On Fresnel Aether Drag, `Moving' Images, and Relativity

TL;DR

The paper clarifies that the observed effects in certain optical experiments are illusions and not evidence of Fresnel drag or relativistic phenomena, emphasizing the distinction between true light drag and visual illusions.

Contribution

It provides a clear theoretical distinction between genuine Fresnel drag and visual illusions in optical experiments, with three proofs demonstrating the non-physical nature of the observed 'movement'.

Findings

Spatial shift in experiments is an illusion, not physical movement.

Relative velocity and spatial shift are zero in the observed effect.

Recorded effects are due to time lag differences, unrelated to relativity.

Abstract

I show the decisive difference between genuine transverse Fresnel drag of light in a moving medium and the "spatial shift" measured with a time dependent interference pattern of light traversing a homogeneous finite medium (J. Leach et al., PRL 100, 153902 (2008)). In the latter case, the relative velocity and spatial shift are in fact zero and the `movement' is an elementary visual illusion, easily made superluminal. Three separate proofs are given for this fact. What is recorded in the experiment is just the difference between a time dependent space-fixed pattern and its time lagged version. This has no relevance to relative motion of any physical entity, Fresnel drag or relativity.

Equations

$$c^{\prime }=\frac{c}{n}\pm v(1-1/n^2)$$

$$\delta x=\frac{vL}{c}(n_g-1/n_p)$$

$$\delta x=v_{d}t=\frac{vL}{c/n_g}(1-\frac{1}{n_{p^2}})\simeq \frac{vL}{c}(n_g-\frac{1}{n_p})$$

$$\delta x^{\prime }=\frac{vL}{c}(n_g-1)$$

$$\frac{d\left(\delta \varphi (x)\right)}{dt}=\delta \nu \frac{4\pi \alpha \delta x}{\lambda }$$

$$\delta I(x)=\delta t\times \frac{\partial I(x)}{\partial t}$$

$$\delta t$$

$$\delta I(x)$$

$$\delta x^{\prime }=v\frac{L_1-L_2}{c}=v\delta t$$

$$\delta {\theta }^{\prime }=\frac{\Omega L}{c}(n_g-1)$$

$$\delta \theta =\frac{\Omega L}{c}(n_g-\frac{1}{n_p})$$

$$\delta f^{\prime }=\frac{\left(df/dt\right)L}{c}(n_g-1)$$

$$\delta {\epsilon }^{\prime }=\frac{\left(d\epsilon /dt\right)L}{c}(n_g-1)$$

Full text

Fresnel drag, first measured interferometrically by Fizeau, is an important pre-relativity result on the propagation of light in moving media with phase refractive index $n$ [1]. When the velocity of the medium is parallel to the propagation direction, the resultant velocity of light is given by

[TABLE]

So, the drag velocity is $v_{d}=v(1-1/n^{2})\text{,}$ with the characteristic and crucial $1/n^{2}$ dependence in the effective drag. The expression can be derived from special relativity’s velocity addition formula as well [4].

When light propagates in a direction transverse to the motion of the (finite) medium, the ‘drag’ should carry light laterally, resulting in a shift of point of emergence of the ray (fig. 1). This transverse Fresnel drag was measured carefully by the master metrologist R. V. Jones and confirmed that it follows the relation

[TABLE]

where $L$ is the thickness of the medium, $n_{g}$ its group refractive index and $n_{p}$ its phase refractive index [2, 3]. Since the time for propagation inside the medium is $L/\left(c/n_{g}\right)\text{,}$ light is dragged by

[TABLE]

This is a difficult measurement compared to the usual Fresnel drag because it is not amenable to any straightforward interferometric scheme. The shift to be measured, taking 10 m/s for the velocity of the medium, with total thickness 5 cm in a double pass configuration, is less than 2 nm. This needs to be measured to the accuracy of 1%, to compare with different theoretical possibilities. This task was admirably achieved by Jones in two experiments, requiring elaborate mechanical and optical arrangement [2, 3].

More recently, Leach et al. considered a special relativistically symmetric situation of the light beam moving transversally in a lab-fixed static medium [5]. The desire was to measure the transverse drag when there was relative motion between the medium and light field, by moving the light field across the medium that was static in the laboratory. However, instead of moving the source of light and field relative to the medium, they used the interference of two optical fields with slightly different frequency, spatially overlapping at an angle. This results in a pattern of fringes that are time dependent, visually mimicking a movement of the pattern, though its spatial envelope is static. They reported a measured drag and ‘spatial shift’ of the pattern of light or the ‘optical image’ that has passed through the medium while transversely moving, following the formula

[TABLE]

instead of the Fresnel relation. In terms of the drag velocity, this is equivalent to setting $v_{d}^{\prime}=v(1-1/n)\text{,}$ in contradiction to the characteristic $1/n^{2}$ Fresnel drag. Quoting Leach et al., “The discrepancy between our results and the work of Jones is intriguing. For each configuration, either moving medium (Jones’s experiment) or moving image (our experiment), the analysis of the phenomenon is explainable in either the rest frame of the medium or the frame in which the medium is moving.”

The unexpected large deviation from the Fresnel drag relation was analyzed in terms the difference between the Poynting vector along energy flows and the wave vector, in an attempt to understand the discrepancy. However, a supplementary experiment in which an optical interference pattern was rotated inside the medium, also gave a result following eq. 4, instead of the Fresnel drag relation that was expected. Such interference fringes are formed by overlapping helically phased optical beams. This was considered ‘puzzling’, by Leach et al.

I now show that what was measured in the experiment by Leach et al. has no relevance for relative motion, Fresnel drag and relativity. All the results of Leach et al. are explained as the comparison of a time dependent intensity pattern with its time delayed copy, without any genuine spatial movement or velocity. A concise version of the essential argument was published as a short comment [6]. Moving an aperture across a light field results in the visual impression of a bright spot of light falling on a transverse surface as ‘moving’ (fig. 2A). The illusion of motion is generated by ‘moving’ images in the static visual frame, as in cinema; there is no transverse motion of light. The method used by Leach et al. is effectively the same kind of visual illusion – animation rather than motion.

Relativity deals with the relations between physical entities in two frames that are relatively moving. One common situation is when the medium is moving relative to the source of the optical field. To see what the reciprocal situation is, one has to just shift to the frame of the moving medium. Relative to that frame, the source of the optical field is moving (even the lab and rest of the world is, but we need not consider that in the present analysis). Then the photons are expected to have a transverse velocity, relative to the medium. This is not what was done in the experiment. Leach et al. created the ‘impression’ of lateral movement of an optical field relative to a medium – a block of glass – by introducing a frequency difference between two beams overlapping at a small angle resulting in a time dependent phase at a transverse field point $(x,y)\text{,}$ but no relative motion at any velocity was involved. When wavefronts overlap at small angle $2\alpha$, parallel fringes with spacing $\Lambda=\lambda/2\alpha$ are formed and a frequency difference $\delta\nu$ between the beams create the impression of moving fringes with apparent velocity $v_{a}=\Lambda\delta\nu\text{,}$without the beams moving (fig. 2B). That it is not a physical motion in the sense of relativity and kinematics is easily seen by considering beams of size 5 cm, with a small angle between them $\alpha\simeq 2\times 10^{-5}$rad, $\Lambda\simeq 2.5$ cm, and a practical $\delta\nu\simeq 20$ GHz. The ‘velocity’ then is an unphysical $5\times 10^{8}$ m/s! (one can also just magnify the fringe pattern and then the ‘velocity’ increases unphysically with magnification).

The fact that the optical impression is not physically relevant motion can be illustrated easily, and in many ways. If the interference fringes are formed such that the visibility is not 100%, then the whole pattern will not change across the field of view. The progressive time dependent phase change ‘moves’ only the partial fringe pattern. The rest of the light forms an incoherent static field. Hence, it is obvious that the ‘movement’ is mere optical illusion.

Another experimental demonstration is the following. If the visual impression were genuine motion, the photons would have a transverse velocity relative to lab-fixed references. With fringe spacing of 1 cm and a frequency difference of 30 MHz, the transverse velocity is $3\times 10^{5}\text{{}}$ m/s. If the light in the pattern is passed through a small aperture, this stream of photons would reach a screen at a distance $L=1\text{{}}$ m with a transverse shift equal to $\Delta x=Lv/c\simeq 1$ mm. This is easily measurable with a CCD camera. However, it may be verified that there is no such shift. Thus, I have mentioned three different proofs for the illusion of relative motion in the experiments by Leach et al.

All these comments apply to the motional animation involving circularly symmetric optical patters as well, used in the experiments by Leach et al. There, the linear patterns looped in a circle generate the visual impression of circular movement.

The relative phase of the two beams at nearby points in the transverse plane change by

[TABLE]

but clearly there is no transverse motion of any physical entity. The optical field at $x^{\prime}$ is a copy of the field at $x$ at an earlier time $\delta t=2\alpha\left(x^{\prime}-x\right)/\lambda\delta\nu$ without any transverse component of velocity imparted to the optical beam. There is no velocity or photon momentum in the transverse direction. In fact, one may consider just one of the interfering beams. If we move a periodic grid in front of the beam, the shadow (the contract between the bright and dark regions) moves across at an apparent velocity but there is no movement of the beam itself nor any relative velocity between the optical beam and the medium in the sense of relativity and kinematics. This will give the same results obtained by Leach* et al*. The case of interference pattern is similar, with one more optical field added to the configuration, since the light beams are not moving relative to the medium or the lab. Now, if we compare this time dependent intensity with itself after a time delay $\delta t\text{,}$ without any spatial drag or displacement, we get the difference

[TABLE]

The time delay in this case is due to the two different paths to the CCD camera, one through free space and another through glass of thickness $L\text{.}$

[TABLE]

So, the total beam at two different times occupies exactly the same spatial region and boundary, but the internal intensity pattern across the beam $I(x)$ differ by the temporal ‘speed’ at which the intensity is modulated, without genuine motion or velocity. This is what Leach et al. saw in their experiments, both in the linear version and in the rotational version. Because the experiment has no relation to movement in relativity or in Fresnel drag, the experiment does not address those relevant issues. The camera is comparing the time dependent spatial intensity pattern with an earlier copy, arrived delayed through glass, giving the false impression of a spatial shift. That there is zero spatial shift can easily be seen from the boundary of the image or the beam, which is part of the optical field and remains static relative to the medium. This lag could have been just an optical delay in free space without a medium, resulting in two paths with a delay between them and the same result would have followed, which is

[TABLE]

The experiment with rotating image is identical. An time dependent, but spatially stationary image is compared with its slightly earlier copy and the difference in angles will of course be

[TABLE]

instead of the Fresnel result

[TABLE]

This completely explains the non-relativity results in the paper by Leach et al., obtained due to the use of motion-animation rather than genuine motion, .

In fact, the apparent movement in space is not important. The modulation can be in any quantity associated with beam, like the frequency, polarization etc., and two measurement with a time delay generated with a medium will differ by the same formula, with the $L(n-1)/c$ dependence. If the image was changing color linearly (frequency of light) at some rate, the difference between the two images will be in frequency space with a shift

[TABLE]

If polarization was changing linearly in time, then ‘shift’ in polarization,

[TABLE]

The fine measurements done by R. V. Jones on the transverse Fresnel drag in 1970s are yet to be surpassed in precision and ingenuity. Measurements using medium with very large refractive index is not very useful because large group refractive index $n_{g}$ renders the crucial ‘relativistic’ term $-1/n_{p}^{2}$ unobservable. Measurements in moving media with anomalously large effective refractive index, like what can be arranged with an EIT (Electromagnetically Induced Transparency) medium, are not relevant to the issues of relativity! This is because, the Fresnel drag term $(1-1/n^{2})$ is totally negligible compared to the anomalous refractive index, $\Delta n=\left(\partial n/\partial\omega\right)\omega/n$. What Fizeau and Jones achieved was not just the measurement of the drag of light by the medium, but the verification of the crucial fact of the ‘partial drag’, which is the relativistic signature. Complete drag is Galilean with no indication of a limiting velocity. Therefore, the measurement of a genuine relativistic Fresnel drag is a problem in the domain of precision measurements requiring considerable ingenuity. The specific issue of whether the drag is symmetric between the movement of the medium and movement of the light field is a more difficult issue to answer experimentally. Possibilities for new measurements will be discussed in another paper, in a wider context covering optics, quantum mechanics and relativity [7].