(ed.). Gravitational waves (IOP, 2001)(422s).

124 LISA: A proposed joint ESA–NASA gravitational-wave missioneach spacecraft. If the transmitted beam gave a perfectly spherical wavefront inthe far field, attitude changes for the transmitting spacecraft would not give phasechanges in the received signal at the distant spacecraft. However, the combinationof diffraction due to the finite size transmitting aperture plus imperfections inthe optical system cause the wavefronts at the distant spacecraft to be somewhatdistorted. Thus, the attitude of the spacecraft and any other sources of jitter in thepointing of the transmitted beam have to be controlled quite closely.There are two methods under consideration for measuring jitter in thespacecraft attitude. In one, perhaps 10% of the received light from the distantspacecraft is picked off and focused on a CCD array or quadrant diode via a fairlylong effective focal length optical system. Changes in attitude then give changesin the position of the focal spot on the detector, which are used in a servo systemto control the spacecraft attitude, along with the similar signals from the secondoptical assembly on the spacecraft. In the second approach, all of the light goesto the main detector for measuring changes in the arm length, but the detectoris replaced by a quadrant device. If the spacecraft tips slightly with respect tothe received wavefronts, the differences in phase of the four detected signals willchange, and these changes are used as the inputs to the attitude control system.Even if the attitude jitter is made small, there still is a need to set themean beam pointing direction carefully. Diffraction plus a defocus of thetransmitted beam would result in the phase of the received wavefront varyingonly quadratically with the angular offset from the optical axis of the transmittingsystem. However, non-axisymmetric defects in the transmitted wavefronts canmake the change in the received phase vary linearly with the attitude change,even on what otherwise would have been the optical axis. To avoid this increasedsensitivity to beam pointing, the pointing along each axis of each optical assemblyis modulated at a known frequency, and the resulting apparent changes in the armlength differences at the modulation frequencies and their second harmonics aredetected. This information permits the outputs at the modulation frequencies tobe minimized by small offsets in the dc pointing directions, which corresponds tohaving just the quadratic variations in the received phase due to attitude jitter.As a measure of the remaining requirement on pointing jitter, a convenienttest case is to assume only astigmatic error in the transmitted wavefronts withan error of a tenth of a wavelength rms. For this example, if the distance errordue to pointing is allowed to be equal to that from shot noise, the errors in thedc pointing offset and in the pointing jitter can be as large as 10 milliarcsec and4 milliarcsec/rtHz, respectively. (Here and later, /rtHz stands for ‘per square rootHz’, and the given error or noise is the spectral amplitude of the error, which isthe square root of the power spectral density.) These error allocations are eachabout three times larger than those given in section 3.1.8 of [6], since the errorallocation in that case was assumed to be only 10% of the error due to shot noise.Further information on the optical path error allocation budget for LISAis given in sections 4.2.1 and 4.2.2 of [6]. The total error allocation for themeasurement of the difference in the round-trip path lengths for two arms of