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4.2 Cancellation of laser phase noise in the unequal-arm interferometer

The use of commutative algebra is very conveniently illustrated with the help of the simpler example of the unequal-arm interferometer. Here there are only two arms instead of three as we have for LISA, and the mathematics is much simpler and so it easy to see both physically and mathematically how commutative algebra can be applied to this problem of laser phase noise cancellation. The procedure is well known for the unequal-arm interferometer, but here we will describe the same method but in terms of the delay opertors that we have introduced.

Let f(t) denote the laser phase noise entering the laser cavity as shown in Figure 4View Image. Consider this light f(t) making a round trip around arm 1 whose length we take to be L1. If we interfere this phase with the incoming light we get the phase f1(t), where

2 f1(t) = f(t - 2L1) - f(t) =_ (D 1- 1)f(t). (21)
The second expression we have written in terms of the delay operators. This makes the procedure transparent as we shall see. We can do the same for the arm 2 to get another phase f (t) 2, where
2 f2(t) = f(t - 2L2) - f(t) =_ (D 2- 1)f(t). (22)
Clearly, if L1 /= L2, then the difference in phase f2(t) - f1(t) is not zero and the laser phase noise does not cancel out. However, if one further delays the phases f1(t) and f2(t) and constructs the following combination,
X(t) = [f2(t- 2L1) - f2(t)]- [f1(t - 2L2) - f1(t)], (23)
then the laser phase noise does cancel out. We have already encountered this combination at the end of Section 2. It was first proposed by Tinto and Armstrong in [29].
View Image

Figure 4: Schematic diagram of the unequal-arm Michelson interferometer. The beam shown corresponds to the term 2 2 (D2 - 1)(D 1- 1)f(t) in X(t) which is first sent around arm 1 followed by arm 2. The second beam (not shown) is first sent around arm 2 and then through arm 1. The difference in these two beams constitutes X(t).
The cancellation of laser frequency noise becomes obvious from the operator algebra in the following way. In the operator notation,
2 2 X(t) = (D 1 - 1)f2(t) - (D2 - 1) f1(t) = [(D21- 1)(D22 - 1)- (D22 - 1)(D21- 1)]f(t) = 0. (24)
From this one immediately sees that just the commutativity of the operators has been used to cancel the laser phase noise. The basic idea was to compute the lowest common multiple (L.C.M.) of the polynomials 2 D 1- 1 and 2 D 2- 1 (in this case the L.C.M. is just the product, because the polynomials are relatively prime) and use this fact to construct X(t) in which the laser phase noise is cancelled. The operation is shown physically in Figure 4View Image.

The notions of commutativity of polynomials, L.C.M., etc. belong to the field of commutative algebra. In fact we will be using the notion of a Gröbner basis which is in a sense the generalization of the notion of the greatest common divisor (gcd). Since LISA has three spacecraft and six inter-spacecraft beams, the problem of the unequal-arm interferometer only gets technically more complex; in principle the problem is the same as in this simpler case. Thus the simple operations which were performed here to obtain a laser noise free combination X(t) are not sufficient and more sophisticated methods need to be adopted from the field of commutative algebra. We address this problem in the forthcoming text.

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