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4.1 Cancellation of laser phase noise

We now only have six data streams s i and s' i, where i = 1, 2,3. These can be regarded as 3 component vectors s and ' s, respectively. The six data streams with terms containing only the laser frequency noise are
s = D p - p , 1 3 2 1 (15) s'1 = D2p3 - p1
and their cyclic permutations.

Note that we have intentionally excluded from the data additional phase fluctuations due to the GW signal, and noises such as the optical-path noise, proof-mass noise, etc. Since our immediate goal is to cancel the laser frequency noise we have only kept the relevant terms. Combining the streams for cancelling the laser frequency noise will introduce transfer functions for the other noises and the GW signal. This is important and will be discussed subsequently in the article.

The goal of the analysis is to add suitably delayed beams together so that the laser frequency noise terms add up to zero. This amounts to seeking data combinations that cancel the laser frequency noise. In the notation/formalism that we have invoked, the delay is obtained by applying the operators Dk to the beams si and s'i. A delay of k1L1 + k2L2 + k3L3 is represented by the operator Dk11 Dk22Dk33 acting on the data, where k 1, k 2, and k 3 are integers. In general a polynomial in D k, which is a polynomial in three variables, applied to, say, s1 combines the same data stream s1(t) with different time-delays of the form k1L1 + k2L2 + k3L3. This notation conveniently rephrases the problem. One must find six polynomials say qi(D1,D2, D3), q'i(D1,D2, D3), i = 1,2,3, such that

sum 3 qisi + q'is'i = 0. (16) i=1
The zero on the right-hand side of the above equation signifies zero laser phase noise.

It is useful to express Equation (15View Equation) in matrix form. This allows us to obtain a matrix operator equation whose solutions are q and q', where qi and q'i are written as column vectors. We can similarly express si, s' i, pi as column vectors s, s', p, respectively. In matrix form Equation (15View Equation) becomes

s = DT .p, s'= D .p, (17)
where D is a 3× 3 matrix given by
( - 1 0 D ) 2 D = D3 - 1 0 . (18) 0 D1 - 1
The exponent ‘T’ represents the transpose of the matrix. Equation (16View Equation) becomes
qT .s + q'T .s'= (qT .DT + q'T .D) .p = 0, (19)
where we have taken care to put p on the right-hand side of the operators. Since the above equation must be satisfied for an arbitrary vector p, we obtain a matrix equation for the polynomials (q,q'):
qT .DT + q'.D = 0. (20)
Note that since the Dk commute, the order in writing these operators is unimportant. In mathematical terms, the polynomials form a commutative ring.
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