In the post-Newtonian (PN) regime, the time scale on which the spins of binary black holes precess is much shorter than the radiation-reaction time scale on which the black holes inspiral to smaller separations. On the precession time scale, the angle between the total and orbital angular momenta oscillates with nutation period τ, during which the orbital angular momentum precesses about the total angular momentum by an angle α. This defines two distinct frequencies that vary on the radiation-reaction time scale: the nutation frequency ω ≡ 2π=τ and the precession frequency Ω ≡ α=τ. We use analytic solutions for generic spin precession at 2PN order to derive Fourier series for the total and orbital angular momenta in which each term is a sinusoid with frequency Ω − nω for integer n. As black holes inspiral, they can pass through nutational resonances (Ω ¼ nω) at which the total angular momentum tilts. We derive an approximate expression for this tilt angle and show that it is usually less than 10 −3 radians for nutational resonances at binary separations r > 10M. The large tilts occurring during transitional precession (near zero total angular momentum) are a consequence of such states being approximate n ¼ 0 nutational resonances. Our new Fourier series for the total and orbital angular momenta converge rapidly with n providing an intuitive and computationally efficient approach to understanding generic precession that may facilitate future calculations of gravitational waveforms in the PN regime.

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High-aspect-ratio microchannels were fabricated by femtosecond-double-pulse-laser-assisted polarization-selective etching. The etching rate and uniformity of the microchannels were mainly determined by the double-pulse polarization and time delay. We found that when the two sub-pulses had a different polarization (one linear, the other circular), the microchannel etching rate increased by a factor of 10 compared to when both sub-pulses were linearly polarized. The maximum etching rate was obtained when the polarization combination was circular for the first sub-pulse and vertical for the second one. In this case, the etching rate was independent from the time delay. Laser confocal microscopy images showed that when the polarization was circular, the area modified by the laser was larger than when the polarization was linear, explaining the higher etching rate value obtained after irradiation with circularly polarized laser light.