Huygens’ Principle

It is a key concept in wave optics that explains how wavefronts propagate. According to this principle, every point on a given wavefront acts as a source of secondary spherical wavelets that spread out in all directions at the same speed as the original wave. The new wavefront is formed by drawing a surface tangent to these secondary wavelets. This principle successfully explains various wave phenomena such as reflection, refraction, and diffraction, which could not be fully understood using particle theory alone. Huygens’ Principle also plays a crucial role in the study of light behavior, supporting the wave theory of light. It is especially important in understanding concepts like interference and diffraction.

1.0Introduction

• When a stone is dropped into still water, circular waves spread out from the point of impact.
• Every point on the water’s surface begins to oscillate, forming circular rings where the disturbance is maximum.
• All points on a ring oscillate in phase, as they are equidistant from the source.

2.0Wavefronts

• A wavefront is defined as a locus of points having the same phase of oscillation.
• In simple terms, it is a surface of constant phase.
• The wave speed is the rate at which the wavefront moves outward from the source.
• Energy carried by the wave travels in a direction perpendicular to the wavefront.

3.0Type of Wavefront

The shape of wavefront depends upon the shape of the light source from which the

wavefront originates. On this basis there are three types of wavefronts.

1. Spherical Wavefront:

• Produced by a point source of light or sound.
• Energy moves outwards in straight lines (rays) from the source.
• Rays are radii of the spherical wavefront.
• Spacing between wavefronts (wavelength) remains constant along any ray.

$Intensity(I)\propto \frac{1}{Area}$

$I\propto \frac{1}{r^2}(Area=4\pi r^2)$

$Intensity\propto (Amplitude{)}^2$

$\frac{1}{r^2}\propto A^2\Rightarrow A\propto \frac{1}{r}$

2. Cylindrical Wavefront:

• Formed by linear sources such as a narrow slit illuminated by a light source behind it.
• At large distances, cylindrical wavefronts also appear planar.

$Intensity(I)\propto \frac{1}{Area}\Rightarrow I\propto \frac{1}{r}(Area=2r\pi h)$

$Intensity\propto (Amplitude{)}^2$

$\frac{1}{r^2}\propto A^2\Rightarrow A\propto \frac{1}{\sqrt{r}}$

3. Plane Wavefront:

• At large distances from a point source, a small portion of a spherical wavefront appears flat.
• In such cases, wavefronts are considered planes.
• Rays are parallel and perpendicular to the wavefronts.

$Intensity(I)\propto \frac{1}{Area}$

I=Constant (Area is constant)

$Intensity\propto (Amplitude{)}^2$

A=Constant

Wavefront	Source	Shape of Wavefront	Amplitude (A)	Intensity (I)
Spherical	Point source		$$A\propto \frac{1}{r}$$	$$I\propto \frac{1}{r^2}$$
Cylindrical	Linear source/ Slit		$$A\propto \frac{1}{\sqrt{r}}$$	$$I\propto \frac{1}{r}$$
Plane	Extended Large source / Point source at very large distance		$$A\propto r^0$$	$$I\propto r^0$$

4.0Characteristics of Wavefront

The phase difference between various particles on the wavefront is zero.

• These wavefronts travel with the speed of light in all directions in an isotropic medium.
•  A point source of light always gives rise to a spherical wavefront in an isotropic medium.
•  Normal to the wavefront represents a ray of light.
•  It always travels in the forward direction in the medium.

5.0Huygens' Wave Theory of Light

The locus of all particles vibrating in the same phase is known as wavefront.

•  Light travels in a medium in the form of a wavefront.
•  When light travels in a medium then the particles of medium start vibrating and consequently a disturbance is created in the medium.
•  Every point on the wavefront becomes the source of secondary wavelets. It emits secondary wavelets in all directions which travel with the speed of light.
• The tangent plane to these secondary wavelets represents the new position of the wavefront

The phenomena explained by this theory

•  Reflection, Refraction, interference and diffraction
•  Rectilinear propagation of light.
•  Velocity of light in the rarer medium being greater than that in denser medium.

The phenomena not explained by this theory

(1) Photoelectric effect (2) Polarisation

Note: Huygens considered that the environment is filled with anisotropic luminiferous ether but later he was proved wrong, later.

6.0Huygens' Construction

Initial Wavefront $(F_1)$

• At time t = 0, the wavefront $F_1$ separates the disturbed region from the undisturbed region.

Secondary Wavelets

• Each point on wavefront $F_1$ emits a spherical secondary wavelet.
• After time t, Each wavelet has a radius of vt, where v is the speed of the wave.

New Wavefront $(F_2)$

• The surface that is tangent to all these wavelets is the new wavefront $F_2$.
• This is called the forward envelope of the wavelets.

Ray Construction

• Join any point $A_1$ on $F_1$ to its corresponding point $A_2$ on $F_2$.
• This line segment $A_1{A}_2$ is a ray, and it is:

a. Perpendicular to both $F_1$ and $F_2$

b. Of length vt, indicating the wave has traveled a distance vt  in time t

• The time taken by light between wavefronts is the same along every ray.

Repetition (Huygens' Construction)

• This process can be repeated from $F_2$ to get $F_3$, then $F_4$, and so on.
• This step-by-step geometrical method is known as Huygens’ Construction.

7.0Law of Reflection by Huygens's Wave Theory

$B{B}^{\prime }=v_{1}t\Rightarrow A{A}^{\prime }=v_{1}t$

$A{A}^{\prime }=B{B}^{\prime }\Rightarrow v_{1}t$

In $\Delta AB{B}^{\prime }$

$sini=\frac{B{B}^{\prime }}{A{B}^{\prime }}$………..(1)

In $\Delta B{A}^{\prime }A$

$sinr=\frac{A{A}^{\prime }}{A{B}^{\prime }}$……….(2)

Divide 1 by 2

$\frac{sini}{sinr}=\frac{B{B}^{\prime }}{A{A}^{\prime }}=1\Rightarrow sini=sinr\Rightarrow i=r$

8.0Law of Refraction by Huygens's Wave Theory

$B{B}^{\prime }=v_{1}t$

$A{A}^{\prime }=v_{2}t$

In $\Delta AB{B}^{\prime }$

$\sin i=\frac{B{B}^{\prime }}{A{B}^{\prime }}$………..(1)

In $\Delta A{A}^{\prime }{B}^{\prime }$

$\sin r=\frac{A{A}^{\prime }}{A{B}^{\prime }}$……….(2)

Divide 1 by 2

$\frac{\sin i}{\sin r}=\frac{B{B}^{\prime }}{A{A}^{\prime }}=\frac{v_{1}t}{v_{2}t}=\frac{v_1}{v_2}=\frac{{\mu }_2}{{\mu }_1}$

${\mu }_1\sin i={\mu }_2\sin r$

Illustration-1. If amplitude of light at 10 m from a small light bulb is A0 then find amplitude of light at a distance 50 m from the same light bulb?

Solution:$A\propto \frac{1}{r}\Rightarrow \frac{A_1}{A_2}=\frac{r_2}{r_1}\Rightarrow \frac{A_0}{A_2}=\frac{50}{10}\Rightarrow A_2=\frac{A_0}{5}$

Illustration-2:A plane wavefront is incident on a given optical device in each case. Draw the correct wavefront after interaction of the light ray with the Optical device.

Solution:

Illustration-3:A plane wavefront of monochromatic light is incident on a given optical device. Draw the correct wavefront after interaction of the light ray with the Optical device.

Solution:

9.0Wavefront Behavior Through Optical Elements

1. Plane Wave Passing Through a Prism

• A plane wave passes through a thin glass prism.
• The part of the wavefront passing through the thickest region of the prism experiences the maximum delay.
• Since light travels slower in glass, this causes the emerging wavefront to tilt.
•  This tilt explains the deviation of light through the prism.

2. Plane Wave Passing Through a Convex Lens

• In a convex lens, the central part of an incoming plane wave travels through the thickest portion.
• This causes a greater delay at the center compared to the edges.
• As a result, the emerging wavefront becomes spherical with a central depression.
• The wavefront converges to a real focus — this is how lenses form focused images.

3. Reflection by a Concave Mirror

• In a concave mirror, the central portion of the wavefront travels a longer path (to the mirror and back) than the edges.
• This extra path length acts like a time delay for central rays.
• The reflected wavefront becomes spherical and converging.
•  This is how concave mirrors form real images by convergence.

4. Concave Lenses and Convex Mirrors

• In a concave lens or a convex mirror, the edges of the wavefront are delayed more than the center.
• This causes the emerging wavefront to become diverging.
• The result is a virtual, diverging image — consistent with the behavior of these optical elements.

5. Equal Time for Image Formation

• The total time taken by light to travel from an object point to the image point is the same along all rays. Example: In a convex lens, central rays travel a shorter geometric path but move slower through thicker glass.
• Edge rays, though longer in distance, travel faster through thinner glass.
•  Both reach the focus at the same time, ensuring image clarity.

Note:

• Optical elements shape wavefronts by introducing controlled time delays across different parts.
• These delays result in bending (refraction) or reflection that forms focused or diverging wavefronts, leading to image formation.