A method of processing phase measurements in a large, unequal-arm laser Michelson interferometer makes it possible to measure phase effects much smaller than the laser phase noise. This method is related to the method reported in "Cancellation of Laser Noise in an Unequal-Arm Interferometer" (NPO-20611) *NASA Tech Briefs*, Vol 24, No. 3 (March 2000), page 22a.

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 a triangular formation with arm lengths of the order of 5 × 10^{6} km. In principle, the method could also be utilized in other applications in which one seeks to measure relative lengths, relative velocities, phases, or frequencies interferometrically with high precision. The method is fundamentally different from conventional interferometric methods in which one relies on equal arm lengths to cancel laser phase noise. The method is also fundamentally different from conventional two-way Doppler interferometry, in which one relies on a precise oscillator to maintain coherence.

In an interferometer of the type to which the method applies, a free-running laser at each corner of the triangular formation transmits a beam to each of the other two corners (see figure). Thus, the interferometer can also be characterized as a closed triangular array of six one-arm delay lines between the corners. The transmitting laser at each corner also serves as a local oscillator for reception of the beams transmitted by the lasers at the other two corners. The raw data output of the receiver at each corner consists of two Doppler time series - one for each of the two interferometer arms that intersect at that corner. Each of the total of six Doppler time series embodies phase and frequency contributions that include laser phase noise, phase noise from secondary sources, and the phase effect of interest. Typically, the phase effect of interest is a Doppler effect caused by changing arm length and/or a gravitational wave that crosses the interferometer. The laser phase noise is the largest source of noise in these measurements.

In this method, the Doppler time series are recorded and postprocessed to cancel all laser phase noise and extract the phase effect of interest. The line of reasoning that leads to the postprocessing algorithm begins with recognition of the following fact concerning each of the delay lines: The Doppler time series generated at the receiving corner at a given time, *t*, contains the difference between (1) the phase or frequency noise of the laser at the receiving corner at time *t* and (2) the phase or frequency noise of the laser at the transmitting corner at time *t - L/c*, where *L* is the length of the delay line and *c *is the speed of light.

By a lengthy but straightforward mathematical derivation, it can be shown that by (1) delaying each Doppler time series by a suitable interval equal to the delay in the same delay line, a different delay line, or some combination of delay lines and (2) forming suitable linear combinations of the variously delayed Doppler time series, one can obtain complete cancellation of the effects of the phase noises in all three lasers. The phase effect of interest (e.g., of a gravitational wave) is not canceled; instead, it appears in the linear combination as a multipulse response that depends on the specific linear combination and the interferometer geometry.

*This work was done by John Armstrong, Frank Estabrook, and Massimo Tinto of Caltech for *NASA's Jet Propulsion Laboratory*.*

*NPO-20739*

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###### Time-Delay Interferometry in an Unequal-Arm Interferometer

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This article first appeared in the February, 2001 issue of *Photonics Tech Briefs* Magazine.

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