# Deriving the Law of Reflection from Huygens Principle

• rtareen
In summary, Fermat's principle is a theorem that states that a ray of light will always reflect off of a mirror in the same direction it came from.
rtareen
Attached is section 33.7 from my book, which introduces Huygen's principle in order to derive the law of reflection. I am more used to the ray model rather than the wave model, so I'm constantly going to try to relate everything back to rays. Making this connection also helps with completeness of understanding. I believe that the model we are using for this derivation is the wave model. I could be wrong.

Firstly, regarding Figure 33.34, I believe the arrows attached the the wave fronts do not represent polarizations but are instead the directions the rays would be if we want to think back to the ray model. I hope this is right because I would like that to be true.

So if I understand this properly, if we apply Huygen's principle, each wave front is made from the envelope of small wavelets emerging from the previous wave front. When the wave front hits the surface, the wavelets change direction. I'm not sure how.

So wave front AA' has not yet been reflected. But the future wave front BOB', made up of the wavelets of AA' has a part that has been bent by the surface. How did the wavelets of BO end up in that exact direction, as if they are emerging from the surface rather than the previous wave front? Is there some sort of geometric explanation, or are there some hidden assumptions about how the wavelets are supposed to behave?One problem I have with this section is the paragraph that starts with "The effect of the reflecting surface". They say the surface changes the direction of the wave, but they don't say how. The wavelets to the left of point O strike the reflecting surface, and then what? And then they say in that same sentence: "so the part of a wavelet that would have penetrated the surface actually lies to the left of it, as shown by the full lines". I have no clue what they mean by "to the left of it" or what "the full lines" are. Does this have to do with the mysterious continuation of the wavefront under the interface (the continuation of the line B'O) , even though we can see the part that has clearly been reflected?

Next, in Figure 33.34 (b) they construct the line OP = vt. Is this decision to start at the starting point P on the wave front AA' rather than some other point arbitrary? Can somebody explain? Well the point is to construct a triangle on both sides for the proof. They finish the proof by constructing another triangle on the left which leads to ##\theta_r = \theta_a##. But they never gave any justification for the orientation of OB that they drew in the figure. So they constructed the picture just right without justification to derive the law of reflection? Is that right?

#### Attachments

• Section33.7.pdf
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Like you I find it difficult to follow the logic of finding the direction of OB.
When a wave front hits a mirror, the wavelets strike the mirror one after the other, with a phase delay (or time delay) between them. This phase delay is correct to produce a reflected beam in the required direction, having an equi-phase wavefront, in the manner of an antenna array.

rtareen
tech99 said:
Like you I find it difficult to follow the logic of finding the direction of OB.
When a wave front hits a mirror, the wavelets strike the mirror one after the other, with a phase delay (or time delay) between them. This phase delay is correct to produce a reflected beam in the required direction, having an equi-phase wavefront, in the manner of an antenna array.

Fermat's principle tends to run in parallel with Huygens - minimal transit time for the path of the 'ray' applies in both theories.

First picture here - except the folding and the different ##v##.

sophiecentaur
rtareen said:
Is this decision to start at the starting point P on the wave front AA' rather than some other point arbitrary?
Yes you can coose any point P along AA'. It won't change the resulting angles, it will just scale the diagram.

Step by step animation (reflection starts at 3:00 min):

## 1. What is Huygens Principle?

Huygens Principle is a theory proposed by Dutch scientist Christiaan Huygens in the 17th century. It states that every point on a wavefront can be considered as a source of secondary spherical wavelets, and the new wavefront is the envelope of these wavelets.

## 2. How does Huygens Principle relate to the Law of Reflection?

Huygens Principle can be used to explain the Law of Reflection, which states that the angle of incidence is equal to the angle of reflection. According to Huygens Principle, the secondary wavelets generated by each point on the original wavefront will all have the same speed and wavelength, resulting in a reflected wave that maintains the same angle as the incident wave.

## 3. What is the process of deriving the Law of Reflection from Huygens Principle?

The process involves considering a plane wavefront incident on a flat surface. Using Huygens Principle, the secondary wavelets generated at each point on the wavefront can be traced to determine the reflected wavefront. By comparing the angles of the incident and reflected wavefronts, it can be shown that they are equal, thus proving the Law of Reflection.

## 4. Why is it important to derive the Law of Reflection from Huygens Principle?

Deriving the Law of Reflection from Huygens Principle provides a deeper understanding of the underlying principles behind the behavior of light. It also allows for a more general and intuitive explanation of the law, as it applies to all types of waves, not just light.

## 5. Are there any limitations to using Huygens Principle to derive the Law of Reflection?

While Huygens Principle is a useful tool for understanding the Law of Reflection, it does have its limitations. For example, it does not take into account the effects of diffraction, which can alter the shape of the wavefront. Additionally, it assumes that all points on the wavefront are in phase, which may not always be the case in real-world scenarios.

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