Magnetic noise around metallic microstructures

Thermal motion of charge carriers in a metallic object creates a randomly fluctuating magnetic field in the object's vicinity. These fields are relevant for many applications like high-precision measurements of biomagnetic signals 1 , nuclear magnetic resonance microscopy 2 , and miniaturized traps for ultra cold atoms 3,4 . The original purpose of Purcell's influential 1946 paper 5 was to point out that these magnetic fields have a spectral density that by far exceeds the Planck law for blackbody radiation at low frequencies. In fact, if only blackbody fields were present, magnetic dipole transitions between atomic or nuclear levels would never happen on laboratory time scales. The near fields sustained by material objects (that play the roles of sources and cavity) give the dominant contribution. Indeed, these near fields contain non-propagating (evanescent) components that are thermally excited as well and that dominate over free space radiation 6,7 . Phrased in another way, the dipole transition rate is enhanced because the elementary excitations in the metal provide additional decay channels 8,9 .

We focus in this paper on accurate calculations of magnetic field noise that are able to describe objects of arbitrary shape. Such objects occur, for example, in magnetic microtraps where complex networks of metallic wires create electromagnetic potentials with typical scales in the micron range 3,4 . The behaviour of the field spectrum, as the metallic geometry is changed, is far from intuitive.The spectral density increases with the material volume for small structures, but this trend saturates as soon as the typical scale gets larger than the penetration length (skin depth) of the fields in the material. It has even been found that a thin metallic layer can produce less noise than a half-space, depending on the ratios between observation distance, layer thickness, and skin depth 7,10,11 . Experiments in the field of biomagnetism have shown significant changes when a metallic film is cut, at constant volume, into stripes 1 . Calculations for these cases necessarily require numerical methods to describe the propagation of magnetic fields both in vacuum and inside a metallic structure. This is the main topic of this paper. We also discuss previously developed approximations for planar structures and within the magnetostatic regime where analytical calculations are possi-ble. Our numerical methods are restricted here to two spatial dimensions (2D) where efficient solutions of Maxwell equations can be found with the help of boundary integral equations 12,13,14,15 . Our approach can also be combined with any other numerical method for field computations, permitting to cover three-dimensional cases as well.

The results we find can be summarized as follows. A planar structure (infinite lateral size) creates equal magnetic noise for all components of the magnetic field vector. This does not apply in three dimensions, but is specific to the two-dimensional setting we focus on here. Finite metallic objects show a strong anisotropy: the noise occurs preferentially along ‘azimuthal’ directions circling around the object. Increasing the amount of metallic material does not always give larger noise, in particular when the geometrical size becomes comparable to the skin depth. We find qualitative agreement with measurements of Ref.1 where thermal field fluctuations are reduced when a metallic object is split into disconnected pieces. The surface impedance approximation, that provides an accurate description of metallic reflectors for far-field radiation, is shown to be not reliable for observation distances shorter than the skin depth. Finally, qualitative (albeit not quantitative) agreement is obtained between our numerical data and an approximation based on the incoherent summation of fields generated by thermal current elements filling up the metallic volume. This method has been used in the interpretation of previous experiments 26,27 . Our results are, to our knowledge, the first quantitative test of this approximation in a nontrivial geometry.

The paper is organized as follows. We first review the link between the thermal radiation spectrum and classical dipole radiation (Sec.II). Planar structures are analyzed in Sec.III using the angular spectrum representation. We demonstrate in particular the isotropy of the magnetic noise spectrum and discuss the accuracy of the surface impedance approximation. Sec.IV is devoted to our numerical scheme and to the results for single and multiple objects of rectangular shape.

The fluctuations of the thermal magnetic field B(r, t) are characterized by their local power spectral density (the Fourier transform of the autocorrelation function)

Higher moments are not needed for our purposes, and the average field (at frequency ω) vanishes as is typical for thermal radiation. We assume the field to be statistically stationary so that the spectrum (1) does not depend on t. The diag

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