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Let's say I shine a laser from a stationary medium into a moving medium (suppose the water is moving very quickly) perpendicular to the interface and back to a stationary medium like this:

Scenarios

(Note: left and right sides of the image are stationary mediums, center is a medium moving in the direction indicated by the arrow)

Which of the above scenarios (A, B, C, or "I'm way off") correctly reflects the path the light will take (even if the translation is incredibly small)?

Edit:

To answer some good questions (and things I left out of the original question):

  • The center (moving) medium is water
  • The left and right medium (stationary) is air
  • The first angle of incidence (on the left) between the air and the water is perpendicular
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  • $\begingroup$ I think, you are considering the incident light to be perpendicular to the interface, but it's not mentioned in your question. Would you like to mention it if it is so indeed? $\endgroup$ – AJS May 19 '14 at 2:45
  • $\begingroup$ @A.J.Shajib - sorry about that! yes, the incident light is perpendicular to the interface. $\endgroup$ – Arsinio May 19 '14 at 2:49
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Light passing through a moving medium undergoes a shift due to the difference in frames between the two media. This problem is quite simple to solve in the frame of the river. In this frame the light is moving at an angle and the river is still. The air is moving relative to the river but since air has an index of refraction of $1$, its movement doesn't have any effect on the behaviour of light. Then you can use the ordinary Snell's law and finally boost back to the original frame.

The only subtlety here is that we are in some sense using both the particle and wave viewpoints of light since we will discuss momenta as well as Snell's law, however I can't see an issue with doing so.

I denote the lab frame without a prime and the river frame with a prime.

The initial light momenta was, \begin{equation} p _i = E \left( 1,1,0,0 \right) ^T \end{equation} Boosting into the river frame we have, \begin{equation} p' _i = E\left( \gamma , 1 , - \beta \gamma , 0 \right) ^T \end{equation} Therefore the angle of incidence is \begin{align} & \tan \theta _i ' = \beta/ \sqrt{ 1 - \beta ^2 } \\ \Rightarrow & \sin \theta _i ' = \beta \end{align} Now using Snell's law we have, \begin{equation} \sin \theta _f '= \frac{ \beta }{ r} \end{equation} where $ r $ is the ratio of indices of refraction, $ n _f / n _i $.

Therefore the momenta of light in the water in the frame of the water is, \begin{equation} p _f ' = E \gamma \left( 1 , \sqrt{ 1 - \beta ^2 / r } , \beta / r , 0 \right) ^T \end{equation}
Boosting back to the lab frame we have, \begin{equation} p _f = E\left( \gamma + \beta ^2 \gamma /r , \sqrt{ 1 - \beta ^2 / r ^2 }, \beta \gamma + \beta \gamma / r , 0 \right) ^T \end{equation}

To find out how the light will behave once it exits the river we note that the angle of incidence at the second interface is the same as the angle of refraction in the water (still in the river frame). We have, \begin{equation} \sin \theta _{ \mbox{out }} '= r \sin \theta _f ' = \beta \end{equation} This is the same as the initial angle in the water frame. After boosting back to the lab frame we should get back the same perpendicular light arrow. In total the journey of the light should take the form,

$\hspace{1cm}$enter image description here

where the shade of red is proportional to the speed of the river, the lightest being $0.1c$ and the darkest being $0$.

As we would expect in the limit that $\beta\rightarrow 0 $ the refraction effect goes to zero and we note that the effect is only significant for huge river speeds.

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    $\begingroup$ Changing the frame of reference is a nice intuitive way to solve this - it is quite convincing. It also says that the angle inside the medium will be on the order of $n\cdot v/c$ - for small angles. This should mean that not only is there a shift, but (easier to observe?) there should be a phase shift when the medium moves (longer path-> more phase shift). I am not aware that this has ever been measured, but I can't say I have looked. Of course the phase shift goes as $\theta ^2$ so it will be hard to observe for small velocities - maybe harder than a beam translation. $\endgroup$ – Floris May 19 '14 at 16:30
  • $\begingroup$ It looks like this answer is getting the most attention...not being an expert in the field, does this take into account the aberration mentioned by others? $\endgroup$ – Arsinio May 20 '14 at 14:08
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The correct scenario here would be B.

When two mediums are in relative motion, the angle of incidence would not be equal in the rest frames of both the mediums due to aberration requiring a modification in Snell's law. You can look here for a fairly detailed derivation of the special relativistic Snell's law which is $$(1+\beta^2\Psi_r)n_i\mathrm{sin}\ \theta_i=\frac{n_r\mathrm{sin}\ \theta_r}{\sqrt{1+\beta^2\Psi_r\mathrm{cos^2}\ \theta_r}}+\beta\Psi_r$$ where $$\Psi_k = \frac{1-n_k^2}{1-\beta^2}$$ and the indices $i$ and $r$ refer to the mediums of incidence and refraction respectively, $\theta$ is the corresponding angle, $n$ is the refractive index of a medium and $\beta=\frac{v}{c}$ where $v$ is the relative speed of the mediums. As a result, the angle of refraction $\theta_r$ can be non-zero for a zero angle of incidence $\theta_i$ when the mediums are in relative motion.

Due to relativistic aberration, the angle of refraction $\theta'_r$ in the moving medium frame would be related to the angle of refraction $\theta_r$ in the stationary medium frame by

$$\mathrm{tan}\ \theta'_r = \frac{\mathrm{sin}\ \theta_r - n_r\beta}{\mathrm{cos}\ \theta_r\sqrt{1-\beta^2}}$$

We can insert this $\theta'_r$ into the relativistic Snell's law to find the angle of emittance $\theta'_e$ in the moving frame for refraction at the second interface. Applying the relativistic aberration formula again on this $\theta'_e$ would produce the angle of emittance $\theta_e$ in the stationary frame. As $\theta'_e=\theta'_i$, because of applying the aberration transformation in the same medium would imply that $\theta_e=\theta_i$, hence the emitted light path would be parallel to the incident light path in both frames.

As an example, if we take $v=0.1c$, $\theta_i=0^{\circ}$, $n_i\simeq1$(air), $n_r=1.33$(water), then the angle of refraction in each frame would be $\theta_r=3.33^{\circ}$ and $\theta'_r=-4.31^{\circ}$. Here, if $\beta$ is taken as positive along $x$-axis and the direction of light is taken to be from the negative half-plane to the positive half-plane along $y$-axis then the corresponding optical angles of a light path with positive slope in $xy$-plane are considered positive. The angle of emittance in each frame would then be $\theta'_e = -5.74^{\circ} = \theta'_i$ and $\theta_e\simeq0=\theta_i$.

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  • $\begingroup$ Be handy to add a numerical example showing, e.g. that hypersonic water (but still less than $\epsilon$ fraction of $c$) changes Snell's law by some microscopic amount :-) $\endgroup$ – Carl Witthoft May 19 '14 at 11:42
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    $\begingroup$ If you are right, the JeffDror's answer below (which makes B a plausible answer) must be wrong. Can you give a counter argument for his "change frame of reference"? I found it quite convincing. $\endgroup$ – Floris May 19 '14 at 16:31
  • $\begingroup$ @Floris I am sorry that I forgot to consider aberration for the second refraction and mistook $\theta'_e$ for $\theta_e$. Indeed, JeffDror's argument is right and greatly convincing for that matter. I edited my answer accordingly. $\endgroup$ – AJS May 20 '14 at 3:27
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The correct answer is B as stated in some of the previous answers. This effect is called the photon drag effect. According to this[1] Science paper (pdf here)

This phenomenon was first considered by Fresnel in 1818 and then, for the longitudinal case, verified [experimentally] by Fizeau (1859), who used water flowing along the light paths within an interferometer as a means of introducing a phase shift.

The transverse drag effect was later verified experimentally by passing light through the edge of a spinning glass plate causing a transverse displacement of the beam[2].

In addition [1] showed that an image propagating through a spinning medium will be rotated. This effect was amplified by the fact that spinning medium was a slow light medium which slowed the light down (thus increasing the effect) by roughly a factor of a million $(10^6)$. An image from the paper is shown below.

enter image description here

Sources

  1. Franke-Arnold, Sonja, Graham Gibson, Robert W. Boyd, and Miles J. Padgett. "Rotary photon drag enhanced by a slow-light medium." Science 333, no. 6038 (2011): 65-67. (pdf)
  2. Jones, R. V. "'Fresnel Aether Drag' in a Transversely Moving Medium." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 328, no. 1574 (1972): 337-352.
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According to me, the path c is correct. The movement of medium should not affect the speed of light and hence path of light. This sort of light-dragging effect was considered in the case of aether hypothesis proposed by physicists in 19th century. It was discarded by Michelson-Morley experiment. And later, special theory of relativity discarded the possibility of the dragging effects as well.

For more discussion, you can read this

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    $\begingroup$ If the movement of medium should not affect the path of light, why should B be correct? Did you actually mean to say C? $\endgroup$ – AJS May 19 '14 at 2:47
  • $\begingroup$ @A.J.Shajib the Snell's law will still hold,right? The medium is changing! $\endgroup$ – puru May 19 '14 at 2:54
  • $\begingroup$ okay I assumed that the stationary medium is air and the moving medium is water. If that's not the case, and the medium is same throughout, according to me, the answer should be path C $\endgroup$ – puru May 19 '14 at 2:56
  • $\begingroup$ I asked OP whether s/he is considering the incident light to be perpendicular to the interface, which was apparent from the drawing. In that case, the light path will not change for different still mediums. $\endgroup$ – AJS May 19 '14 at 2:59
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    $\begingroup$ puru, the ether hypothesis was that there was a medium for propogation of light even in a vacuum. Most certainly the behaviour of light in a material differs from that in a vacuum(and far from gravitational effects, etc.) $\endgroup$ – Carl Witthoft May 19 '14 at 11:44

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