College Physics 150. Chapter 25 Interference and Diffraction

Transcription

1 College Physics 50 Chapter 5 Interference and Diffraction Constructive and Destructive Interference The Michelson Interferometer Thin Films Young s Double Slit Experiment Gratings Diffraction Resolution of Optical Instruments

2 Constructive and Destructive Interference Two waves are coherent if they maintain a fixed phase relationship (waves from the same source). Two waves are incoherent otherwise (waves from different sources). Constructive interference occurs when two waves are in phase. To be in phase, the points on the wave must have Δφ(π)m, where m is an integer. When coherent waves are in phase, the resulting amplitude is just the sum of the individual amplitudes. The energy content of a wave depends on A. Thus, I A.

3 Constructive interference occurs when two waves are in phase. To be in phase, the points on the wave must have Δφ(π)m, where m is an integer. When coherent waves are in phase, the resulting amplitude is just the sum of the individual amplitudes. The energy content of a wave depends on A. Thus, I A. The resulting amplitude and intensity are: A I I A + + I A + I I

4 Destructive interference occurs when two waves are a half cycle out of phase. To be out of phase the points on the wave must have Δφ(π)(m+½), where m is an integer. The resulting amplitude and intensity are: A I I A + I A I I

5 Coherent waves can become out of phase if they travel different distances to the point of observation. P S d θ Δl d sinθ S This represents the extra path length (Δl) that the wave from S must travel to reach point P. When both waves travel in the same medium the interference conditions are: For constructive interference where m an integer. Δl mλ For destructive interference Δl m + λ where m an integer.

6 Example (text problem 5.): A 60.0 khz transmi]er sends an EM wave to a receiver km away. The signal also travels to the receiver by another path where it reflects from a helicopter. Assume that there is a 80 phase shift when the wave is reflected. (a) What is the wavelength of this EM wave? λ f c km/sec Hz 5.0 km (b) Will this situation give constructive interference, destructive inference, or something in between? The path length difference is Δl 0 km λ, a whole number of wavelengths. Since there is also a 80 phase shift there will be destructive interference.

7 Michelson Interferometer In the Michelson interferometer, a beam of coherent light is incident on a beam spli]er. Half of the light is transmi]ed to mirror M and half is reflected to mirror M. The beams of light are reflected by the mirrors, combined together, and observed on the screen. If the arms are of different lengths, a phase difference between the beams can be introduced.

8 Example (text problem 5.): A Michelson interferometer is adjusted so that a bright fringe appears on the screen. As one of the mirrors is moved 5.8 µm, 9 bright fringes are counted on the screen. What is the wavelength of the light used in the interferometer? Moving the mirror a distance d introduces a path length difference of d. The number of bright fringes (N) corresponds to the number of wavelengths in the extra path length. Nλ d λ d N 0.56µ m

9 Thin Films When an incident light ray reflects from a boundary with a higher index of refraction, the reflected wave is inverted (a 80 phase shift is introduced). A light ray can be reflected many times within a medium.

10 Example (text problem 5.7): A thin film of oil (n.50) of thickness 0.40 µm is spread over a puddle of water (n.33). For which wavelength in the visible spectrum do you expect constructive interference for reflection at normal incidence? Water Oil Air Consider the first two reflected rays. r is from the air- oil boundary and r is from the oil- water boundary. Incident wave r has a 80 phase shift (n oil >n air ), but r does not (n oil <n water ).

11 To get constructive interference, the reflected waves must be in phase. For this situation, this means that the wave that travels in oil must travel an extra path equal to multiples of half the wavelength of light in oil. The extra path distance traveled is d, where d is the thickness of the film. The condition for constructive interference here is: d λ air m + dnoil m + λ oil m + Make a table: m λ air (µm) λ n air oil Only the wavelengths that satisfy this condition will have constructive interference All of these wavelengths will show constructive interference, but it is only this one that is in the visible portion of the spectrum.

12 Young s Double- Slit Experiment Place a source of coherent light behind a mask that has two vertical slits cut into it. The slits are L tall, their centers are separated by d, and their widths are a.

13 The slits become sources of waves that, as they travel outward, can interfere with each other.

14 The pa]ern seen on the screen There are alternating bright/ dark spots. An intensity trace The bright spots occur where there is constructive interference: The dark spots occur where there is destructive interference: Δl d sinθ mλ where m is an integer and is called the order. Δl d sinθ m + λ

15 Example (text problem 5.8): Show that the interference fringes in a double- slit experiment are equally spaced on a distant screen near the center of the interference pa]ern. The condition for constructive interference is: From the geometry of the problem, tanθ h D Δl sinθ d sinθ mλ d mλ The screen is far away compared to the distance between the slits (D>>d) so tanθ sinθ θ. Here, sinθ θ mλ h d D mλd h d mλ and d tanθ θ The distance between two adjacent maxima is: h h D λd h d ( m m ) λd d

16 Gratings A grating has a large number of evenly spaced, parallel slits cut into it.

17 Example (text problem 5.38): Red light with λ 650 nm can be seen in three orders in a particular grating. About how many rulings per cm does this grating have? d sinθ For each of the maxima 0 d sinθ d sinθ 3 d sinθ 4 0 d sinθ λ λ 3λ 4λ Third order is observed. d sinθ mλ This order is not observed. Since the m 4 case is not observed, it must be that sinθ 4 >. We can then assume that θ This gives 6 and N d d 3λ ,000 lines/m m 500 lines/cm.

18 Diffraction Using Huygens s principle: every point on a wave front is a source of wavelets; light will spread out when it passes through a narrow slit. Diffraction is appreciable only when the slit width is nearly the same size or smaller than the wavelength.

19 The intensity pa]ern on the screen.

20 The minima occur when: a sinθ mλ where m ±, ±,

21 Resolution of Optical Instruments The effect of diffraction is to spread light out. When viewing two distant objects, it is possible that their light is spread out to where the images of each object overlap. The objects become indistinguishable. For a circular aperture, the Rayleigh criterion is: asin Δθ. λ Δθ where a is the aperture size of your instrument, λ is the wavelength of light used to make the observation, and Δθ is the angular separation between the two observed bodies. asinδθ.λ To resolve a pair of objects, the angular separation between them must be greater than the value of Δθ.

22 Example (text problem 5.55): The radio telescope at Arecibo, Puerto Rico, has a reflecting spherical bowl of 305 m diameter. Radio signals can be received and emi]ed at various frequencies at the focal point of the reflecting bowl. At a frequency of 300 MHz, what is the angle between two stars that can barely be resolved? asinδθ.λ sinδθ.λ a sinδθ radians Δθ 0.3 degrees (.) ( m/s ) Hz 305 m