The ray angles q1 and q2 for incident and refracted optical rays at a plane boundary between regions of constant indices of refraction n1 and n2 are related according to Snells law
However, this formula applies only if the media (which are assumed to have isotropic index of refraction with respect to their rest frames) are at rest relative to each other. If the media are in relative transverse motion, it is necessary to account for the effect of aberration on the ray angles relative to the rest frames of the respective media. The result is that the effective refraction is a function of the relative transverse velocity of the media. Thus, measurements of the optical refraction could (in principle) be used to determine the velocity of a moving volume of fluid. Unlike Doppler shift measurement techniques, this approach does not rely on the presence of discrete particles in the fluid, and involves only measurements of direct, rather than reflected, light signals.
Since the amount of refraction at a boundary depends on the angle of incidence with respect to the rest frames of the media, it follows that if the media have different rest frames the simple form of Snells law does not apply directly, because it will be necessary to account for aberration. To derive the law of refraction for transversely moving media, consider the arrangement shown in Figure 1, drawn with respect to a system of coordinates (x,y,t) relative to which the medium with refractive index n1 is at rest.
In these coordinates the medium with index n2 is moving transversely with a speed v. By both Fermats principle of "least time" and the principles of quantum electrodynamics, we know that the path of light from point P0 to point P2 is such that the travel time is stationary (which, in this case, means minimized), so if we express the total travel time as a function of the x coordinate of the "corner point" P1, we can differentiate to find the position that minimizes the time, and from this we can infer the angles of incidence and refraction.
With respect to the xyt coordinates in which the n1 medium is at rest, the squared spatial distance from P0 to P1 is x12 + y12, so the time required for light to traverse that distance is
On the other hand, for the trip from point P1 to point P2 we need to know the distance traveled with respect to the coordinates x'y't' in which the n2 medium is at rest. If we define
then the Lorentz transformation gives us the corresponding increments in the primed coordinates
Therefore, the squared spatial and temporal distances from P1 to P2 in the n2 rest coordinates are given by
Since the ratio of these increments equals the square of the speed of light in the n2 medium, which is 1/n22, we have
Solving this quadratic for Dt, which equals tC - tB, gives
Differentiating with respect to Dx, and noting that d(Dx)/dx1 = - 1, we can minimize the total travel time t2 - t0 by adding the derivatives of Dt and t1 - t0 with respect to x1, and setting the result to zero. This leads to the condition
Making the substitutions
we arrive at the equation for refraction at the plane boundary between transversely moving media
As expected, this reduces to Snells law for stationary media if we set v = 0. Also, if the moving medium has a refractive index of n2 = 1, this equation again reduces to Snells law, regardless of the velocity, because the concept of speed doesnt apply to the vacuum.
If we define the parameter
then the refraction equation can be written more compactly as
This can be solved explicitly for sin(q2) to give the result
with the appropriate sign for the square root. Taking n1 = 1.2 and n2 = 1.5, the figure below shows the angle of refraction q2 as a function of the transverse speed v of the medium with various angles of incidence q1 ranging from -3p/8 to +3p/8 radians.
Incidentally, when plotting these lines it is necessary to take the positive root when v is above the zero-crossing speed, and the negative root when v is below. The zero-crossing speed (i.e., the speed v when the refracted angle is zero) is
The figure shows that at high relative speeds and high angle of incidence we can achieve total internal reflection, even though the downstream medium is more dense than the upstream medium. The critical conditions occur when the squared quantity in parentheses in the preceding equation reaches 1, which implies
Solving these two quadratics for v (remembering that y2 is a function of v), we have the four distinguished speeds
The two speeds given by 1/n2 (which are just the speeds of light in the moving medium) generally correspond to removable singularities, because both the numerator and denominator of the expression for sin(q2) vanish. At these speeds the values of q2 can be assigned continuously as
It isnt clear what, if any, optical effects would appear at these two removable singularities. The other two distinguished speeds represent the onset of total internal reflection if their values fall in the range from -1 to +1. For example, the figure above shows that total internal reflection for an incident angle of q1 = 3p/8 with n1 = 1.2 and n2 = 1.5 begins when the speed v exceeds
Notice that for an incidence angle of zero, this speed is simply n2, which is ordinarily greater than 1, and thus outside the range of achievable speeds (since we assume the medium itself is moving through a vacuum). However, for non-zero angles of incidence it is possible for one of these two critical speeds to lie in the achievable range. In fact, for certain values of n1, n2, and q1, it is possible for all four of the critical speeds to lie within the achievable range, leading to some interesting phenomena. For example, with n1 = n2 = 2.5 and with q1 = 45 degrees, the refracted angle as a function of medium speed is as shown below.
In this case the distinguished speeds are -0.4, +0.203, +0.4, and +0.783. This suggests that as the transverse speed of the medium increases from 0, the refracted ray becomes steeper until reaching 90 degrees at v = +0.203, at which point there is total internal reflection. This remains the case until achieving a speed of +0.783, at which point some refraction is re-introduced, and the refracted angle sweeps back from +90 to about +80 degrees (relative to the stationary frame), and then back to +90 degrees as speed continues to increase to 1. This can be explained in terms of the variations in the effective critical angle and the aberration angle. As speed increases, the effective critical angle for total internal reflection initially increases faster than the aberration angle, pushing the ray into total internal reflection. However, eventually (at close to the speed of light) the aberration effect brings the incident ray back into the refractive range.
For an alternative derivation that leads to a different, but equivalent, relation, suppose the index of refraction of the stationary region is n1 = 1, which implies this region is a vacuum. If we let d1 denote the spatial distance from P0 to P1 with respect to the rest frame, then we have
These are the components of the interval P0 to P1 with respect to the rest frame of n1, and they can be converted to the frame of n2 (denoted by upper case letters) using the Lorentz transformation
Letting Q1 denote the angle q1 with respect to the moving n2 coordinate system, we can express the tangent of this angle as
Taking the sine of the inverse tangent of both sides gives the familiar aberration formula
Since we are assuming the n1 medium is a vacuum, we are free to treat the entire configuration as being at rest in the n2 coordinates, with the angle of incidence as defined above. Therefore, Snells law for stationary media can be applied to give the refracted angle relative to these coordinates
Now, if D2 is the spatial distance from P1 to P2 with respect to the moving coordinates, we have
Also, the Lorentz transformation gives the coordinates of points P1 and P2 in the rest frame in terms of the coordinates in the moving frame as follows:
From these we can construct the tangent of q2 with respect to the rest coordinates
Substituting for the coordinate differences gives
We saw previously that
so we can explicitly compute q2 from q1. It can be shown that this solution is identical to the solution (with n1 = 1) derived previously on the basis of Fermat's principle. Furthermore, we can solve these equations for sin(q1) as a function of q2 and then by equating this sin(q1) with n3 sin(q3) for a stationary medium neighboring the vacuum region, we again have the general solution for two refractive media in relative transverse motion. A plot of q2 from q1 for various values of v is shown below:
