Huygens' Principle: Wave Optics

• A wavefront is a continuous set of medium particles vibrating in the same phase at any instant.
• A ray is a line drawn perpendicular to a wavefront that shows the wave's direction of travel.

Huygens' Principle

• Every point on a wavefront acts as a source of secondary waves. These waves combine to form a new wavefront later in time.
• Huygens' Principle is based on the following:
• Each point on a wavefront is a new source of disturbance, creating secondary waves or wavelets.
• Secondary wavelets spread in all directions at the speed of light in the medium.
• A new wavefront at a later time is the forward envelope (tangential surface) of the secondary wavelets at that time.

Laws of Reflection Using Huygens' Wave Theory

• When a plane wavefront AB hits a reflective surface XY, it first touches at point B, then at successive points toward C.
• From each point on BC, secondary wavelets grow at speed c, according to Huygens' Principle.
• The disturbance from A reaches C in time t, so wavelets from B spread over a hemisphere of radius BD = AC = ct.
• The tangent plane CD from point C is the new reflected wavefront.
• The angle of incidence (i) equals the angle of reflection (r), proving the first law of reflection.
• Incident ray SB, normal BN, and reflected ray BD are perpendicular to wavefronts AB, XY, and CD, respectively, confirming they lie in the same plane and proving the second law of reflection.

Laws of Refraction Using Huygens' Wave Theory

• Consider a plane wavefront AB incident on surface XY, separating media 1 and 2 with light velocities v1 and v2 (v2 < v1).
• The wavefront first hits point A, then successive points toward C.
• Wavelets from AC grow in the second medium at speed v2, according to Huygens' Principle.
• In time t, the disturbance from B reaches C, so BC = v1t.
• Meanwhile, the disturbance from A spreads over a hemisphere of radius AD = v2t.
• The tangent plane CD from point C is the new refracted wavefront.
• The angles of incidence and refraction are i and r, respectively.
• Snell's law of refraction states sini/sinr = v1/v2 = constant = 1µ2 (refractive index of the second medium relative to the first).
• Incident ray SA, normal AN, and refracted ray AD are perpendicular to wavefronts AB, XY, and CD, so they lie in the same plane.
• Frequency remains constant as light travels from one medium to another.
• Wavelength is directly proportional to phase speed and inversely proportional to the refractive index.
• Light speed is greater in an optically rarer medium than in a denser one.

Superposition of Waves

• When multiple waves superpose, the resultant displacement at any point is the vector sum of individual displacements: ŷ = ŷ1 + ŷ2 + ŷ3 + ... + ŷn.

Interference of Light

• When light waves of same frequency and constant phase difference superpose, intensity is redistributed, creating maximums and minimums (interference).

Conditions for Constructive and Destructive Interference

• Displacements of two light waves from coherent sources S1 and S2 at point P are y1 = a1sin(ωt) and y2 = a2sin(ωt + φ), where φ is the constant phase difference.
• The superposition principle gives resultant displacement y = a1sin(ωt) + a2sin(ωt + φ).
• The intensity at any point in an interference pattern
• Squaring and adding equations leads to the resultant wave's intensity, dependent on the amplitudes and phase difference of the original waves.

Constructive Interference

• Maximum intensity occurs when cos φ = 1, or φ = 0, 2π, 4π,...
• Path difference p = nλ, where n = 0, 1, 2, 3,...
• Constructive interference happens when the phase difference is a multiple of 2π, or the path difference is an integer multiple of the wavelength λ.

Destructive Interference

• Minimum intensity occurs when cos φ = -1, or φ = π, 3π, 5π,...
• Path difference p = (2n - 1)λ/2 = λ/2, 3λ/2, 5λ/2,..., where n = 1, 2, 3, 4,...
• Destructive interference happens when the phase difference is an odd multiple of π, or the path difference is an odd multiple of λ/2.

Coherent and Incoherent Sources

• Coherent sources emit light of same frequency and constant phase difference.
• Incoherent sources emit light with no constant phase difference.

Theory of Interference Fringes

• A narrow slit S is illuminated by monochromatic light of wavelength λ.
• Two narrow slits S1 and S2 are equidistant from S and separated by a small distance d; they act as coherent sources.
• The point P on a screen is a distance x from the center O.
• The path difference p = S2P - S1P.

Positions of Bright Fringes

• For constructive interference, p = xd/D = nλ.
• Positions of bright fringes are xn = nDλ/d, where n = 0, 1, 2, 3...

Positions of Dark Fringes

• For destructive interference, p = (2n - 1)λ/2.
• Positions of dark fringes are xn = (2n - 1)Dλ/(2d), where n = 1, 2, 3...

Fringe Width

• Fringe width is the separation between two successive bright or dark fringes.
• Hence the expression for the fringe width in Young's double slit experiment is β = Dλ/d
• All fringes are of equal width.
• λ is small, D should be much larger than d, making β appreciable.

Conservation of Energy in Interference

• In an interference pattern, Imax α (a1 + a2)² and Imin α (a1 - a2)².
• If the intensity at every point is the same with or without interference, then I = I1 + I2 α a1² + a2².
• There is no violation of the law of conservation of energy

Comparison of Intensities at Maxima and Minima

• Let a1 and a2 be the amplitudes and I1 and I2 be the intensities of light waves from two different sources.
• As intensity ∝ Amplitude² the ratio 11/12 = a12/a22
• Amplitude at a maximum = a1 + a2
• Amplitude at a minimum = a1 - a2

Ratio of Intensities at Maxima and Minima

• The ratio of intensities at maxima and minima: Imax/Imin = [(a1 + a2)/(a1 - a2)]² or [(r + 1)/(r - 1)]², where r = a1/a2 = √I1/√I2 (amplitude ratio).

Diffraction of Light

• Diffraction is the bending of light around corners of obstacles or apertures, causing it to spread into geometric shadows.

Diffraction at a Single Slit

• A source of monochromatic light shines through a narrow rectangular slit of width d.
• According to Huygens' theory, each part of the slit becomes a source of secondary wavelets.
• A convex lens focuses the diffraction pattern onto a screen.

Central Maximum

• Wavelets from corresponding points of the slit's two halves reach the central point in phase, creating a bright fringe.

Path Difference Calculation

• For wavelets diffracted at angle θ and focused at point P, the path difference is p = d sin θ.

Positions of Minima

• At point P on the screen: d sin θ1 = λ.

• The slit is divided into halves to calculate the result

• First dark fringe's condition: d sin θ1 = λ.

• Second dark fringe's condition: d sin θ2 = 2λ.

• The condition for nth dark fringe can be written as d sin θn = nλ, n = 1, 2, 3, .

• The directions of secondary maxima are given by θn = sin θn=(2n+1) .

Positions of Secondary Maxima

• Assume point P is situated so that p = 3λ/2.
• If θ = θ1, then d sin θ1 = 3λ/2.
• The slit is dividing into three equal components for analysis
• The first secondary maximum is the condition d sin θ1 = 3λ/2
• Similarly, the condition for the second secondary maximum is d sin θ2 = 5λ/2

Intensity Distribution

• The intensity is graphed with respect to the diffraction angle θ.
• Maxima are on either side of the central maxima and it has decreasing intensity at minima, with positions , θ = ±λ/d

Angular Width of Central Maximum

• The angular width of the central maximum refers to the angular separation between the first minima's directions on either side of the central max.
• The first minima's directions are provided by θ = λ/d
• This is known as the maximum's half angular width.
• The angular width of central maximum = DA/d

Linear Width of Central Maximum

• Let D represent the screen's distance from the single slit; the linear width of center maximum becomes B = D×20= 2DA/d

Linear Width of a Secondary Maximum

• The nth secondary maximum's angular width corresponds to the angular separation between the nth and (n + 1)th minima.
• Angular width of nth secondary maximum, equals Angular width × D or DA/d
• Thus central maximum/slight width is twice as wide as any secondary.

Difference between Interference and Diffraction

• Interference results from the superposition of secondary waves originating from two different wavefronts
• All bright and dark fringes are of equal width,
• All bright fringes are of same intensity.
• Regions of dark fringes are perfectly dark
• Diffraction results from the superposition of secondary waves starting from different parts of the same wavefront.
• The central bright fringe has twice the width of any secondary maximum.
• As one moves away from the center, the intensity of bright fringes diminishes on either side.
• Dark fringes are not areas in the regions dark

Limit of Resolution

• The limit of resolution defines the ability to distinguish objects as distinct entities

Rayleigh's Criterion

• According to Rayleigh's criterion, images of two point objects can be resolved when the central maximum of one's diffraction pattern falls over the other's first minimum.

Resolving Power of a Microscope

• Resolving power is the reciprocal of the shortest resolvable distance: d = λ/(2μsinθ).

Resolving Power of a Telescope

• The telescope's resolving power is the distance at which images of far-off objects can be resolved
• Limit of resolution: dθ = 1.22(λ/D)..

Law of Malus

• When polarized light passes through an analyzer, the intensity I varies as the square of the cosine of the angle θ between the polarizer and analyzer: I = I0cos²θ.

Brewster's Law

• The angle of incidence at which unpolarized light reflects as completely polarized light is the polarizing or Brewster angle (ip).
• The refractive index μ = tan ip.

Polaroids

• Made by means of selective absorption of linearly polarized light

Explore Huygens' Principle. Learn about wavefronts, secondary wavelets, reflection, and refraction. Understand how the principle explains light propagation and the laws of reflection and refraction.