[Physics Class Notes] on Huygens Principle Pdf for Exam

Huygens Principle is also known as the Huygens–Fresnel Principle. This principle was named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel. It is a method of analysis that applies to problems of wave propagation both in the near-field and far-field limit and in near-field diffraction and also in reflection. 

 

Huygens’ Principle Statement is that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets arising from different points mutually interfere. The total of these spherical wavelets forms the wavefront.

 

In this article, we will discuss the Huygens wave theory of light and Huygens Principle Derivation.

 

State Huygens Principle

In the year 1678, Huygens assumed that every point to which a luminous vibration reaches becomes a source of a spherical wave, and therefore, the sum of these secondary waves helps us determine the form of the wave at any subsequent time. 

 

He presumed that the secondary waves travelled in the forward direction only and it is not explained in the Huygens Wave Theory why this is the case. 

 

Huygens Theory of Light

Huygens Principle is used to provide a qualitative explanation of both rectilinear and spherical wave propagation, and to derive the laws of reflection and refraction using the Huygens Principle of secondary wavelets principle, but could not explain the rectilinear propagation derivations that occur when light encounters edges, apertures, and screens, commonly known as effects of diffraction. The resolution of the error mentioned above was then explained by David A. B. Miller in 1991. The resolution is that the source is a dipole, i.e., not the monopole as per the assumptions of Huygens that cancels in the reflected direction.

 

What After Huygens’ Principle Resolution?

In 1818, Fresnel showed that Huygens’s principle, together with his interference principle could explain both the linear propagation of light and the diffraction effects. To achieve accurate experimental results, he included the additional arbitrary assumptions about the phase and amplitude of the secondary waves and introduced an obliquity factor. Though these assumptions have no obvious firm foundation but have led to predictions that agreed with many experimental observations, including the Poisson spot.

 

Huygens Principle of Secondary Wavelets

 

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Huygens principle of secondary wavelets states that every point on a given wavefront is a secondary wavelet/disturbance. Further, the disturbances originating from the secondary source scatters in all directions in the way when originated from the primary source.

This principle further highlights the following things:

• Secondary sources make their wavelets, and these waves are similar to that of the primary source.

• At the instant, a common tangent drawn on the wavelets in the forward direction points to the new wavefront.

• The spherical wavelets together form a wavelet. 

In conclusion, Huygen’s principle is a comprehensive method of analysis that we can use to understand the problems of wave propagation both in diffraction and reflection.

Now, let’s perform the Huygens principle derivation.

 

Huygens Principle Derivation

AA’ is the wavefront that is incident on a reflecting surface XY having an angle of incidence i. Following Huygen’s principle, every point on AA’ acts as a source of secondary wavelets.

 

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Time taken from A’ to D = time taken from B’ to C

A’D / v = B’C / v

A’D = B’C

A’C sin (i) = A’C sin (r)

Hence, i = r

Here, 

The angle of incidence equals the reflecting angle. This is also stated in the first law of reflection. Also, as the incident wavefront AB, the normal and reflected wavefront are on the same plane, we can also verify the second law of reflection.

 

In this case, we have the following interpretation:

Time taken from A’ to R is equal to the time taken from P to B₁

 

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A₁P / v₁ = B₁R / v₂

A₁B₁ sin (i) /v1 = A₁B₁ sin(r) / v₂

n₁ sin (i) = n₂  sin (r)

Now, let’s understand a few applications of Huygens Principle.

 

Huygens Principle Definition

By using Huygen’s Wave Theory after time t, we can determine the new position of the wavefront. Also, we can mathematically calculate the new position of the wavefront after time t.

1. Let the initial time be zero.

2. Distance travelled by the individual wave equals the radius of the sphere.

3. Hence sphere radius = v * t₁

4. Common tangent joining all of the spheres gives the location of the new wavefront at time t = t₁

5. There are two choices, viz: the inner tangent and outer tangent.

6. The back wave amplitude is zero. So the back wave is ignored, and attention is given to the forward wave only.

7. From every point on the outer wavefront comes out a new wavefront.

8. The distance travelled by these wavefronts is equal to vt₂.

9. Now, again spheres will be obtained and after time t₂ the position of all the new wavefronts is obtained by drawing a common tangent.

10.  Again, the forward wavefront is taken into consideration and the back wavefront is ignored.