Huygen's Principle doesn't necessarily imply the law of reflection

In summary, the conversation discusses the relationship between constructive interference in wavelets and the law of reflection. It is mentioned that the zeroth order is the only maximum included in the law of reflection, and diffraction gratings can cause higher order reflections at other angles. It is also noted that Huygen's principle implies that the diffraction grating principle is correct for a smooth plane mirror, but in any other direction the addition of all the Huygens sources gives zero. The concept of underfilled arrays and side lobes is also mentioned in relation to diffraction gratings, and the conversation concludes with a reference to antenna theory and the sin(x)/x distribution in reflected beams.
  • #1
etotheipi
There are many diagrams like the following

1576424174582.png


which demonstrate that if A acts as a point source of wavelets, then we will have constructive interference in the case that ##i=r## as shown, as we can show with a little geometry that the path difference is zero.

However, surely this isn't the only option, since we can let the path difference between the two waves shown equal any multiple of ##\lambda## a consequently obtain many different orders. If we accept Huygens principle and apply it to a smooth reflector, we should obtain reflection across many different angles.

So why is it that only the zeroth order is included in the law of reflection?

Previously, I thought that we could only get the "scattering" required with a reflective diffraction grating, however Huygen's principle seems to imply that the diffraction grating principle is correct even for a smooth plane mirror - is this right?
 

Attachments

  • 1576424084675.png
    1576424084675.png
    10.1 KB · Views: 201
Science news on Phys.org
  • #2
If we had just two rays, on the extremities of the incident beam, they would induce current in two places (like dipoles) on the reflector surface and give rise to multiple re-radiated lobes, in the manner of Young's Slits. But the incident beam is solid, so across the surface of the reflector we have a row of points which will re-radiate. The phasing of these is such as to create a parallel reflected beam in the direction i = r. The radiation in other directions tends to average to zero.
Notice, incidentally, that one cannot have a parallel beam with finite diameter except close to a source.
 
  • Like
Likes Ibix and etotheipi
  • #3
tech99 said:
But the incident beam is solid, so across the surface of the reflector we have a row of points which will re-radiate.

Thank you, I forgot to consider this.

If the incident angle is ##i## and the reflected is ##r##, then for any two rays to interfere constructively we require $$d(\sin{i}-\sin{r})= n\lambda$$Now suppose two rays separated by a distance ##d## interfere constructively along their common first order, so that ##d(\sin{i}-\sin{r}) = \lambda##. The ray in between them at a distance ##\frac{d}{2}## will be such that its path difference from the two previous rays is $$\frac{d}{2}(\sin{i}-\sin{r}) = \frac{\lambda}{2}$$ and we will get destructive interference.

So the result is the only maximum is where ##i=r##. Is this right?
 
  • #4
etotheipi said:
So why is it that only the zeroth order is included in the law of reflection?
Only the zeroth order works for all wavelengths. You can build reflectors called diffraction gratings that cause the higher order reflections you are describing. They do reflect at other angles, but are very sensitive to wavelength.
 
  • Like
Likes sophiecentaur
  • #5
etotheipi said:
however Huygen's principle seems to imply that the diffraction grating principle is correct even for a smooth plane mirror - is this right?
Yes it is but, for a uniform reflector, the equivalent diffraction slits have zero spacing. The limit as the spacing goes to zero gives only the zeroth order beam. In any other direction the addition of all the Huygens sources gives zero.
Take any diffraction grating and see where the second order beam occurs. Then double the pitch of the grating lines and the second order beam will be greater (double, in fact, for a wide spaced grid). Repeat the process and the second order beam is deflected more. Carry on with that process and the second order beam will be deflected by more than the angle of the grating / mirror. So only the zeroth order beam will remain.
 
  • Like
Likes tech99 and etotheipi
  • #6
Yes, a diffraction grating is an underfilled array, so we have side lobes.
 
  • Like
Likes sophiecentaur
  • #7
tech99 said:
Yes, a diffraction grating is an underfilled array, so we have side lobes.
I just love it when people bring it all back to antenna theory.
 
  • Like
Likes vanhees71
  • #8
sophiecentaur said:
I just love it when people bring it all back to antenna theory.
And on that theme, you have to remember that there will be a sin(x)/x distribution over the width of the reflected beam. You can't avoid that one.
 

FAQ: Huygen's Principle doesn't necessarily imply the law of reflection

1. What is Huygen's Principle?

Huygen's Principle states that every point on a wavefront can be considered as a source of secondary spherical wavelets that spread out in all directions with the same speed as the original wave.

2. How does Huygen's Principle relate to the law of reflection?

Huygen's Principle is often used to explain the law of reflection, which states that the angle of incidence is equal to the angle of reflection. This is because the secondary wavelets from each point on the wavefront follow the same path and create a reflected wavefront that appears to have originated from a single point.

3. Why doesn't Huygen's Principle necessarily imply the law of reflection?

While Huygen's Principle can be used to explain the law of reflection, it is not the only explanation. Other principles, such as Fermat's principle, can also be used to derive the law of reflection.

4. What are some examples of situations where Huygen's Principle does not apply to the law of reflection?

One example is when light reflects off of a rough or irregular surface. In this case, the secondary wavelets from each point on the wavefront do not follow the same path, resulting in a distorted reflection.

5. How does Huygen's Principle differ from the law of reflection?

Huygen's Principle is a theoretical concept that describes the behavior of waves, while the law of reflection is an empirical observation that describes the behavior of light. Huygen's Principle can be used to derive the law of reflection, but it is not the only explanation for this phenomenon.

Similar threads

Back
Top