Huygens Principle

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Huygens Principle Statement

Huygens Principle is also known as the Huygensâ€“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.Â

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.Â

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.Â

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 Huygen's 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

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 own 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.Â

In conclusion, Huygenâ€™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, letâ€™s perform the Huygens principle derivation.

Huygens Principle Derivation

• Proof of Reflection By Huygens Principle

If you look at the above figure, AAâ€™ is the wavefront that is incident on a reflecting surface XY having an angle of incidence i. Following Huygenâ€™s principle, every point on AAâ€™ acts as a source of secondary wavelets.

Time is taken from Aâ€™ to D = time is taken from Bâ€™ to C

Aâ€™D / v = Bâ€™C / v

Aâ€™D = Bâ€™C

Aâ€™C sin (i) = Aâ€™C sin (r)

Hence, i = r

Here,Â

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 is on the same plane, we can also verify the second law of reflection.

• Proof of Refraction by Huygens Principle

In this case, we have the following interpretation:

Time taken from Aâ€™ to R is equal to the time taken from P to B1

A1P / v1 = B1R / v2

A1B1 sin (i) / v1 = A1B1 sin(r) /v2

n1 sin (i) = n2 sin (r)

Now, letâ€™s understand a few applications of Huygenâ€™s Principle:

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Huygens Principle Definition

By using Huygenâ€™s Wave Theory after time t, we can determine the new position of the wavefront. Also, we can mathematically calculate the new position of wavefront after time t.

1. Let the initial time ti be zero.

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

3. Hence sphere radius = v *Â  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. Â Again, the forward wavefront is taken into consideration and the back wavefront is ignored.

Q1: What Does Wavefront Mean?

Ans: Wavefront is the locus of all the points lying in the same phase.

There are three following types of wavefronts:

• Spherical wavefront

• Planar wavefront

• Cylindrical wavefront

Spherical Wavefront

After the point source gains radiating energy, the particle around the energy starts oscillating, these waves travel in all directions and form a spherical wavefront.

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Planar Wavefront

If the source is at infinity, the waves emerging from the source become parallel to each other, i.e., light rays coming from the sun are parallel to each other on earth. In such cases, a planer wavefront forms.

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Cylindrical Wavefront

A cylindrical wavefront forms only when the light source is linear. Here, all the points become equidistant from the source and we find them on the cylindrical surface.

Q2: Describe Some Interpretations of Modern Physics in Huygens Principle.

Ans: Huygens theory of light included amplitudes only but it does not include phase, waves propagating at varying speeds because of the diffraction within continuous medium and therefore does not involve interference.

• The Huygens analysis didnâ€™t include polarization for the light that implies a vector potential, instead, sound waves can be described with a scalar potential and there is no unique and natural translation between the two.

Q3: Describe the Outcomes of the Huygens Principle.

Ans: The Huygens Principle proved the following concepts:

1. Reflection

2. Refraction

3. Diffraction and Interference of light