### 4.1 Cancellation of laser phase noise

We now only have six data streams and , where . These can be regarded as 3 component vectors and , respectively. The six data streams with terms containing only the laser frequency noise are
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 to the beams and . A delay of is represented by the operator acting on the data, where , , and are integers. In general a polynomial in , which is a polynomial in three variables, applied to, say, combines the same data stream with different time-delays of the form . This notation conveniently rephrases the problem. One must find six polynomials say , , , such that

The zero on the right-hand side of the above equation signifies zero laser phase noise.

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

where is a matrix given by
The exponent ‘’ represents the transpose of the matrix. Equation (16) becomes
where we have taken care to put on the right-hand side of the operators. Since the above equation must be satisfied for an arbitrary vector , we obtain a matrix equation for the polynomials :
Note that since the commute, the order in writing these operators is unimportant. In mathematical terms, the polynomials form a commutative ring.