A method of processing phase measurements in an unequal-arm laser Michelson interferometer makes it possible to detect phase effects much smaller than the laser phase noise. In the original application for which the method has been proposed, the interferometer, used to detect gravitational waves, would be based on three spacecraft flying in an approximately equilateral triangular formation with arm lengths of the order of 5 × 106 km. In principle, the method could also be utilized in other applications in which one seeks to measure relative lengths interferometrically with high precision and the interferometer arm lengths cannot be made equal.
In an interferometer of the type to which the method applies, a laser at corner A of the triangular formation transmits a beam of nominal frequency n0along leg 1 (of length L1) to corner B and along leg 2 (of length L2) to corner C. Lasers at B and C use the phase of the light arriving from A for coherent transmission of light back to A. For each leg, the phase or frequency change in the light returning to A is measured. This measurement includes contributions of laser phase noise, phase noise from secondary sources, and the phase effect of interest. Typically, the phase effect of interest is associated with a Doppler effect caused by changing arm length and/or a gravitational wave that crosses the interferometer.
The laser is the main source of phase noise. Conventionally, one desires equal arm lengths because in that case, the laser-phase-noise components of the measurements for the two arms are equal, making it possible to cancel the effect of laser phase noise by subtraction. The resulting relative-phase information can be much more precise than the raw laser phase noise would otherwise allow. If the arm lengths are not equal, then simple subtraction does not result in cancellation of the laser phase noise and, as a consequence, the desired measurement can be severely degraded.
In the present method, one records the interference of the outgoing and incoming light for each of the two arms as a function of time. One also takes account of the fact that for each arm i (i = 1 or 2), the laser phase noise in the light returning at time t equals the phase noise in the light that was transmitted at time t - 2Li/c, where c is the speed of light. Let the time series of phase-difference measurements for the ith arm be denoted by zi. One can synthesize a double-difference time series Z(t) in which the data for each arm are time-shifted by the round-trip propagation time in the other arm:
Z(t) = [z1(t - 2L2/c) - z1(t)] - [z2(t- 2L1/c) - z2(t)].
By inserting the explicit time dependences for the two arms in this equation, one can readily show that the laser-phase-noise terms cancel exactly, even whenL1 "` L2; the only terms that remain are those for the phase effect of interest plus noise from secondary sources.
Of course, the success of this approach depends on the approximate knowledge of L1 and L2. Provided that L1and L2 are known with sufficient accuracy, the precision of the phase measurement is limited only by the phase noise from secondary sources, which can be reduced as much as 100 to 200 dB below the laser phase noise.
This work was done by Massimo Tinto and John Armstrong of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com under the Physical Sciences category.
NPO-20611
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Cancellation of laser noise in an unequal-arm interferometer
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Overview
The document is a technical support package from NASA detailing a method for canceling laser noise in an unequal-arm interferometer, developed by Massimo Tinto and John W. Armstrong. The focus of the research is on improving the detection of gravitational waves using a formation of three spacecraft arranged in an equilateral triangle. Each spacecraft acts as a free-falling test particle, continuously tracking one another through coherent laser light.
The paper outlines the challenges posed by frequency fluctuations of the laser, which cannot be eliminated by simply differencing the data from the two arms of the interferometer. To address this, the authors present a technique that synthesizes the unequal-arm interferometer's response in the time domain. This method involves a carefully chosen linear combination of the Doppler data collected from the spacecraft, which allows for the cancellation of laser noise to a level below secondary noise sources.
In the second section, the authors state the problem and derive the two Doppler responses from the unequal arms. They introduce Cartesian coordinates centered on one of the spacecraft to analyze the effects of gravitational waves on the frequency and phase of the laser signals. The gravitational wave's perturbations are replicated in the data collected from each arm of the interferometer.
The document also provides mathematical expressions for the low-frequency response of the interferometer, particularly for an equal-arm, one-bounce Michelson interferometer detector of gravitational radiation. The response is influenced by the arm lengths and the frequency of the gravitational waves, with specific attention given to the sensitivity range of the LISA (Laser Interferometer Space Antenna) mission.
The final sections of the document include a comparison of the proposed method with previous techniques and outline the conclusions drawn from the research. The authors emphasize the importance of accurately estimating arm-lengths to effectively cancel laser noise and enhance the interferometer's sensitivity to gravitational waves.
Overall, this technical support package presents a significant advancement in the field of gravitational wave detection, offering a novel approach to overcoming the challenges of laser noise in unequal-arm interferometers, which could have far-reaching implications for future space-based observatories.
