Design and performance of a sub-nanoradian resolution autocollimating optical lever.

Precision goniometry using optics has the advantage that it does not impose much stress on the object of investigation and, as such, is adopted extensively in gravitational wave detection, in torsion balances investigating fundamental forces, in specialized studies of biological samples, and it has potential applications in condensed matter physics. In this article we present the considerations that go into designing optical levers and discuss the performance of the instrument we have constructed. We motivate the design by considering an idealized setup and the limitations to the angular resolution induced by statistical fluctuations of the photon count rate and diffraction at the apertures. The effects of digitization of the count rate and of the spatial location of the photons on the image plane motivating the actual design are discussed next. Based on these considerations, we have developed an autocollimating optical lever which has a very high resolution and dynamic range. An array of 110 slits, of 90 microm width and a pitch of 182 microm, is located in the focal plane of a field lens, of focal length 1000 mm, and is illuminated by a CCFL tube. This array is imaged back onto the focal plane after retroreflection from a mirror placed just beyond the lens. The image is recorded on a linear charge-coupled device array at the rate of 1000 images/s and is processed through a special algorithm to obtain the centroid. The instrument has a centroid stability of approximately 3 x 10(-10) rad Hz(-1/2) and a dynamic range of approximately 10(7).

An instrument for direct observations of seismic and normal-mode rotational oscillations of the Earth.

The rotations around the vertical axis associated with the normal mode oscillations of the Earth and those induced by the seismic and other disturbances have been very difficult to observe directly. Such observations will provide additional information for 3D modeling of the Earth and for understanding earthquakes and other underground explosions. In this paper, we describe the design of an instrument capable of measuring the rotational motions associated with the seismic oscillations of the Earth, including the lowest frequency normal mode at nu approximately 3.7 x 10(-4) Hz. The instrument consists of a torsion balance with a natural frequency of nu(0) approximately 1.6 x 10(-4) Hz, which is observed by an autocollimating optical lever of high angular resolution and dynamic range. Thermal noise limits the sensitivity of the apparatus to amplitudes of approximately 1.5 x 10(-9) rad at the lowest frequency normal mode and the sensitivity improves as nu(-3/2) with increasing frequency. Further improvements in sensitivity by about two orders of magnitude may be achieved by operating the balance at cryogenic temperatures. Alternatively, the instrument can be made more robust with a reduced sensitivity by increasing nu(0) to approximately 10(-2) Hz. This instrument thus complements the ongoing effort by Igel and others to study rotational motions using ring laser gyroscopes and constitutes a positive response to the clarion call for developments in rotation seismology by Igel, Lee, and Todorovska [H. Igel, W.H.K. Lee and M.I. Todorovska, AGU Fall Meeting 2006, Rotational Seismology Sessions: S22A,S23B, Inauguration of the International Working Group on Rotational Seismology (IWGoRS)].