Historical research guide · 0881 · 2006
The Casimir effect within scattering theory
Astrid Lambrecht · Paulo A Maia Neto · Serge Reynaud
In one minute
Astrid Lambrecht, Paulo Maia Neto and Serge Reynaud review how the Casimir force — the attraction between two mirrors produced by the vacuum’s own electromagnetic fluctuations — is calculated when the mirrors are real rather than ideal. Their tool is scattering theory: describe each mirror by how it reflects light, and the whole force follows from those reflection amplitudes. Casimir’s textbook formula assumes perfect reflection at every frequency, so it depends only on the gap, the area and two constants of nature. Real metal mirrors reflect well only below their plasma frequency, so at separations of a few hundred nanometres the finite conductivity of gold or aluminium matters, and so does surface roughness. The authors show that the widely used proximity force approximation, which treats a bumpy surface as a stack of flat ones, systematically underestimates the roughness correction — by about sixty per cent at a two-hundred-nanometre gap. They propose the sideways Casimir force between corrugated plates as the cleanest test, and name the force’s temperature dependence as the open experimental question.
Connection
Chapter 2 rests on the Casimir force being the vacuum’s most accessible macroscopic effect, and this is the paper that says exactly how accurately we can predict it: a single formula that turns any mirror’s measured reflectivity into a force. That is the calculation any vacuum-energy device — Moddel’s resonator, White’s Casimir cell — has to be designed against, and it is also the calculation that sets how tightly a Casimir experiment can bound new short-range forces on the evidence ladder of chapter 1.
From the source
Casimir’s 1948 result for two perfectly reflecting parallel plates depends only on the separation, the area and two constants of nature, the speed of light and Planck’s constant — about 0.1 micronewton of attraction for one square centimetre at one micrometre — and it is independent of the fine-structure constant, because perfect reflection is a saturated response to the field.Section I, equation 1
Written in scattering form, the Casimir force between two mirrors is an integral over imaginary frequencies of the mirrors’ own reflection amplitudes; this reproduces the Lifshitz expression as the special case of thick bulk mirrors and stays valid for slabs, multilayers and any mirror whose reflection differs from that model.Section II, equations 10 to 13
No real mirror reflects perfectly at all frequencies: metals reflect well only below a plasma frequency whose wavelength is about 107 nanometres for aluminium and 137 nanometres for gold, so at gaps of that order the finite conductivity changes the force substantially, and the residual theoretical uncertainty comes from not knowing the optical properties of the particular samples used.Section II A, finite conductivity correction
A non-specular scattering formula, in which the reflection amplitudes become matrices that mix wavevectors and polarisations, extends the calculation to rough and corrugated surfaces; against it the proximity force approximation systematically underestimates the roughness correction, by roughly sixty per cent for a two-hundred-nanometre gap and a three-hundred-nanometre roughness wavelength.Section III A, equation 28 and figure 8
For the corrugated plane-and-sphere experiment of Chen and colleagues the full calculation gives a lateral Casimir force of 0.20 piconewton where the proximity force approximation gives 0.28 piconewton; measuring that difference would be the first unambiguous evidence of the approximation’s limits and the first observation of a non-trivial geometry effect on the Casimir force.Section III C, figure 12
The temperature dependence of the Casimir force between dissipative mirrors remains a matter of debate in the literature, and the authors state that the thermal effect has not been unambiguously proven in experiment — its observation is one of the most urgent challenges for experimenters.Abstract and Section IV, conclusion
The way in
https://doi.org/10.1088/1367-2630/8/10/243How to cite it
Astrid Lambrecht, Paulo A Maia Neto, Serge Reynaud (2006) The Casimir effect within scattering theory doi:10.1088/1367-2630/8/10/243