Lesson 8 Light and Optics The Nature of Light Properties of Light: Reflection Refraction Interference Diffraction Polarization Dispersion and Prisms Total Internal Reflection Huygens s Principle
The Nature of Light Light exhibits the characteristics of a wave in some situations and the characteristics of a particle in other situations. The quantization model assumes that the energy of a light wave is present in particles called photons; hence, the energy is said to be quantized. According to Einstein s theory, the energy of a photon is proportional to the frequency of the electromagnetic wave:
The Nature of Light
Reflection of Light When light ray traveling in a medium encounters a boundary leading into a second medium, part or the entire incident ray is reflected back into the first medium. The reflected rays are parallel to each other. Reflection of light from a smooth surface is called specular reflection. If the surface is rough the reflected rays are not parallel but set into various directions. Reflection from rough surface is known as diffuse reflection. A surface behaves as a smooth surface as long as the surface variations are much smaller than the wavelength of the incident light.
Reflection of Light Schematic representation of (a) specular Reflection, where the reflected rays are all parallel to each other, and (b) diffuse reflection, where the reflected rays travel in random directions. (c) and (d) Photographs of specular and diffuse reflection using laser light.
Reflection of Light According to the law of reflection, i. The angle of reflection equals the angle of incidence ii. The incident ray, the reflected ray, and the normal all lie in the same plane.
Image in a Concave Mirror
Refraction of Light When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent medium, part of the energy is reflected and part enters the second medium. The ray that enters the second medium is bent at the boundary and is said to be refracted. The incident ray, the reflected ray, and the refracted ray all lie in the same plane. The angle of refraction, depends on the properties of the two media and on the angle of incidence through the relationship The equation is called Snell s law
Refraction of Light (a) A ray obliquely incident on an air glass interface. The refracted ray is bent toward the normal because v 2 < v 1. All rays and the normal lie in the same plane. (b) Light incident on the Lucite block bends both when it enters the block and when it leaves the block
Refraction of Light (a) When the light beam moves from air into glass, the light slows down on entering the glass and its path is bent toward the normal. (b) (b) When the beam moves from glass into air, the light speeds up on entering the air and its path is bent away from the normal
Refraction of Light Index of Refraction The speed of light in any material is less than the speed in vacuum except near very strong absorption bands. n speed of light in vacuum speed of light in a medium c v From this definition, index of refraction is dimensionless number usually greater than unity, because v < c.
Refraction of Light As light travels from one medium to another, its frequency does not change but its wavelength does Therefore, because the relationship must be valid in both media and because Therefore,
Refraction of Light Example : An Index of Refraction Measurement
Refraction of Light Example: Angle of Refraction for Glass A light ray of wavelength 589 nm traveling through air is incident on a smooth, flat slab of crown glass at an angle of 30.0 to the normal. Find the angle of refraction. Solution
Refraction of Light Example: Laser Light in a Compact Disc A laser in a compact disc player generates light that has a wavelength of 780 nm in air. (i) Find the speed of this light once it enters the plastic of a compact disc (n = 1.55). (ii) What is the wavelength of this light in the plastic? (i) (ii)
Refraction of Light Example: Light Passing Through a Slab A light beam passes from medium 1 to medium 2, with the latter medium being a thick slab of material whose index of refraction is n 2 (see Fig. below). Show that the emerging beam is parallel to the incident beam.
Thin Lenses Lenses are commonly used to form images by refraction in optical instruments, such as cameras, telescopes, and microscopes. We can use what we just learned about images formed by refracting surfaces to help us locate the image formed by a lens. We recognize that light passing through a lens experiences refraction at two surfaces. The development we shall follow is based on the notion that the image formed by one refracting surface serves as the object for the second surface. Refraction of Light
Interference of Light Waves
Interference of Light Waves (a) If light waves did not spread out after passing through the slits, no interference would occur. (b) The light waves from the two slits overlap as they spread out, filling what we expect to be shadowed regions with light and producing interference fringes on a screen placed to the right of the slits.
Interference of Light Waves Young s double-slit experiment. Interference in light waves from two sources was first demonstrated by Thomas Young in 1801. The schematic diagram from the experiment is as shown in the diagram to the left. Light is incident on a screen in which there is a narrow slit not shown in the diagram. The waves emerging from this slit arrive at a second screen, which contains two narrow parallel slits S 1 and S 2. These two slits serve as a pair of coherent light sources because waves emerging from them originate from the same wavefront and therefore maintain constant phase relationship. The light from the two slits produces on screen a visible pattern of bright and dark parallel bands called fridges. Schematic diagram of Young s double-slit experiment. Slits S1 and S2 behave as coherent sources of light waves that produce an interference pattern on the viewing screen (drawing not to scale).
Interference of Light Waves (a) Geometric construction for describing Young s double-slit experiment (not to scale). (b) When we assume that r 1 is parallel to r 2, the path difference between the two rays is r 2 - r 1 = d sinθ. For this approximation to be valid, it is essential that L >>d.
Interference of Light Waves
Interference of Light Waves Example: Separating Double-Slit Fringes of Two Wavelengths
Huygens s Principle Huygens s construction for (a) a plane wave propagating to the right and (b) a spherical wave propagating to the right Huygens principle is a construction for using knowledge of an earlier wavefront to determine the position of a new wavefront at some instant. In Huygen s construction, all points on a given wavefront are taken as point sources for the production of spherical secondary waves called wavelets, which propagate outward with speeds characteristic of waves in that medium. After sometime has elapsed, the new position of the wavefront is the surface tangent to the wavelets.
Diffraction Diffraction refers to the general behavior of waves spreading out as they pass through a slit. The diffraction pattern seen on a screen when a single slit is illuminated is really another interference pattern. The interference is between parts of the incident light illuminating different regions of the slit. When plane light waves pass through a small aperture in an opaque barrier, the aperture acts as if it were a point source of light, with waves entering the shadow region behind the barrier. This phenomenon, known as diffraction, can be described only with a wave model for light. We now investigate the features of the diffraction pattern that occurs when the light from the aperture is allowed to fall upon a screen. Electromagnetic waves are transverse. That is, the electric and magnetic field vectors associated with electromagnetic waves are perpendicular to the direction of wave propagation. Diffraction indicates that light, once it has passed through a narrow slit, spreads beyond the narrow path defined by the slit into regions that would be in shadow if light traveled in straight lines. Other waves, such as sound waves and water waves, also have this property of spreading when passing through apertures or by sharp edges.
Diffraction Diffraction Patterns from Narrow Slits Let us consider a common situation, that of light passing through a narrow opening modeled as a slit, and projected onto a screen. To simplify our analysis, we assume that the observing screen is far from the slit, so that the rays reaching the screen are approximately parallel. This can also be achieved experimentally by using a converging lens to focus the parallel rays on a nearby screen. In this model, the pattern on the screen is called a Fraunhofer diffraction pattern. (a) Fraunhofer diffraction pattern of a single slit. The pattern consists of a central bright fringe flanked by much weaker maxima alternating with dark fringes. (Drawing not to scale.) (b) Photograph of a single-slit Fraunhofer diffraction pattern.
Diffraction A diffraction pattern consisting of light and dark areas is observed, somewhat similar to the interference patterns. For example, when a narrow slit is placed between a distant light source (or a laser beam) and a screen, the light produces a diffraction pattern. The pattern consists of a broad, intense central band (called the central maximum), flanked by a series of narrower, less intense additional bands (called side maxima or secondary maxima) and a series of intervening dark bands (or minima).
Diffraction To analyze the diffraction pattern, it is convenient to divide the slit into two halves, as shown in the Figure. Keeping in mind that all the waves are in phase as they leave the slit, consider rays 1 and 3. As these two rays travel toward a viewing screen far to the right of the figure, ray 1 travels farther than ray 3 by an amount equal to the path difference (a/2)sinθ, where a is the width of the slit. Similarly, the path difference between rays 2 and 4 is also (a/2) sin θ, as is that between rays 3 and 5. If this path difference is exactly half a wavelength (corresponding to a phase difference of 180 ), then the two waves cancel each other and destructive interference results. If this is true for two such rays, then it is true for any two rays that originate at points separated by half the slit width because the phase difference between two such points is 180.
Diffraction Therefore, waves from the upper half of the slit interfere destructively with waves from the lower half when
Diffraction Intensity distribution for a Fraunhofer diffraction pattern from a single slit of width a. The positions of two minima on each side of the central maximum are labeled. (Drawing not to scale.)
Diffraction Diffraction of X-Rays by Crystals A two-dimensional description of the reflection of an x-ray beam from two parallel crystalline planes separated by a distance d. The beam reflected from the lower plane travels farther than the one reflected from the upper plane by a distance 2d sinθ. This condition is known as Bragg s law
Example: The Orders of a Diffraction Grating
Polarization of Light Waves An ordinary beam of light consists of a large number of waves emitted by the atoms of the light source. Each atom produces a wave having some particular orientation of the electric field vector E, corresponding to the direction of atomic vibration. The direction of polarization of each individual wave is defined to be the direction in which the electric field is vibrating. The plane formed by E and the direction of propagation is called the plane of polarization of the wave. Under certain conditions these transverse waves with electric field vectors in all possible transverse directions can be polarized in various ways. This means that only certain directions of the electric field vectors are present in the polarized wave. (a) A representation of an unpolarized light beam viewed along the direction of propagation (perpendicular to the page). The transverse electric field can vibrate in any direction in the plane of the page with equal probability. (b) A linearly polarized light beam with the electric field vibrating in the vertical direction.
Polarization of Light Waves The Figure represents an unpolarized light beam incident on a first polarizing sheet, called the polarizer. Because the transmission axis is oriented vertically in the figure, the light transmitted through this sheet is polarized vertically. A second polarizing sheet, called the analyzer, intercepts the beam. In Figure, the analyzer transmission axis is set at an angleθ to the polarizer axis. We call the electric field vector of the first transmitted beam E 0. The component of E 0 perpendicular to the analyzer axis is completely absorbed. The component of E 0 parallel to the analyzer axis, which is allowed through by the analyzer, is E 0 cosθ. Because the intensity of the transmitted beam varies as the square of its magnitude, we conclude that the intensity of the (polarized) beam transmitted through the analyzer varies as
Polarization of Light Waves (a) When unpolarized light is incident on a reflecting surface, the reflected and refracted beams are partially polarized. (b) The reflected beam is completely polarized when the angle of incidence equals the polarizing angle θ p, which satisfies the equation n=tanθ p. At this incident angle, the reflected and refracted rays are perpendicular to each other.
Polarization of Light Waves Using Snell s law of refraction This expression is called Brewster s law, and the polarizing angleθ p is sometimes called Brewster s angle, after its discoverer, David Brewster (1781 1868). Because n varies with wavelength for a given substance, Brewster s angle is also a function of wavelength.
Dispersion Dispersion is the separation of white light by a prism into bands of colors red, orange, yellow, green, blue and violet. The spectrum is due to the difference in the velocities and wavelength of the spectral colors. Violet is bent most and is slowed down more than the red light.
Dispersion When light passes through a prism, the angle of deviation δ is different for the different wavelengths. This property can be put to use in the form of a prism spectrometer:
Dispersion Example: Measuring n Using a Prism The minimum angle of deviation δ min for a prism occurs when the angle of incidence θ 1 is such that the refracted ray inside the prism makes the same angle with the normal to the two prism faces, as shown in the Figure. Obtain an expression for the index of refraction of the prism material. Using the geometry shown in the Figure, we Hence, knowing the apex angle φ of the prism and measuring δ min, we can calculate the index of refraction of the prism material.
Total Internal Reflection We can use Snell s law of refraction to find the critical angle. When Then (a) Rays travel from a medium of index of refraction n 1 into a medium of index of refraction n 2, where n 2 < n 1. As the angle of incidence θ 1 increases, the angle of refraction θ 2 increases until θ 2 is 90 (ray 4). For even larger angles of incidence, total internal reflection occurs (ray 5). (b) The angle of incidence producing an angle of refraction equal to 90 is the critical angle θ c. At this angle of incidence, all of the energy of the incident light is reflected