Time-delay interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. In a previous publication we derived TDI combinations that exactly cancel the laser phase fluctuations up to first order in the interspacecraft velocities. This was done by interfering two digitally synthesized optical beams propagating a number of times clockwise and counterclockwise around the array. Here we extend that approach by showing that the number of loops made by each beam before interfering corresponds to a specific higher-order TDI space. In it the cancellation of laser noise terms that depend on the acceleration and higher-order time derivatives of the interspacecraft light-travel times is achieved exactly. Similarly to what we proved for the second-generation TDI space, elements of a specific higher-order TDI space can be obtained by first “lifting” the basis ($\alpha$, $\beta$, $\gamma$, $X$) of the first-generation TDI space to the higher-order space of interest and then taking linear combinations of them with coefficients that are polynomials of the six delays operators. Higher-order TDI might be required by future interplanetary gravitational wave missions whose interspacecraft distances vary appreciably with time, in particular, relative velocities are much larger than those of currently planned arrays.

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