1) Huygens' principle states that every point on a wavefront can be considered a source of secondary wavelets, and the new wavefront is the envelope of these secondary wavelets. Fresnel built on this by considering the interference of these wavelets.
2) Snell's law describes how the wavelength and speed of light change when passing from one medium to another with a different refractive index, with the frequency remaining the same.
3) Fermat's principle states that between two points, the path taken by a ray of light is the path that can be traversed in the least time, explaining the bending of light at interfaces.
DISPERSION OF WHITE LIGHT THROUGH A GLASS PRISMSethu Ram
The document discusses the dispersion of white light through a prism and the formation of rainbows. It explains that white light consists of all visible colors, and when passed through a prism or water droplets, the different colors disperse at different speeds and angles. This causes the colors to separate, forming a visible spectrum. Newton's experiment showed that passing the dispersed light through a second prism recombines the colors back into white light. Rainbows form when sunlight enters water droplets in the atmosphere and disperses internally before refracting back out.
Lasers emit a focused, intense beam of coherent light through stimulated emission of radiation. The first laser was created in 1960 by Theodore Maiman using a ruby crystal. Lasers have common components including an active medium to emit light, an excitation mechanism to energize atoms, and mirrors to reflect the beam through the medium. Laser light is useful for applications like barcodes, surgery, manufacturing, and communications due to its monochromatic, coherent, and collimated properties.
Coherent light refers to light rays that travel closely packed in straight parallel lines, like in a sunbeam. Examples of coherent light include lasers, which emit visible light beams that diverge very little over long distances. Automobile headlights and spotlights also emit coherent light by directing rays into a narrow, well-defined beam. Intense direct sunlight passing through a small opening also forms a coherent light beam. Coherent light waves are "in phase" with one another, meaning the crests and troughs of each wave are aligned.
This document summarizes the ruby laser. It begins by explaining that a ruby laser uses a synthetic ruby crystal as its laser medium, which was the first successful laser developed in 1960. It emits deep red light at a wavelength of 694.3 nm. The ruby crystal is doped with small amounts of chromium ions, which provide the necessary population inversion to achieve lasing. When optically pumped by a flash lamp, chromium ions are excited to higher energy states and decay to a metastable state, building up population inversion between that state and the ground state. Stimulated emission then produces coherent red light that is amplified as it reflects within the ruby crystal's resonance cavity and emerges through the partially reflective end.
- Diffraction occurs when waves pass through small openings, around obstacles, or by sharp edges. This causes the waves to spread out after passing through the openings.
- A single slit placed between a light source and screen produces a diffraction pattern of alternating bright and dark bands called interference fringes. The spacing and intensity of the fringes depends on the wavelength of light and the width of the slit.
- In single-slit diffraction, each part of the slit acts as a secondary source, and the light interferes depending on the path differences between waves, causing constructive and destructive interference at different angles.
The document describes an experiment to determine the separation between the plates of a Fabry Perot etalon. It provides background on the Fabry Perot interferometer and the principle of interference in the etalon. The experimental setup involves illuminating the etalon with a laser and measuring the angular diameters of interference fringes observed on a screen. By plotting the order of interference versus the cosine of the fringe angles and determining the slope, the separation between the etalon plates is calculated as approximately 2-3 mm, remaining constant despite varying the screen distance.
1. Diffraction refers to the bending of light around obstacles or through openings, and results in interference patterns.
2. There are two main types of diffraction: Fresnel diffraction occurs when light passes near an obstacle, while Fraunhofer diffraction occurs when light passes through or around objects and the observation is made far from the obstacle.
3. Diffraction gratings consist of many parallel slits and cause light to diffract into several beams. The angles and intensities of these beams can be determined through analysis of interference from the multiple slits.
The document discusses wave propagation and Huygens' principle. It begins by describing circular waves that propagate outward from a point source on the water's surface. It then describes spherical waves, which propagate outward in three dimensions from a point source, like a sound wave from a microphone. Huygens' principle is introduced as the idea that every point on a wavefront can be considered a secondary source of spherical waves, and that the resulting wave can be determined by considering the sum of waves from all of these secondary sources. The document provides steps for applying Huygens' principle to model wave propagation.
DISPERSION OF WHITE LIGHT THROUGH A GLASS PRISMSethu Ram
The document discusses the dispersion of white light through a prism and the formation of rainbows. It explains that white light consists of all visible colors, and when passed through a prism or water droplets, the different colors disperse at different speeds and angles. This causes the colors to separate, forming a visible spectrum. Newton's experiment showed that passing the dispersed light through a second prism recombines the colors back into white light. Rainbows form when sunlight enters water droplets in the atmosphere and disperses internally before refracting back out.
Lasers emit a focused, intense beam of coherent light through stimulated emission of radiation. The first laser was created in 1960 by Theodore Maiman using a ruby crystal. Lasers have common components including an active medium to emit light, an excitation mechanism to energize atoms, and mirrors to reflect the beam through the medium. Laser light is useful for applications like barcodes, surgery, manufacturing, and communications due to its monochromatic, coherent, and collimated properties.
Coherent light refers to light rays that travel closely packed in straight parallel lines, like in a sunbeam. Examples of coherent light include lasers, which emit visible light beams that diverge very little over long distances. Automobile headlights and spotlights also emit coherent light by directing rays into a narrow, well-defined beam. Intense direct sunlight passing through a small opening also forms a coherent light beam. Coherent light waves are "in phase" with one another, meaning the crests and troughs of each wave are aligned.
This document summarizes the ruby laser. It begins by explaining that a ruby laser uses a synthetic ruby crystal as its laser medium, which was the first successful laser developed in 1960. It emits deep red light at a wavelength of 694.3 nm. The ruby crystal is doped with small amounts of chromium ions, which provide the necessary population inversion to achieve lasing. When optically pumped by a flash lamp, chromium ions are excited to higher energy states and decay to a metastable state, building up population inversion between that state and the ground state. Stimulated emission then produces coherent red light that is amplified as it reflects within the ruby crystal's resonance cavity and emerges through the partially reflective end.
- Diffraction occurs when waves pass through small openings, around obstacles, or by sharp edges. This causes the waves to spread out after passing through the openings.
- A single slit placed between a light source and screen produces a diffraction pattern of alternating bright and dark bands called interference fringes. The spacing and intensity of the fringes depends on the wavelength of light and the width of the slit.
- In single-slit diffraction, each part of the slit acts as a secondary source, and the light interferes depending on the path differences between waves, causing constructive and destructive interference at different angles.
The document describes an experiment to determine the separation between the plates of a Fabry Perot etalon. It provides background on the Fabry Perot interferometer and the principle of interference in the etalon. The experimental setup involves illuminating the etalon with a laser and measuring the angular diameters of interference fringes observed on a screen. By plotting the order of interference versus the cosine of the fringe angles and determining the slope, the separation between the etalon plates is calculated as approximately 2-3 mm, remaining constant despite varying the screen distance.
1. Diffraction refers to the bending of light around obstacles or through openings, and results in interference patterns.
2. There are two main types of diffraction: Fresnel diffraction occurs when light passes near an obstacle, while Fraunhofer diffraction occurs when light passes through or around objects and the observation is made far from the obstacle.
3. Diffraction gratings consist of many parallel slits and cause light to diffract into several beams. The angles and intensities of these beams can be determined through analysis of interference from the multiple slits.
The document discusses wave propagation and Huygens' principle. It begins by describing circular waves that propagate outward from a point source on the water's surface. It then describes spherical waves, which propagate outward in three dimensions from a point source, like a sound wave from a microphone. Huygens' principle is introduced as the idea that every point on a wavefront can be considered a secondary source of spherical waves, and that the resulting wave can be determined by considering the sum of waves from all of these secondary sources. The document provides steps for applying Huygens' principle to model wave propagation.
The document discusses lasers, providing details on:
1. How lasers work through the process of stimulated emission of radiation, using a pumping mechanism to create population inversion in the active medium.
2. The key characteristics of laser light being monochromatic, coherent, and highly directional.
3. Examples of common laser types like Ruby and Nd:YAG lasers, describing their construction and working.
4. Applications of lasers in various fields like industry, medicine, communication, and more.
This document describes Newton's rings experiment to observe the interference of light. When a plano-convex lens is placed on a glass slide, a thin air film is formed of varying thickness. Circular interference fringes called Newton's rings are seen when monochromatic light is shone on the setup. The rings appear as alternating bright and dark circles whose diameters are used to determine the wavelength of light through mathematical formulas derived from light interference principles.
A laser is a device that emits light through stimulated emission of radiation, as described by Theodore Maiman who built the first laser in 1960. Lasers produce coherent, monochromatic, collimated light that is useful for applications like barcodes, surgery, welding, and fiber optics. Laser light is more powerful and focused than ordinary light. Lasers are classified based on their hazard levels, with class 4 lasers most dangerous. While lasers have advantages like precision cutting, they also have disadvantages like high costs and safety risks if not properly handled.
This document discusses interference, which occurs when two or more waves overlap. There are two types of interference: constructive and destructive. Constructive interference occurs when waves are displaced in the same direction and amplitudes add, while destructive interference occurs when they are displaced in opposite directions and amplitudes subtract. The document provides examples of interference in light, radio, acoustic, and water waves. It describes Young's double-slit experiment, which demonstrated that light behaves as waves that can interfere and was evidence against the particle theory of light.
The document discusses the phenomenon of interference of light. It explains the conditions required for interference, including coherent sources, monochromatic light, and a constant path difference. It describes several classic interference experiments, including Young's double slit experiment, Fresnel's bi-prism, Newton's rings, and Michelson's interferometer. It discusses how interference patterns are used to determine properties like wavelength and refractive index.
The document summarizes key concepts in optics and optical properties of materials. It discusses topics like electromagnetic radiation spectrum, optical classifications of materials as transparent, translucent or opaque. It also covers concepts like reflection, refraction, absorption, transmission and how they relate to the band structure and band gaps of materials. Specific phenomena like fluorescence, phosphorescence, photoelasticity and their working principles are defined. Applications of optics like lasers, optical data storage are also briefly mentioned.
Light and matter exhibit wave-particle duality, behaving as both particles and waves. When light passes through two slits, it creates an interference pattern like a wave. However, when using a sensitive film, tiny light particles are observed, suggesting particle behavior. Einstein acknowledged two necessary but logically unconnected theories of light. The double slit experiment results cannot be fully explained by treating light solely as particles or waves. While one theory was that photons interacted to cause interference, experiments making the light extremely dim found it was virtually impossible for two photons to be present at the same time. Thus light and matter demonstrate both wave and particle properties and cannot be described by only one model.
Carbon dioxide lasers produce a beam of infrared light with wavelengths of 9.6 and 10.6 micrometers. They work by using an electric discharge to excite carbon dioxide molecules and create a population inversion between vibrational energy levels. This leads to stimulated emission and laser action. Carbon dioxide lasers are efficient and can produce high powers, making them useful for applications like material processing, welding, communication, remote sensing, and surgery.
Optical interferometry uses light interference to provide extremely precise measurements. When two light waves are combined, they can produce interference fringes of light and dark bands that contain information about the optical path differences between the waves. Recent advances in lasers, fiber optics, and digital processing have expanded applications of optical interferometry from measuring molecular sizes to diameters of stars.
The document discusses the principles of superposition and interference of waves. Constructive interference occurs when wave crests and troughs overlap, increasing amplitude, while destructive interference occurs when crests and troughs cancel out. Interference is responsible for the colors seen in soap bubbles and is exploited in holography and interferometry. Examples of interference include light waves, water waves, radio waves, and sound waves.
Reflection and refraction are changes in the direction of a wave at an interface between two media. Reflection occurs when a wave returns to its original medium at the interface, with the angle of the reflected wave being equal to the angle of the incident wave. Refraction occurs when a wave bends as it passes from one medium to another where its speed changes, with the angle of refraction determined by Snell's law. Examples of reflection and refraction include the bending of light, sound, and water waves at boundaries between different materials or densities.
1. Resolution refers to the ability to distinguish between two point sources or details of an object. The resolving power depends on the wavelength of light and the diameter of the aperture.
2. According to Rayleigh's criterion, two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.
3. For the human eye, the resolving power is about 2x10^-4 rad, meaning two sources can be distinguished when separated by an angle of at least this amount.
1. The document discusses the working principles of lasers, including the key components of a laser system and the processes of stimulated emission and population inversion that enable laser action.
2. It specifically describes different laser types such as ruby lasers, He-Ne lasers, semiconductor diode lasers, and their applications. Ruby was the first laser invented and produces red light, while He-Ne lasers emit visible light in the red and infrared spectrum.
3. The document provides detailed explanations of laser concepts like optical pumping, energy level diagrams, cavity mirrors, and continuous wave versus pulsed operation.
This document describes Newton's rings experiment. When a plano-convex lens is placed on a glass plate, it forms a wedge-shaped air film between them whose thickness increases outward from the point of contact. Light incident on this film produces concentric alternating bright and dark rings when viewed through a microscope. The interference is caused by the path difference between light rays partially transmitted through the upper and lower surfaces of the air film. The diameters of the rings are directly proportional to the thickness of the air film. The central spot is dark due to destructive interference when the path difference is half the wavelength of light.
The document discusses solid-state lasers, which use a crystalline solid as the amplifying medium doped with ions that emit light through stimulated emission. Common solid materials used include ruby, titanium sapphire, and neodymium-doped crystals. The doped ions are chosen from rare-earth, transition metal, and actinide elements for their radiative properties. Solid-state lasers provide higher gain density and good thermal and optical qualities compared to other laser types.
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and
offering a wide range of dental certified courses in different formats.for more details please visit
www.indiandentalacademy.com
Christiaan Huygens was a 17th century Dutch scientist who made important contributions to physics, including developing Huygens' Principle which describes wave propagation. Huygens' Principle represents each point on a wave front as a point source of secondary spherical wavelets, with the new wave front being the tangent lines to these secondary wavelets. Plane waves have parallel wave fronts, while spherical waves have expanding spherical wave fronts due to the point source nature of the secondary wavelets.
Geometrical optics is concerned with how light propagates, reflects, and refracts using a ray model of light. There are four main postulates: (1) light travels in straight lines in a homogeneous medium, (2) the angle of reflection equals the angle of incidence, (3) Snell's law governs refraction, and (4) independent light beams do not interact. Geometrical optics is used to understand image formation by lenses and mirrors. Real images are formed when light rays actually intersect, while virtual images are formed by rays that appear to intersect if extended.
This PPT gives an elementary idea about dispersion. The dispersion through prism is discussed in some details & combination of prisms are made to make either dispersion or deviation to be equal to zero.
Study material 12th Physics - Wave Theory of LightEdnexa
The document outlines Christiaan Huygens' wave theory of light from the 17th century. It proposes that light consists of longitudinal waves that propagate in a straight line through a hypothetical medium called the luminiferous ether. According to the theory, each point on a wavefront acts as a secondary source of waves, and the movement of the wavefront over time can be determined using Huygens' principle and construction. The wave theory was able to successfully explain several optical phenomena like reflection, refraction, and interference of light. However, it could not explain some observations like the rectilinear propagation of light and the photoelectric effect.
This document discusses interference patterns produced by double slit diffraction of laser light with a wavelength of 400 nm.
It provides calculations to determine: 1) the angular separation between the m=0 and m=1 bright fringes is 1.15 degrees, 2) the distance between the m=0 and m=1 fringes on the screen is 0.015 m, and 3) the distance between the m=1 bright fringe and the dark fringe between m=1 and m=2 is 0.0075 m.
The document discusses lasers, providing details on:
1. How lasers work through the process of stimulated emission of radiation, using a pumping mechanism to create population inversion in the active medium.
2. The key characteristics of laser light being monochromatic, coherent, and highly directional.
3. Examples of common laser types like Ruby and Nd:YAG lasers, describing their construction and working.
4. Applications of lasers in various fields like industry, medicine, communication, and more.
This document describes Newton's rings experiment to observe the interference of light. When a plano-convex lens is placed on a glass slide, a thin air film is formed of varying thickness. Circular interference fringes called Newton's rings are seen when monochromatic light is shone on the setup. The rings appear as alternating bright and dark circles whose diameters are used to determine the wavelength of light through mathematical formulas derived from light interference principles.
A laser is a device that emits light through stimulated emission of radiation, as described by Theodore Maiman who built the first laser in 1960. Lasers produce coherent, monochromatic, collimated light that is useful for applications like barcodes, surgery, welding, and fiber optics. Laser light is more powerful and focused than ordinary light. Lasers are classified based on their hazard levels, with class 4 lasers most dangerous. While lasers have advantages like precision cutting, they also have disadvantages like high costs and safety risks if not properly handled.
This document discusses interference, which occurs when two or more waves overlap. There are two types of interference: constructive and destructive. Constructive interference occurs when waves are displaced in the same direction and amplitudes add, while destructive interference occurs when they are displaced in opposite directions and amplitudes subtract. The document provides examples of interference in light, radio, acoustic, and water waves. It describes Young's double-slit experiment, which demonstrated that light behaves as waves that can interfere and was evidence against the particle theory of light.
The document discusses the phenomenon of interference of light. It explains the conditions required for interference, including coherent sources, monochromatic light, and a constant path difference. It describes several classic interference experiments, including Young's double slit experiment, Fresnel's bi-prism, Newton's rings, and Michelson's interferometer. It discusses how interference patterns are used to determine properties like wavelength and refractive index.
The document summarizes key concepts in optics and optical properties of materials. It discusses topics like electromagnetic radiation spectrum, optical classifications of materials as transparent, translucent or opaque. It also covers concepts like reflection, refraction, absorption, transmission and how they relate to the band structure and band gaps of materials. Specific phenomena like fluorescence, phosphorescence, photoelasticity and their working principles are defined. Applications of optics like lasers, optical data storage are also briefly mentioned.
Light and matter exhibit wave-particle duality, behaving as both particles and waves. When light passes through two slits, it creates an interference pattern like a wave. However, when using a sensitive film, tiny light particles are observed, suggesting particle behavior. Einstein acknowledged two necessary but logically unconnected theories of light. The double slit experiment results cannot be fully explained by treating light solely as particles or waves. While one theory was that photons interacted to cause interference, experiments making the light extremely dim found it was virtually impossible for two photons to be present at the same time. Thus light and matter demonstrate both wave and particle properties and cannot be described by only one model.
Carbon dioxide lasers produce a beam of infrared light with wavelengths of 9.6 and 10.6 micrometers. They work by using an electric discharge to excite carbon dioxide molecules and create a population inversion between vibrational energy levels. This leads to stimulated emission and laser action. Carbon dioxide lasers are efficient and can produce high powers, making them useful for applications like material processing, welding, communication, remote sensing, and surgery.
Optical interferometry uses light interference to provide extremely precise measurements. When two light waves are combined, they can produce interference fringes of light and dark bands that contain information about the optical path differences between the waves. Recent advances in lasers, fiber optics, and digital processing have expanded applications of optical interferometry from measuring molecular sizes to diameters of stars.
The document discusses the principles of superposition and interference of waves. Constructive interference occurs when wave crests and troughs overlap, increasing amplitude, while destructive interference occurs when crests and troughs cancel out. Interference is responsible for the colors seen in soap bubbles and is exploited in holography and interferometry. Examples of interference include light waves, water waves, radio waves, and sound waves.
Reflection and refraction are changes in the direction of a wave at an interface between two media. Reflection occurs when a wave returns to its original medium at the interface, with the angle of the reflected wave being equal to the angle of the incident wave. Refraction occurs when a wave bends as it passes from one medium to another where its speed changes, with the angle of refraction determined by Snell's law. Examples of reflection and refraction include the bending of light, sound, and water waves at boundaries between different materials or densities.
1. Resolution refers to the ability to distinguish between two point sources or details of an object. The resolving power depends on the wavelength of light and the diameter of the aperture.
2. According to Rayleigh's criterion, two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.
3. For the human eye, the resolving power is about 2x10^-4 rad, meaning two sources can be distinguished when separated by an angle of at least this amount.
1. The document discusses the working principles of lasers, including the key components of a laser system and the processes of stimulated emission and population inversion that enable laser action.
2. It specifically describes different laser types such as ruby lasers, He-Ne lasers, semiconductor diode lasers, and their applications. Ruby was the first laser invented and produces red light, while He-Ne lasers emit visible light in the red and infrared spectrum.
3. The document provides detailed explanations of laser concepts like optical pumping, energy level diagrams, cavity mirrors, and continuous wave versus pulsed operation.
This document describes Newton's rings experiment. When a plano-convex lens is placed on a glass plate, it forms a wedge-shaped air film between them whose thickness increases outward from the point of contact. Light incident on this film produces concentric alternating bright and dark rings when viewed through a microscope. The interference is caused by the path difference between light rays partially transmitted through the upper and lower surfaces of the air film. The diameters of the rings are directly proportional to the thickness of the air film. The central spot is dark due to destructive interference when the path difference is half the wavelength of light.
The document discusses solid-state lasers, which use a crystalline solid as the amplifying medium doped with ions that emit light through stimulated emission. Common solid materials used include ruby, titanium sapphire, and neodymium-doped crystals. The doped ions are chosen from rare-earth, transition metal, and actinide elements for their radiative properties. Solid-state lasers provide higher gain density and good thermal and optical qualities compared to other laser types.
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and
offering a wide range of dental certified courses in different formats.for more details please visit
www.indiandentalacademy.com
Christiaan Huygens was a 17th century Dutch scientist who made important contributions to physics, including developing Huygens' Principle which describes wave propagation. Huygens' Principle represents each point on a wave front as a point source of secondary spherical wavelets, with the new wave front being the tangent lines to these secondary wavelets. Plane waves have parallel wave fronts, while spherical waves have expanding spherical wave fronts due to the point source nature of the secondary wavelets.
Geometrical optics is concerned with how light propagates, reflects, and refracts using a ray model of light. There are four main postulates: (1) light travels in straight lines in a homogeneous medium, (2) the angle of reflection equals the angle of incidence, (3) Snell's law governs refraction, and (4) independent light beams do not interact. Geometrical optics is used to understand image formation by lenses and mirrors. Real images are formed when light rays actually intersect, while virtual images are formed by rays that appear to intersect if extended.
This PPT gives an elementary idea about dispersion. The dispersion through prism is discussed in some details & combination of prisms are made to make either dispersion or deviation to be equal to zero.
Study material 12th Physics - Wave Theory of LightEdnexa
The document outlines Christiaan Huygens' wave theory of light from the 17th century. It proposes that light consists of longitudinal waves that propagate in a straight line through a hypothetical medium called the luminiferous ether. According to the theory, each point on a wavefront acts as a secondary source of waves, and the movement of the wavefront over time can be determined using Huygens' principle and construction. The wave theory was able to successfully explain several optical phenomena like reflection, refraction, and interference of light. However, it could not explain some observations like the rectilinear propagation of light and the photoelectric effect.
This document discusses interference patterns produced by double slit diffraction of laser light with a wavelength of 400 nm.
It provides calculations to determine: 1) the angular separation between the m=0 and m=1 bright fringes is 1.15 degrees, 2) the distance between the m=0 and m=1 fringes on the screen is 0.015 m, and 3) the distance between the m=1 bright fringe and the dark fringe between m=1 and m=2 is 0.0075 m.
This document provides an overview of the principles of laser operation. It discusses:
- Laser cavities consisting of an amplifying medium between two mirrors that provide feedback.
- Fabry-Perot resonators and the standing wave patterns that form from interference between waves moving in opposite directions within the cavity.
- Population inversion being necessary for stimulated emission to exceed absorption, allowing amplification of light passing through the active medium.
- Optical pumping being used to invert the population by exciting atoms to a long-lived excited state, building up a population there.
- Stimulated emission causing photons to be emitted in phase with the stimulating photon, allowing amplification through an avalanche effect within the inverted medium.
The document summarizes the operating principles of phototransistors and photoconductive detectors.
- Phototransistors are bipolar junction transistors that use the photocurrent generated in the base-collector junction to inject a multiplied current into the emitter circuit, similar to a common emitter transistor. The photocurrent acts as the base current.
- Photoconductive detectors have two electrodes attached to a light-absorbing semiconductor. Absorbed photons increase conductivity and the external photocurrent. With ohmic contacts, multiple electrons enter the semiconductor for each hole, producing photoconductive gain.
- The main sources of noise in photodetectors are shot noise from the dark current and photocurrent. The total noise
The document discusses wave optics and electromagnetic waves. It defines key concepts like wavefronts, which connect points of equal phase, and rays, which describe the direction of wave propagation perpendicular to wavefronts. It explains Huygens' principle, which states that each point on a wavefront acts as a secondary source of spherical wavelets to determine the new wavefront position. The principle of superposition states that multiple waves add linearly at each point in space to determine the resulting disturbance. Interference occurs when waves are out of phase and their amplitudes diminish or vanish.
This document summarizes key concepts in geometrical optics, including:
- Ray optics approximates light propagation using rays and geometric rules. Reflection and refraction at an interface follow laws like Snell's law.
- Plane mirrors form virtual, erect images. Spherical mirrors form real images that are inverted with magnification determined by the mirror equation.
- Lenses are analyzed similarly using conjugate planes and the lensmaker's equation. They can form real or virtual images, magnified or demagnified, depending on object and image distances.
This document provides information about optical components and their properties. It discusses plane and spherical surfaces, Snell's law, and thin lenses. The key points are:
1) It defines optical terms like object and image conventions, focal length conventions, and radius of curvature conventions.
2) It explains how to model optical components like plane and spherical surfaces, thin lenses, and thick lenses using matrix methods. The matrix for a single component can be derived and components can be combined by multiplying their matrices.
3) Examples are given for calculating properties of simple lens systems like a thin lens in air using matrices and lensmaker's equation. Ray tracing is also demonstrated through matrix methods.
Sound Waves: Relating Amplitude, Power and Intensitylyssawyh
My LO addresses the relationship between displacement amplitude, power and intensity of sound waves. I made a PowerPoint with a couple of problems that shows and works with this relationship to further understand it.
1) An AC generator with an RMS voltage of 110 V is connected in series with a 35-Ω resistor and 1-μF capacitor. To maintain a current of 1.2 A, the generator must operate at a frequency of 1.9 kHz.
2) An AC generator with an emf of 22.8 V at 353 rad/s is connected to a 17.3 H inductor. When the current is maximum, the emf is 22.8 V. When the emf is -11.4 V and increasing, the current is 4.11 A.
3) A series RLC circuit with a 148-Ω resistor, 1.50-μF capacitor
Graded index (GRIN) optical fibers have a refractive index that decreases continuously from the core center to the cladding. This results in curved ray paths inside the core rather than straight lines, reducing intermodal dispersion. The optimal refractive index profile for minimizing dispersion is parabolic. Attenuation in optical fibers is due to various factors including material absorption, scattering, and bending losses. Rayleigh scattering increases at shorter wavelengths, while absorption peaks exist for hydroxyl and metal impurities.
This document provides an overview of key electronics concepts and components. It discusses basic concepts like capacitors, inductors, and measurements tools. It also covers topics such as types of circuits, Kirchhoff's laws, Ohm's law, resistors, and power sources. The document aims to introduce fundamental electronics principles and components.
Light has several properties that make it useful for information processing and optical communication systems. It can be transmitted without interference from electrical signals or other light beams crossing its path. Optical signals also allow high parallelism and bandwidth exceeding 1013 bits per second. Radiation sources can be classified by their flux output and spectrum. Light behaves as an electromagnetic wave that propagates through space as oscillating electric and magnetic fields. In a material medium, the light's phase velocity decreases and is characterized by the medium's refractive index. Crystalline materials exhibit anisotropic refractive indices depending on the propagation and polarization directions.
The document describes the operation of pn-junction and pin photodiodes. Pn-junction photodiodes convert light to electrical signals by separating electron-hole pairs generated by photon absorption in the depletion region. The quantum efficiency and responsivity characterize a photodiode's performance. Pin photodiodes have wider depletion widths than pn-junctions, allowing detection at higher frequencies and wavelengths. The intrinsic region in pin diodes provides a uniform electric field for carrier separation and drift, improving efficiency.
1. Young's experiment demonstrated interference using a single wavefront that was split into two coherent secondary sources by passing the wavefront through two slits. The overlapping waves from the two slits interfered and produced an interference pattern.
2. Thin film interference occurs when a beam of light is split by reflection and transmission at the interfaces of a thin film. The optical path difference between the reflected and transmitted beams depends on factors like the film thickness and refractive indices, leading to constructive or destructive interference and the appearance of colored fringes.
3. Interferometers like Michelson's use arrangements of mirrors and beamsplitters to split a light beam into two paths that recombine to produce interference patterns, which can
1. Light has both wave and particle properties, though historically there were separate theories proposing one or the other.
2. Thomas Young's double slit experiment provided early evidence of light's wave nature by producing interference patterns. Other experiments like thin film interference and diffraction around obstacles further supported this.
3. Albert Einstein explained the photoelectric effect by proposing light also behaves as discrete packets of energy called photons, providing evidence of its particle nature.
This document provides an overview of equivalent circuits and circuit analysis techniques including node-voltage analysis, mesh analysis, and dealing with dependent and independent sources. It defines equivalent circuits as circuits that can replace one another without changing the external behavior of the overall circuit. It also describes node-voltage and mesh analysis, specifying how to write equations for each method by applying Kirchhoff's laws. Techniques for handling dependent sources and circuits with no path to ground are discussed. Examples demonstrate transforming between delta-wye configurations and using the different analysis methods to solve for voltages and currents.
- Ray theory explains many optical phenomena by considering light to travel in narrow paths called rays. Rays obey simple rules like reflection and Snell's laws.
- Optical resonators like Fabry-Perot cavities allow only certain resonant wavelengths/frequencies to propagate through interference of reflected waves. The modes are separated by the free spectral range and have a spectral width that depends on the finesse.
- Diffraction occurs when light passes through an aperture or obstruction, causing the beam to diverge and form an intensity pattern due to interference between wavelets from each point in the aperture. This pattern is explained by Huygens' principle and diffraction theory.
This document discusses various methods for measuring medium resistances between 1-0.1M Ω. It describes four common methods: the voltmeter-ammeter method, substitution method, ohmmeter method, and bridge circuit method. The bridge circuit method operates on the principle of null comparison using a Wheatstone bridge configuration with ratio and standard arms. Sources of error in resistance measurements include thermal EMF and contact resistance. Worked examples are provided to calculate resistance values and measurement uncertainties.
When two light waves pass through the same point in space simultaneously, interference occurs. Constructive interference happens when the waves are in phase and add to produce a larger wave, while destructive interference occurs when they are out of phase and cancel each other out. The intensity of the resulting interference pattern depends on the phase difference between the waves. In a double slit experiment, the phase difference and resulting interference is determined by the path length difference between waves passing through each slit.
Interference and diffraction are phenomena that occur when two waves overlap and interact. Young's double-slit experiment demonstrated interference by shining light through two slits and observing alternating bright and dark bands in the interference pattern on a screen. The intensity of the bands is determined by the path difference between waves from the two slits, which causes constructive or destructive interference. This experiment provided evidence that light behaves as a wave and established the foundation for optics.
The document discusses various topics related to wave optics and the physics of light, including:
- The wave nature of light and how it explains phenomena like reflection, refraction, the formation of shadows and spectra.
- Huygens' principle which states that each point on a wavefront is the source of secondary wavelets and the new wavefront is the tangent to these wavelets.
- The laws of reflection which state that the angle of incidence equals the angle of reflection.
- Refraction and how the speed and wavelength of light changes when passing from one medium to another.
- Interference and coherence - the addition of waves to form a resultant wave, and how coherent sources are required
MAHARASHTRA STATE BOARD
CLASS XI AND XII
PHYSICS
CHAPTER 7
WAVE OPTICS
CONTENT:
Huygen's principle.
Huygen's principles & proof of laws of reflection/refraction.
Condition for construction & destruction of coherent waves.
Young's double slit experiment.
Modified Young's double slit experiment.
Intensity of light in Y.D.S.E.
Diffraction due to single slit.
Polarisation & doppler effect.
This document provides a history of theories about the nature of light and summarizes key concepts about reflection and refraction of light. It describes how light was initially thought to consist of particles, then was explained as a wave by Huygens, and was later found to have both wave-like and particle-like properties. The document outlines the laws of reflection and refraction, including Snell's law, total internal reflection, and their explanations via Huygens' principle. It also discusses applications such as fiber optics, rainbows, and dispersion.
1. Huygen's wave theory proposes that light propagates as waves, with each point on a wavefront acting as a secondary source of spherical wavelets.
2. The wave theory can explain phenomena like reflection, refraction, and interference through Huygen's principle and the laws of reflection and refraction.
3. Maxwell's electromagnetic theory unified electric and magnetic fields and correctly predicted the speed of light, providing strong evidence for the wave nature of light.
This document provides a summary of key concepts in reflection and refraction of light:
- Light was originally thought to consist of particles (1000 AD) but was later explained as a wave by Huygens in the 1600s and Maxwell in 1865. Planck later showed it has particle-like properties as well.
- Reflection follows the law that the angle of incidence equals the angle of reflection. Refraction follows Snell's law, which relates the indices of refraction and angles of the materials. Dispersion is the dependence of the index of refraction on wavelength.
- Huygen's principle treats each point on a wavefront as a secondary source, and the new wavefront is tangent to these secondary
Light is an electromagnetic wave that exhibits properties of both waves and particles. As a wave, it propagates at a constant speed of about 3x108 m/s in a vacuum. It has characteristics such as wavelength, frequency, and amplitude. Light also behaves as discrete particles called photons, with energy levels dependent on frequency. The electromagnetic spectrum encompasses all different wavelengths of electromagnetic radiation including visible light, which humans can see. Refractive index is the ratio of light's speed in a vacuum to its speed in a material, and determines how much its path is bent upon entering the material.
Waves can be categorized as mechanical or electromagnetic. Mechanical waves require a medium to travel through, while electromagnetic waves do not. Waves can also be transverse or longitudinal depending on the direction of particle oscillation relative to wave propagation. Important wave properties include amplitude, wavelength, frequency, and speed. Reflection, refraction, diffraction, interference, and polarization are key wave phenomena. Reflection follows the laws of reflection, while refraction follows Snell's law. Diffraction and interference result in constructive and destructive patterns. Polarization occurs when waves vibrate in a single plane. Waves have many applications including ultrasound imaging, fiber optics, and 3D displays.
Measurement of physical optics and microwavesSubhasis Shit
This document provides an overview of physical optics and microwaves. It discusses several topics including interference using a Michelson interferometer, diffraction of light and microwaves, photoconductivity, and polarization of light. For interference, it describes types of interference and performing measurements using a Michelson interferometer to determine the wavelength of a laser and refractive index of a glass plate. For diffraction, it discusses Fresnel and Fraunhofer diffraction and performing experiments to observe diffraction patterns from microwaves and laser light. It also presents the results of experiments measuring photoconductivity and verifying Malus' law of polarization.
This document discusses transverse wave motion. It defines transverse waves as disturbances that occur perpendicular to the direction of propagation. Transverse waves include electromagnetic waves and waves on strings. The document covers characteristics of waves like wavelength and frequency. It derives the one-dimensional wave equation and explores solutions and properties of transverse waves, including phase velocity, group velocity, and impedance. Key concepts covered are the definitions of progressive and standing waves, and the distinction between particle/oscillator velocity and wave/phase velocity.
Paras Sundriyal presented on the topic of interference to Mrs. Ramna Tripathi. They discussed key concepts of interference including coherent sources, conditions for interference, and types of interference like constructive and destructive. Specific experiments were covered like Young's double slit experiment, fringe width, displacement of fringes, Stokes treatment, and Newton's rings experiment using a plano-convex lens and glass plate to form interference patterns. The presentation aimed to provide a clearer understanding of interference beyond the typical syllabus.
The document discusses the wave properties of particles. Some key points:
1) Louis de Broglie hypothesized in 1924 that matter has an associated wave-like nature with a wavelength given by Planck's constant divided by momentum.
2) A particle can be represented as a localized "wave packet" resulting from the interference and superposition of multiple waves with slightly different wavelengths and frequencies.
3) Davisson and Germer's electron diffraction experiment in 1927 provided direct evidence of the wave nature of electrons and supported de Broglie's hypothesis by measuring electron wavelengths matching those expected.
This document discusses Huygen's principle and its applications. Huygen's principle states that each point on a wavefront can be considered a secondary source of spherical wavelets. The position and shape of subsequent wavefronts can be determined by the interference of these secondary wavelets. The document shows how Huygen's principle can be used to derive the laws of reflection and refraction. It also discusses applications such as total internal reflection and diffuse reflection.
This document provides information on physics and aerodynamics topics including optics, light, reflection, refraction, lenses, wave motion, and sound. It discusses key concepts such as the properties of light, dispersion of light, reflection and refraction of light by curved mirrors and lenses, characteristics and types of mechanical waves, interference and standing waves, sound propagation, the Doppler effect, and resonance. The document is prepared by a student for a class on these physics topics relating to aviation technology.
This document discusses the phenomenon of diffraction, which refers to the bending of waves around obstacles. It provides explanations of diffraction from both classical physics and quantum mechanics perspectives. Examples of diffraction effects in everyday life are given, such as the rainbow pattern seen on CDs/DVDs. The document also covers the history of diffraction studies, analytical models used to calculate diffraction, and the role of coherence in diffraction.
1. Physical Optics deals with the wave nature of light, specifically electromagnetic waves described by Maxwell's equations, whereas Geometrical Optics deals with the particle nature of light.
2. Maxwell established that light is an electromagnetic wave that propagates through space at a constant speed. Hertz later produced electromagnetic waves experimentally.
3. Interference and diffraction of light can be explained using Huygens' principle that each point on a wavefront acts as a secondary source emitting spherical wavelets. This allows prediction of phenomena like interference patterns, reflection and refraction of light.
This document discusses several key concepts in waves and optics:
- Interference occurs when two waves pass through the same space and can be constructive or destructive depending on the relative phases of the waves.
- Diffraction causes waves to bend around obstacles, with more bending for smaller obstacles or shorter wavelengths.
- Dispersion of light occurs because the refractive index varies with wavelength, causing different colors to refract differently.
- Reflection and refraction change the direction of waves at material interfaces due to changes in speed of light and refractive index.
- Mirrors and lenses use reflection and refraction to focus or diffuse light waves.
This document discusses several key concepts in waves and optics:
- Interference occurs when two waves pass through the same space and can be constructive or destructive depending on the relative phases of the waves.
- Diffraction causes waves to bend around obstacles, with more bending for smaller obstacles or shorter wavelengths.
- Dispersion of light occurs because the refractive index varies with wavelength, causing different colors to refract differently.
- Reflection and refraction change the direction of waves at material interfaces due to changes in speed, governed by Snell's Law and the refractive index.
- Mirrors and lenses use reflection and refraction to focus or diffuse light rays using their focal points and lengths.
The document discusses wave behavior and reflection and refraction of waves. It provides examples of reflection at fixed and free boundaries and how this causes inversion or no inversion of pulses. It introduces the law of reflection where the angle of incidence equals the angle of reflection. Refraction is discussed where the speed and wavelength change upon entering a new medium. Snell's law is derived relating the sines of the angles of incidence and refraction to the refractive indices of the media. Total internal reflection at the critical angle is also mentioned.
The document summarizes key concepts about the particle and wave properties of light. It discusses (1) Newton's corpuscular theory of light and the establishment of the wave theory by Huygens, (2) wave phenomena such as reflection, refraction, diffraction and interference, (3) the photoelectric effect and how Einstein's photon theory explained experimental observations, and (4) provides an example calculation of determining the work function of a metal from photoelectric emission data.
Superconducting qubits for quantum information an outlookGabriel O'Brien
The document discusses the progress and future directions of quantum information processing using superconducting qubits. It describes the stages needed to build a functional quantum computer, from controlling individual qubits to implementing error correction. Superconducting qubits are well-suited for this task as their Hamiltonians can be designed using circuit elements like inductors and Josephson junctions. While full fault-tolerant quantum computing has yet to be achieved, the performance of superconducting qubits has improved dramatically in recent years, suggesting the goals may be within reach this century.
This document describes a student's graduation project on using superconducting circuits for quantum computation. It provides an introduction to the topic and outlines the structure of the project. The project will first introduce the concept of a qubit and basics of quantum computation. It will then describe different types of qubit technologies before focusing on superconducting circuits. The document will explain the necessary quantum phenomena like coherence and noise. It will explore superconducting qubits in detail and how to couple them. Finally, it will demonstrate how to perform logical operations using superconducting qubits.
Ion traps offer a possible solution to the challenge of building a quantum computer by isolating ions as physical qubits. Ion traps use oscillating electric fields to confine ions in a line, isolating them from the environment while still allowing for manipulation and interaction through laser beams and the Coulomb force. The linear Paul trap in particular was inspired by Wolfgang Paul observing eggs on a tray and forms the basis for trapped ion quantum computation.
This document discusses the use of trapped atomic ions for quantum information processing and the creation of entangled states. It describes how ions can be trapped and laser cooled to suppress environmental perturbations and allow for coherent manipulation over long durations. Recent experiments have successfully generated entanglement between the internal states of pairs of trapped ions, implemented quantum logic gates like CNOT, and improved tools for high-precision measurement. Trapped ions provide a promising system for studying and applying concepts of quantum information processing.
1) Laser cooling uses lasers to cool atoms to very low temperatures, creating clouds of cold atoms. It involves using counter-propagating laser beams slightly detuned from the atomic resonance to create friction and cool atomic motion.
2) In 1985, an optical molasses was first observed, cooling atoms to below 1 mK. However, optical molasses have low density due to the lack of a restoring force. Jean Dalibard then proposed using polarized light and a magnetic field gradient to create a restoring force in a magneto-optical trap (MOT), greatly increasing atom density.
3) Now, researchers have demonstrated an efficient way to generate the laser beam configuration for a MOT or molasses using a
This document summarizes the current state of semiconductor qubits for quantum applications. It discusses different types of semiconductor qubits including charge qubits in gate-controlled quantum dots, spin qubits in quantum dots, dopants, and color centers. For each type of qubit, it evaluates their potential for applications in quantum sensing, simulation, computation, and communication. Overall, the review finds that semiconductor qubits show promise for diverse applications depending on their specific material properties and degrees of freedom, such as charge, spin, or photon interfaces.
A silicon based nuclear spin quantum computerGabriel O'Brien
This document describes a proposed scheme for building a quantum computer using the nuclear spins of phosphorus donor atoms embedded in silicon. Key points:
- Information would be encoded in the nuclear spins of phosphorus atoms acting as quantum bits (qubits).
- Logical operations on the qubits would be performed by manipulating the hyperfine interaction between electron and nuclear spins using electric fields from nearby gates.
- Measurements of the nuclear spin states could be made by transferring spin polarization to electrons and detecting the effect on electron orbital wavefunctions using capacitance measurements.
- Silicon is proposed as the host semiconductor because it contains only spin-0 isotopes, isolating the phosphorus donor nuclear spins from decoherence caused by host
Spin qubits for quantum information processingGabriel O'Brien
This chapter reviews manipulating spin qubits for quantum information processing. It describes the history of spin manipulation techniques dating back to the 1940s. An electron spin can be manipulated faster than a nuclear spin, making it suitable for a quantum processor qubit, while a nuclear spin has a longer coherence time, making it suitable for a quantum memory qubit. The chapter discusses how to manipulate single electron and nuclear spins with alternating magnetic fields and transfer information between them using hyperfine coupling.
1. The Stern-Gerlach experiment discovered that silver atoms split into two beams, indicating the presence of an intrinsic "spin" angular momentum of 1/2 beyond orbital angular momentum.
2. Elementary particles are classified as fermions, with half-integer spin, and bosons, with integer spin. The spin of the electron is represented by a two-component spinor.
3. In a magnetic field, the spin precesses around the field direction at the Larmor frequency, independent of initial spin orientation. This principle underlies paramagnetic resonance and nuclear magnetic resonance spectroscopy.
Quantum computer based on color centers in diamondGabriel O'Brien
This document summarizes research on developing a quantum computer based on color centers in diamond. Key points include:
- Nitrogen vacancy (NV) centers in diamond can function as qubits at room temperature and have demonstrated quantum properties needed for a quantum computer.
- While progress has been made in areas like quantum sensing, scaling up NV centers to build a full quantum computer remains challenging due to fabrication and control limitations.
- The document reviews various proposals for controlling and entangling NV center spins, performing readout, and fabrication methods, but acknowledges that most proposals are still in early stages and scaling up faces significant hurdles. It provides an overview of the status and challenges of developing a diamond-based quantum computer.
This document discusses electron spin resonance (ESR), which is similar in principle to nuclear magnetic resonance (NMR). It derives the magnetic moment of an electron's spin and shows that a spin placed in a constant magnetic field will precess around the field at the Larmor frequency. When an additional oscillating magnetic field is applied at the Larmor frequency, the spin will rotate at the Rabi frequency in the rotating frame. This rotation appears as oscillations between the spin states on the Bloch sphere in the lab frame.
1. The Stern-Gerlach experiment demonstrated the quantization of electron spin by splitting a beam of silver atoms into two distinct beams based on their magnetic orientation.
2. This was represented mathematically using a two-level quantum system and Bloch sphere, where the spin states are written as vectors and manipulated using Pauli matrices.
3. The Stern-Gerlach device is modeled as a Hermitian operator whose eigenstates correspond to the "aligned" and "anti-aligned" spin measurements, and different device orientations are represented by rotating the Pauli matrices.
Fox m quantum_optics_an_introduction_photon antibunching1Gabriel O'Brien
This chapter discusses photon antibunching and the Hanbury Brown-Twiss experiments which helped develop modern quantum optics. It introduces the second-order correlation function g(2)(τ) which can be used to classify light as antibunched, coherent, or bunched. The chapter then discusses how the Hanbury Brown-Twiss experiments measured intensity fluctuations in light beams and how this led to defining g(2)(τ). It explores how g(2)(τ) can take different values for classical versus quantum light, with antibunched light only possible due to quantum effects.
Fox m quantum_optics_an_introduction_optical cavitiesGabriel O'Brien
The document discusses atoms placed inside optical cavities. It begins by summarizing the key properties of optical cavities, including resonant modes, finesse, quality factor, and photon lifetime. It then introduces the interaction between atoms and cavities, which is determined by three parameters: the atom-cavity coupling strength, the photon decay rate from the cavity, and the non-resonant decay rate of the atom. The interaction is strongly affected when the atomic transition frequency matches a resonant cavity mode frequency.
This document discusses light-matter interaction using a two-level atom model. It describes how an atom with only two energy levels can be modeled as a two-dimensional quantum mechanical system. The interaction of such a two-level atom with an electromagnetic field is then derived, leading to Rabi oscillations between the atomic energy levels driven by the field. Dissipative processes require a statistical description using the density operator formalism.
This document reviews single-photon sources and detectors. It discusses the characteristics of ideal single-photon sources and describes current technologies for deterministic and probabilistic single-photon sources, including quantum dots, color centers, and parametric downconversion. It also provides a brief history of single-photon detectors from photomultiplier tubes to avalanche photodiodes and reviews their applications in quantum information science and other fields.
Invited review article single photon sources and detectorsGabriel O'Brien
This document reviews the current state of single-photon sources and detectors operating from ultraviolet to infrared wavelengths. It discusses how these technologies are driven by applications in quantum communication and quantum information science which require single photons. Specifically, technologies for quantum key distribution and quantum computation rely on single photons and single-photon detectors. The review provides a brief history of the development of single-photon detectors from early photomultiplier tubes to modern solid-state devices like avalanche photodiodes and superconducting nanowire single-photon detectors.
This document provides an introduction to quantizing the electromagnetic field. It begins with a classical description of the electromagnetic field using Maxwell's equations. It then shows that the classical electromagnetic field can be described as an infinite collection of independent harmonic oscillators. The document proceeds to quantize these harmonic oscillators by promoting the classical variables to quantum operators. This leads to a description of the electromagnetic field in terms of photon creation and annihilation operators. The quantized electromagnetic field gives rise to phenomena like zero-point energy and the Casimir effect that cannot be explained classically.
Quantum jumps of light recording the birth and death of a photon in a cavityGabriel O'Brien
This document summarizes an experiment that observed quantum jumps in the photon number inside a superconducting cavity. Key points:
- Microwave photons were stored in a superconducting cavity for up to half a second and repeatedly probed by non-absorbing atoms passing through.
- An atom interferometer measured the atomic phase shift induced by the non-resonant cavity field, revealing the presence or absence of a single photon.
- Sequences of hundreds of correlated atom measurements were interrupted by sudden changes, recording the creation and destruction of individual photons over time.
- This realized a quantum non-demolition measurement of the photon number in the cavity in real time, allowing observation of its
This document discusses entangled states and quantum teleportation. It begins by describing how entangled photon pairs are generated in the laboratory using atomic cascades in calcium. It then explains the concept of entanglement through Einstein-Podolsky-Rosen experiments using correlated photon pairs. The document introduces Bell states and discusses how entanglement implies that measuring one photon determines the result of measuring the other. It also discusses Schrodinger's cat paradox to illustrate entanglement. Finally, it discusses experiments that have tested ideas of entanglement and quantum teleportation.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Equivariant neural networks and representation theory
Topic 7 wave_interference(latest)
1. Topic 4.1 Waves, Interference and Optics
1
UEEP1033 Oscillations and Waves
Topic 7:
Interference and Diffraction
2. Topic 4.1 Waves, Interference and Optics
2
UEEP1033 Oscillations and Waves
• When a wavefront encounters an
aperture in an opaque barrier, the
barrier suppresses all propagation of
the wave except through the aperture
• Following Huygen’s principle, the
points on the wavefront across the
aperture act as sources of secondary
wavelets
• When the width of the aperture is
comparable with the wavelength, the
aperture acts like a point source and
the outgoing wavefronts are
semicircular
Huygen’s Principle
3. Topic 4.1 Waves, Interference and Optics
3
UEEP1033 Oscillations and Waves
3
• Ignores most of each secondary wavelet and only retaining the
portions common to the envelope
• As a result, Huygens’s principle by itself is unable to account
for the details of the diffraction process
• The difficulty was resolved by Fresnel with his addition of the
concept of interference
Huygens’s Principle
4. Topic 4.1 Waves, Interference and Optics
4
UEEP1033 Oscillations and Waves
Augustin Jean Fresnel
• 1818, Fresnel brought together the ideas of Huygens and Young
and by making some arbitrary assumptions about the amplitude
and phases of Huygens’ secondary sources
• Fresnel able to calculate the distribution of light in diffraction
patterns with excellent accuracy by allowing the various
wavelet to mutually interfere
Huygens-Fresnel Principle
5. Topic 4.1 Waves, Interference and Optics
5
UEEP1033 Oscillations and Waves
Huygens-Fresnel Principle
Every unobstructed point of a wavefront, at given instant, serves
as a source of spherical secondary wavelets
(with the same frequency as that of the primary wave)
The amplitude of the optical field at any point beyond is the
superposition of all these wavelets
(considering their amplitudes and relative phases)
6. Topic 4.1 Waves, Interference and Optics
6
UEEP1033 Oscillations and Waves
Christian Huygens
Huygens’s Principle
Each point on the wavefront of a disturbance were considered to be a
new source of a “secondary” spherical disturbance, then the
wavefront at a later instant could be found by constructing the
“envelope” of the secondary wavelets”
7. Topic 4.1 Waves, Interference and Optics
7
UEEP1033 Oscillations and Waves
Huygens’s Principle
Every point on a propagation wavefront serves as the source of
spherical secondary wavelets, such that the wavefront at some
later time is the envelope of these wavelets
Plane wave Spherical wave
8. Topic 4.1 Waves, Interference and Optics
8
UEEP1033 Oscillations and Waves
9. Topic 4.1 Waves, Interference and Optics
9
UEEP1033 Oscillations and Waves
Huygens’s Principle
Plane wave
Spherical wave
Every point on a propagation wavefront serves as the
source of spherical secondary wavelets
the wavefront at some
later time is the
envelope of these
wavelets
10. Topic 4.1 Waves, Interference and Optics
10
UEEP1033 Oscillations and Waves
ri
Law of Reflection
Law of Refraction (Snell’s law)
ttii nn sinsin
Interface
Incident
medium ni
Refracting
medium ni
Surface
normal
11. Topic 4.1 Waves, Interference and Optics
11
UEEP1033 Oscillations and Waves
Law of Reflection
When a ray of light is reflected at an interface dividing two
uniform media, the reflected ray remains within the plane of
incidence, and the angle of reflection equals the angle of
incidence. The plane of incidence includes the incident ray and
the normal to the point of incidence
Law of Refraction (Snell’s law)
When a ray of light is refracted at an interface dividing two
uniform media, the transmitted ray remains within the plane of
incidence and the sine of the angle of refraction is directly
proportional to the sine of the angle of incidence
12. Topic 4.1 Waves, Interference and Optics
12
UEEP1033 Oscillations and Waves
Huygens’ construction to prove the law of reflection
Narrow,
parallel ray
of light
Plane of
interface XY
Angle of
incidence
Angle of
reflection
13. Topic 4.1 Waves, Interference and Optics
13
UEEP1033 Oscillations and Waves
Huygens’ construction to prove the law of reflection
• Since points along the plane wavefront do not arrive at the
interface simultaneously, allowance is made for these
differences in constructing the wavelets that determine the
reflected wavefront
• If the interface XY were not present, the Huygens
construction would produce the wavefront GI at the instant
ray CF reached the interface at I
• The intrusion of the reflecting surface, means that during the
same time interval required for ray CF to progress from F to
I, ray BE has progressed from E to J and then a distance
equivalent to JH after reflection
14. Topic 4.1 Waves, Interference and Optics
14
UEEP1033 Oscillations and Waves
Huygens’ construction to prove the law of reflection
• Wavelet of radius JN = JH centered at J is drawn above the
reflecting surface
• Wavelet of radius DG is drawn centered at D to represent the
propagation after reflection of the lower part of the light
• The new wavefront, which must now be tangent to these
wavelets at points M and N, and include the point I, is shown
as KI in the figure
• A representative reflected ray is DL, shown perpendicular to
the reflected wavefront
• The normal PD drawn for this ray is used to define angles of
incidence and reflection for the light
15. Topic 4.1 Waves, Interference and Optics
15
UEEP1033 Oscillations and Waves
The Law of Refraction
Use Huygen’s principle to derive the
law of refraction
The refraction of a plane wave at an
air-glass interface
Figures show three successive stages
of the refraction of several
wavefronts at a plane interface
between air (medium 1) and glass
(medium 2)
1 = wavelength in medium 1
v1 = speed of light in medium 1
v2 = speed of light in medium 2 < v1
1 = angle of incidence
16. Topic 4.1 Waves, Interference and Optics
16
UEEP1033 Oscillations and Waves
As the wave moves into the glass, a
Huygens wavelet at point e will
expand to pass through point c, at a
distance of 1 from point e.
The time interval required for this
expansion is that distance divided by
the speed of the wavelet = 1/v1
In the same time interval, a Huygens
wavelet at point h will expand to pass
through point g, at the reduced speed
v2 and with wavelength 2, i.e. the
time interval = 2/v2
2
2
1
1
vv
2
1
2
1
v
v
17. Topic 4.1 Waves, Interference and Optics
17
UEEP1033 Oscillations and Waves
According to Huygens’ principle, the
refracted wavefront must be tangent
to an arc of radius 2 centered on h,
say at point g
the refracted wavefront must also be
tangent to an arc of radius 1 centered
on e, say at point c
2 = angle of refraction
h c
e
h c
g
hc
1
1sin
hc
2
2sin
2
1
2
1
2
1
sin
sin
v
v
18. Topic 4.1 Waves, Interference and Optics
18
UEEP1033 Oscillations and Waves
Define: refraction index for a medium
c = speed of light
v = speed of light in the medium
Speed of light in any medium depends on the index of
refraction of the medium
1
1
v
c
n e.g.
2
2
v
c
n
v
c
n
1
2
2
1
2
1
2
1
/
/
sin
sin
n
n
nc
nc
v
v
2211 sinsin nn
19. Topic 4.1 Waves, Interference and Optics
19
UEEP1033 Oscillations and Waves
The wavelength of light in any medium depends on the index
of refraction of the medium
Let a certain monochromatic light:
Medium refraction index wavelength speed
vacuum 1 c
medium n n v
2
1
2
1
v
v
From slide-8:
c
v
n
The greater the index of refraction of a medium, the smaller the
wavelength of light in that medium
n
n
20. Topic 4.1 Waves, Interference and Optics
20
UEEP1033 Oscillations and Waves
21. Topic 4.1 Waves, Interference and Optics
21
UEEP1033 Oscillations and Waves
Frequency Between Media
• As light travels from one
medium to another, its
frequency does not change.
– Both the wave speed
and the wavelength do
change.
– The wavefronts do not
pile up, nor are they
created or destroyed at
the boundary, so ƒ must
stay the same.
22. Topic 4.1 Waves, Interference and Optics
22
UEEP1033 Oscillations and Waves
n
n
v
f
Frequency of the light in a medium with index of refraction n
fv
f
c
n
nc
fn
/
/
f = frequency of the light in vacuum
The frequency of the light in the medium is the same as it is in
vacuum
23. Topic 4.1 Waves, Interference and Optics
23
UEEP1033 Oscillations and Waves
The fact that the wavelength of light depends on the index of
refraction is important in situations involving the interference
of light waves
Example:
Two light rays travel through two media having different
indexes of refraction
• Two light rays have identical wavelength
and are initially in phase in air (n 1)
• One of the waves travels through medium
1 of index of refraction n1 and length L
• The other travels through medium 2 of
index of refraction n2 and the same
length L
24. Topic 4.1 Waves, Interference and Optics
24
UEEP1033 Oscillations and Waves
• When the waves leave the two media, they will have the same
wavelength – their wavelength in air
• However, because their wavelengths differed in the two media,
the two waves may no longer be in phase
The phase difference between two light waves can change if
the waves travel through different materials having different
indexes of refraction
How the light waves will interfere if they reach some common
point?
25. Topic 4.1 Waves, Interference and Optics
25
UEEP1033 Oscillations and Waves
Number N1 of wavelengths in the length L of medium 1
11 / nn wavelength in medium 1:
1
1
1
LnL
N
n
wavelength in medium 2: 22 / nn
2
2
2
LnL
N
n
)( 1212 nn
L
NN
Phase difference
between the waves
21 nn
26. Topic 4.1 Waves, Interference and Optics
26
UEEP1033 Oscillations and Waves
Example:
phase difference = 45.6 wavelengths
•i.e. taking the initially in-phase waves and shifting one of them
by 45.6 wavelengths
•A shift of an integers number of wavelengths (such as 45)
would put the waves back in phase
•Only the decimal fraction (such as 0.6) that is important
•i.e. phase difference of 45.6 wavelengths 0.6 wavelengths
•Phase difference = 0.5 wavelength puts two waves exactly out
of phase
•If the two waves had equal amplitudes and were to reach some
common point, they would then undergo fully destructive
interference, producing darkness at that point
27. Topic 4.1 Waves, Interference and Optics
27
UEEP1033 Oscillations and Waves
• With the phase difference = 0 or 1wavelengths, they would
undergo fully constructive interference, resulting brightness
at that common point
• In this example, the phase difference = 0.6 wavelengths is an
intermediate situation, but closer to destructive interference,
and the wave would produces a dimly illuminated common
point
28. Topic 4.1 Waves, Interference and Optics
28
UEEP1033 Oscillations and Waves
Example:
= 550 nm
Two light waves have equal
amplitudes and re in phase before
entering media 1 and 2
Medium 1 = air (n1 1)
Medium 2 = transparent plastic (n2 1.60, L = 2.60 m)
Phase difference of the emerging waves:
o
9
6
1212
1020rad17.8
swavelength84.2
)00.160.1(
10550
1060.2
)(
nn
L
NN
29. Topic 4.1 Waves, Interference and Optics
29
UEEP1033 Oscillations and Waves
Effective phase difference = 0.84 wavelengths = 5.3 rad 300o
• 0.84 wavelengths is between 0.5 wavelength and 1.0
wavelength, but closer to 1.0 wavelength.
• Thus, the waves would produce intermediate interference that is
closer to fully constructive interference,
• i.e. they would produce a relatively bright spot at some
common point.
30. Topic 4.1 Waves, Interference and Optics
30
UEEP1033 Oscillations and Waves
Fermat’s Principle
• The ray of light traveled the
path of least time from A to B
• If light travels more slowly in
the second medium, light
bends at the interface so as to
take a path that favors a
shorter time in the second
medium, thereby minimizing
the overall transit time from
A to B
Construction to prove the law of
refraction from Fermat’s principle
31. Topic 4.1 Waves, Interference and Optics
31
UEEP1033 Oscillations and Waves
Interference
Young’s Double-Slit Experiment
32. Topic 4.1 Waves, Interference and Optics
32
UEEP1033 Oscillations and Waves
Fermat’s Principle
• Mathematically, we are required to minimize the total time:
ti v
OB
v
AO
t
22
xaAO 22
)( xcbOB
ti v
xcb
v
xa
t
2222
)(
33. Topic 4.1 Waves, Interference and Optics
33
UEEP1033 Oscillations and Waves
Fermat’s Principle
0
)( 2222
xcbv
xc
xav
x
dx
dt
ti
• minimize the total time by setting dt / dx = 0
22
sin
xa
x
i
• From diagram:
22
)(
sin
xcb
xc
t
0
sinsin
t
t
i
i
vvdx
dt
0
/
sin
/
sin
t
t
i
i
ncnc ttii nn sinsin
34. Topic 4.1 Waves, Interference and Optics
34
UEEP1033 Oscillations and Waves
Interference
two waves are
out of phase
destructive
interference
two waves are
in phase
constructive
interference
amplitude of their
superposition is zero
amplitude of the
superposition
(ψ1 + ψ2) = 2A
A is the amplitude of the
individual waves
35. Topic 4.1 Waves, Interference and Optics
35
UEEP1033 Oscillations and Waves
Figure (a)
• Two monochromatic waves ψ1 and ψ2 at a particular point
in space where the path difference from their common
source is equal to an integral number of wavelengths
• There is constructive interference and their superposition
(ψ1 + ψ2) has an amplitude that is equal to 2A where A is
the amplitude of the individual waves.
Figure (b)
• The two waves ψ1 and ψ2 where the path difference is
equal to an odd number of half wavelengths
• There is destructive interference and the amplitude of
their superposition is zero
Interference
36. Topic 4.1 Waves, Interference and Optics
36
UEEP1033 Oscillations and Waves
37. Topic 4.1 Waves, Interference and Optics
37
UEEP1033 Oscillations and Waves
Light source
Aperture
Observation
plane
Screen
Arrangement used for observing
diffraction of light
Corpuscular Theory
shadow behind the screen
should be well defined, with
sharp borders
Observations
• The transition from light
to shadow was gradual
rather than abrupt
• Presence of bright and
dark fringes extending far
into the geometrical
shadow of the screen
38. Topic 4.1 Waves, Interference and Optics
38
UEEP1033 Oscillations and Waves
Young’s Double-Slit Experiment
L >> a
d
d = slits separation
d
39. Topic 4.1 Waves, Interference and Optics
39
UEEP1033 Oscillations and Waves
• A monochromatic plane wave of wavelength λ is incident upon
an opaque barrier containing two slits S1 and S2
• Each of these slits acts as a source of secondary wavelets
according to Huygen’s Principle and the disturbance beyond
the barrier is the superposition of all the wavelets spreading out
from the two slits
• These slits are very narrow but have a long length in the
direction normal to the page, making this a two-dimensional
problem
• The resultant amplitude at point P is due to the superposition
of secondary wavelets from the two slits
Young’s Double-Slit Experiment
40. Topic 4.1 Waves, Interference and Optics
40
UEEP1033 Oscillations and Waves
• Since these secondary wavelets are driven by the same incident
wave there is a well defined phase relationship between them
• This condition is called coherence and implies a systematic
phase relationship between the secondary wavelets when they
are superposed at some distant point P
• It is this phase relationship that gives rise to the interference
pattern, which is observed on a screen a distance L beyond the
barrier
Young’s Double-Slit Experiment
41. Topic 4.1 Waves, Interference and Optics
41
UEEP1033 Oscillations and Waves
The secondary wavelets from S1 and S2 arriving at an arbitrary
point P on the screen, at a distance x from the point O that
coincides with the mid-point of the two slits
Distances: S1P = l1 S2P = l2
Since L >> d it can be assumed that the secondary wavelets
arriving at P have the same amplitude A
The superposition of the wavelets at P gives the resultant
amplitude:
Young’s Double-Slit Experiment
)cos()cos( 21 kltkltAR
ω = angular frequency
k = wave number
(5)
d= slits separation
42. Topic 4.1 Waves, Interference and Optics
42
UEEP1033 Oscillations and Waves
This result can be rewritten as:
Since L >> d, the lines from S1 and S2 to P can be assumed to be
parallel and also to make the same angle θ with respect to the
horizontal axis
Young’s Double-Slit Experiment
2/)(cos[]2/)(cos2 1212 llkllktAR
The line joining P to the mid-point of the slits makes an angle θ
with respect to the horizontal axis
21 cos/ lLl
cos/212 Lll
(6)
d = slits separation
43. Topic 4.1 Waves, Interference and Optics
43
UEEP1033 Oscillations and Waves
When the two slits are separated by many wavelengths, θ is very
small and cos θ 1. Hence, we can write the resultant amplitude
as:
Young’s Double-Slit Experiment
)2/cos()cos(2 lkkLtAR
= path difference of the secondary wavelets
The intensity I at point P = R2
12 lll
)2/(cos)(cos4 222
lkkLtAI
This equation describes the instantaneous intensity at P
The variation of the intensity with time is described by the
cos2(ωt − kL) term
(7)
(8)
44. Topic 4.1 Waves, Interference and Optics
44
UEEP1033 Oscillations and Waves
• The frequency of oscillation of visible light is of the order of
1015 Hz, which is far too high for the human eye and any
laboratory apparatus to follow.
• What we observe is a time average of the intensity
• Since the time average of cos2(ωt − kL) over many cycles = 1/2
the time average of the intensity is given by:
Young’s Double-Slit Experiment
)2/(cos2
0 lkII
2
0 2AI = intensity observed at a maximum of the interference pattern
described how the intensity varies with l)2/(cos2
lk
(9)
45. Topic 4.1 Waves, Interference and Optics
45
UEEP1033 Oscillations and Waves
I = maximum whenever l = n (n = 0,±1, ±2, …)
I = 0 whenever l = (n + ½)
Young’s Double-Slit Experiment
From figure on slide-25: l a sin θ
Substituting for l in Equation (9), we obtain:
(10))2/sin(cos)( 2
0 kdII
When θ is small so that sinθ θ, we can write:
)/(cos)(
)2/(cos)(
2
0
2
0
dII
kdII
(11)
/2where k
d = slits separation
46. Topic 4.1 Waves, Interference and Optics
46
UEEP1033 Oscillations and Waves
If there were no
interference, the
intensity would be
uniform and equal
to Io/2 as indicated
by the horizontal
dashed line
Young’s Double-Slit Experiment
Light intensity I (θ) vs angle θ
d = slits separation
L/d
separation of the
bright fringes
47. Topic 4.1 Waves, Interference and Optics
47
UEEP1033 Oscillations and Waves
Young’s Double-Slit Experiment
Intensity maxima: .....,2,1,0, n
d
n
.....,2,1,0, n
d
L
nLx
(12)
(13)
(14)
(15)
The bright fringes occur at distances from the point O given by:
Minimum intensity occur when:
The distance between adjacent bright fringes is:
.....,2,1,0,
2
1
n
d
L
nx
d
L
xx nn
1
d = slits separation
48. Topic 4.1 Waves, Interference and Optics
48
UEEP1033 Oscillations and Waves
Point source of light is
illuminating an opaque object,
casting a shadow where the
edge of the shadow fades
gradually over a short distance
and made up of bright and dark
bands, the diffraction fringes.
Shadow fades gradually
>> Bright and Dark Bands
= Diffraction Fringes
Diffraction
49. Topic 4.1 Waves, Interference and Optics
49
UEEP1033 Oscillations and Waves
Francesco Grimaldi
in 1665 first accurate report
description of deviation of light from
rectilinear propagation (diffraction)
The effect is a general characteristics of wave phenomena
occurring whenever a portion of a wavefront is
obstructed in some way
Diffraction
50. Topic 4.1 Waves, Interference and Optics
50
UEEP1033 Oscillations and Waves
Plane wavefronts approach a barrier with an opening or an
obstruction, which both the opening and the obstruction are
large compared to the wavelength
Opening
(size = a)
Obstruction
(size = a)
wavelength, a >>
51. Topic 4.1 Waves, Interference and Optics
51
UEEP1033 Oscillations and Waves
• If the size of the opening or obstruction becomes comparable to the
wavelength
• The waves is not allowed to propagate freely through the opening or past the
obstruction
• But experiences some retardation of some parts of the wavefront
• The wave proceed to "bend through" or around the opening or obstruction
• The wave experiences significant curvature upon emerging from the opening
or the obstruction
curvaturea
52. Topic 4.1 Waves, Interference and Optics
52
UEEP1033 Oscillations and Waves
As the barrier or opening size gets smaller,
the wavefront experiences more and more curvature
More curvature
Diffraction
a
53. Topic 4.1 Waves, Interference and Optics
53
UEEP1033 Oscillations and Waves
Fraunhofer and Fresnel
Diffraction
54. Topic 4.1 Waves, Interference and Optics
54
UEEP1033 Oscillations and Waves
Observation
screen
Fraunhofer and Fresnel Diffraction
S
Lens
Plane
waves
Opaque shield , with a single
small aperture of width a is
being illuminated by plane wave
of wavelength from a distant
point source S
Case-1
observation screen is very
close to
Image of aperture is projected
onto the screen
55. Topic 4.1 Waves, Interference and Optics
55
UEEP1033 Oscillations and Waves
Observation
screen
Fraunhofer and Fresnel Diffraction
S
Lens
Plane
waves
Case-2
observation screen is moved farther
away from
Image of aperture become
increasingly more structured as the
fringes become prominent
Fresnel or Near-Field
Diffraction
56. Topic 4.1 Waves, Interference and Optics
56
UEEP1033 Oscillations and Waves
Fraunhofer and Fresnel Diffraction
S
Lens
Plane
waves
Case-3
observation screen is at very
great distance away from
Projected pattern will have spread
out considerably, bearing a little or
no resemblance to the actual
aperture
Observation
screen
Thereafter moving the screen
away from the aperture change
only the size of the pattern and not
its shape
Fraunhofer or Far-Field
Diffraction
57. Topic 4.1 Waves, Interference and Optics
57
UEEP1033 Oscillations and Waves
Fraunhofer and Fresnel Diffraction
S
Lens
Plane
waves
Case-4
If at that point, the wavelength of
the incoming radiation is reduce
Observation
screen
the pattern would revert back
to the Fresnel case
If were decreased even more, so that → 0
The fringes would disappear, and the image
would take on the limiting shape of the aperture
58. Topic 4.1 Waves, Interference and Optics
58
UEEP1033 Oscillations and Waves
Fraunhofer and Fresnel Diffraction
If a point source S and the
observation screen are very far
from
S
Lens
Plane
waves
Observation
screen
Fraunhofer Diffraction
If a point source S and the
observation screen are
too near
Fresnel Diffraction
59. Topic 4.1 Waves, Interference and Optics
59
UEEP1033 Oscillations and Waves
Fraunhofer and Fresnel Diffraction
S
Lens
Plane
waves
Observation
screen
Fraunhofer Diffractiona
R R
R is the smaller of the two
distances from S to and to
2
a
R
d = slit width
60. Topic 4.1 Waves, Interference and Optics
60
UEEP1033 Oscillations and Waves
Practical realization of the Fraunhofer condition
F1 F2
61. Topic 4.1 Waves, Interference and Optics
61
UEEP1033 Oscillations and Waves
Diffraction
• Any obstacle in the path of the wave affects the way it spreads out; the
wave appears to ‘bend’ around the obstacle
• Similarly, the wave spreads out beyond any aperture that it meets. such
bending or spreading of the wave is called diffraction
• The effects of diffraction are evident in the shadow of an object that is
illuminated by a point source. The edges of the shadow are not sharp but
are blurred due to the bending of the light at the edges of the object
• The degree of spreading of a wave after passing through an aperture
depends on the ratio of the wavelength λ of the wave to the size d of the
aperture
• The angular width of the spreading is approximately equal to λ/d; the
bigger this ratio, the greater is the spreading
62. Topic 4.1 Waves, Interference and Optics
62
UEEP1033 Oscillations and Waves
The Mechanism of Diffraction
• Diffraction arises because of the way in which waves propagate as
described by the Huygens-Fresnel Principle
• The propagation of a wave can be visualized by considering every point
on a wavefront as a point source for a secondary radial wave
• The subsequent propagation and addition of all these radial waves form
the new wavefront
• When waves are added together, their sum is determined by the relative
phases as well as the amplitudes of the individual waves, an effect
which is often known as wave interference
• The summed amplitude of the waves can have any value between zero
and the sum of the individual amplitudes
• Hence, diffraction patterns usually have a series of maxima and minima
63. Topic 4.1 Waves, Interference and Optics
63
UEEP1033 Oscillations and Waves
• A monochromatic plane wave is incident
upon an opaque barrier containing a single
slit
• Replace the relatively wide slit by an
increasing number of narrow subslits
• Each point in the subslits acts as a point
source for a secondary radial wave
• When waves are added together, their sum is
determined by the relative phases and the
amplitudes of the individual waves, an effect
which is often known as wave interference
• The summed amplitude of the waves can
have any value between zero and the sum of
the individual amplitudes
• Hence, diffraction patterns usually have a
series of maxima and minima
Single Slit Diffraction
64. Topic 4.1 Waves, Interference and Optics
64
UEEP1033 Oscillations and Waves
One can find the second dark fringes above and below the central axis as the
first dark fringes were found, except that we now divide the slit into four
zones of equal widths a/4, as shown in Fig. 36-6a.
In general,
65. Topic 4.1 Waves, Interference and Optics
65
UEEP1033 Oscillations and Waves
Example, Single Slit Diffraction Pattern with White Light:
66. Topic 4.1 Waves, Interference and Optics
66
UEEP1033 Oscillations and Waves
36.4: Intensity in Single-Slit Diffraction Pattern, Qualitatively:
67. Topic 4.1 Waves, Interference and Optics
67
UEEP1033 Oscillations and Waves
Fig. 36-8 The relative intensity in single-slit diffraction for three values of the
ratio a/. The wider the slit is, the narrower is the central diffraction maximum.
The intensity pattern is: where
For intensity minimum,
68. Topic 4.1 Waves, Interference and Optics
68
UEEP1033 Oscillations and Waves
36.5: Intensity in Single-Slit Diffraction Pattern, Quantitatively:
From the geometry, f is also the angle between the
two radii marked R. The dashed line in the figure,
which bisects f, forms two congruent right
triangles.
69. Topic 4.1 Waves, Interference and Optics
69
UEEP1033 Oscillations and Waves
Example, Intensities of the Maximum in a Single Slit Interference Pattern:
70. Topic 4.1 Waves, Interference and Optics
70
UEEP1033 Oscillations and Waves
36.6: Diffraction by a Circular Aperture:
71. Topic 4.1 Waves, Interference and Optics
71
UEEP1033 Oscillations and Waves
36.6: Diffraction by a Circular Aperture, Resolvability:
Fig. 36-11 At the top, the images of two point sources (stars) formed by a
converging lens. At the bottom, representations of the image intensities. In (a)
the angular separation of
the sources is too small for them to be distinguished, in (b) they can be
marginally distinguished, and in (c) they are clearly distinguished. Rayleigh’s
criterion is satisfied in (b), with the central maximum of one diffraction pattern
coinciding with the first minimum of the other.
Two objects that are
barely resolvable
when the angular
separation is given by:
72. Topic 4.1 Waves, Interference and Optics
72
UEEP1033 Oscillations and Waves
73. Topic 4.1 Waves, Interference and Optics
73
UEEP1033 Oscillations and Waves
Example, Pointillistic paintings use the diffraction of your eye:
74. Topic 4.1 Waves, Interference and Optics
74
UEEP1033 Oscillations and Waves
Example, Rayleigh’s criterion for resolving two distant objects:
75. Topic 4.1 Waves, Interference and Optics
75
UEEP1033 Oscillations and Waves
Fig. 36-15 (a) The intensity plot to be
expected in a double-slit interference
experiment with vanishingly narrow
slits.
(b) The intensity plot for diffraction by
a typical slit of width a (not
vanishingly narrow).
(c) The intensity plot to be expected
for two slits of width a. The curve of
(b) acts as an envelope, limiting the
intensity of the double-slit fringes in
(a). Note that the first minima of the
diffraction pattern of (b) eliminate the
double-slit fringes that would occur
near 12° in (c).
The intensity of a double slit pattern
is:
76. Topic 4.1 Waves, Interference and Optics
76
UEEP1033 Oscillations and Waves
Example, Double slit experiment,
with diffraction of each slit
included:
77. Topic 4.1 Waves, Interference and Optics
77
UEEP1033 Oscillations and Waves
Example, Double slit experiment,
with diffraction of each slit
included, cont. :
78. Topic 4.1 Waves, Interference and Optics
78
UEEP1033 Oscillations and Waves
Diffraction Grating
Definition
A repetitive array of diffracting elements that has the effect
of producing periodic alterations in the phase, amplitude, or
both of an emergent wave
An idealized grating
consisting of only
five slits
Opaque surface with
narrow parallel grooves
e.g. made by ruling or
scratching parallel notches
into the surface of a flat,
clean glass plate
Each of the scratches
serves as a source of
scattered light, and
together they form a
regular array of parallel
line sources
79. Topic 4.1 Waves, Interference and Optics
79
UEEP1033 Oscillations and Waves
Diffraction Grating
Grating Equation: d sinm = m
m = specify the order of the
various principal maxima
The intensity plot produced by a
diffraction grating consists of narrow
peaks, here label with their order
number m
The corresponding bright fringes seen
on the screen are called lines
The maxima are very narrow and they
separated by relatively wide dark
region
d = grating spacing (spacing
between rulings or slits)
N rulings occupy a total
width w, then d = w/N
80. Topic 4.1 Waves, Interference and Optics
80
UEEP1033 Oscillations and Waves
36.8: Diffraction Gratings:
81. Topic 4.1 Waves, Interference and Optics
81
UEEP1033 Oscillations and Waves
36.8: Diffraction Gratings, Width of the Lines:
82. Topic 4.1 Waves, Interference and Optics
82
UEEP1033 Oscillations and Waves
Diffraction Grating
Application: Grating Spectroscope
collimator
Plane wave
Diffraction grating
telescope
Visible emission lines of cadmium
Visible emission lines from hydrogen
The lines are farther apart at greater angles
83. Topic 4.1 Waves, Interference and Optics
83
UEEP1033 Oscillations and Waves
36.9: Gratings, Dispersion and Resolving Power:
A grating spreads apart the diffraction lines associated with the various
wavelengths.
This spreading, called dispersion, is defined as
Here is the angular separation of two lines whose wavelengths differ by .
Also,
To resolve lines whose wavelengths are close together, the line should also
be as narrow as possible. The resolving power R, of the grating is defined as
It turns out that
84. Topic 4.1 Waves, Interference and Optics
84
UEEP1033 Oscillations and Waves
Gratings, Dispersion and Resolving Power, proofs:
The expression for the locations of the lines in the diffraction pattern of a
grating is:
Also, If is to be the smallest angle that will permit the two lines to be
resolved, it must (by Rayleigh’s criterion) be equal to the half-width of each
line, which is given by :
85. Topic 4.1 Waves, Interference and Optics
85
UEEP1033 Oscillations and Waves
36.9: Gratings, Dispersion and Resolving Power Compared:
86. Topic 4.1 Waves, Interference and Optics
86
UEEP1033 Oscillations and Waves