Wave Optics Class 12 Physics Chapter 10 Notes

Wave Optics Class 12 Physics Chapter 10 Notes

Huygens’ Principle

1. Wavefront Definition:
   • A wavefront is a surface of constant phase in a wave. It represents the locus of points where oscillations are in phase.
   • Circular rings in a calm pool after a disturbance illustrate this concept.
2. Wavefront Speed:
   • The speed at which a wavefront moves away from its source is the wave’s speed.
   • Energy in a wave travels perpendicular to the wavefront.
3. Spherical Waves:
   • A point source emits spherical waves, with waves propagating uniformly in all directions.
   • At a distance from the source, a portion of the sphere behaves like a plane wave.
4. Huygens’ Principle:
   • Huygens’ principle allows the determination of a wavefront’s shape at a later time, given its shape at t = 0.
   • It is a geometrical construction approach.
5. Wavefront Evolution:
   • Consider a diverging spherical wave with portions F1F2 at t = 0.
   • Each point on the wavefront acts as a source for secondary wavelets that propagate in all directions at the wave speed.
   • These secondary wavelets form new wavefronts at a later time when their tangents are connected.
   • For a wavefront at t = t, draw spheres of radius vt from each point on the wavefront.
   • The new wavefront’s shape is determined by connecting tangents to these spheres.
6. Backwave Issue:
   • Huygens proposed that secondary wavelets have maximum amplitude in the forward direction and zero in the backward direction, explaining the absence of a backwave.
   • However, this ad hoc assumption is not entirely satisfactory and is better justified by rigorous wave theory.
7. Plane Waves:
   • Huygens’ principle can also be applied to determine the shape of wavefronts for plane waves propagating through a medium.

Huygens’ principle provides a useful geometric method to understand the propagation of waves and predict the shape of wavefronts at different times.

Refraction of Plane Waves

1. Setup:
   • Consider an interface PP’ separating two mediums: medium 1 (with speed v1) and medium 2 (with speed v2).
   • A plane wavefront AB travels in direction A’ A, making an angle of incidence (i) with the normal to the interface.
2. Wavefront Propagation:
   • Time taken by the wavefront to travel BC is t, where BC = v1 * t.
   • A sphere with radius v2 * t is drawn from point A in medium 2, and a tangent plane CE is drawn from point C to the sphere.
   • CE represents the refracted wavefront.
3. Deriving Snell’s Law:
   • By considering triangles ABC and AEC, we obtain:
     • sin(i) = BC / AC = v1t / AC
     • sin(r) = AE / AC = v2t / AC
4. Relating Speeds and Angles:
   • Combining the above results, we have: sin(i) / sin(r) = (v1 / v2)
   • This equation indicates that if the ray bends toward the normal (r < i), the speed of light in medium 2 (v2) is less than that in medium 1 (v1).
5. Refractive Indices:
   • The speed of light in vacuum (c) is related to the refractive indices of the mediums:
     • n1 = c / v1 and n2 = c / v2
   • Thus, n1 and n2 represent the refractive indices of medium 1 and medium 2, respectively.
6. Snell’s Law:
   • In terms of refractive indices, Snell’s law of refraction is given by: n1 * sin(i) = n2 * sin(r)
7. Wavelength and Frequency:
   • If l1 and l2 are the wavelengths in medium 1 and medium 2, respectively, and BC = l1:
     • AE will be equal to l2, as waves maintain their frequency.
   • This leads to the relationship: v1 / λ1 = v2 / λ2
   • When a wave is refracted into a denser medium (v1 > v2), the wavelength and speed decrease, while frequency remains the same.

Snell’s law and the relationships between refractive indices, speeds, wavelengths, and frequencies provide a comprehensive explanation of the refraction of plane waves at the interface between two mediums.

Refraction in Rarer Medium

1. Rarer Medium:
   • Consider the case where medium 2 (v2) is rarer than medium 1 (v1), which means v2 > v1.
2. Construction of Refracted Wavefront:
   • Similar to the previous case, a refracted wavefront is constructed.
   • The angle of refraction (r) will be greater than the angle of incidence (i).
3. Snell’s Law for Rarer Medium:
   • Despite the increase in the angle of refraction, Snell’s law still holds: n1 * sin(i) = n2 * sin(r).
4. Critical Angle:
   • An angle i_c is defined as follows: sin(i_c) = n2 / n1.
   • When i equals i_c, sin(r) becomes 1, and r equals 90°.
   • For angles of incidence greater than i_c, there is no refracted wave.
5. Total Internal Reflection:
   • The angle i_c is known as the critical angle.
   • For all angles of incidence greater than the critical angle, there is no refracted wave.
   • Instead, the wave undergoes total internal reflection.

Total internal reflection occurs when a wave, traveling from a denser medium to a rarer medium at an angle greater than the critical angle, is reflected entirely within the denser medium. This phenomenon is essential in optical devices like fiber optics and prisms.

Reflection of Plane Waves at a Plane Surface

1. Incident Wave:
   • Consider a plane wave AB incident at an angle (i) on a reflecting surface MN.
   • The wave travels at speed (v) in the medium, and it takes time (t) for the wavefront to advance from point B to point C.
   • The distance BC = vt.
2. Construction of Reflected Wavefront:
   • To construct the reflected wavefront, draw a sphere with a radius of vt from point A.
   • Let CE represent the tangent plane drawn from point C to the sphere.
   • It is evident that AE = BC = vt.
3. Law of Reflection:
   • By considering triangles EAC and BAC, it is found that they are congruent.
   • As a result, the angles i (incident) and r (reflected) are equal.
   • This is known as the law of reflection.
4. Understanding Optical Phenomena:
   • Understanding the laws of reflection and refraction is crucial for comprehending the behavior of optical devices such as prisms, lenses, and mirrors.
   • These devices were previously explained based on the rectilinear propagation of light.
5. Reflection and Refraction:
   • a. Light passing through a thin prism causes a tilt in the emerging wavefront due to the difference in the speed of light in glass.
   • b. A plane wave incident on a thin convex lens results in a converging, spherical wavefront that focuses at a point called the focus.
   • c. A plane wave incident on a concave mirror leads to a spherical wavefront converging at the focal point F.
   • Similar principles apply to reflection and refraction by concave lenses and convex mirrors.
6. Equal Time for All Rays:
   • The total time taken from a point on the object to the corresponding point on the image is the same along any ray.
   • For instance, when a convex lens forms a real image, even though the ray passing through the center of the lens travels a shorter path, the slower speed in the glass ensures that the time taken is the same as for rays near the lens’s edge.

Understanding these principles of reflection and refraction is fundamental to optics and the behavior of light when interacting with various optical elements.

Coherent and Incoherent Addition of Waves

1. Superposition Principle:
   • Based on the superposition principle, the resultant displacement at a point is the vector sum of displacements produced by multiple waves.
2. Coherent Sources:
   • Consider two identical sources, S1 and S2, both producing periodic waves.
   • Coherent sources maintain a constant phase relationship over time, making their waves interfere in a predictable manner.
3. Constructive Interference:
   • At a point P, where the path difference between S1P and S2P is equal to an integer multiple of the wavelength (nl), constructive interference occurs.
   • Resultant displacement is in phase, leading to maximum intensity (4I0).
4. Destructive Interference:
   • At a point Q, where the path difference between S1Q and S2Q is equal to half-integer multiples of the wavelength ((n + 1/2)l), destructive interference occurs.
   • Resultant displacement is out of phase, leading to zero intensity.
5. General Interference:
   • For an arbitrary point G with a phase difference (f) between the displacements, the resultant displacement is given by: y = 2a cos(f/2) cos(wt + f/2).
   • The intensity at this point is: I = 4I0 cos^2(f/2).
   • Maximum intensity occurs when f = 0, ±2π, ±4π, etc., corresponding to constructive interference.
   • Zero intensity occurs when f = ±π, ±3π, ±5π, etc., corresponding to destructive interference.
6. Stable Interference Pattern:
   • When two coherent sources maintain a constant phase difference over time, the interference pattern remains stable, with fixed positions of maxima and minima.
7. Incoherent Sources:
   • Incoherent sources do not maintain a constant phase difference over time.
   • Rapidly changing phase differences lead to varying interference patterns and time-averaged intensity distributions.
   • Incoherent sources exhibit an average intensity where the intensities just add up.
8. Incoherent Light Sources:
   • When two separate light sources illuminate a wall, the intensities add up due to incoherence, resulting in an average intensity.

Interference of Light Waves & Young’s Experiment

1. Incoherent Light Sources:
   • When using two sodium lamps to illuminate two pinholes, no interference fringes are observed.
   • Light waves from ordinary sources like sodium lamps undergo rapid phase changes (on the order of 10^-10 seconds).
   • Light waves from independent sources lack a fixed phase relationship and are incoherent, resulting in intensity addition.
2. Young’s Experiment:
   • British physicist Thomas Young devised an experiment to create coherent sources.
   • He placed two pinholes, S1 and S2, very close to each other on an opaque screen. These pinholes were illuminated by another pinhole, which, in turn, was lit by a bright source.
   • Light waves from the bright source spread out and fall on both S1 and S2.
   • S1 and S2 behave like coherent sources because any abrupt phase changes in the original source are mirrored in both S1 and S2.
   • S1 and S2 are locked in phase, making them coherent sources, similar to the vibrating needles in the water wave example.
3. Interference Fringes:
   • Spherical waves emanating from S1 and S2 create interference fringes on the screen GG’.
   • Positions of maximum and minimum intensities can be calculated using the principles of interference.
4. Constructive Interference:
   • Constructive interference occurs, creating a bright region when the path difference xdD is equal to an integer multiple of the wavelength (xdD = nl).
   • This condition is expressed as x = xn = nDd/λ, where n = 0, ±1, ±2, ….
5. Destructive Interference:
   • Destructive interference results in a dark region when the path difference xdD is equal to a half-integer multiple of the wavelength (xdD = (n + 1/2)λ).
   • This condition is expressed as x = xn = (n + 1/2)Dd/λ, where n = 0, 1, 2, ….
6. Interference Fringes:
   • As a result of these conditions, dark and bright bands, known as fringes, appear on the screen.
   • The dark and bright fringes are equally spaced.

Thomas Young’s experiment demonstrated that coherent sources can create interference fringes, leading to a pattern of bright and dark bands known as fringes, with regular spacing. This experiment played a significant role in the development of wave theory in optics.

Diffraction

1. General Phenomenon:
   • Diffraction is a general characteristic exhibited by all types of waves, including sound waves, light waves, water waves, and matter waves.
2. Phenomenon Close to Geometrical Shadow:
   • Close to the region of the geometrical shadow cast by an opaque object, there are alternating dark and bright regions, similar to interference.
   • This phenomenon is a result of diffraction.
3. Wavelength and Diffraction:
   • Diffraction occurs because the wavelength of a wave is comparable to or larger than the dimensions of the obstacle or aperture.
   • In the case of light, the wavelength is much smaller than most everyday objects, so diffraction effects are not typically observed.
4. Practical Applications:
   • Diffraction plays a crucial role in various practical applications:
     • The finite resolution of the human eye and optical instruments like telescopes and microscopes is limited by diffraction.
     • Diffraction effects are responsible for the colors observed when viewing a CD.
     • It impacts the behavior of waves in various real-world situations.

Diffraction is a fundamental phenomenon that occurs when waves encounter obstacles or apertures, leading to characteristic patterns of alternating dark and bright regions, which are vital in various aspects of wave theory and everyday observations.

The Single Slit and Diffraction

1. Historical Observations:
   • Early experimenters, including Newton, noticed that light spreads out from narrow holes and slits.
   • Light seemed to turn around corners and enter regions where shadows were expected.
   • These effects, known as diffraction, can be best understood using wave principles.
2. Analogous to Sound Waves:
   • Just as we hear sound waves from someone talking around a corner, light exhibits similar behavior in certain situations.
3. Single Narrow Slit Diffraction:
   • When a single narrow slit (illuminated by a monochromatic source) is used in an experiment, a broad pattern with a central bright region is observed.
   • On both sides of the central maximum, there are alternating dark and bright regions, with the intensity decreasing away from the center.
4. Single Slit Diffraction Pattern:
   • Consider a parallel beam of light falling normally on a single slit LN of width “a.”
   • The diffracted light meets a screen, and the midpoint of the slit is M.
   • A straight line through M perpendicular to the slit plane meets the screen at point C.
   • Intensity at any point P on the screen can be determined by considering the angle “θ” formed by the straight line from P to points L, M, N, etc.
5. Treatment as Secondary Sources:
   • The basic idea is to divide the slit into smaller parts and add their contributions at point P with the proper phase differences.
   • Different parts of the wavefront at the slit act as secondary sources.
   • Since the incoming wavefront is parallel to the plane of the slit, these sources are in phase.
6. Intensity Pattern:
   • The intensity pattern exhibits a central maximum at θ = 0 and secondary maxima at θ = (n + 1/2)λ/a, with n representing increasing integers.
   • The minima (zero intensity) occur at θ = nλ/a, where n = ±1, ±2, ±3, and so on.
   • The intensity pattern typically shows a central maximum with alternating dark and bright regions, with the brightness diminishing as n increases.
7. Interference vs. Diffraction:
   • The distinction between interference and diffraction has been a subject of discussion among scientists.
   • According to Richard Feynman, there is no specific, important physical difference between them.
   • Usage generally dictates the terminology, with interference often used for a few sources and diffraction for a large number of sources.
   • In the double-slit experiment, the pattern on the screen is a combination of single-slit diffraction from each slit or hole and double-slit interference patterns.

Diffraction patterns from single slits provide insights into wave behavior and are often integral to understanding light’s interactions with objects and apertures.

Seeing the Single Slit Diffraction Pattern

1. Equipment Needed:
   • To observe the single-slit diffraction pattern, you can use everyday items that can be found in most homes.
   • You’ll need two razor blades and a clear glass electric bulb with a straight filament.
2. Setting Up the Experiment:
   • Hold the two razor blades so that their edges are parallel and create a narrow slit between them.
   • This can be easily achieved by using your thumb and forefingers to adjust the blades.
   • Ensure that the slit remains parallel to the filament inside the bulb.
3. Observation:
   • Position the slit right in front of your eye, keeping it parallel to the filament.
   • If you wear glasses, use them as well for a clearer view.
   • By slightly adjusting the width of the slit and the parallelism of the edges, you should be able to observe the diffraction pattern with its alternating bright and dark bands.
4. Color in the Pattern:
   • The position of all the bands, except the central one, depends on the wavelength of light, resulting in a display of colors.
   • You can enhance the visibility of the fringes by using a red or blue filter.
   • When both filters are available, you’ll notice that the fringes are wider for red light compared to blue.
5. Experiment Explanation:
   • In this experiment, the straight filament inside the bulb simulates the role of the first slit (S) in the theoretical setup.
   • The lens of your eye focuses the observed diffraction pattern onto your retina.
6. Alternative Experiment:
   • With some effort, you can also cut a double slit in aluminum foil using a blade.
   • This homemade double-slit setup allows you to repeat Young’s double-slit experiment.
   • During the daytime, you can find another suitable bright source by looking at the reflection of the Sun in any shiny convex surface (e.g., a cycle bell). Be cautious not to look at direct sunlight, which can damage your eyes and won’t produce fringes due to the Sun’s large angular size.
7. Conservation of Energy:
   • In both interference and diffraction, light energy is redistributed.
   • When it decreases in one region, resulting in a dark fringe, it increases in another region, producing a bright fringe.
   • Importantly, there is no net gain or loss of energy, which is consistent with the principle of the conservation of energy.

This simple experiment allows you to directly observe the fascinating behavior of light in the presence of a single slit, producing a diffraction pattern with alternating bands of color and darkness.

Polarization of Light Waves

In the context of polarization, the behavior of light waves is discussed:

1. Transverse Waves:
   • Light waves are described as transverse waves. In a transverse wave, the displacement is perpendicular to the direction of wave propagation.
2. Linear Polarization:
   • When the displacement of each point of a wave is along a straight line and always at right angles to the direction of propagation, it is referred to as a linearly polarized wave.
   • A string moved up and down generates a transverse wave, and this displacement is linearly polarized. Such a wave remains confined to the x-y plane and is called a plane polarized wave.
   • Similarly, if the string vibrates in the x-z plane, it generates a z-polarized wave, which is also linearly polarized.
3. Unpolarized Waves:
   • If the plane of vibration of the wave (e.g., the string’s motion) changes randomly in very short intervals of time, it is referred to as an unpolarized wave.
   • In an unpolarized wave, the displacement is constantly changing with time but always perpendicular to the direction of propagation.
4. Polaroids:
   • Polaroids are sheets made of long chain molecules aligned in a particular direction. They can be used to demonstrate the polarization of light.
   • When unpolarized light is incident on a polaroid, it gets linearly polarized, with the electric vector oscillating along a direction perpendicular to the aligned molecules (known as the pass-axis).
   • A polaroid will absorb electric vectors in the direction of the aligned molecules, allowing only the perpendicular components to pass through.
5. Malus’ Law:
   • Malus’ law explains the behavior of polarized light intensity as it passes through polaroids.
   • If unpolarized light passes through a polaroid (let’s call it P1), its intensity is reduced by half. The rotation of P1 does not affect the transmitted intensity.
   • When a second identical polaroid (P2) is placed before P1, the intensity is further reduced. The angle between the pass-axes of P1 and P2 plays a significant role.
   • Rotating P1 causes variations in the intensity of light emerging from P2. This variation is described by Malus’ law: I = I0 cos^2(q), where I0 is the intensity after P1, and q is the angle between the pass-axes of P1 and P2.
6. Applications of Polaroids:
   • Polaroids are used in various applications such as sunglasses, windowpanes, photographic cameras, and 3D movie cameras to control and manage light intensity.

Polarization of light plays a crucial role in a range of technologies and applications, allowing control and manipulation of light waves for specific purposes.