonic waves n Interference pattern from multiple sources n Single slit diffraction pattern n Interference-diffraction pattern Gratings n Diffraction gratings n Spectroscope n Interferometer

3 Addition of two harmonic waves and phosors 3 Addition of two waves with different phases can be graphically expressed with phasors. Each wave function is represented by the y component of a phasor. E " = A " sin α, where α = ωt. E + = A + sin α + δ E " + E + = A sin α + δ.

4 Three slit interference 4 Assume the rays hitting the screen are parallel. Interference patters appear due to the path length differences of the rays. The wave at a point on the screen, P, is the sum of the three waves. E " = A / sin α E + = A / sin α + δ E 0 = A / sin α + 2δ δ = 2π

5 5 Quiz: 1

6 Multi-slit interference patterns 6 The principal maxima occur at the same θ as for two slit interference. The more slits there are, the brighter and narrower the principal maxima. The first minimum occurs at δ = 360 N (N phasors form a closed polygon of N sides). There are N - 2 secondary maxima between each pair of principal maxima. There are N - 1 zeros between each pair of principal maxima. 5 I N /I θ (radians)

7 Diffraction patterns 7 The slit of width a is divided into N (N >> 1) equal intervals. The rays from the sources to a point P on the far screen are parallel. The diffraction pattern arises from the path length difference between any two adjacent sources resulting in the phase difference given by d sin θ δ = 2π λ Let A 0 denote the amplitude due to a single source, then d E B = A / sin α + nδ To screen at point P

8 Diffraction pattern and phasors 8 Intensity I 0

9 Interference-diffraction pattern 9 Interference only: I = 4I / cos + H +I , where δ = + 8 I / is the intensity from one slit. Diffraction only: Interference only a d a +, where φ = +IL J + I = I / J + I / is the intensity at the center. Combined: + sin φ 2 I = 4I / cos φ + δ 2 2 I / is the intensity at the center from one slit. 8 Diffraction only Both q (degrees) q (degrees) q (degrees)

10 Demo 1 10 Multiple Slit Interference/Diffraction

11 Example 1 11 Light that has a wavelength equal to 550 nm illuminates two slits that both have widths equal to mm and separations equal to 0.18 mm. This creates 11 fringes in the central diffraction maximum. What is the ratio of the intensity of the third interference maximum to the intensity of the center interference maximum? I max I 3

12 Diffraction gratings 12 A diffraction grating consists of a large number (thousands per mm) of equally spaced lines or slits on a flat surface. Reflection grating: light is reflected from the ridges between the lines or grooves. Transmission grating: the light passes through the gaps between the rulings. Diffraction gratings are used to disperse light into separate colors.

13 Transmission grating 13 The interference maxima are given by d sin θ M = mλ, m = 0,1,2, y M = L tan θ M The intensity of central maximum is N + I /, where N 1 is the number of slits illuminated, and I / is the intensity due to one slit. The intensity is zero when the path-length difference for the light from the first slit and that from the Nth slit (~Nd) is λ. θ X56 sin θ X56 = λ Nd

14 14 Quiz: 2

15 Example 2 15 Light from a sodium lamp passes through a diffraction grating having 1000 slits per millimeter. The interference pattern is viewed on a screen m behind the grating. Two bright yellow fringes are visible at distances of cm and cm from the central maximum. Assuming that m = 1, what are the wavelengths of these two fringes?

16 Quiz: 3

17 Demo 2 17 Transmission Gratings White light and two lasers (one red, one green) are passed through four diffraction gratings of various line densities.

18 Grating spectroscope 18 Atoms in a low pressure electrical discharge produce light with characteristic wavelengths. A grating spectroscope provides a way of making precise measurements of wavelength by noting the angular positions at which interference maxima occur. Light from the source passes through a narrow collimating slit and is made parallel by a lens. Parallel light from the lens is incident on the grating. The parallel light from the grating is focused by a telescope and viewed by the eye. By measuring the angle of the interference maximum, θ, you can measure the wavelength. d sinθ M = mλ, m = 0,1,2,

19 Spectral lines 19 Each wavelength emitted by the source produces a separate spectral line (the image of the collimating slit). The set of lines corresponding to m = 1 is called the first-order spectrum. The set of lines corresponding to m = 2 is called the second-order spectrum. d sin θ M = mλ, m = 0,1,2, Light with λ 1 = 400 nm and λ 2 = 700 nm.

20 Resolving power 20 If two lines from two wavelengths do not overlap, these wavelengths are resolved. Let Δλ denote the smallest observable difference between two nearby wavelengths. The resolving power of a diffraction grating, R, is defined to be λ/δλ. R λ λ = mn m = order of spectrum N = number of slits illuminated

21 Quiz: 4

22 Example 3 22 Mercury has several stable isotopes, among them 198 Hg and 202 Hg. The strong spectral line of mercury, at about nm, is a composite of spectral lines from the various mercury isotopes. The wavelengths of the line for 198 Hg and 202 Hg are nm and nm, respectively. a) What must be the resolving power of a grating capable of resolving the two isotopic lines? b) If the grating is illuminated over a 2.00-cmwide region, what must be the number of lines per centimeter of the grating to resolve the isotopic lines in the third-order spectrum?

23 Reflection gratings 23 Ruling parallel grooves in a mirror-like metal surface produces a reflection grating. The angles of the grooves can be chosen so that light of particular order is reflected in a particular direction, enhancing the intensity of that order. This is called blazing. Some of the most colorful bird feathers, butterfly wings, insect shells, etc are, in effect, blazed reflection gratings.

24 24 Radio telescope arrays A line of parabolic dish radio receivers behaves like a diffraction grating in reverse, giving high-precision locations of radio sources in the sky. Radio signals from distant galaxies add constructively when the distance between two adjacent telescopes, d, satisfies d sin θ M = mλ, m = 0,1,2, Saturn Su n Mars

25 Michelson interferometer 25 An interferometer is used to measure change in lengths with great accuracy. Change in the path length difference between A and B can be determined by the change in the interference pattern.

26 Laser Interferometer Gravitational-wave Observatory (LIGO) 26 Interferometer is used in detection of gravitational-wave induced motion, predicted by Einstein as part of the theory of general relativity. Each arm is 4 km. Hanford, WA