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 × 10^{6} 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 n_{0}along leg 1 (of length *L*1) to corner B and along leg 2 (of length *L*2) 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* - 2*L _{i}*/

*c*, where

*c*is the speed of light. Let the time series of phase-difference measurements for the

*i*th arm be denoted by

*z*. One can synthesize a double-difference time series

_{i}*Z*(

*t*) in which the data for each arm are time-shifted by the round-trip propagation time in the other arm:

*Z*(*t*) = [*z*_{1}(*t* - 2*L _{2}*/

*c*) -

*z*

_{1}(

*t*)] - [

*z*

_{2}(

*t*- 2

*L*

_{1}/

*c*) -

*z*

_{2}(

*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 when*L*_{1} "` *L*_{2}; 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 *L*_{1} and *L*_{2}. Provided that *L*_{1}and *L*_{2} 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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This article first appeared in the March, 2000 issue of *Photonics Tech Briefs* Magazine.

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