Optometry Semester 2

OPTOMETRY · SEMESTER 2 Additional Study Notes: Optics Calculation Workbook Physical and Geometric Optics Additional Study Notes — newly authored explanations and examples. These sections supplement the supplied course material. START READING NOTES Contents of This Topic Learning objectives Units and optical power Vergence convention Real-is-positive thin-lens convention Refraction and plausibility checks Learning objectives Select one sign convention, convert units before substitution and interpret a calculated image or vergence physically. All examples below were created for this notes package. Units and optical power In air, thin-lens power F = 1/f when f is in metres. A +0.25 m focal length corresponds to +4.00 D; a −0.50 m focal length corresponds to −2.00 D. Millimetres must be divided by 1000 and centimetres by 100 before using these relations. A positive converging lens and a negative diverging lens must not be assigned the same sign. Vergence convention For this example, light travels left to right, distances to points on the right are positive, and distances to points on the left are negative. Vergence L = n/l, with l in metres. At a thin lens, L′ = L + F. An object is 0.50 m to the left of a +4.00 D thin lens in air. L = 1/(−0.50) = −2.00 D. Therefore L′ = −2.00 + 4.00 = +2.00 D, and the image is 1/2 = 0.50 m to the right of the lens. The positive emergent vergence indicates converging rays. Real-is-positive thin-lens convention A different common convention uses 1/f = 1/do + 1/di, with positive do for a real object and positive di for a real image. It gives the same physical prediction when used consistently. Do not insert a signed Cartesian object distance into this alternative equation without translating conventions. Example: f = +0.20 m and do = +0.60 m. Then 1/di = 5 − 1.667 = 3.333 m⁻¹, giving di = +0.30 m. Magnification m = −di/do = −0.50. The image is real, inverted and half the object height. Refraction and plausibility checks Snell’s law is n1 sin θ1 = n2 sin θ2, with angles measured from the normal. Example: light passes from air, n1 = 1.00, into a medium with n2 = 1.50 at θ1 = 30°. Then sin θ2 = (1/1.50) × 0.5 = 1/3 and θ2 ≈ 19.47°. The ray bends toward the normal, as expected on entry into the higher-index medium. Before accepting an answer, state its unit, sign and physical meaning. For a sine result exceeding 1, reconsider whether total internal reflection is possible and whether the incident and transmitted indices were assigned correctly. Study References Source notes: Vergence; Refraction; Thin Lenses I and II OpenStax: Thin Lenses External references checked 13 September 2026. Worked numerical examples and teaching activities are original. ← PREVIOUS TOPICVIEW MODULE NOTESVIEW SEMESTER NOTESALL OPTOMETRY NOTES Need These Notes as PDF? Request a formatted copy for offline study, printing or revision. GET PDF NOTES ON WHATSAPP

OPTOMETRY · SEMESTER 2 Diffraction and polarisation Physical and Geometric Optics START READING NOTES Contents of This Topic Diffraction and polarisation DIFFRACTION POLARISATION Figure 10.2: Polarisation POLARISATION BY SELECTIVE ABSORPTION POLARISATION BY REFLECTION POLARISATION BY DOUBLE REFRACTION Tourmaline 1.669 1.638 POLARISATION BY SCATTERING Diffraction and polarisation DIFFRACTION AND POLARISATION CHAPTER CONTENTS DIFFRACTION ……………………………………………………………………………………………………………………………………………… 1 POLARISATION……………………………………………………………………………………………………………………………………………. 0 POLARISATION BY SELECTIVE ABSORPTION ……………………………………………………………………………………………… 1 POLARISATION BY REFLECTION…………………………………………………………………………………………………………………. 3 POLARISATION BY DOUBLE REFRACTION ………………………………………………………………………………………………….. 6 POLARISATION BY SCATTERING ………………………………………………………………………………………………………………… 7 DIFFRACTION Think back to Young’s double-slit experiment. If the light truly travelled in straight-line paths after passing through the slits, the waves would not overlap and no interference pattern would be seen. In other words, the light deviates from a straight-line path and enters the region that would otherwise be shadowed. This deviation of light from its initial line of travel (in the same medium) is called diffraction. It can also be referred to as the bending of light round the edges of an obstacle. When light passes through an opening that is large compared to the wavelength of light, it casts a shadow, as shown on the left in figure 10.1. But if we pass light through a thin slit we see that the light diffracts. The sharp boundary between the light and the dark area disappears, and the light spreads out like a fan to produce a bright area that fades into darkness without sharp edges. This can be seen on the right below. Diffraction and polarisation Figure 10.1: Representation of the principle of diffraction Diffraction is not confined by narrow slits or to openings in general but can be seen for all shadows. On close examination, even the sharpest shadow is blurred slightly at the edge. Under ordinary conditions, we seldom notice the diffraction of light. Light sources such as incandescent lamps or the sun are not monochromatic point sources, and the diffraction patterns due to different parts of the source and to different wavelengths usually overlap and obscure each other. Diffraction is a problem when we wish to see very small objects with microscopes. If the size of the object is about the same as the wavelength of light, the image of the object will be blurred by diffraction. If the object is smaller than the wavelength of light, no structure can be seen. The entire image is lost due to diffraction. No amount of magnification or perfection of microscope design can defeat this fundamental diffraction limit. One of the goals of an optical design is to achieve resolution as close to the theoretical diffraction limit as possible. This is true not only of clinical instruments, but also corrective lenses. Larger objects viewed with visible light can be seen clearly. A similar situation occurs with very small objects if the wavelength of the waves used to view the object is made very small. This can be done by using a beam of electrons instead of a light beam. POLARISATION An ordinary beam of light consists of a large number of waves emitted by the atoms or molecules of the light source. Each atom produces a wave with its own orientation of E, the electric field vector, corresponding to the direction of atomic vibration. However, because all directions of vibration are possible, the resultant electromagnetic wave is a superposition of waves produced by the individual atomic sources. The result is an unpolarised light wave. The direction of polarisation of the electromagnetic wave is defined to be the direction in which E is vibrating.Diffraction and polarisation LEFT: Light waves having all possible orientations of E, RIGHT: Light waves having only ONE orientation of E Figure 10.2: Polarisation Note that all directions of the electric field vector are equally probable and all lie in a plane perpendicular to the direction of propagation. At any given point and at some instant of time, there is only one resultant electric field, which can be resolved into a vertical and a horizontal component. You should not be misled into thinking that the figure is showing that the electric field vector has a number of directions at a particular time. A wave is said to be linearly polarised if E vibrates in the same direction at all times at a particular point. Such a wave is described as plane-polarised, or simply polarised. The phenomenon of polarisation provides firm evidence of the transverse nature of electromagnetic waves. All waves show interference and diffraction but only transverse waves can be polarised. It is possible to obtain a linearly polarised beam from an unpolarised beam by removing from it all components except those whose electric field vectors oscillate in a single plane. We shall now discuss some physical processes for producing polarised light from unpolarised light. POLARISATION BY SELECTIVE ABSORPTION The most common technique for obtaining polarised light is to use a material that transmits waves whose electric field vectors vibrate in a plane parallel to a certain direction and absorbs those waves whose electric field vectors vibrate in Diffraction and polarisation other directions. Any substance that has the property of transmitting light with the electric field vector vibrating in only one direction is called a dichroic substance. Examples of these are tourmaline, and quinine iodosulfate. Tourmaline has the disadvantage of producing coloured polarised light. In 1932, E. H. Land prepared the first version of Polaroid by embedding a thin layer of quinine iodosulfate in a plastic sheet and stretching the plastic. An improved version followed in 1938 which used long polymeric molecules of polyvinyl alcohol that polarized light through selective absorption by oriented molecules. The sheets are stretched during manufacture so that the long-chain molecules align. After such a sheet is dipped into a solution containing iodine, the molecules become conducting. However, the conduction takes place primarily along the hydrocarbon chains since the valence electrons of the molecules can move easily only along the chains (recall that valence electrons are “free” electrons that can readily move through the conductor). As a result, the molecules readily absorb light

OPTOMETRY · SEMESTER 2 Optical instruments Physical and Geometric Optics START READING NOTES Contents of This Topic Optical instruments INTRODUCTION CONTROL OF LIGHT IN THE CAMERA SPEED OF A CAMERA LENS DEPTH OF FIELD COMPARISON OF THE CAMERA TO THE EYE LENS ABERRATIONS CHROMATIC ABERRATION THE SIMPLE MAGNIFIER YOU ARE NOT REQUIRED TO KNOW THESE DERIVATIONS. THE COMPOUND MICROSCOPE THE TELESCOPE REFLECTING TELESCOPE TERRESTRIAL TELESCOPE Optical instruments OPTICAL INSTRUMENTS CHAPTER CONTENTS INTRODUCTION…………………………………………………………………………………………………………………………………………… 1 THE CAMERA………………………………………………………………………………………………………………………………………………. 1 LENS ABERRATIONS…………………………………………………………………………………………………………………………………… 5 THE SIMPLE MAGNIFIER……………………………………………………………………………………………………………………………… 8 THE COMPOUND MICROSCOPE…………………………………………………………………………………………………………………11 THE TELESCOPE ……………………………………………………………………………………………………………………………………….13 INTRODUCTION An understanding of the optics of basic optical instruments is important since the applications are to be found in many different clinical instruments as well as the therapeutic devices that are used by practicing optometrists. THE CAMERA CONSTRUCTION The single-lens photographic camera is a simple optical instrument. It consists of a light-tight box, a converging lens that produces a real image, and a film behind the lens to receive the image. Unlike the eye, this type of camera lens has a fixed focal length. Focusing is accomplished (where possible) by varying the distance between lens and film. For proper focusing, which leads to sharp images, the lens-to-film distance will depend on the object distance as well as on the focal length of the lens. Optical instruments The shutter, located behind the lens, is a mechanical device that is opened for selected time intervals. To take a photograph, you first adjust the lens position so that a real, inverted image of the object is in focus in the plane of the film. Then, when you press a button, the shutter momentarily opens and an image forms on the film. This image is stored in the light-sensitive material for later chemical processing, yielding a reproduction of the scene. Digital cameras replace the photographic film with an electronic sensor array, the sensitivity of which can be adjusted to resemble the response of film. The effective lens opening is controlled by an iris diaphragm, the size of which can be varied. It is given this name because it performs a similar function to the iris of the eye. In some cameras, the iris diaphragm also acts as the shutter; while normally closed, the iris opens to a pre-determined size in order to exposure the film. CONTROL OF LIGHT IN THE CAMERA With modern photographic films and with adequate lighting, the shutter only needs to be open for a small fraction of a second to successfully record an image on film. Many cameras have a range of available shutter speeds – that is, lengths of time during which the shutter is open. Shutter speeds of 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000, and 1/2000 second are standard. Note that each of these is approximately half as long (or, in photographic terminology, “twice as fast”) as the preceding one. The faster the shutter speed is, the faster the object can be moving and still produce a sharp image. With this arrangement, moving objects can be photographed with the use of short exposure times, and dark scenes (low light levels) with the use of long exposure times. Without this control, it would be impossible to take stop-action photographs. For example, a speeding race car would move far enough while the shutter was open to produce a blurred image. Stationary objects can be shot with a shutter speed of 1/60 s since there is no chance of motion blurring, although the camera must be held steady. Older cameras had semi-circular shutters, but later models usually had diaphragm shutters. The amount of light reaching the film must be carefully controlled to avoid underexposure (too little light for any but the brightest objects to show up) or overexposure (too much light, so that all bright objects look the same, with a consequent lack of contrast). To control the exposure, a “stop” or iris diaphragm, whose opening is of variable diameter, is placed behind the lens. The size of the opening is varied to compensate for different lighting conditions, the sensitivity of the film used, and for different shutter speeds. Getting the “correct exposure” in a photograph corresponds to allowing the appropriate amount of light to strike the film. Therefore “correct exposure” means the correct luminous energy per unit area of the film. This amount is different for different types of film. The amount of light that strikes the film is determined not only by how long the shutter stays open, but also by how large the effective lens opening is. Thus it is analogous to the amount of water flowing from a tap, which depends on both the length of time the tap stays open and the cross-sectional area of the opening. We can deduce a relationship between “amount of light” and camera parameters as follows: (See the photometry section of chapter 8 for detail on terms used here.) 1. The amount of luminous power falling on the image in a camera is proportional to the area of the lens aperture, or to d2 , where d is the diameter of the aperture. This follows from the fact that the area of a circle is given by πr 2 (πd2 /4). 2. The area of the image formed is proportional to f2 , where f is the focal length of the lens, since the length of the image formed is proportional to the focal length. It therefore follows that the luminous power per unit area of the image, or illuminance of the image, is proportional to d2 /f2. The time of exposure (shutter speed), t, for activating the chemicals on the given negative is inversely proportional to the illuminance (the greater the amount of light falling on unit area per unit time, the less time will be needed to get the “correct amount of light” onto the film). Hence: t∝ f 2 /d2 OR t ∝ (f/d)2Optical instruments The size of the lens opening, (often referred to as the aperture or relative aperture), is usually measured by what is called

OPTOMETRY · SEMESTER 2 Photometry Physical and Geometric Optics START READING NOTES Contents of This Topic Photometry INTRODUCTION SOLID ANGLES LUMINOUS POWER AND LUMINOUS INTENSITY LUMINOUS EFFICIENCY AND LUMINOUS EFFICACY RADIOMETRIC AND PHOTOMETRIC UNITS ILLUMINANCE FUNDAMENTAL LAWS OF PHOTOMETRY Law 2. LIGHT SOURCES GAS DISCHARGE TUBES Low Pressure Sodium Lamps High Pressure Sodium Lamps FLUORESCENT LAMPS LASERS STIMULATED EMISSION LASER ACTION MEDICAL APPLICATIONS LASER SAFETY LIGHT EMITTING DIODES HOW AN LED WORKS ADVANTAGES DISADVANTAGES Photometry PHOTOMETRY CHAPTER CONTENTS INTRODUCTION………………………………………………………………………………………………………………………………………………… 2 SOLID ANGLES…………………………………………………………………………………………………………………………………………………. 2 LUMINOUS POWER AND LUMINOUS INTENSITY……………………………………………………………………………………………….. 3 LUMINOUS EFFICIENCY AND LUMINOUS EFFICACY…………………………………………………………………………………………. 4 RADIOMETRIC AND PHOTOMETRIC UNITS……………………………………………………………………………………………………….. 4 ILLUMINANCE …………………………………………………………………………………………………………………………………………………… 4 FUNDAMENTAL LAWS OF PHOTOMETRY …………………………………………………………………………………………………………. 5 LIGHT SOURCES………………………………………………………………………………………………………………………………………………. 6 LASERS ………………………………………………………………………………………………………………………………………………………….. 10 LIGHT EMITTING DIODES………………………………………………………………………………………………………………………………… 13Photometry INTRODUCTION Photometry may be described as the study and measurement of light in terms of the visual response it produces. The eye does not respond with equal sensitivity to light of different colours; the sensitivity is a maximum for green light and it decreases towards both ends of the visible spectrum. Consequently, although they produce significant damage to the eye, IR and UV light do not produce that much of a visual response compared to some visible light. The sensitivity curve, sometimes referred to as the luminosity curve, gives the relative brightness as assessed by the average eye of the different colours of the spectrum. For the average light adapted eye at moderate intensities (photopic vision) the maximum visual effect is obtained with light of wavelength 555nm (yellow-green). This maximum shifts to 500nm in scotopic vision (for the average dark adapted eye). This is known as the Purkinje shift. The electromagnetic spectrum covers a wide range of wavelengths. However, in photometry we restrict ourselves to visible light (officially the region from 360 to 830 nm). This is done because the aim of photometry is to measure light in such a way that the results correlate with human vision. (The study of radiant energy, including visible light, without regard to its visual response is known as radiometry.) Occupational Safety legislation puts a duty on employers to provide sufficient lighting of suitable standard for specific purposes. By ensuring that photometric quantities conform to standards, mankind benefits in terms of: reduction of eyestrain fewer accidents better working conditions greater productivity improved leisure facilities etc. Most practical light sources are incandescent or fluorescent sources which emit energy over a wide range of wavelengths, mainly as wasted heat in the infrared part of the spectrum. The distribution of luminous power is not uniform in all directions and light is emitted from different points on the source. However, we simplify matters by assuming that: (i) the dimensions of the source are negligible compared to its distance from a surface or object and (ii) the light is emitted equally in all directions. The first assumption amounts to saying that we will consider all sources to be point sources. The second assumption is of course equivalent to saying that we consider all sources to be isotropic. SOLID ANGLES The unit solid angle, or steradian, is the solid angle of a cone that, having its apex at the centre of a sphere, cuts off an area of the sphere’s surface equal to the square of its radius. Photometry Figure 8.1: Representation of a steradian The number of steradians in any solid angle is given by the area of the sphere’s surface that is included in the angle divided by the square of the radius, i.e.: 22 area Area at cone opening solid angle = = r (cone length) LUMINOUS POWER AND LUMINOUS INTENSITY The term central to photometry is luminous power, Φ. This is because the visual sensation produced depends on the rate at which light energy reaches the retina and not on the total energy. Hence, a dimly lit object will not continue to grow brighter and brighter as it is observed. The rate at which light energy flows is called the luminous power (flux). The luminous power is therefore the luminous energy flowing per second and this could be expressed in joules per second (watts). However, in photometry, we use the unit called the lumen. Definition: One lumen is the luminous power emitted into a unit solid angle (1 steradian) from a point source of intensity 1 candela. The total number of lumens emitted in all directions from a source is thus the luminous power. If we consider the luminous power emitted per unit solid angle, we are now dealing with the luminous intensity. This is indicated by the symbol I and has as unit the candela (cd). We saw previously that there are 4π steradians round a point source. Therefore, I is related to Φ by the relationship : (Φ is the Greek letter phi. ) Φ = 4πIPhotometry LUMINOUS EFFICIENCY AND LUMINOUS EFFICACY A light may produce a great deal of radiant energy, (energy in the form of electromagnetic radiation) but relatively little luminous energy (visible light). To be useful for lighting purposes, we need most of the output to be in the visible region. Two concepts are used to indicate the effectiveness of a source. The luminous efficiency indicates the fraction of the total power produced which is actually visible. The units will be lumens per lumen. luminous flux emitted Luminous efficiency = total radiant power emitted For example, a 60 watt incandescent light bulb produces the same luminous power as a 15 W fluorescent bulb. This is because much of the incandescent light bulb output is invisible infrared. Its luminous efficiency is much less since it is converting a large portion of the power input into wasted heat. It should be noted that the power output at each wavelength is multiplied by a factor determined by the visual response of the eye to light of that wavelength. The human eye is most sensitive to green light with a wavelength of 555 nm. One watt of such light is “worth” 683 lumens. In contrast, infrared and ultraviolet radiation are invisible and therefore count zero

OPTOMETRY · SEMESTER 2 Interference Physical and Geometric Optics START READING NOTES Contents of This Topic Interference INTERFERENCE COMPLETE CONSTRUCTIVE INTERFERENCE. YOUNG’S DOUBLE-SLIT EXPERIMENT Bright Fringes Dark Fringes CHANGE OF PHASE DUE TO REFLECTION EXAMPLES ANTI-REFLECTION COATINGS ANTI-REFLECTIVE COATINGS ON LENSES MATERIAL REFRACTIVE INDEX Interference INTERFERENCE CHAPTER CONTENTS INTERFERENCE ………………………………………………………………………………………………………………………………………….. 1 YOUNG’S DOUBLE-SLIT EXPERIMENT…………………………………………………………………………………………………………. 1 INTERFERENCE IN THIN FILMS……………………………………………………………………………………………………………………. 4 ANTI-REFLECTION COATINGS …………………………………………………………………………………………………………………….. 7 INTERFERENCE When two or more waves arrive at a point in a medium simultaneously, the principle of superposition can be used to find the resultant displacement. The superposition principle states that when two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. This interaction of waves that exist at a point at the same time is known as interference. The same holds for electromagnetic waves moving through a medium. The amplitudes can add together either constructively or destructively. In constructive interference, the amplitude of the resultant wave is greater than that of either of the individual waves, while in destructive interference, the resultant amplitude is less than that of either of the individual waves. In the case of two waves of equal amplitude, complete constructive and complete destructive interference may occur and will be illustrated first (figure 7.1). Fundamentally, all interference associated with light waves arises as a result of combining the fields that constitute the individual waves.Interference COMPLETE CONSTRUCTIVE INTERFERENCE. COMPLETE DESTRUCTIVE INTERFERENCE. INTERFERENCE DUE TO WAVES PARTIALLY IN PHASE. Figure 7.1: Constructive and destructive interference Interference effects in light waves are not easy to observe because of the short wavelengths involved. These short wavelengths mean that they have very high frequencies and therefore there are many waves interacting per second. Interference In order to observe sustained interference in light waves from more than one source, the following conditions must be met: 1. The sources must be coherent, that is, they must maintain a constant phase with respect to each other. 2. The sources must emit identical wavelengths. (This condition is easiest to meet if the sources are monochromatic.) If two separate light sources are placed side by side, no interference effects are observed because in this case the light waves emitted by each of the sources are emitted independently of the other source. Hence, their emissions do not maintain a constant phase relationship with each other over the time of observation. Light from an ordinary light source undergoes such random changes about once every 10-8 seconds. Therefore the conditions for constructive interference, destructive interference, or some intermediate state last for times of the order of 10-8 seconds. The result is that no interference effects are observed since the eye cannot follow such short-term changes. Such light sources that do not maintain a constant phase relationship are said to be incoherent. Please note that interference phenomenon can be observed with incoherent “white” (i.e. polychromatic) light (e.g. colour patterns on soap bubbles, anti reflections coatings on spectacles, etc.) but the conditions for them to be observed are much more strict (and difficult to put in place) than with coherent light. YOUNG’S DOUBLE-SLIT EXPERIMENT The phenomenon of interference in light waves from two sources was first demonstrated by Thomas Young in 1801. Figure 7.2: Young’s double slit experimentInterference Sunlight is incident on a screen, which is provided with a narrow slit S0. The waves emerging from this slit arrive at a second screen, which contains two narrow, parallel slits, S1 and S2. Since these slits are equidistant from the first slit, they serve as a pair of coherent light sources because waves emerging from them originate from the same wavefront and therefore maintain a constant phase relationship. Furthermore, they emit identical wavelengths since the waves arise from the same source. In this way the conditions for sustained interference are met. The light from the two slits produces a visible pattern on a target screen. The pattern consists of a series of alternating bright and dark parallel bands called fringes surrounding a fairly broad central light band. Young accounted for the interference pattern generally as follows: When the light from slits S1 and S2 arrives at a point on the target screen such that constructive interference occurs at that location, a bright line appears. When light from the two slits interferes destructively at any location on the screen, a dark line results. Two waves (that necessarily leave the slits in phase) strike the screen at the central spot. Since they travel an equal distance, they arrive in phase and as a result complete constructive interference occurs here and a bright spot is observed. Figure 7.3: Constructive interference in Young’s experiment Figure 7.4 illustrates how the first dark band away from the central spot is formed. The waves leaving the two slits once again start out being in phase but travel different distances. As a result, one wave is half a wavelength behind the other so that the crest of one wave overlaps with the trough of the other. Complete destructive interference results and a dark fringe is observed.Interference Figure 7.4: Destructive interference in Young’s experiment In the following diagram, the waves again start out in phase and travel different distances, but the difference in path length is exactly one wavelength and the waves thus reach the screen in phase. Complete constructive interference results and another (higher order) bright band is observed. Figure 7.4: Constructive interference of higher order in Young’s experiment The foregoing can be summarized as follows: If the path difference is either zero or some integral multiple of the wavelength, the two waves are in phase at the screen and a bright fringe results. When the path difference is an odd multiple of λ/2, the two waves arriving at the screen will be 180° out of phase and a dark fringe will result. What is the effect of wavelength on the interference pattern? The shorter the wavelength of the incident light, the closer the fringes are to each other, and the

OPTOMETRY · SEMESTER 2 Reflection Physical and Geometric Optics START READING NOTES Contents of This Topic Reflection REFLECTING SURFACES REFLECTION AT PLANE SURFACES LAWS OF SPECULAR REFLECTION IMAGE FORMED BY A PLANE MIRROR ROTATION OF REFLECTED RAY MULTIPLE IMAGES REFLECTION AT CURVED MIRRORS IMAGE FORMATION BY SPHERICAL MIRRORS THE MIRROR EQUATION USES OF MIRRORS Reflection REFLECTION CHAPTER CONTENTS REFLECTING SURFACES…………………………………………………………………………………………………………………………….. 1 REFLECTION AT PLANE SURFACES ……………………………………………………………………………………………………………. 0 LAWS OF REGULAR REFLECTION……………………………………………………………………………………………………………….. 1 IMAGE FORMED BY A PLANE MIRROR ………………………………………………………………………………………………………… 1 ROTATION OF REFLECTED RAY………………………………………………………………………………………………………………….. 2 MULTIPLE IMAGES………………………………………………………………………………………………………………………………………. 2 REFLECTION AT CURVED MIRRORS …………………………………………………………………………………………………………… 4 IMAGE FORMATION BY SPHERICAL MIRRORS…………………………………………………………………………………………….. 6 THE MIRROR EQUATION……………………………………………………………………………………………………………………………… 6 USES OF MIRRORS …………………………………………………………………………………………………………………………………….. 7 REFLECTING SURFACES A ray of light may be turned through an angle by being reflected at a suitable surface. (Bear in mind that some reflection occurs together with any refraction.) Highly polished glass will reflect only approximately 4% of the incident light near to normal incidence, whereas highly polished silver reflects about 95% of the incident light. Many other reflecting surfaces may be used (even obsidian!), but in all cases, some loss of energy occurs. The original optical reflecting surface was a chemical deposit of silver on glass. The silver was on the front surface (front-reflecting mirror), in contrast to reflectors produced by depositing a layer of aluminium or tin amalgam on the back of a sheet of glass (back-reflecting mirror).Reflection One of the main drawbacks to using such back-reflecting mirrors is the formation of multiple images. For this reason, back-reflecting mirrors are never used in high-grade optical equipment and instruments. The glass used for front-reflecting mirrors need not be of the best quality because the light does not actually pass through it. The glass is simply a carrier for the reflecting surface. However, the deposit may not be able to be protected against corrosion and may have to be renewed at frequent intervals. REFLECTION AT PLANE SURFACES All real objects reflect a certain proportion of the light falling upon them, and it is this reflected light that enables us to see them. In most cases, the surface of the object has irregularities that spread out an initially parallel beam of light in all directions to produce diffuse reflection. Figure 6.1: Irregular surface causing diffuse reflection A surface so smooth that any irregularities in it are small relative to the wavelength of light falling on it behaves differently. When a parallel pencil of rays of light is incident on such a surface, it is reflected in a single definite direction. Such reflection in one definite direction is termed regular or specular reflection. Figure 6.2: Regular surface causing specular reflection For instance, consider the two types of reflection from a road surface that one sees while driving at night. When the road is dry, light from oncoming vehicles is scattered off the road in different directions (diffuse) and the road is quite visible. This diffuse reflection is also responsible for the perception of depth which makes the raised road markings visible. On a rainy night, when the road is wet, the road irregularities are filled with water. Because the wet surface is quite smooth, the light undergoes specular reflection. This means that the light is reflected straight ahead and the driver only sees what is directly in front. Our concern in this course is only with this specular reflection.Reflection LAWS OF SPECULAR REFLECTION MIRROR Figure 6.3: Incident and reflected angle on a mirror surface This diagram shows light being reflected from the surface of a smooth plane mirror. Note, once again, that the angles are measured between the ray and the normal. Law 1: The angle of incidence is equal to the angle of reflection. Law 2: The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane. One of the consequences of these laws is a common occurrence in photographs of individuals – their eyes appear to be glowing red. This occurs when a flash unit is used and the unit is very close to the camera lens. The light from the flash unit will enter the eye normally and will thus be reflected back along its original path from the retina. This reflected light can enter the lens. Most of the light reflected from the retina is red, mainly due to the blood vessels at the back of the eye, giving the “red-eye effect”. IMAGE FORMED BY A PLANE MIRROR The image formed by a plane mirror can be analyzed by graphical construction in a similar fashion to what we did with lenses. The results of this analysis give the characteristics of the image formed by a plane mirror. The image is: 1. as far behind the mirror as the object is in front. 2. the same size as the object. 3. virtual. 4. erect. 5. laterally inverted. Figure 6.4: Image formed by a plane mirror i r NormalReflection Lateral inversion refers to the phenomenon that the image is reversed from left to right in comparison to the object. For example, when you look in a mirror and touch your right ear, it is the left ear being touched in the image. Figure 6.5: Lateral inversion in mirrors ROTATION OF REFLECTED RAY If a ray is incident on a plane mirror at an angle i, the angle of reflection is also equal to i. The angle between the two rays will thus be 2i. Suppose now that the mirror is rotated through an angle θ. The angle of incidence will increase by θ and so will the angle of reflection. The total angle between the two rays is now 2i+ 2θ. Consequently, the angle of reflection will rotate through an angle of 2θ. Thus, the reflected ray rotates through twice the angle through which the mirror rotates. This important effect may be used as a magnifying device for small rotations in applications known as optical levers. MULTIPLE IMAGES With a thick back-silvered plane mirror, some light is always reflected at

OPTOMETRY · SEMESTER 2 Thin lenses 2 Physical and Geometric Optics START READING NOTES Contents of This Topic Thin lenses 2 SUMMARY OF IMAGE FORMATION DIVERGING LENSES THE LENS EQUATION L'−L = F NEWTON’S LENS EQUATION LINEAR MAGNIFICATION POWER OF A LENS LENS SYSTEMS F = F1 + F2 Thin lenses 2 THIN LENSES II CHAPTER CONTENTS SUMMARY OF IMAGE FORMATION………………………………………………………………………………………………………………. 1 THE LENS EQUATION………………………………………………………………………………………………………………………………….. 2 NEWTON’S LENS EQUATION……………………………………………………………………………………………………………………….. 3 LINEAR MAGNIFICATION……………………………………………………………………………………………………………………………… 3 POWER OF A LENS……………………………………………………………………………………………………………………………………… 4 LENS SYSTEMS…………………………………………………………………………………………………………………………………………… 4 LENSES IN CONTACT………………………………………………………………………………………………………………………………….. 4 LENS-MAKER’S EQUATION………………………………………………………………………………………………………………………….. 5 SUMMARY OF IMAGE FORMATION CONVERGING LENSES N.B. When referring to an object position, F should be understood to be the first focal point. Similarly, F should be understood to be the second focal point when referring to an image position. We can use the graphical construction techniques of the previous chapter to analyze the nature and the position of the images obtained for various object positions. The results obtained are as follows.Thin Lenses II 5. When the object is between the lens and the focal point′, the image is formed behind the object. It is virtual, erect and larger than the object. This is how a converging lens can be used as a magnifier. Figure 5.1: Converging lenses: object between lens and focal point B. An object placed at the focal point produces an image at infinity. Figure 5.2: Converging lenses: object at focal point C. When the object is between F and 2F, the image is formed beyond 2F. The image is real, inverted and larger than the object. Figure 5.3: Converging lenses: object between focal point and twice the focal length 2F F F 2F 2F F F 2F F FThin Lenses II If we compare the situations in A and C, we see that in both cases an enlarged image is produced. However, the situation in A is preferred for the use of the lens as a magnifier, because there an ERECT image is produced. D. When the object is at 2F, we find the image at 2F. It is real, inverted and the same size as the object. (See number 5 in Problem Set 4.) Figure 5.4: Converging lenses: object distance at twice the focal length E. An object placed beyond 2F forms an image between F and 2F. It is real, inverted and smaller than the object. Figure 5.5: Converging lenses: object distance further than twice the focal length F. When the object is at infinity, the image is at F. It is again real, inverted and smaller than the object. Recall that this is how the focal point of a converging lens was defined in the previous chapter. DIVERGING LENSES In the case of a converging lens, the image is sometimes real and sometimes virtual. In a diverging lens, the image is always virtual, erect and diminished. The image is situated between the object and the lens. 2F F F 2F 2F F F 2FThin Lenses II Figure 5.6: Diverging lenses, for all object positions THE LENS EQUATION Instead of carrying out ray-tracing for each situation, we can determine the nature and position of an image by calculation instead. A simple equation relates the positions of the image and the object of a thin lens to the lens’s focal length, f. o i + = d df 1 11 Where do is the position (distance) of the object, di is the position of the image and f is the focal length. We can rewrite the equation in terms of vergence as L'−L = F L is the object vergence L’ is the image vergence F is the focal power of the lens Sign Convention Converging lenses have a positive focal length and diverging lenses a negative focal length. F FThin Lenses II NEWTON’S LENS EQUATION Newton showed that the conjugate points O and I obey the relation: x1x2 = f2 This formula states that the product of the distances of two conjugate points from the respective foci of a lens or mirror is equal to the square of the focal length. From this, it follows that x2 decreases as x1 increases. This implies that the image recedes from the focus on the right (away from the lens) when the object approaches the lens. This can be confirmed with the ray diagrams at the start of this chapter. Figure 5.7: Newton’s lens equation LINEAR MAGNIFICATION As previously stated in chapter 4, the linear magnification (m) of an optical system is the ratio between the size of the image and that of the object. A magnification of exactly 1 means that the image and the object are the same size. By “size” we mean any linear dimension such as height or width. Image Size (I) Object Size (O) m = A relationship can also be derived relating magnification to image and object distances: i o d d = = Image Size (I) m – Object Size (O) If we use the sign convention specifying that distances above the optical axis are considered positive and that distances below are considered negative, we obtain the following result: If the image is erect with respect to the object, the magnification is positive. If the image is inverted with respect to the object, the magnification is negative. The magnification is sometimes referred to as the relative size. Thin Lenses II POWER OF A LENS The power of a lens, F, is defined as the reciprocal of the focal length, in metres. The unit of power, m-1 , will be recognized as being the same as the unit of vergence, the dioptre (D). The power of a lens is thus its ability to change the vergence of a pencil of rays. The fact that vergence at a point and the power of a lens share the same units, does NOT mean that the vergence at a point equals the power of a lens. Expressed in symbols: 1 F = f LENS SYSTEMS In general, when light passes through more than one lens,