Test of the one-way speed of light and the first-order experiment of Special Relativity using phase-conjugate interferometers

Scientific Paper
Title	Test of the one-way speed of light and the first-order experiment of Special Relativity using phase-conjugate interferometers
Read in full	Link to paper
Author(s)	Ruyong Wang
Keywords	speed of light, Special Relativity, Sagnac effect
Published	2003
Journal	ArXiv
Volume	arXiv:physics/0609202
No. of pages	11

Read the full paper here

Abstract

With a Michelson interferometer using a phase-conjugate mirror (PCM) that reverses the uniform phase shift in a light path, we can conduct a first-order experiment of Special Relativity. Utilization of the PCM changes the basic concepts of an interference experiment. Placing a conventional partially reflecting mirror just in front of the PCM at the end of a light path, we can test the isotropy of the one-way speed of light in a system moving uniformly in a straight line and conduct the one-way Sagnac experiment. According to the reported phase-conjugate Sagnac experiment using a segment light path, we can expect that the phase shift is phi = 4pivL/clambda in the one-way Sagnac experiment with path length L and speed v, even with an increasingly larger radius of the rotation. Based on these and the experimental fact of the generalized Sagnac effect, it is very important to examine whether there is the same phase shift for the test of the one-way speed of light and the first-order experiment using the PCM in a system in straight-line uniform motion. The sensitivities of these experiments are very high.

Overview

This is a proposal paper, not a report of measurement. Ruyong Wang, with co-authors Yi Zheng and Aiping Yao of St. Cloud State University, argues that the standard reason interferometry cannot measure the one-way speed of light — that the light path must close on itself, so that any first-order term picked up going out is cancelled coming back — fails once a phase-conjugate mirror (PCM) is put at the far end of the arm. A PCM returns a wave whose phase is the negative of the incident phase. The outbound and inbound first-order terms then add instead of cancelling, and what was a second-order experiment in v/c becomes a first-order one.

The departure from the textbook account is not in the optics of the PCM, which is standard nonlinear optics, but in what Wang expects the experiment to show. He takes his own published measurements of the "generalized Sagnac effect" — phase differences produced by linearly moving fibre segments in conveyor-belt loops — as evidence that a light path of length L moving at speed v produces a phase difference 4πvL/cλ regardless of whether the motion is circular or straight. If that carries over to an apparatus in straight-line uniform motion, the phase-conjugate interferometer would return a non-null first-order result, in conflict with the principle of relativity and with the isotropy of the one-way speed of light. Wang does not assert that it will; he argues that the question is now experimentally open and that the sensitivity available is enormous.

The argument

Why the Michelson-Morley experiment is second order

Wang begins from the classical expansion. For an arm of length L along the motion, the outbound and return times are t₁ = L/(c+v) and t₂ = L/(c−v), giving

t = t₁ + t₂ = 2L/c + 2(L/c)(v/c)²

to second order. In phase terms, with φ = 2πct/λ, the outbound leg contributes φ₁ = 2πL/λ − (2πL/λ)(v/c) + … and the return leg φ₂ = 2πL/λ + (2πL/λ)(v/c) + …. The first-order terms are equal and opposite, so the round-trip total

φ = φ₁ + π + φ₂ = π + 4πL/λ + (4πL/λ)(v/c)²

retains only the second-order piece. This is why, as Wang notes, Lorentz concluded that interference experiments cannot detect first-order effects at all.

What the phase-conjugate mirror changes

The PCM's distinctive property is phase reversal. Wang illustrates it with two published bench experiments: in a Michelson interferometer with a conventional mirror, a uniform phase shift Δφ introduced into an arm appears doubled at the detector, whereas with a PCM it is exactly cancelled, because the mirror turns Δφ into −Δφ before the beam retraces the same path. Conversely, an arrangement that puts −Δφ into the incident path and +Δφ into the reflected path gives zero with a conventional mirror and 2Δφ with a PCM.

Applied to the moving Michelson interferometer, the round-trip phase becomes

φ = −φ₁ + φ₂ = (4πL/λ)(v/c)

The constant 4πL/λ term drops out entirely, which Wang points out removes sensitivity to vibration and to path-length drift in the horizontal arm. Comparing the reading at speed v with the reading at rest gives φ(v) − φ(0) = 4πvL/cλ; rotating the apparatus through 180° doubles this to 8πvL/cλ. With fibre optics, L = 5 m, λ = 0.5 μm and a phase resolution of 10⁻⁷ radians, the threshold speed is v = 10⁻⁷cλ/8πL ≈ 0.12 μm/s. Wang stresses that exact reversal is not required: any response other than the conventional φ₁ + constant — including −0.5φ₁, zero, or 2φ₁ — leaves a first-order signal.

The phase-conjugate Sagnac experiment

For a rotating arc segment AB of length L at radius R and rate Ω, the Sagnac contributions are φ₁ = kL − 2πRΩL/cλ outbound and φ₂ = kL + 2πRΩL/cλ on return, so with phase conjugation

φ = −φ₁ + φ₂ = 4πRΩL/cλ = 4πvL/cλ

Wang emphasises that this has been done — on an arc segment, not a closed loop — by Yeh, McMichael and Khoshnevisan (1986) and by McMichael and Yeh (1986). The rhetorical force of the result comes from the limit argument: hold v = RΩ fixed and let R grow without bound. Every other rotational effect, centripetal acceleration included, goes to zero, the arc straightens into a segment in linear motion, and yet the predicted phase shift stays at 4πvL/cλ.

The generalized Sagnac effect

Wang supports the extrapolation with his own measurements (Wang, Zheng, Yao and Langley, Phys. Lett. A 312 (2003) 7; Wang, Zheng and Yao, Phys. Rev. Lett. 93 (2004) 143901), in which any moving segment of a fibre loop contributes φ = 4πv·L/cλ, the relevant quantity being the projection of the segment on its direction of motion. The result was reproduced across circles, two- and three-wheel conveyors, figure-8 and zero-area conveyors, and parallelograms — configurations chosen precisely so that the enclosed area, the usual Sagnac variable, is not what the signal tracks.

The one-arm interferometer

The final construction abandons the closed loop altogether. A partially reflecting mirror placed just in front of the PCM — or a PCM with a partially reflecting coating — makes one physical path serve as both a conventional-mirror path and a phase-conjugate path. A phase shift Δφ accumulated on the one-way trip is preserved by the one and reversed by the other, so the detector sees −2Δφ between the two returning beams. A one-way shift of −2πvL/cλ therefore registers as 4πvL/cλ. Wang concludes that the two received principles of interferometry — that the effective paths must form a closed loop, and that the leg from source to beam splitter is irrelevant — both fail for a phase-conjugate interferometer.

Choosing the mirror

Not every PCM will do. Self-starting devices such as stimulated Brillouin scattering carry overall phase ambiguities and cannot supply reversal. For the externally pumped PCM using four-wave mixing, Wang works through the case honestly and against his own interest: if the pump waves travel with the signal, each acquires the same −2πvL/cλ, and the conjugate output is 2πvL/cλ − 2πvL/cλ − 2πvL/cλ = −2πvL/cλ — identical to the incident shift, so the reversal is lost and the experiment gives nothing. He therefore requires either a separate pump laser or a mutually pumped PCM, for which two configurations are given.

Assessment

The optical idea is genuinely good, and it is not fringe: phase conjugation does reverse a uniform phase shift, and a nonlinear element does sit outside the assumptions of Lorentz's first-order theorem, which was proved for linear optics and closed circuits. Wang is careful to present this as a case the theorem does not cover rather than as a refutation of it. The arithmetic is correct throughout. The expansions of t₁ and t₂, the cancellation and non-cancellation of the first-order terms, the Sagnac time difference Δt = 2RΩL/c² and its phase equivalent 4πRΩL/cλ, and the four-wave-mixing phase bookkeeping all check out; so does the quoted sensitivity, 10⁻⁷ × (3×10⁸ × 5×10⁻⁷)/(8π × 5) = 1.2×10⁻⁷ m/s. Nothing here is a units artefact or a mis-scaled constant. The self-critical treatment of the pumped PCM, where he shows his own preferred configuration fails, is a mark of care.

The weight of the paper rests entirely on one step, and that step is asserted rather than derived: that a segment in uniform straight-line motion behaves like an arc segment at very large radius. The large-R limit removes centripetal acceleration, but it does not make the apparatus inertial — however slowly it turns, the loop still closes on itself and the light still traverses a rotating circuit, and rotation is absolutely detectable in a way uniform translation is not. The generalized-Sagnac measurements share this feature in a more important way: in the conveyor experiments the fibre segment moves relative to the source, the detector and the rest of the loop. That relative motion is a physically real, frame-independent quantity, and it is what the 4πvL/cλ formula tracks. In the proposed first-order experiment there is no such relative motion — source, splitter, fibre, PCM and detector all move together at v. The extrapolation therefore requires the phase to depend on motion with respect to some preferred frame, which is the very thing at issue. Wang does not supply an argument bridging the two cases; he asks that the experiment be done, which is a defensible position, but the reader should not mistake the published conveyor results for evidence that the co-moving experiment will be non-null. Special relativity predicts a strict null, and does so consistently with those same conveyor results.

Two smaller difficulties. The configuration with a separate pump laser (Fig. 13) is offered without the phase bookkeeping that condemned the single-laser version — yet the pump beams must reach the PCM along paths of their own, in the same moving frame, and whether their contributions cancel is exactly the question that had to be worked out in the previous case. And the paper never confronts the accumulated first-order null results from other methods: Kennedy-Thorndike-type and modern optical-cavity and resonator tests constrain isotropy violations far below the level a 0.12 μm/s threshold implies for Earth's motion, and any framework predicting 4πvL/cλ for a co-moving apparatus would have to explain why those experiments see nothing. As a proposal the paper is clear, honest and cheap to test; as an argument that the result will be non-null it is a conjecture resting on an analogy.

See also

• Ruyong Wang
• Sagnac Effect
• Speed of Light
• Simultaneity
• Michelson interferometer
• Michelson–Morley experiment
• Kennedy-Thorndike experiment
• Georges Sagnac