Finite mass beam splitter in high power interferometers

The beam splitter in high-power interferometers is subject to significant radiation-pressure fluctuations. As a consequence, the phase relations which appear in the beam splitter coupling equations oscillate and phase modulation fields are generated which add to the reflected fields. In this paper, the transfer function of the various input fields impinging on the beam splitter from all four ports onto the output field is presented including radiation-pressure effects. We apply the general solution of the coupling equations to evaluate the input-output relations of the dual-recycled laser-interferometer topology of the gravitational-wave detector GEO 600 and the power-recycling, signal-extraction topology of advanced LIGO. We show that the input-output relation exhibits a bright-port dark-port coupling. This mechanism is responsible for bright port contributions to the noise density of the output field and technical laser noise is expected to decrease the interferometer’s sensitivity at low frequencies. It is shown quantitatively that the issue of technical laser noise is unimportant in this context if the interferometer contains arm cavities.

Figure 1

The noise density of the output field $\overline{\mathbf{o}}$ determines the noise of the gravitational-wave detection. All ${\overline{\mathbf{c}}}^i$ and the input field $\overline{\mathbf{i}}$ are propagating towards the beam splitter. The fields ${\overline{\mathbf{d}}}^i$, $\overline{\mathbf{o}}$ propagate away from it. The asymmetric beam splitter reflects with a minus sign on the side where it is indicated in the picture. We demand that the classical carrier amplitudes ${\Lambda }^i$ in each port are the same for the incoming and outgoing beams which is a valid approximation for low loss interferometers. The south port does not contain any carrier field at dark fringe.

Figure 2

Power-recycled interferometer. Both arms are described by the same transfer matrix $A$. The west port contains a power-recycling mirror with amplitude reflectivity ${\rho }_{\mathrm{p}\mathrm{r}}$. It forms the power-recycling cavity with the endmirrors of the Michelson interferometer. The mirror’s distance to the beam splitter is set to be an integer multiple of the carrier wavelength. The same holds for the pathlength of the light inside the two interferometer arms.

Figure 3

GEO 600 is a dual-recycled Michelson interferometer with a power-recycling mirror in the bright port that enhances the light power within the Michelson arms and a signal-recycling mirror in the dark port that can be tuned to a specific signal frequency. Since the arms are folded once, the effective armlength is doubled to 1200 m. The distance between the beam splitter and the so-called far mirrors of the Michelson arms is 600 m, whereas the so-called near mirrors which form the end of the arms are placed very close to the beam splitter.

Figure 4

Single-sided noise spectral density of GEO 600 with $P=300\mathrm{W}$ at the beam splitter. The spectral density of the bright port vacuum field is lying below the dark port noise spectral density throughout the entire detection band. However, the technical noise from the bright port is dominating the spectral density up to 10 Hz where it is more than 1 order of magnitude higher than the vacuum noise density. The technical noise corresponds to an input laser field with power $P_{\mathrm{i}\mathrm{n}}=1.5\mathrm{W}$.

Figure 5

Single-sided spectral density of the dark port field for the GEO 600 topology with design power $P=10\mathrm{k}\mathrm{W}$ at the beam splitter assuming the same relative technical noise here as for the low power interferometer underlying Fig. 4. However, the absolute technical noise of the input field is increased due to the higher input power of the light $P_{\mathrm{i}\mathrm{n}}=10\mathrm{W}$.

Figure 6

The advanced LIGO configuration is a power-recycling, signal-extraction Michelson interferometer with arm cavities. The armlength is 4 km. Each arm cavity is formed by an input test mass and an end test mass. The signal-extraction cavity is formed by a mirror in the dark port and the interferometer which has the conventional LIGO topology.

Figure 7

Single-sided spectral density for the LIGO topology with $P=10\mathrm{k}\mathrm{W}$ at the beam splitter. We chose the same spectral density of relative technical laser noise as for the GEO 600 configurations. The bright port fluctuations are negligible.