Wavelength Crest. Amp litude. is the highest point of that portion of a transverse wave above the equilibrium position.

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1 Chapter Twenty-Four: Interference of Light interference patterns are prouce when two (or more) coherent sources prouce waves of the same frequency an amplitue which superimpose in the same region of space (ie they overlap). sources are coherent when there is a constant phase relationship between the waves emitte by the sources. Page 1of 15 Wavelength Crest +ve Amp litue Trough crest is the highest point of that portion of a transverse wave above the equilibrium position. trough is the lowest point of that portion of a transverse wave below the equilibrium position. estructive interference occurs if the amplitue of the resultant of two interfering waves is smaller than the amplitue of either wave. constructive interference occurs if the amplitue of the resultant of two interfering waves is larger than the amplitue of either wave. iffraction occurs when waves ben an sprea out as they pass an obstacle or through a narrow opening. The amount of iffraction epens on the wavelength of the waves relative to the size of the obstacle. In the case of a narrow opening, the amount of bening increases as the size of the opening ecreases. iffraction grating consists of a large number of closely space parallel slits which iffract light incient on the grating. The iffracte light will exhibit a pattern of light an ark spaces ue to constructive interference an estructive interference. polarization is a property of light that inicates that light is a transverse wave phenomenon. A transverse wave is a wave in which the oscillations or vibrations of the wave are at right angles to the irection of motion of the wave. An electromagnetic wave in which the electric vector is vibrating in only one plane is sai to be plane polarize. Displacement -ve

2 Spee, Frequency an Wavelength The back an forth vibratory motion (often calle oscillatory motion) of a swinging penulum or mass suspene on a spring prouces a sine curve. This curve epicts a wave: Page 2of 15 Wavelength Crest +ve Amp litue Trough Displacement -ve Displacements are measure from the mean position. The wave comprises of a series of crests an troughs. Wavelength (λ) is the istance from one crest to another crest, more generally the shortest istance between successive ientical parts of the wave that are in phase. λ is measure in metres. Amplitue (A) is the istance of the maximum isplacement point of the wave from the mean position. A is measure in metres. Frequency (f) is the number of to-an-fro vibrations in time. f is measure in cycles per secon or hertz. where Perio (T) is the time taken to complete one oscillation. 1 frequency perio Wave motion involves a transfer of energy from one point to another, with no net transfer of matter. Wave Equation v fλ where: v spee or velocity of the wave (in m s -1 ) f frequency in hertz (Hz) λ wavelength in metres (m). Transverse Waves A wave where the motion of the oscillations of the wave are at right angles (perpenicular) to the irection of travel. It consists of a series of crests an troughs. v

3 Longituinal Waves The meium moves back an forth parallel to the irection in which the wave travels. The wave consists of a series of compressions an rarefactions. Page 3of 15 C R R rarefaction C comp ression C R C R C R λ Particle Vibration Note: Transverse waves an longituinal waves transfer the energy along the meium. The matter oes not transfer but moves to an fro about a fixe position. This relationship hols for all kins of waves whether they are water waves, soun waves, raio waves, or light waves. The spee of light is a universal constant, c m s -1 in a vacuum. Note that a beam of light coul: travel aroun the Earth 7.5 times in 1 secon take 8 minutes to travel from the Sun to Earth take 4 years to reach the nearest star, Alpha Centauri. Electromagnetic Waves Light is energy that is emitte by accelerating electric charges often electrons in atoms. This energy travels in a wave that is partly electric an partly magnetic. Such a wave is an electromagnetic wave. Light is a small portion of the broa family of electromagnetic waves that inclue raio waves, microwaves an x-rays. The range of the electromagnetic waves is calle the electromagnetic spectrum. The range of wavelengths in the visible spectrum is shown below. (1 nm m)

4 Wave Fronts A wave front is a line on a surface on which all points are in phase at any instant. This means that when one point is at the crest, so are the others. Huygens Principle Every point on a wave front can be consiere as a source of tiny wavelets that sprea out in the forwar irection at the spee of the wave itself. The new wave front is the envelope (ie. the common tangent to) of all the wavelets. Note: This principle is useful in preicting the future position of a wave front when an earlier position is known. Suppose that AB is a wave front for waves create from a point source S (ie. the wave is travelling away from S at a spee v). In the wave front AB which is travelling away from a source S with a spee v. To fin the wave front after a short time t tiny circles, centre on wavefront AB, raius r vt represent the Huygens (imaginary) wavelets. The tangent to all these wavelets, the curve CD, is the new position of the wavefront. Diffraction Effect Huygens principle is particularly useful when waves impinge on an obstacle an the wavefronts are partially interrupte. Huygens principle preicts that waves ben in behin an obstacle (as water waves o). The bening of waves behin obstacles into the shaow-region is known as iffraction. The iffraction effect is not significant when the size of the opening or obstacle is much larger than the wavelength of the light. Page 4of 15

5 Interference A monochromatic light source emits a wave of a single colour an therefore has single frequency an wavelength in a vacuum(in air). e.g. Re light has wavelength of about m an hence a frequency of about Hz. Two sources are coherent if the waves create by them have a constant phase relationship. An interference pattern is observe only when the sources are coherent. If two tiny light bulbs are use as sources, an interference pattern woul not be seen. The light emitte by one light globe woul have a ranom phase with respect to the secon light bulb. Two such sources, whose output waves bear no fixe relationship to each other, are calle incoherent sources. Page 5of 15 Wave Aition Wave Aition is the superposition of wave forms. Crest + Crest + Trough + Trough + Crest + Trough + When waves overlap, interference is prouce. C o n stru ctiv e Interference Destructive Interference

6 Young s Double Slit Interference Convincing evience for the wave nature of light was provie by Thomas Young who was able to measure the wavelength of visible light using an interference effect. Page 6of 15 If light from a single monochromatic source falls onto a screen containing two closely space slits S 1 an S 2, bright an ark fringes are seen on the screen behin. Young realise that the bright fringes of light resulte from light waves from both holes (or slits) arriving crest to crest (constructive interference - more light). Similarly, the ark areas resulte from light waves arriving trough to crest (estructive interference - no light). Young s experiment can one in the laboratory with two closely space holes (or slits) with a soium vapour lamp, light from which is passe through a single slit proviing the monochromatic coherent source (a laser is even better because no first single slit is require only a ouble slit). The arrangement above is equivalent to two rocks being thrown into a lake, or when soun from two louspeakers interferes. These two slits have circular waves leaving them because of the iffraction effect. Using the iagrams below we can see how an interference pattern is prouce on the screen. (a) (b) The waves of wavelength λ are shown entering the slits S 1 an S 2 which are a istance apart. The waves sprea out in all irections after passing through the slits. The iagrams show only three ifferent angles. Diagram (a) shows that at the centre of the screen (θ0 ) the waves from each slit travel the same istance an are in phase. ie. a crest of one wave arrives at the same time as the crest of the other wave. Hence the amplitue of both waves a to form a larger amplitue an this is known as CONSTRUCTIVE INTERFERENCE an there is a bright spot in the centre of the screen.

7 Diagram (b) shows that at an angle θ the lower wave travels an extra istance of one whole wavelength, an the waves are in phase. Note that from the shae triangle the extra istance equals sinθ. At the point on the screen constructive interference occurs again since the paths of the two rays iffers by one wavelength (or any whole number of wavelengths). Diagram (c) shows that at this angle of θ the lower wave travels an extra istance equal to one-half wavelength, so that the two waves arrive at the screen fully out of phase. At this point on the scree since the extra istance travelle is one half wavelength (or 3 / 2 λ, 5 / 2 λ an so on) the two waves are exactly out of phase when they reach the screen: the crest of one wave arrives at the same time as the trough of the other an so prouce zero amplitue. This is DESTRUCTIVE INTERFERENCE an the screen is ark. The pattern on the screen is a series of bright an ark lines (or fringes) as shown below. Young s Double Slit Interferometer The coherent sources prouce a pattern of uniformly space bright (constructive interference) an ark (estructive interference) fringes. The point O is a maximum intensity or reinforcement since the wavelets from the sources S 1 an S 2 meet at O at the same time. The waves travelling from S 1 an S 2 to the screen will have a path ifference. If the path ifference (the extra istance travelle by one wavelet to reach the same point on the screen) is: an o number of half wavelengths then DESTRUCTIVE INTERFERENCE an even number of half wavelengths then CONSTRUCTIVE INTERFERENCE. A constant phase relationship between the sources is a necessary conition for observable interference to occur, they are COHERENT (implies same λ). To achieve maximum contrast the amplitues must be equal. Page 7of 15

8 Single Slit Double Slit SCREEN P x Page 8of 15 S S 1 S 2 M θ C θ L O A single source S prouces wavelets that meet S 1 an S 2 at the same time (since SS 1 SS 2 ). Therefore the sources S 1 an S 2 have a constant same phase relationship (in fact the same phase) so they are coherent. From the iagram below S C sin θ (path ifference) 2 sinθ tanθ x L PD S If consier a position of constructive interference then 2 C x PD S2C whole number of λ L nλ (where n 0,1,2,3,4,...) For a particular constructive ban (fringe) The path ifference for the next ban (fringe) is x1 S 2C nλ L S2 C x 2 L x L ( n + 1) (since θ small) λ

9 Rearranging these two equations so that x is the subject of the formulae gives: L x n λ 1 x 2 W x x1 ( n + 1) λl n + 1 λl nλl ( ) 2 Page 9of 15 W λl Note: W is the fringe separation or ban with. since λ is small we nee L to be large an small to see W. W is proportional to λ if L an are constant. W RED > W VIOLET since λ RED >λ VIOLET. Above is a iagram of the intensity pattern of interference fringes prouce by the ouble slit experiment. The arrow marks the central fringe. No. 7 in problem set: 0.1 mm L 50 cm λ I W x λ R 700nm W R 3.5cm λ B 450nm W B 2.25cm λ L W λl W 22 5mm B. λl W R 35.mm

10 Diffraction Gratings Transmission Diffraction Gratings We have iscusse iffraction before with regar to water waves as well as light an we now have seen that it refers to the spreaing or bening of light aroun eges. Diffraction patterns of (a) a penny, (b) a razor blae an (c) a slit, each illuminate by a (nearly) point source of monochromatic light. Page 10of 15 A iffraction pattern exists aroun any sharp object illuminate by a point source. This resembles the interference fringes of a ouble slit an is ue to the waves being iffracte aroun the object. Diffraction Grating A large number of equally space parallel slits is calle a iffraction grating, although it may be more appropriate to use the term interference grating. The grating can be mae by precision machinery by scratching very fine parallel lines on a glass plate. The untouche spaces between the lines serve as the slits. Photographic transparencies of the original grating serve as an inexpensive grating. Transmission Grating A transmission grating is a glass slab on which a large number of lines have been scratche so that the spacings of scratches is very small (eg. 10,000 scratches per cm). The light is passe through the transparent part thus supplying a many slit source. Reflection Grating A reflection iffraction grating is a mirror surface on which a large number of lines have been scratche. The light is reflecte from the smooth reflecting surface, eg. compact isc. Both operate similarly except that the reflection grating is preferre as less energy from the incient light is absorbe. These are also the most expensive. Reflection gratings are also useful as ultra-violet an infrare light are absorbe by glass.

11 The analysis of the iffraction grating is much like that of Young s ouble-slit experiment. We assume parallel rays of light are incient on the grating. Since this light is parallel there is a nee for it to be focusse by a convex lense or even the lense in our eye. Page 11of 15 We also assume that the slits are narrow enough so that iffraction by each of them spreas light over a very wie angle on a screen an interference can occur with light from all other slits. Light rays that pass through each slit without eviation (θ0 ) interfere constructively to prouce a bright line at the centre of the screen. Constructive interference also occurs at an angle θ such that the rays from ajacent slits travel an extra istance of l mλ, where m is an integer. For constructive interference sinθ l mλ (m 0, 1, 2,...) These are the principal maxima an this is the criterion to have a maximum brightness. Note previously in interference: S 2 C path ifference sin θ m is the orer of the pattern eg m 1 sinθmλλ m 2 sinθmλ2λ etc.

12 When white light is passe through the grating, each wavelength prouces constructive interference at a ifferent angle an so a series of spectra are prouce. Page 12of 15 (a) a line spectrum with two wavelengths of light, 400 nm an 700 nm. (b) Diffraction prouce by white light. the shorter wavelengths (blue) are iffracte through smaller angles than the longer wavelengths (re), an the 1 st orer spectrum will show less ispersion (sprea) than the 2 n orer. the intensity of the 1 st orer spectrum is greater than the intensity of the 2 n orer because of iffraction causing greater ispersion. Diffraction Grating: Lines Calculate the first an secon orer angles for light of wavelength 400 nm an 700 nm if the grating contains 10,000 line/cm. The grating contain 10 4 lines/cm10 6 lines/m, which means the separation between slits is (1/10 6 )m m. In first orer (m1), the angles are: -7 m λ sin θ m sin θ so In sin sin so θ λ orers secon θ θ θ nm will an orer m λ f 1 because appear. θ but the m secon sin θ cannot orer oes excee 1. not No exist higher for

13 Spectra Overlap White light containing wavelengths from 400 nm to 750 nm strikes a grating containing 4000 lines/cm. Show that the blue at λ450 nm of the thir orer spectrum overlaps the re at 700 nm of the secon orer. The grating spacing is (1/4000) cm m. The blue of the thir orer occurs at an angle θ given by: -7 m λ sin θ m Page 13of 15 Re in secon orer occurs at: mλ sin θ m which is a greater angle; so the secon orer overlaps into the beginning of the thir-orer spectrum. Activities 1. Consier Summary (pg. 679) 2. Questions (pg. 680: no.1, 2, 3, 11) 3. Problems (pg. 681: no 10, 11, 12, 13, 14, 15, 16, 17) 4. Look over Summary (pg. 751) 5. Look over Problem Solving: Interference (pg. 750) 6. Sample Questions 7. Questions (pg. 752: no.1, 2, 4, 5, 6, 9) 8. Problems (pg 752-3: no 2, 3, 4, 5) 9. Questions (pg. 752: no.13, 14) 10. Problems (pg 752-3: no.18, 19, 26, 27, 28, 29, 30, 31, 32.)

14 SPECKLE Speckle is prouce whenever a laser beam is reflecte from a rough surface. Each point on the surface reflects light in a ifferent irection, as shown in the iagram below. The light, which reaches each observer, will therefore have travelle through many ifferent paths. Because the surface is rough, the angle of incience an hence that of reflection of the coherent light "rays" in the incient beam will vary. At one location (P for example) the interference may be mostly constructive, epening on the path ifference of the rays, an prouce a bright speckle. The interference at P' may be mostly estructive an thus prouce a ark speckle. Page 14of 15 Note that even the iffraction (interference) patterns ue to ouble slits an gratings can be prouce using a Na lamp (for example), speckle interference patterns can only be prouce using a laser. Why? Because the rays which interfere at each observation, travel along paths which are significantly ifferent in length. The roughness of the surface shoul be comparable with (or greater than) the wavelength of the laser light. Wavelets coming (reflecte) from ifferent sections of the rough surface will overlap to give an interference pattern that will be seen on a screen as an irregular array of ot-like "speckles". Whenever the screen is place, the speckles will be seen, as the interference effects occurring within the total region in which the reflecte wavelets overlap (this assumes the knowlege of Huygen's wavelets)

15 Demonstration of Speckle Pass a Helium-Neon (He-Ne) laser beam through a converging lens. The laser beam will expan after the focus is prouce by the lens. Choose the focal length so that iameter of the beam on the rough surface is 5-10 cm - if the surface is about 4 m from the lens, then the focal length shoul be mm. Most surfaces are suitable - even a black one! Speckle shoul be evient in the reflection. Depening on the surface an the power of your (He-Ne) laser, you may nee to either leave the room lights on to prevent being azzle, or to im the room lights. Page 15of 15 A simple emonstration of speckle can be achieve as escribe below. Speckle pattern on the screen Short focal length lens (to sprea laser light) wavelets Generation of laser speckle pattern by scattering from a rough surface Light from a laser (He-Ne, 1mW) light is brought to a focus by by the lens (then it spreas as shown) NEVER LOOK DIRECTLY INTO A LASER NOR THAT REFLECTED FROM A MIRROR View the screen, from a istance with one eye covere, while you are still. A stuent shoul be able to change the nature of the surface, looking with both eyes, moving the screen (both from laser light itself an between screen an eye) an escribe an try to explain these phenomena. Looking at the pattern through a ark screen with a small hole might change things. Does it? See if you can iscover why. It is also worthwhile looking at the surface (with one eye only) through small holes of varying size in a black screen from a carboar or thick paper. Does the pattern change. See if you can see any change in the resolution of the pattern as the hole size changes. (Resolution means the ability to istinguish fine etail in a "picture")