Class 11 > Unit # 13: Physical Optics > Huygens Principle & Interference of Waves

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Wavefront, Huygen’s Principle & Wave Interference

Wavefront

“Whenever a wave passes through a certain medium, its particles execute simple harmonic motion. The locus of all the points in the medium having the same phase of vibration is called a wavefront.”

Types of Wavefronts:

• Spherical Wavefront: In the case of a point source of light in a homogeneous medium, the wavefronts form concentric spheres centered at the source $S$.
• Plane Wavefront: At a very large distance from the source, a small portion of a spherical wavefront becomes nearly flat. Such a portion is called a plane wavefront.

In optics, a ray represents the direction in which a wave propagates and is always normal (perpendicular) to the wavefront. Thus, a plane wavefront represents a parallel pencil of rays.

Fig: Spherical wavefronts from a point source and plane wavefronts at a large distance.

Huygen’s Principle

Huygen’s principle consists of two fundamental postulates:

1. Every point on a given wavefront acts as a source of secondary spherical wavelets.
2. The secondary wavelets spread out forward with a speed equal to the propagation speed of the primary waves in that medium.

Geometrical Construction: In the figure below, let $AB$ represent the position of a wavefront at a given instant. To find the new position of the wavefront after time $t$:

• Select points on the wavefront $AB$ acting as secondary sources.
• Draw secondary wavelets (arcs) with radius $r = vt$, where $v$ is the wave speed.
• Draw a surface $CD$ tangential to the forward envelope of these secondary wavelets. The new surface $CD$ represents the new wavefront after time $t$.

Fig: Huygen’s construction showing propagation from wavefront $AB$ to $CD$.

Interference of Waves

“When two waves superpose one another, they either enhance or reduce their net effect at that point. This phenomenon is called interference of waves.”

Constructive vs. Destructive Interference

1. Constructive Interference

When two waves meet such that the crest of one wave coincides with the crest of another, and their troughs coincide, the resulting amplitude and loudness are enhanced. This is called constructive interference.

For constructive interference, the path difference between the waves must be an integral multiple of wavelength ($\lambda$):

$\text{Path Difference} = m\lambda$

where $m = 0, \pm 1, \pm 2, \pm 3, \dots$

2. Destructive Interference

When two waves meet such that the crest of one wave coincides with the trough of another, the resulting amplitude and loudness are reduced or cancelled out. This is called destructive interference.

For destructive interference, the path difference between the waves must be an odd multiple of half-wavelength ($\frac{\lambda}{2}$):

$\text{Path Difference} = \left(m + \frac{1}{2}\right)\lambda = (2m + 1)\frac{\lambda}{2}$

where $m = 0, \pm 1, \pm 2, \pm 3, \dots$

Fig: Constructive and destructive interference patterns resulting from wave superposition.

Conditions for Interference of Light

To observe a stable and sustained interference pattern of light, the following conditions must be satisfied:

1. Monochromatic and Coherent Sources: The sources of light must emit light of a single frequency (monochromatic) and maintain a constant phase relationship (coherent).
2. Narrow Slits: The emitting slits must be extremely narrow, on the order of the wavelength of light used.
3. Close Separation: The interfering slits must be separated by a very small distance to ensure observable fringe spacing.

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