Geometrical Optics Lecture Notes PDF

Lecture 35 – GEOMETRICAL OPTICS This lecture focuses on geometrical optics, the study of light as rays traveling in straight lines. While light is fundamentally a wave, exhibiting interference and diffraction, these wave properties are not considered in geometrical optics. Instead, we observe that l...

Lecture 35 – GEOMETRICAL OPTICS This lecture focuses on geometrical optics, the study of light as rays traveling in straight lines. While light is fundamentally a wave, exhibiting interference and diffraction, these wave properties are not considered in geometrical optics. Instead, we observe that light, such as from a candle, casts shadows when obstructed. Shadows form in the direction opposite to the light source, a phenomenon observable both on Earth and in the celestial bodies we see in the sky. For instance, the moon also casts its shadow on the Earth, leading to solar eclipses when the Sun's rays are blocked. This shadow has two distinct parts: a completely dark region, and the region where the darkness is partial. As the Moon orbits, these shadow regions move along with it. Similarly, lunar eclipses occur when the Sun's rays are blocked by the Earth and cast its shadow on the Moon. Figure 35. 1. (a) Solar eclipse (b) Lunar eclipse Reflection and Mirrors 1. Plane Mirrors When you look into a mirror, the image you see is known as a virtual image. It's formed by the way light rays reflect off the mirror. Imagine a candle in front of a mirror emitting light rays as shown in fig 35.2. The rays that hit the mirror reflect at the same angle. If we extend these reflected rays backward using dashed lines, they appear to converge at a point behind the mirror. This point, where the extended rays meet, is where the virtual image of the candle appears. Figure 35. 2. A virtual image of a candle forms behind a mirror where reflected light rays appear to converge. The characteristics of a virtual image in a plane mirror are: The rays appear to converge behind the mirror, although they do not physically do so.  The image appears to be the same distance behind the mirror as the object is in front of it. The image is upright and the same size as the object. The image is laterally inverted, meaning left and right are switched. This behavior of light is fundamental to the definition of a plane mirror, which reflects light in such a way that the virtual image seems to exist behind the mirror's surface. The image created by a mirror is known as a virtual image because it forms where the extended rays appear to converge, but no actual object exists there. Conversely, a real image is formed when light rays converge at a point. In a plane mirror, the angle of incidence equals the angle of reflection, meaning the light rays reflect symmetrically. This holds true for any smooth section of the mirror. However, if the mirror's surface is rough, the light rays scatter in various directions, preventing the formation of a clear image (fig 35.3). Figure 35. 3. (a) Smooth surfaces cause specular reflection with clear images, while (b) rough surfaces cause diffuse reflection, scattering light. For a mirror to reflect light properly, its surface must be extremely smooth. Given that the wavelength of visible light ranges from 4000 to 7000 Å (or atomic lengths), the surface roughness must be significantly smaller, ideally at least eight times smaller than these wavelengths. This ensures that the light reflects accurately, maintaining the equality of the angle of incidence and the angle of reflection. On the other hand, consider satellite dishes, which use microwaves with much larger wavelengths compared to visible light. Despite the use of a wire mesh in these dishes, they still function effectively because the large wavelengths of microwaves can reflect off a less smooth surface. Another example of forming an image involves a source of light placed in front of a bi-convex lens which bends the light as shown in fig 35.4 (a). The light rays from the bulb pass through the lens and reach our eyes. If we trace these rays backward, they converge at a single point, forming an image. A partially silver mirror allows light to both reflect off it and pass through it. When light from a source is directed at such a mirror, it can create images in the same location, regardless of the source's position (see fig 35.4 (c)). If we position the same light source in different locations, its image will still be formed at the same point. For instance, fig 35.4 (a) shows light originates from the source, and when traced back, converges at a point, creating a virtual image. fig 35.4 (b) depicts identical object and image positions, while the third employs a mirror, resulting in identical mirror images. Thus, discerning between the three is challenging.  Figure 35. 4. (a) Light bending through a bi-convex lens to form an image as rays converge at a single point. (b) Identical object and image positions showing consistent image formation. (c) A partially silvered mirror creating images in the same location regardless of the source’s position. To determine if an image is real or virtual, trace the light rays backward to their apparent origin. If they seem to emanate from a point behind the mirror, it's a virtual image. It's important to note that the distance from the object to the mirror is equal to the distance from the mirror to the image. 2. Spherical Mirrors Not all reflective surfaces are flat like glass. For example, by cutting a sphere, we can create curved mirrors that form images with different properties. Imagine cutting a portion of a spherical object and coating its surface with reflective material illustrated in fig 35.5. Depending on whether the coating is applied to the inner or outer surface, two types of mirrors are created: concave and convex mirrors. Each mirror has unique reflective properties. Figure 35. 5. (a) concave and (b) convex mirrors created by coating the inner and outer surfaces of a cut spherical object, respectively. Let's focus on the concave mirror, which is relatively inexpensive. When positioned at various distances, it produces different images of surrounding lights on a screen. As the mirror is brought closer, the image clarity improves until reaching a specific position, beyond which the image becomes blurry again. This phenomenon can be explained through a diagram, where parallel lines originating from a distant source interact with the mirror. Figure 35. 6. (a) Parallel light rays converge at the focal point (F) after reflection from a concave mirror, (b) the focal point is located at R/2 from the center of the mirror, known as the focal length. Parallel light rays that strike a mirror converge at a single point after reflection, known as the focal point (F) as shown in fig 35.6. For a concave mirror, the focal point is located at a distance of R/2 from the centre of mirror, known as the focal length. When parallel rays strike a convex mirror, they appear to originate from a specific point behind the mirror known as the virtual focus. Despite seeming to originate from this point, the rays reflect off different portions of the mirror (fig 35.7). In optics, we use a convention where the focal length is negative if the focus is behind the mirror. Thus, for a Figure 35. 7. Parallel rays striking a convex convex mirror, the focal length is negative, mirror appear to converge at a virtual focus point behind the mirror, reflecting off different portions while for a concave mirror, it's positive, at of the surface. R/2. Image formation in concave mirror A concave mirror reflects light in a way that forms images with varying sizes. Three principal rays are considered: P-ray: Travels parallel to the axis, reflects through the focal point. F-ray: Strikes the focal point, reflects parallel to the axis. C-ray: Passes through the center, reflects on itself. Figure 35. 8. Principal rays for a concave mirror: P-ray reflects through the focal point, F-ray reflects parallel after striking the focal point, and C-ray reflects through the center, illustrating image formation. When an object, like a pencil with a lit tip, is placed before a concave mirror, the light rays it emits reflect off the mirror. If the object is placed beyond the center (C), the reflected rays, P and F, intersect, forming a real axis where the image appears smaller than the object shown in fig 35.9 (a). If the object is between the center (C) and the focal point (F), the reflected rays, P and F, intersect, forming a real axis where the image appears larger than the object shown in fig 35.9 (b). This magnification property is why concave mirrors are used in applications like shaving mirrors, where an enlarged view is beneficial. Figure 35. 9. Image formation by a concave mirror: (a) a smaller image formed when the object is placed beyond the center (C), while fig (b) illustrates a larger image formed by the object placed between the center (C) and the focal point (F), highlighting the mirror's magnification property. In a concave mirror, an object like a pencil creates a real image. The ratio of the image size (h') to the object size (h) is known as the magnification factor. Mathematically, it's expressed as h'/ h or -q / p, where q is the distance from the mirror to the image, and p is the distance between the object and the mirror. Image formation in convex mirror When an object, such as a pencil, is placed in front of a convex mirror, the reflected rays— P-rays, F-rays, and C-rays—diverge. P-rays reflect upward, C-rays reflect along the same path, and F-rays reflect parallel to the axis. Extending these rays backward, they seem to converge at a point behind the mirror, forming a diminished, upright image of the object. Unlike concave mirrors, which produce inverted images (hence the negative magnification factor), convex mirrors always create upright images. The focal length for a convex mirror is represented as -R/2, indicating the virtual nature of the focus.  Figure 35. 10. A virtual, upright and diminished image formed by a convex mirror with diverging reflected rays appearing to converge behind the mirror. In convex mirrors, reflected light rays diverge, creating a scattered effect. Yet, when these rays are extended backward, they seem to converge at a point, forming a virtual image. The magnification factor, (M), is the ratio of the image height to the object height, expressed as (M = h'/ h), which is also (q / p). Unlike images formed by concave mirrors, images in convex mirrors remain upright, hence the absence of a negative sign in the magnification formula. Refraction and Lenses Lenses focus or disperse light due to the change in speed caused by the difference in refractive index between the lens material and the surrounding medium, like air or vacuum. The refractive index (n) is defined as the ratio of the speed of light in a vacuum to its speed in the medium. When light enters a lens, it slows down and bends towards normal due to refraction. In the case of a convex lens, the light converges towards the center, while in a concave lens, it diverges. In a convex lens, parallel light rays converge to a point, focusing the light. When a light source is positioned on the focal point of a lens, the emitted rays are parallel. Figure 35. 11. Diagrams can help visualize how convex lenses (a) converge light to a focal point, (b) Light emitted from the focal point of a lens emerges as parallel rays, while (c) concave lenses cause light to diverge, affecting the size and focus of the resulting image. If parallel light rays encounter a diverging (concave) lens, they scatter. Extending these diverging rays backward, they seem to meet at a virtual focal point, denoted as negative for concave lenses, while it remains positive for convex lenses. Lensmaker's formula for Spherical Single Lens The focal length of a spherical lens is determined using the lensmaker's formula. It accounts for the lens's curvature on both sides, which may differ, leading to different radii of curvature (R1 and R2). A larger radius corresponds to a flatter lens, whereas a smaller radius indicates a more curved and thus more powerful lens. The formula is: 1 1 1  = ( n − 1)  −  f  R1 R2  Here, n is the refractive index of the lens material, and 1 is the refractive index of air. R1 and R2 are the radii of curvature for the lens's surfaces. R2 is considered negative when the lens's curvature is inward from the light source's perspective. The power of a lens increases as the radii of curvature decreases. Since R2 is negative, its magnitude contributes positively to the lens's power. Different lens shapes include biconvex (curved outward on both sides), biconcave (curved inward on both sides), or a combination of convex and concave, including plano-convex (flat on one side and curved outward on the other), and plano-concave (flat on one side and curved inward on the other) configurations. Each type has unique properties affecting how light is refracted through them (fig 35.12).  Figure 35. 12. Various lens shapes have distinct properties influencing light refraction. Diopter The diopter is the reciprocal of a lens's focal length (f) in meters, expressed as Diopter = 1/f The lens's power, measured in diopters, depends on its refractive index (n) and the radii of curvature (R1 and R2) on each side. A lens with greater power can focus or diverge light more effectively. Image formation by convex Lens To understand how a lens shapes light, we trace the path of light rays through the lens. For a convex or converging lens, the image formed is always inverted. If an object, like a pencil, is placed close to the lens, the image appears larger (magnified) and virtual because the light rays diverge and do not meet after passing through the lens indicating magnification greater than 1. As the object moves further from the lens as shown in fig 35.13, the image becomes smaller and real, as the light rays converge to a point. The magnification changes with the object's distance from the lens, affecting the size of the image produced. As the distance between the object and the lens changes, the nature of the image also changes.  Figure 35. 13. "Figures illustrate the behavior of convex lens: close objects create larger, virtual, and magnified images, while distant objects form smaller, real images due to light ray convergence, demonstrating varying magnification and image properties. Image formation by concave Lens On the other hand, concave lenses produce a smaller, virtual image as they diverge from light rays. While concave lenses reduce the size of images, they are beneficial for correcting vision and other applications. Lens Defects: i. A concave lens's primary function is to diverge rays coming from a distant source, such as parallel rays from the Sun. However, lens makers face challenges like chromatic aberration (Aberration refers to imperfection), where different colors focus at different points due to varying refractive indices. This means that red and blue light, for instance, will converge at slightly different focal points, leading to a blurry image. This issue is tackled by employing various techniques, including applying specialized coatings to the lenses to minimize the difference in focal points for various colors, ensuring clearer images. ii. Spherical aberration occurs when rays striking different parts of a wide lens do not converge at a single point, leading to a distorted image. This defect is particularly pronounced in wide lenses where parallel rays don't meet at one focal  point. Reducing the lens's aperture can minimize aberration, allowing parallel rays to converge more effectively and produce a clearer image. Let's explore why do objects appear larger or smaller at different distances, and how can we calculate this? Optical Instruments The simple Magnifier Consider a convex lens, when it is held close to the thumb, the thumb appears its normal size. As the lens moves away from the thumb, the size of image changes, becoming larger, until it eventually defocuses. Figure 35. 14. A simple magnifier To calculate this, we use magnification, defined as the ratio of θ' to θ.  h/ f m = =  h / 25cm 25cm m = f 25 cm is the typical closest distance at which the human eye can focus comfortably, about 8 inches. Closer than this, objects defocus. A shorter focal length results in greater magnification, making small objects appear larger. However, there's a limit to magnification with a single lens, as too short a focal length can cause aberrations. Thus, there are practical limitations to how much a single lens can magnify. Compound Microscope In a compound microscope shown in fig 35.15, magnification is achieved through two lenses: the objective and the eyepiece. The object's height is denoted as h. Light rays originating from the object pass through the objective lens, creating an intermediate image of height h’ at F1’. This image is then magnified by the eyepiece lens, resulting in a larger final image.  Figure 35. 15. A compound microscope, featuring multiple lenses, magnifies small objects by combining the magnification powers of its objective and eyepiece lenses. The total magnification M of the microscope is the product of the magnifications by the objective (m) and the eyepiece (mθ) which is calculated as: s 25cm M = mm = − f ob f ey Here, s is the tube length, fob is the focal length of the objective lens, and fey is the focal length of the eyepiece. The negative sign indicates that the intermediate image is inverted. The dual-lens system allows for greater magnification than a single lens. Refracting Telescope A binocular or refracting telescope can also be constructed using two lenses to enhance the apparent size of distant objects (fig 35.16). The parallel rays are focused to form a small angle since the object is far away. This angle is the size of the object divided by the distance, a very small quantity. We aim to increase this angle. The objective lens captures the object at a distance, creating an image at the focal point of size h', which is then enlarged by the eyepiece for clearer viewing. The overall design allows for an increased angle of view, making distant objects appear closer and larger to the observer. Figure 35. 16. A refracting telescope comprised of two lenses to gather and focus light from distant objects, providing magnified views of celestial bodies. Human Eye In a normal eye, light focuses precisely on the retina, creating a clear image. This sharp focus allows the retina to transmit electrical signals to the brain, enabling us to perceive the image. People with poor vision particularly benefit from corrective lenses. Figure 35. 17. Image formation in normal human eye. Eye Defects and their Correction There are two common types of vision problems: Farsightedness (Hypermetropia): Difficulty seeing near objects because the image forms behind the retina. To correct hypermetropia, a convex lens is used, which adjusts the focal point, so the image falls on the retina. Figure 35. 18. Eye defects and their correction using spherical lenses: Hypermetropia (left), Myopia (right) Nearsightedness (Myopia): Difficulty seeing distant objects because the image forms in front of the retina. For myopia, a concave lens is employed to diverge the light rays, moving the focal point forward onto the retina. Thus, lenses help restore clear vision by compensating for the eye's focusing deficiencies. Prism and total Internal Reflection Prisms are fascinating optical devices that not only disperse light into its constituent colors but also enable total internal reflection. This phenomenon occurs when the angle of incidence exceeds a critical value, causing light to reflect entirely within the prism rather than passing through it. When light enters a prism and reflects internally, it can create an image that is the same size as the object if the rays are parallel. If the prism is rotated, the image can be inverted (see fig 35.19). This principle is utilized in binoculars to correct the orientation of the observed image, ensuring that it appears right-side-up to the viewer. Figure 35. 19. Total internal reflection within a prism creates an image the same size as the object with parallel rays, capable of inversion when the prism is rotated. In binoculars, prisms play a crucial role. They are used to extend the optical path without increasing the physical length of the binoculars, which keeps them compact. The prisms inside binoculars cause light to undergo total internal reflection, effectively lengthening the path the light travels before reaching the eyes. This design allows for greater magnification power while maintaining a portable size. Figure 35. 20. The diagram illustrates light reflecting through multiple prisms, effectively extending the path length within a shorter physical space. Prisms can be used as mirrors through the phenomenon of total internal reflection. When parallel rays strike the prism, it can be reflected off the internal surfaces if the angle of incidence is greater than the critical angle for the material. This reflection can effectively turn the prism into a mirror without the need for a reflective coating.  Figure 35. 21. Prisms utilizing total internal reflection to act as mirrors for parallel rays, reflecting internally without external coatings. Advanced Optics: Gravitational Lensing Light typically bends when it passes through materials with different refractive indices, such as air or glass. However, in space, light can also bend due to gravity, a phenomenon predicted by Einstein's general theory of relativity. This bending occurs near massive objects like stars or black holes having strong gravitational fields, creating what is known as a gravitational lens. Gravitational lensing allows us to observe distant objects like quasars, as their light is focused by these massive celestial bodies. This effect has been observed and helps astronomers study the distribution of matter in the universe.