Physics
Wave Optics
Monochromatic Light
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Light with a dominant wavelength and a small spread of wavelengths.
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Strictly monochromatic light is impossible to obtain; there's always a spread in wavelength. LASERs are the best approximation. We use 'monochromatic light' to mean light with a dominant wavelength and a small spread.
Diffraction and Wavelength
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When a wave encounters an obstacle or opening, it bends. Diffraction is negligible if the obstacle/opening is much larger than the wavelength, leading to straight-line propagation.
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When a wave encounters an obstacle or opening, it diffracts. A plane wave becomes a spherical wave after passing through a small opening. If the obstacle/opening is much larger than the wavelength, diffraction is negligible, and rays go along straight lines. For light (380-780 nm), common obstacles are millimeters or larger, so diffraction is often negligible.
Geometrical Optics Approximation
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Treating light as rays traveling in straight lines; valid when diffraction is negligible.
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When diffraction is negligible, light can be treated as rays: straight lines from the source that terminate at opaque surfaces or pass undeflected through openings. This is the Geometrical Optics approximation.
Geometrical Optics Rules
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The three major rules governing geometrical optics are rectilinear propagation, reflection, and refraction.
sin(i) / sin(r) = v1 / v2
1. Rectilinear Propagation: Light travels in straight lines unless reflected or the medium changes.
2. Reflection: Angle of incidence equals angle of reflection. Incident ray, reflected ray, and normal are coplanar.
3. Refraction: sin(i)/sin(r) = v1/v2. Incident ray, refracted ray, and normal are coplanar.
2. Reflection: Angle of incidence equals angle of reflection. Incident ray, reflected ray, and normal are coplanar.
3. Refraction: sin(i)/sin(r) = v1/v2. Incident ray, refracted ray, and normal are coplanar.
Wavefront
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A surface where the wave disturbance is in the same phase at all points.
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(a) A surface on which the wave disturbance is in the same phase at all points is called a wavefront. (b) The direction of propagation of a wave at a point is perpendicular to the wavefront through that point. (c) The wavefronts of a wave originating from a point source are spherical. (d) The wavefronts for a wave going along a fixed direction are planes perpendicular to that direction.
Huygens' Principle
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Every point on a wavefront acts as a secondary source of spherical wavelets. The superposition of these wavelets determines the wavefront's future shape.
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Huygens considered light a mechanical wave in ether. Each point on a wavefront acts as a secondary source emitting spherical wavelets. The disturbance beyond results from the superposition of these wavelets.
Huygens' Principle: Various points of an arbitrary surface, when reached by a wavefront, become secondary sources of light emitting secondary wavelets. The disturbance beyond the surface results from the superposition of these secondary wavelets.
Huygens' Principle: Various points of an arbitrary surface, when reached by a wavefront, become secondary sources of light emitting secondary wavelets. The disturbance beyond the surface results from the superposition of these secondary wavelets.
Reflection of Light (Huygens' Principle)
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Huygens' principle explains reflection by considering each point on a wavefront as a source of secondary wavelets. The reflected wavefront is the envelope of these wavelets, and the law of reflection (angle of incidence equals angle of reflection) can be derived geometrically.
PR = v2*t(1-x); QP = x*v1*t
Huygens' construction is applied to the surface σ. The point A of the surface is reached by the wavefront AB at t = 0. This point becomes the source of secondary wavelet which expands in medium 2 at velocity v. At time t, this takes the shape of a hemisphere of radius v t centred at A. The point C of the surface is just reached by the wavefront at time t and hence, the wavelet is a point at C itself. Draw the tangent plane CD from C to the wavelet originating from A.The plane CD is, therefore, the geometrical envelope of all the secondary wavelets at time t. It is, therefore, the position of the wavefront AB at time t. The reflected rays are perpendicular to CD.
Refraction of Light (Huygens' Principle)
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Huygens' principle explains refraction by considering the change in speed of light as it enters a new medium. Each point on the wavefront acts as a source of secondary wavelets, and the refracted wavefront is the envelope of these wavelets. Snell's law can be derived geometrically.
sin i / sin r = v1 / v2
σ represents the surface separating two transparent media, medium 1 and medium 2 in which the speeds of light are v1 and v2 respectively. A parallel beam of light moving in medium 1 is incident on the surface and enters medium 2.The point P becomes a source of secondary wavelets at time t1. The radius of the wavelet at time t, a = v(t − t1) = v(t − xt) = vt(1 − x).
Young's Double Slit Experiment
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Deals with interference pattern formed by two coherent light sources. Explains bright and dark fringes, fringe width, and relation between wavelength, slit separation, and screen distance.
['∆x = d = nλ (for bright fringes)', 'y = nDλ/d (position of bright fringes)', '∆x = d = (n + 1/2)λ (for dark fringes)', 'y = (n + 1/2)Dλ/d (position of dark fringes)', "I = 4I'cos²(δ/2) (intensity in interference pattern)", 'w = Dλ/d (fringe width)', 'δ = (ω/c) * µ∆x (phase change)']
- Conditions for bright fringes: The path difference between the waves from the two slits is an integer multiple of the wavelength (∆x = nλ). The position of bright fringes is given by y = nDλ/d, where n is an integer, D is the distance between the slits and the screen, d is the separation between the slits, and λ is the wavelength of light.
- Conditions for dark fringes: The path difference between the waves from the two slits is an odd multiple of half the wavelength (∆x = (n + 1/2)λ). The position of dark fringes is given by y = (n + 1/2)Dλ/d.
- Fringe Width (w): The distance between two consecutive bright or dark fringes. w = Dλ/d. Fringe width decreases as the separation between the slits (d) increases.
- Intensity Variation: The intensity at a point in the interference pattern is given by I = 4I'cos²(δ/2), where I' is the intensity due to a single slit and δ is the phase difference. The intensity is maximum (4I') at the center of bright fringes and zero at the center of dark fringes.
- Optical Path: For a light wave travelling in a medium of refractive index µ, the phase changes by ω/c * µ∆x after travelling a distance ∆x.
Thin Film Interference
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When light falls on a thin film, it undergoes multiple reflections and refractions. These waves interfere, leading to constructive or destructive interference, resulting in colored patterns when white light is used.
['Optical path difference: ∆x = 2µd', 'Phase difference: δ = (2π/λ) * 2µd', 'Constructive interference (transmitted light): 2µd = nλ', 'Destructive interference (transmitted light): 2µd = (n + 1/2)λ', 'Minimum illumination in reflection: 2µd = nλ']
When light is incident on a thin film, it gets divided into reflected and refracted parts at both the upper and lower surfaces. This results in multiple reflected and transmitted waves. The interference of these waves determines the intensity of the reflected and transmitted light. The optical path difference between consecutive transmitted waves is 2µd, where µ is the refractive index of the film and d is its thickness. For normal incidence (i=0), the phase difference between the waves is also related to 2µd. Constructive interference occurs when 2µd = nλ, leading to strong transmission. Destructive interference occurs when 2µd = (n + 1/2)λ, leading to weak transmission. When white light is used, different wavelengths interfere constructively or destructively depending on the film thickness, resulting in a colored appearance. For reflected light, the optical path difference between consecutive reflected waves is also 2µd. The conditions for maximum and minimum illumination in reflection are opposite to those in transmission. Minimum illumination in reflection occurs when 2µd = nλ.
Coherence and Incoherence
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Ordinary light sources emit light in short bursts (wavetrains) with random phases, making them incoherent. Lasers emit long, coherent wavetrains due to cooperative behavior of atoms.
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- Ordinary light sources emit light in discrete steps by atoms.
- Atoms emit short light pulses and become inactive for a time before emitting another pulse.
- Active time is on the order of 10-8 s, during which a wavetrain of several meters is emitted.
- Ordinary light is a collection of several wavetrains with no fixed phase relation.
- These sources are incoherent.
- Different wavetrains are emitted by different groups of atoms acting independently.
- A narrow aperture is used to select a particular wavetrain to ensure the initial phase difference of wavelets does not change with time.
- Path difference between coherent sources should be kept small for a distinct interference pattern because wavetrains are finite in length.
- Incoherent nature of light emission leads to a spread in wavelength.
- Strictly monochromatic light requires an infinite sine wave.
- Shorter wavetrains imply a larger spread in wavelength.
- Laser sources emit very long wavetrains (hundreds of meters).
- Laser light is coherent because atoms behave cooperatively.
- Independent laser sources can produce interference fringes with path differences of several meters.
Diffraction of Light
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Diffraction is the bending of waves around obstacles. It's explained by Huygens' principle, where wavelets from exposed parts of a wavefront superpose. This results in interference patterns with varying intensity. Fraunhofer diffraction is a special case where the light source and screen are far from the obstacle, using plane waves.
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- Diffraction: Bending of waves around corners of an obstacle.
- Explanation: Huygens' principle. Wavelets from exposed parts of a wavefront superpose.
- Effect: Formation of fringe patterns (maxima and minima of intensity).
- Experimental Setup: Narrow light source, diffracting element (obstacle or opening), screen.
- Fraunhofer Diffraction: A special case where the source and screen are far from the diffracting element.
- Characteristics of Fraunhofer Diffraction: Plane waves are incident on the diffracting element, and interfering waves travel parallel to each other.
- Observation of Fraunhofer Diffraction: Achieved using converging lenses before and after the diffracting element, with the source and screen in their focal planes.
Diffraction
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⚡ Quick Summary
Diffraction is the bending of waves around obstacles or through apertures. The amount of bending depends on the wavelength of the wave and the size of the obstacle or aperture.
['I = I0 (sin(β)/β)^2 where β = (πb sinθ)/l (Intensity in single slit diffraction)', 'sin(θ) ≈ l/b (Angular position of first minima in single slit diffraction)', 'sin(θ) ≈ 1.22 (l/b) (Angular position of first dark ring in circular aperture diffraction)', 'R ≈ 1.22 (l*f)/b (Radius of first dark ring/Airy disk in circular aperture diffraction)']
Diffraction by a Single Slit
When light passes through a narrow slit of width 'b', the diffraction pattern consists of a central bright maximum flanked by alternating dark and bright fringes of decreasing intensity. The angular positions of the minima (dark fringes) are given by:
sin(θ) ≈ l/b
where:
* l is the wavelength of light.
* b is the width of the slit.
* θ is the angle of diffraction.
The intensity 'I' at an angle θ is given by:
I = I0 (sin(β)/β)^2, where β = (πb sinθ)/l
Fraunhofer Diffraction by a Circular Aperture
When a parallel beam of light passes through a circular aperture of diameter 'b', the diffraction pattern consists of a bright central disc (Airy disc) surrounded by concentric dark and bright rings. The first dark ring occurs at an angle θ given by:
sin(θ) ≈ 1.22 (l/b)
If a converging lens is used to focus the diffracted light onto a screen at the focal plane of the lens, the radius 'R' of the first dark ring (Airy disc radius) is:
R ≈ 1.22 (l*f)/b
where 'f' is the focal length of the lens.
Resolution and Rayleigh Criterion
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The Rayleigh criterion defines the limit of resolution for two point sources. Two images are considered just resolved when the center of one bright disc (diffraction pattern) falls on the edge of the second.
No specific formulas are mentioned directly for the Rayleigh criterion in this passage. It's a qualitative criterion based on the diffraction pattern.
Rayleigh Criterion: Two images are said to be just resolved when the center of the bright disc of one image falls on the periphery of the second image. This implies that the radius of each bright disc equals the separation between their centers. At this point, the resultant intensity between the images has a minimum.
Scattering of Light
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Scattering of light is the phenomenon where light deviates from its straight path due to interactions with particles in a medium. The intensity and color of scattered light depend on the wavelength of light and the size of the scattering particles.
Rayleigh's Law: Scattering is proportional to 1/λ<sup>4</sup>, where λ is the wavelength of light.
Scattering of Light: When light passes through a gas, a portion of it is redirected in different directions. This occurs due to the absorption of light by molecules followed by re-radiation. Scattering differs from absorption, where light energy converts into internal energy.
The intensity of scattering depends on wavelength and particle size. For particles smaller than the wavelength of light, the scattering is proportional to 1/λ4 (Rayleigh's Law). This means shorter wavelengths are scattered more strongly.
The intensity of scattering depends on wavelength and particle size. For particles smaller than the wavelength of light, the scattering is proportional to 1/λ4 (Rayleigh's Law). This means shorter wavelengths are scattered more strongly.
Applications of Scattering: Blue Sky and Red Sunset
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The blue color of the sky is due to Rayleigh scattering of sunlight by air molecules, which scatters blue light more than other colors. Sunsets appear red because blue light is scattered away as sunlight travels through a longer path in the atmosphere, leaving predominantly red light to reach the observer.
No specific formulas.
Blue Sky: Shorter wavelengths (like blue) are scattered more by air molecules, causing the sky to appear blue when looking away from the sun.
Red Sunset/Sunrise: At sunrise and sunset, sunlight travels a greater distance through the atmosphere. Blue light is scattered away, leaving the longer wavelengths (red) to dominate the light reaching the observer.
Red Sunset/Sunrise: At sunrise and sunset, sunlight travels a greater distance through the atmosphere. Blue light is scattered away, leaving the longer wavelengths (red) to dominate the light reaching the observer.
Factors Affecting Sky Color
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The presence of water droplets, dust, and pollutants can affect the color of the sky. Larger particles can scatter colors other than blue, leading to a lighter blue or even a grayish appearance.
No specific formulas mentioned.
Water droplets and dust particles, especially when larger than the wavelength of light, scatter light differently than predicted by Rayleigh's Law. This can result in a light blue sky on humid days. Pollution can also lead to a hazy, greyish sky due to the presence of suspended particles.
Polarization of Light
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Light is linearly polarized if the electric field oscillates along a single fixed direction perpendicular to the direction of propagation.
No specific formulas provided in this context.
Polarization of Light: The electric field in a light wave is perpendicular to the direction of propagation. If the electric field at a point always remains parallel to a fixed direction, the light is linearly polarized along that direction.
Polarization by Reflection and Refraction
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Light can be polarized by reflection and refraction. The degree of polarization depends on the angle of incidence. At Brewster's angle, the reflected light is completely polarized. Polarizing sunglasses utilize this principle to reduce glare.
["tan i = μ (Brewster's Law)"]
- When unpolarized light is incident on a transparent medium, the reflected and refracted rays are partially polarized.
- The plane containing the incident ray, reflected ray, and refracted ray is called the plane of incidence.
- The electric field of unpolarized light can be resolved into two components: one in the plane of incidence and the other perpendicular to it.
- Light with an electric field perpendicular to the plane of incidence is more strongly reflected.
- Brewster's Angle: The angle of incidence at which the reflected light is completely polarized. It is given by tan i = μ, where μ is the refractive index of the medium.
- Brewster's Law: The relationship between the angle of incidence and the refractive index (tan i = μ).
- Polarizing sunglasses use polaroids to absorb the polarized reflected light, reducing glare.
Polarization by Scattering
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Scattered light by small particles is partially polarized, like the blue light from the sky.
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- When unpolarized light is scattered by small particles, the scattered light becomes partially polarized.
- The blue light received from the sky is an example of partially polarized light due to scattering.
- Bees can detect the difference between unpolarized and polarized light and determine the direction of polarization.
Wavelength in a medium
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When light travels through a medium, its wavelength changes depending on the refractive index.
['λ = λ₀/μ']
- When a light having wavelength λ₀ in vacuum goes through a medium of refractive index μ, the wavelength in the medium becomes λ = λ₀/μ.
Optical Path
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Optical path is the product of the refractive index and the distance traveled.
['Optical Path = μx']
- When light travels through a distance x in a medium of refractive index μ, its optical path is μx.
Fringe Shift due to Transparent Sheet
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When a transparent sheet is introduced in the path of one of the interfering beams in Young's double slit experiment, the fringe pattern shifts. The number of fringes shifted depends on the thickness and refractive index of the sheet.
Number of fringes shifted = (μ - 1)t / λ
When light travels through a sheet of thickness t, the optical path traveled is μt, where μ is the refractive index. When one of the slits is covered by the sheet, air is replaced by the sheet, and hence, the optical path changes by (μ-1)t. One fringe shifts when the optical path changes by one wavelength.
Condition for Strong Reflection
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Light is strongly reflected from a thin film when the condition for constructive interference is met. This condition depends on the refractive index and thickness of the film, as well as the wavelength of the light.
2μd = (n + 1/2)λ
The condition for strong reflection (constructive interference) for a thin film of thickness 'd' and refractive index 'μ' is given by 2μd = (n + 1/2)λ, where n is a non-negative integer and λ is the wavelength of light.
Young's Double Slit Experiment - Fringe Position
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In Young's double slit experiment, bright fringes (maxima) are formed at specific distances from the central maximum. The position of these fringes depends on the wavelength of light, the distance between the slits, and the distance to the screen.
y = (nλD)/d
The center of the nth bright fringe is at a distance y = (nλD)/d from the central maximum, where n is the order of the fringe, λ is the wavelength of light, D is the distance between the slits and the screen, and d is the distance between the slits.
Single Slit Diffraction - Minima Position
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In single-slit diffraction, minima (dark fringes) occur at specific angular deviations. The position of these minima depends on the wavelength of light and the width of the slit.
['b sinθ = nλ', 'x ≈ Dsinθ ≈ (λD)/b']
The minima occur at an angular deviation θ given by b sinθ = nλ, where n is an integer (n = ±1, ±2, ...), λ is the wavelength of light, and b is the width of the slit. For small angles, sinθ ≈ θ. The linear distance from the central maximum to the minima is x ≈ Dsinθ, where D is the distance to the screen.
Wave Theory of Light
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Light can travel in vacuum and material medium. Wave theory of light is supported by interference, laws of reflection. Huygens' principle helps find wavefront positions and explain Snell's law. Speed of light is constant in vacuum regardless of observer motion. Frequency remains the same when light travels from one medium to another.
None explicitly mentioned in the provided text.
- Light Propagation: Light waves can travel in vacuum and material mediums.
- Wave Theory Support: Properties like interference support the wave theory of light.
- Huygens' Principle: This principle is used to find the new position of a wavefront and explain Snell's law.
- Speed of Light in Vacuum: The speed of light in vacuum is constant for all observers, regardless of their motion relative to the source.
- Frequency Invariance: The frequency of light remains constant when it travels from one medium to another.
Interference
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Coherent light sources produce interference patterns. Fringe width changes with the medium. In Young's double-slit experiment, changing the light source affects fringe brightness and spacing. Using white light results in a white central fringe and colored fringes next to it. Interference occurs between waves with the same frequency.
None explicitly mentioned in the provided text.
- Coherent Sources: Interference patterns are produced by coherent light sources.
- Fringe Width: The fringe width in Young's double-slit experiment changes depending on the medium in which the experiment is performed.
- Young's Double Slit: Changing the light source (e.g., from red to violet) in Young's double-slit experiment affects the brightness and spacing of the fringes.
- White Light: When white light is used, the central fringe is white, and the fringes next to it are colored.
- Frequency and Interference: Interference can be observed due to the superposition of waves with the same frequency.
Refraction and Refractive Index
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Refractive index is the ratio of the speed of light in vacuum to its speed in a medium. The wavelength of light changes when it enters a different medium, but the frequency remains constant.
None explicitly mentioned in the provided text.
- Refractive Index: The refractive index of a medium is defined as the ratio of the speed of light in vacuum to its speed in that medium.
- Wavelength Change: The wavelength of light changes when it enters a different medium.
- Frequency Conservation: The frequency of light remains constant when it travels from one medium to another.
Interference
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Interference is the phenomenon where two or more waves superimpose to form a resultant wave of greater, lower, or the same amplitude. This section deals with interference patterns produced by slits and mirrors.
['Fringe width (β) = λD/d, where λ is the wavelength of light, D is the distance between the slits and the screen, and d is the separation between the slits.']
- Fringe Width: The distance between two consecutive bright or dark fringes in an interference pattern.
- Young's Double Slit Experiment: A classic experiment demonstrating interference of light waves. Two coherent light sources (slits) are used to create an interference pattern on a screen.
Interference
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This section provides answers to problems related to interference of light waves, including calculations of fringe width, wavelength, refractive index, and intensity.
['Fringe width (β) = λD/d (where λ is wavelength, D is the distance to the screen, and d is the slit separation)', 'Optical path difference = (µ - 1)t (where µ is refractive index and t is thickness)', 'Condition for constructive interference: 2µt = nλ (where n is an integer)', 'Condition for destructive interference: 2µt = (n + 1/2)λ']
This section primarily contains numerical answers to problems. However, we can infer the underlying concepts and formulas based on the context of Wave Optics and Interference. These include:
* **Fringe Width (β):** The distance between two consecutive bright or dark fringes in an interference pattern.
* **Wavelength (λ):** The distance between two successive crests or troughs of a wave.
* **Refractive Index (µ):** The ratio of the speed of light in a vacuum to its speed in a medium.
* **Intensity (I):** The power per unit area carried by a wave.
Diffraction
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The bending of waves around obstacles or through apertures.
Single Slit Diffraction Minima: a sin θ = nλ; Resolving Power: R = λ/Δλ
Covers diffraction by single slits, circular apertures, and straight edges. Also covers Fraunhofer and Fresnel diffraction.
Interference
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The superposition of waves, resulting in constructive or destructive interference.
Path difference = nλ (constructive interference); Path difference = (n + 1/2)λ (destructive interference)
Deals with coherent sources, fringes, and interference patterns.
Refraction
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The bending of light as it passes from one medium to another.
Snell's Law: n1 sin θ1 = n2 sin θ2
Includes critical angle and dispersion.