Chapter 27: Wave Optics PDF

CHAPTER 27: WAVE OPTICS Mrs. Pooja Brahmaiahchari Introduction Interesting phenomena, such as the dispersion of white light into a rainbow of colors when passed through a narrow slit, cannot be explained fully by geometric optics. In these cases, light interacts with small...

CHAPTER 27: WAVE OPTICS Mrs. Pooja Brahmaiahchari Introduction Interesting phenomena, such as the dispersion of white light into a rainbow of colors when passed through a narrow slit, cannot be explained fully by geometric optics. In these cases, light interacts with small objects and exhibits its wave characteristics. The branch of optics that considers the behavior of light when it exhibits wave characteristics called as wave optics. Wave optics is essentially the way light interactis with obects, it is being treated as a wave not a ray The Wave Aspect of Light: Interference We know that visible light is the type of electromagnetic wave to which our eyes respond. Like all other electromagnetic waves, it obeys the equation 𝑐 = 𝑓𝜆 where c is the speed of light in vacuum, f is the frequency of the electromagnetic waves, and λ is its wavelength. The range of visible wavelengths is approximately 380 to 760 nm. As is true for all waves, light travels in straight lines and acts like a ray when it interacts with objects several times as large as its wavelength.  Interference is the hallmark of a wave; both the ray and wave characteristics of light can be seen. The laser beam emitted by the observatory epitomizes a ray, traveling in a straight line. When light goes from a vacuum to some medium, like water, its speed and wavelength change, but its frequency remains the same. The speed of light in a medium is v = c/n, where n is its index of refraction.  If we divide both sides of equation c =fλ by n we get c/n = v= fλ/n. This implies that 𝑣 = 𝑓𝜆𝑛 where 𝜆𝑛 is the wavelength in a medium and that, 𝜆 𝜆𝑛 = 𝑛 where λ is the wavelength in vacuum and n is the medium’s index of refraction. Therefore, the wavelength of light is smaller in any medium than it is in vacuum. Huygens's Principle: Diffraction A transverse wave, such as an electromagnetic wave like light, as viewed from above and from the side. The direction of propagation is perpendicular to the wavefronts (or wave crests) and is represented by an arrow like a ray.  The Dutch scientist Christian Huygens (1629–1695) developed a useful technique for determining in detail how and where waves propagate. Know huygens principle Starting from some known position, Huygens’s principle states that: Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets. Each wavelet in the figure was emitted when the wavefront crossed the interface between the media.  Since the speed of light is smaller in the second medium, the waves do not travel as far in a given time, and the new wavefront changes direction as shown. This explains why a ray changes direction to become closer to the perpendicular when light slows down.  (a) Light passing through a doorway makes a sharp outline on the floor. Since light’s wavelength is very small compared with the size of the door, it acts like a ray. (b) Sound waves bend into all parts of the room, a wave effect, because their wavelength is similar to the size of the door.The wavelength of the wave must be smaller than the width of the door If we pass light through smaller openings, often called slits, we can use Huygens’s principle to see that light bends as sound does (see Figure). The bending of a wave around the edges of an opening or an obstacle is called diffraction. Diffraction is a wave characteristic and occurs for all types of waves. If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a wave.  Young’s Double Slit Experiment The acceptance of the wave character of light came many years later when, in 1801, the English physicist and physician Thomas Young (1773– 1829) did his now-classic double slit experiment. Young first passed light from a single source (the Sun) through a single slit to make the light somewhat coherent. coherent waves are used when interference patterns are needed By coherent, we mean waves are in phase or have a definite phase relationship. know these definitions Incoherent means the waves have random phase relationships.  Why did Young then pass the light through a double slit? The answer to this question is that two slits provide two coherent light sources that then interfere constructively or destructively. Q, what type of wave do you use coherent or incohenrent When light passes through narrow slits, it is diffracted into semicircular waves, as shown in Figure (a). Pure constructive interference occurs where the waves are crest to crest or trough to trough. Pure destructive interference occurs where they are crest to trough. The light must fall on a screen and be scattered into our eyes for us to see the pattern.  An analogous pattern for water waves is shown in Figure(b). Note that regions of constructive and destructive interference move out from the slits at well-defined angles to the original beam. These angles depend on wavelength and the distance between the slits.  Double slits produce two coherent sources of waves that interfere. (a) Light spreads out (diffracts) from each slit, because the slits are narrow. These waves overlap and interfere constructively (bright lines) and destructively (dark regions). We can only see this if the light falls onto a screen and is scattered into our eyes. The dark parts is the pure destructive interferance where the crests and troughs align, the light parts is the pure constructive interferance where the crest and crest align (b) Double slit interference pattern for water waves are nearly identical to that for light. Wave action is greatest in regions of constructive interference and least in regions of destructive interference. (c) When light that has passed through double slits falls on a screen, we see a pattern such as this.  (a) Destructive interference occurs here, because one path is a half wavelength longer than the other. The waves start in phase but arrive out of phase. (b) Constructive interference occurs here because one path is a whole wavelength longer than the other. The waves start out and arrive in phase.  To obtain constructive interference for a double slit, the path length difference must be an integral multiple of the wavelength 𝑑 sin 𝜃 = 𝑚𝜆, 𝑓𝑜𝑟 𝑚 = 0,1, −1, 2, −2 …. 𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑣𝑒 Similarly, to obtain destructive interference for a double slit, the path length difference must be a half-integral multiple of the wavelength, or 1 𝑑 sin 𝜃 = 𝑚 + 𝜆, 𝑓𝑜𝑟 𝑚 = 0,1, −1, 2, −2 …. (𝑑𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑣𝑒) 2 Where λ is the wavelength of the light, d is the distance between slits, and θ is the angle from the original direction of the beam. We call m the order of the interference. For example, m =4 is fourth- order interference. Be able to identify which order of interfeance it would be.  The equations for double slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes, illustrated in Figure below. The intensity of the bright fringes falls off on either side, being brightest at the center. The closer the slits are, the more is the spreading of the bright fringes. Need to be able to know what would happen depending on if you change the distance between the slits increasing the slit separation in the double-slit experiment results in a closer spacing of fringes, creating a denser interference pattern decreasing the slit separation results in a wider and more spread-out interference pattern, with increased fringe spacing  We can see this by examining the equation. 𝑑 sin 𝜃 = 𝑚𝜆, 𝑓𝑜𝑟 𝑚 = 0,1, −1, 2, −2 For fixed λ and m, the smaller d is, the larger θ must be, since sin θ = 𝑚𝜆Τ𝑑 This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance d apart) is small. Small d gives large θ,hence a large effect. 1. At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm? 2. Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm. 3. What is the wavelength of light falling on double slits separated by 2.00µm if the third-order maximum is at an angle of 60o? Multiple Slit Diffraction An interesting thing happens if you pass light through a large number of evenly spaced parallel slits, called a diffraction grating. A diffraction grating can be manufactured by scratching glass with a sharp tool in a number of precisely positioned parallel lines, with the untouched regions acting like slits. These can be photographically mass produced rather cheaply. Diffraction gratings work both for transmission of light, as in Figure, and for reflection of light as on butterfly wings and the Australian opal.  A diffraction grating is a large number of evenly spaced parallel slits. (a) Light passing through is diffracted in a pattern similar to a double slit, with bright regions at various angles. (b) The pattern obtained for white light incident on a grating. The central maximum is white, and the higher-order maxima disperse white light into a rainbow of colors.  Idealized graphs of the intensity of light passing through a double slit (a) and a diffraction grating (b) for monochromatic light. Maxima can be produced at the same angles, but those for the diffraction grating are narrower and hence sharper. The maxima become narrower and the regions between darker as the number of slits is increased. Know the key difference that with diffraction grating the maxima are more narrow and more extreme  Need to know the natural occurance of diffraction gratings and the applications of them Natural diffraction gratings occur in the feathers of certain birds. Tiny, finger-like structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colors not solely due to their pigmentation. This is called iridescence. Where are diffraction gratings used? Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples.  Know applications A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected frequency of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting specific wavelengths for such use.  Know what happens if slit gets wider/ narrower Single Slit Diffraction a wider slit reduces the spread of the diffraction pattern, while a narrower slit increases it, If wider, central max is more extreme/narrower and there is less diffraction, less intense and closer fringe if narrower, central max is less extreme and there is more diffraction, more intense and further fringe Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings. (a) Single slit diffraction pattern. Monochromatic light passing through a single slit has a central maximum and many smaller and dimmer maxima on either side. The central maximum is six times higher than shown. (b) The drawing shows the bright central maximum and dimmer and thinner maxima on either side. 4. At what angle is the first minimum for 550-nm light falling on a single slit of width 1.00µm? 5. Calculate the angle at which a 2.00µm -wide slit produces its first minimum for 410-nm violet light. 6. White light containing wavelengths from 400nm to 750 nm strikes a grating containing 4000 lines /cm. Show that the blue at λ =450 nm of the third order spectrum overlaps the red at 700 nm of the second order. 7. An aircraft maintenance technician walks past a tall hangar door that acts like a single slit for sound entering the hangar. Outside the door, on a line perpendicular to the opening in the door, a jet engine makes a 600-Hz sound. At what angle with the door will the technician observe the first minimum in sound intensity if the vertical opening is 0.800 m wide and the speed of sound is 340 m/s? THANK YOU