Reflection, refraction, and Snell's law

[Ahmed1029]

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

 

[tech99]

I think you can derive all these using a ruler and protractor. For Snell's Law you need first to find out the relative velocities of light in the media, using knowledge of the refractive indices.

 

[Ibix]

I think you need the principle that light follows the path that minimises travel time. Then you fix two points and a flat surface, and draw a reflected/refracted ray between the points with the reflection/refraction event lying somewhere on the surface. Write down the travel time, and calculus will then find you the location of the event and you can derive the angles you want from there.

 

[Andy Resnick]

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

I derive them in class as originating from conservation of (linear) momentum.

 

[vela]

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

Whenever you have waves, you'll get reflection and refraction. You can derive these laws from analyzing how waves propagate through media in general. The others have indicated some approaches to doing this. Using Huygen's principle is another way.

At a little deeper level, the laws are a consequence of the boundary conditions the wave must satisfy where the two media meet. For the case of a wave propagating down a string connected to a different string, one condition arises because the strings must stay tied together, and another from the fact that the strings exert equal and opposite forces on each other, i.e., Newton's third law. For light, the electric and magnetic fields have to satisfy similar boundary conditions to satisfy Maxwell's equations.

 

[Ahmed1029]

Whenever you have waves, you'll get reflection and refraction. You can derive these laws from analyzing how waves propagate through media in general. The others have indicated some approaches to doing this. Using Huygen's principle is another way.

At a little deeper level, the laws are a consequence of the boundary conditions the wave must satisfy where the two media meet. For the case of a wave propagating down a string connected to a different string, one condition arises because the strings must stay tied together, and another from the fact that the strings exert equal and opposite forces on each other, i.e., Newton's third law. For light, the electric and magnetic fields have to satisfy similar boundary conditions to satisfy Maxwell's equations.

So is geometric optics derived from the physics of waves? which one comes first?

 

[vela]

Waves. Classically, light is an electromagnetic wave, after all.

Geometric optics is the limit where we can neglect the waviness of light and model light as rays, which travel in straight lines through a uniform medium.

 

[binis]

I think you need the principle that light follows the path that minimises travel time.

https://www.physicsforums.com/threads/how-is-phase-velocity-related-to-deflection-angle.1010744/ # 8 Snell's law is not derived from Fermat's principle.

 

[Ibix]

https://www.physicsforums.com/threads/how-is-phase-velocity-related-to-deflection-angle.1010744/ # 8 Snell's law is not derived from Fermat's principle.

That's not what I said. In that thread I said "what's wrong with the explanation on Wikipedia", to which you replied "you mean Fermat's principle", to which I replied (correctly) that Fermat's principle is not mentioned on the page about refraction (although I didn't look at the page on Snell's law, which does, so perhaps we were looking at different pages). At the time, I was not explaining, I was attempting to get you to tell us what explanations you were aware of and say what you didn't understand about them because there was no point repeating stuff you already knew.

 

[vanhees71]

So is geometric optics derived from the physics of waves? which one comes first?

From a fundamental-physics point of view "optics" is just a special application of Maxwellian electrodynamics (or even quantum electrodynamics, given that nowadays "quantum optics" is ubiquitous), i.e., it's wave optics. Geometrical optics can be derived from wave optics using the socalled eikonal approximation, which is valid if the typical scale of spatial variations of the matter around are small on the scale of a typical wavelength of the light under consideration.

 

[binis]

(although I didn't look at the page on Snell's law, which does, so perhaps we were looking at different pages). At the time, I was not explaining, I was attempting to get you to tell us what explanations you were aware of and say what you didn't understand about them because there was no point repeating stuff you already knew.

I am skeptical like many wiki readers in Talk:Snell's law page contents 19 & 25.

 

[difalcojr]

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

...................................

All of the methods of derivation explained in this post up to this time seem valid, creative, and eloquently stated, I think, in answer to your question. There are probably even other derivations too.
One point, though, the refractive ratio, Snel's law, can indeed be derived from Fermat's minimum conditions. Tech99's reply nails it. Ruler and protractor was first. Geometrical optics. Vela too refers to a Huygens method for ratios of line lengths. Here's his diagram of that. When he found out that light traveled at a constant speed in a homogeneous medium, he used the refractive ratio as travel time, line lengths, as 3 to 2, for a 1.5 index. He found that AC:CX equaled 3:2, additionally.
So, Snel's law in geometric optics can be derived from the modern physics of waves, and vice-versa, and waves do come first in classical optics. Waves come first in modern physics, too, but geometric optics came first, historically, as was indirectly said.
Look at Descartes' diagrams for good sine ratio (line lengths) diagrams. Before him Snel derived the law of refraction. Before him, Harriot had done it. It has been written that, though I have not confirmed it myself, that the law of refraction goes way back to either ibn Sina or Alkindi, I forget which now.

 

[difalcojr]

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

Here's where Snel's law comes from in geometrical optics. It comes from the ancient hidden coin trick, shown in the first 2 diagrams. Magician places cup so viewer can see into it but not see the coin at the bottom. Then pours water to fill the cup, and Voila!, a coin appears in the very same place where there was just nothing there! Magic. This was the example to illustrate refraction in optics texts from Euclid on.

Other two are close copies of Huygens' later summary drawings for both Snel's and Descartes' diagrams. Lettering same, just added angle designators and what the media are. Snel used the ratio of cosecants, initially, Descartes the ratio of sines. The second equations for each diagram are first, a sines ratio for Snel's diagram, also possible, and then a cosines ratio for Descartes' diagram, which is also valid. If you measure the different line lengths yourself, you will see that 4:3 is a constant, the refractive ratio of water to air.

 

[binis]

Still sceptical at #11 page contents 19 Is it really a law? & 25 Huygens derivation

 

[vanhees71]

Hm, I'd say Hugens's principle is a mathematical fact about the solutions of the wave equation in odd space dimensions (it's valid in 3D but not in 2D!). Snell's Law of refraction as well as the Law of reflection can all be derived from Maxwell's equations and wave optics, using the usual linear constitutive relations for electrodynamics in a medium. This leads to Fresnel's laws, which give not only the direction but also the detailed polarization properties of the reflected and refracted light.

 

[binis]

Α "duality" series episode. Debate "wave (spread as Huygens) or particle"
(lines as & Descartes diagrams) goes on.

 

[vanhees71]

Of course ray optics is also fine within its realm of applicability. It's derived from wave mechanics by the eikonal approximation (also known as WKB approximation in quantum wave mechanics or in the most general expression: "singular perturbation theory").

 

[difalcojr]

Still sceptical at #11 page contents 19 Is it really a law? & 25 Huygens derivation

Well, an amateur here, I can only give you a very basic, simple answer why I think it is a law. Now this is only for a 2D refraction model of two homogeneous media. For one frequency only and for one theoretical ray only. As in the above diagrams.
First, light travels in straight lines as we see it, both in air and in water. Obvious. Second, its different speeds in other media are constant. Huygens proved it. Third, an index value of 1 is initially set for air (vacuum). The unknown index is that of water. So, everyone tested and measured over the centuries. With the cup empty and full. With a hemispheric glass bowl empty and full. At different incident angles. The line ratios were always a constant value.
Then it's just geometry fitting nature. Similar triangles having one angle equal, one side equal, and sharing another side. Equal ratios. This led to all the possible trig function ratios that were used over the years.

 

[vanhees71]

As I said, in 2D Huygens's principle is invalid. This you can easily see by throughing a stone into a pond. You don't see a single ring-like wave going radially out from the impact but an entire wave pattern, i.e., the wave from a $\delta$-distribution source (i.e., the Green's function) in 2D is non-zero in the entire cirlce $r \leq c t$, where $c$ is the speed of the waves.

 

[difalcojr]

As I said, in 2D Huygens's principle is invalid. This you can easily see by throughing a stone into a pond. You don't see a single ring-like wave going radially out from the impact but an entire wave pattern, i.e., the wave from a $\delta$-distribution source (i.e., the Green's function) in 2D is non-zero in the entire cirlce $r \leq c t$, where $c$ is the speed of the waves.

Well, I might like to disagree a bit, maybe, for another post. Now I want to agree with you though, and quote what a historian of science wrote on this matter that might be beneficial here to everyone. Alan E. Shapiro, in "Huygens' kinematic theory of light", says this so much better than me.

""Huygens' principle" can be viewed in two ways: as a definition and method of construction of a wave front, and as a physical explanation of rectilinear propagation. Only the former kinematic interpretation is capable of confirmation." And, further,
"Huygens himself was aware of the limited scope of his "Traite de la Lumiere" and had no pretensions to its being a comprehensive theory of light, as he had not considered color, diffraction, interference phenomena, or even an explanation of polarization, which he had discovered. The investigation of these phenomena was left to succeeding generations, and they ultimately led to the success of the wave theory of light in the nineteenth century."

 

[vanhees71]

Well, I might like to disagree a bit, maybe, for another post. Now I want to agree with you though, and quote what a historian of science wrote on this matter that might be beneficial here to everyone. Alan E. Shapiro, in "Huygens' kinematic theory of light", says this so much better than me.

You cannot disagree to mathematical facts. Just calculate the 2D retarded Green's function of the D'Alembert operator!

""Huygens' principle" can be viewed in two ways: as a definition and method of construction of a wave front, and as a physical explanation of rectilinear propagation. Only the former kinematic interpretation is capable of confirmation." And, further,
"Huygens himself was aware of the limited scope of his "Traite de la Lumiere" and had no pretensions to its being a comprehensive theory of light, as he had not considered color, diffraction, interference phenomena, or even an explanation of polarization, which he had discovered. The investigation of these phenomena was left to succeeding generations, and they ultimately led to the success of the wave theory of light in the nineteenth century."

That's not a very clear definition of Huygens's principle.

 

[difalcojr]

Wait! Hold your horses! There's a misunderstanding here. My bad. I do disagree a bit but not to your above post. Of course, your math and physics methods are facts and very informative, and I don't want to be a 'retarded' function in this discussion. Agree too on Shapiro, but I didn't bring him in for his definition of Huygens' principle. Rather, only to confirm that Huygens' methods were strictly geometrical, 2D, kinematic models, and that they could only go so far. As you have also explained. My bit of disagreement is about what is the correct answer to the original question posed.

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

and

So is geometric optics derived from the physics of waves? which one comes first?

And also waves vs. rays was brought up in discussion here. Rays came first. Waves later. Sure, the laws of reflection and refraction can be derived from the physics of waves. You all have provided elegant methods and examples, large in number and impressive. However, originally, the laws were derived from real experiments, real measurements, ray models, and a whole lot of Euclid, Book 6. The laws derived using geometry only. Other 'physical optics' studies and derivations came later. The law of reflection and Snell's law of refraction are the two basic laws of physics in optics, i.e., geometrical optics. Derived from ray diagrams and geometry only. That's my take on a fuller answer to these particular historical questions brought up.

 

[vanhees71]

Well, then we have to get into business. As you'll see in a moment the most simple approach to calculate the Green's function of the D'Alembert operator in (1+2)D is to do first the calculation in (1+3)D.

We want to solve
$$\Box G(x)=\delta^{(4)}(x),$$
where I use relativistic notation, i.e., $x=(t,\vec{x})$ (setting $c=1$), $x \cdot y=x_{\mu} y^{\mu} =\eta_{\mu \nu} x^{\mu} y^{\nu}=t_x t_y -\vec{x}\cdot \vec{y}$.
Then we want a solution with
$$G(x) \propto \Theta(t),$$
defining the "retarded Green's function". The most simple way is to use a 4D Fourier transformation,
$$G(x)=\int_{\mathbb{R}^4} \frac{\mathrm{d}^4 k}{(2 \pi)^4} \tilde{G}(k) \exp(-\mathrm{i} k \cdot x).$$
This gives
$$-k^2 \tilde{G}(k)=1 \; G(k)=-\frac{1}{k^2}=-\frac{1}{\omega^2-\vec{k}^2}.$$
Now to transform this to time-position space we need a description, how to deal with the poles in our integration over $\omega$.

To calculate the $\omega$ integral it's most convenient to use integration in the complex plane. We can close the path along the real axis by a large semi-circle in the lower (upper) plane for $t>0$ ($t<0$) since then for the radius going to infinity the additional path doesn't contribute, and we can thus use the theorem of residues to evaluate the integral. To get $G(x) \propto \Theta(t)$ we have to shift both poles of $\tilde{G}$ a bit to the lower half-plane, i.e., we write
$$\tilde{G}(k)=-\frac{1}{(\omega + \mathrm{i} 0^+)^2-\vec{k}^2}.$$
Then the integral over $\omega$ gives
$$G(x)=\Theta(t) \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 k}{(2 \pi)^3} \frac{\mathrm{i}}{2 K} \left \{ \exp[\mathrm{i} (\vec{k} \cdot \vec{x}-K t)] - \exp [\mathrm{i} (\vec{k} \cdot \vec{x}+K t)] \right\}.$$
Here $K=|\vec{k}|$. Now we can assume $\vec{x}=r \vec{e}_3$ and introduce spherical coordinates. Doing the $\varphi$-integral then gives (with the substitution $u=\cos \vartheta$)
$$G(x)=\Theta(t) \frac{\mathrm{i}}{8 \pi^2} \int_0^{\infty} \mathrm{d} K \int_{-1}^1 \mathrm{d} u K \exp(\mathrm{i} K r u) [\exp(-\mathrm{i} K t)-\exp(\mathrm{i} K t)].$$
Now the $u$-integral is straight-forward, giving
$$G(x)=\Theta(t) \frac{1}{8 \pi^2 r} \int_0^{\infty} \mathrm{d} K [\exp(\mathrm{i} K r)-\exp(-\mathrm{i} K r)][\exp(-\mathrm{i} \omega t) - \exp(\mathrm{i} K t)].$$
Since the integrand is even in $K$ we can write
$$G(x)=\Theta(t) \frac{1}{16 \pi^2 r} \int_{\mathbb{R}} \mathrm{d} K [\exp(\mathrm{i} K r)-\exp(-\mathrm{i} K r)][\exp(-\mathrm{i} \omega t) - \exp(\mathrm{i} K t)],$$
multiply out the exponential functions and finally get
$$G(x)=\frac{\delta(t-r)}{4 \pi r}.$$
This is indeed Huygens's principle. That's most easily seen when Fourier transforming only with respect to time,
$$\tilde{G}(\omega,\vec{x})=\int_{\mathbb{R}} \mathrm{d} t \exp(\mathrm{i} \omega t) G(x) = \frac{\exp(\mathrm{i} \omega r)}{4 \pi r},$$
which is a outgoing spherical wave.

Now to get the retarded Green's function in (1+2)D we simply have to integrate $G(x)$ over $z=x^3$, which gives (with $R=x^2+y^2$)
$$G^{(2)}(t,R)=\int_{\mathbb{R}} \mathrm{d} z \frac{\delta(t-\sqrt{R^2+z^2})}{4 \pi \sqrt{R^2+z^2}} = \frac{\Theta(t-R)}{2 \pi \sqrt{t^2-R^2}},$$
which clearly shows that now the contribution from the sources do not only originate from the boundary of the disk $R=t$ form the entire disk $R<t$.

 

[sophiecentaur]

Where do the laws of reflection, refraction, and Snell's law come from in geometric optics? Are they derivable from basic laws of physics?

https://www.physicsforums.com/threads/how-is-phase-velocity-related-to-deflection-angle.1010744/ # 8 Snell's law is not derived from Fermat's principle.

I was looking at the time line of optical Science and it seems that Snell's law came in the 1620s but details of the wave theory of light seems to have arrived with Young (slits) around 1800. So it looks like Fermat's principle, based on rays and time was responsible for Snells Law (although the wave theory tidied things up somewhat).
But I wonder how Fermat would have established the relative speeds in vacuum and glass. So it may have been a bit of a circular argument without using a wave explanation in 1600.