Geometric approach for the modified second generation time delay interferometry

For convenience, in this paper, we utilize subscript with a single index to indicate the physical quantities, such as the arm length and data stream [15]. The choice of the conventions is further discussed in Appendix A. The layout of the space-borne GW detector is illustrated in Fig. 1. The constellation consists of three spacecraft (S/C), denoted by S/C $1$, S/C $2$, and S/C $3$ in the clockwise direction. An individual spacecraft carries two almost identical optical bench, represented by $i$ and $i^{\prime}$ where $i=1,2,3$. The two opposite armlengths of S/C $i$ are denoted by $L_{i}$ and $L_{i^{\prime}}$, respectively, for counter-clockwise and clockwise propagation directions. The information of the GW signals is primarily captured through the interference between the local and distant laser beams, known as the science data stream. Based on the principle of three-segment laser interferometry [34], the above measurement is further complemented by the test mass data stream and reference data stream. In this study, we consider the laser frequency noise, test mass noise, and shot noise.

The data streams recorded at the optical benches ${i}$ and ${i^{\prime}}$ are

and

In the above equations, $s_{i}$ and $s_{i^{\prime}}$ with ${i=1,2,3}$ and ${i^{\prime}=1^{\prime},2^{\prime},3^{\prime}}$ are the science data stream, which represent the interference between the laser beams from the local and distant spacecraft. ${\varepsilon_{i}}$ and $\varepsilon_{i^{\prime}}$ are the test mass data stream, where the light beams are bounced off from test mass deliberately in order to capture its mechanical motion. $\tau_{i}$ and $\tau_{i^{\prime}}$ are the reference data stream, which represents the direct interference between two adjacent laser beams from the same spacecraft. The possible GW signal, which demonstrates itself as a Doppler frequency shift, is embedded in the incident laser beam from the distant spacecraft. ${h_{i}}$ and ${h_{i^{\prime}}}$ represent such contributions due to the possible presence of a transverse-traceless GW. ${p_{i}}$ and $p_{i^{\prime}}$ denote either the laser frequency noise or laser phase noise. The latter two physical quantities, namely, the laser frequency and phase noise, are by and large equivalent, but in the case of the laser frequency noise, there will be an additional Doppler shift, and the delay operator needs to be corrected (also see the discussions in Ref. [35]). In the remainder of the present paper, the calculations will be carried out by considering the laser frequency noise, which perturb around the center frequency ${\nu_{i}\approx 282~{}\rm{THz}}$. ${N_{i}^{opt}}$ and ${N_{i^{\prime}}^{opt}}$ are the shot noise, ${\vec{\delta}_{i}}$ and ${\vec{\delta}_{i^{\prime}}}$ are associated with the mechanical vibrations of test mass with respect to the local inertial reference frame. ${{\cal D}_{i}}$, ${{\cal D}_{i^{\prime}}}$ represent six time-delay operators.

The signal and noise embedded in the laser beam that propagates along a specific detector armlength $i$ are subjected to a time delay associated with the optical path connecting the two spacecraft. Mathematically, the effect can be formulated by the action of a time-delay operator ${\cal D}_{i}$ on the relevant data streams ${x(t)}$. To be specific, we have [34]

where $c$ is the speed of light. Successive applications of the time-delay operators can be simplified to

To eliminate primed laser frequency noise and optical bench noise, one introduces intermediate variables [15, 36]. The process gives rise to the following six observables

Moreover, the TDI algorithm constructs virtual equal armlength interference by linear combination of the above variables, namely [29],

where ${P_{i}}$ and ${P_{i^{\prime}}}$ are polynomials of the time-delay operators. The laser frequency noise is expected to be eliminated from the above resultant expression.

The geometric TDI is an algorithm that constitutes appropriate TDI combinations in the form of Eq. (6) by the method of exhaustion [23]. For example, a typical geometric TDI solution is illustrated by the space-time diagram in Fig. 2. The diagram is featured by discrete grids. The grids in the horizontal direction are spatial, and they correspond to the index numbers of the three spacecraft. While the vertical ones are temporal, the time propagation is upward. A TDI combination can be seen as being composed of two optical paths, as illustrated by Fig. 2. Both paths start from a given S/C $i$ and eventually terminate at a second S/C $j$. For instance, in Fig. 2, the two optical paths start from S/C 1 and cease at S/C 1, which is the third intersection point from top to bottom. Equivalently, one can visualize the TDI combination to construct a loop that eventually returns the starting spacecraft by propagating successively forward and backward in time. To show the successive propagation of the laser between the spacecraft more transparently, when it is necessary, the grid indices can be periodically expanded in the horizontal direction (such as those shown in the diagrams Fig. 4 (c) and (d).) An optical path may also propagate in the opposite direction of time, which gives rise to a time-advance operation. Apparently, since the latter is the inverse action of the time delay, successive application of the time delay and time advance operations along the same armlength, known as the null bigrams, should not be considered due to its redundancy. Subsequently, the algorithm exhausts all possible diagrams for a given number of total links and searches for valid solutions that satisfy the criterion discussed below.

Following the spirit of the geometric TDI, a valid TDI combination implies that the summations of optical paths along the two synthesized routes must be identical [23]. Equivalently, if one assign a negative sign if the the time propogation is backward, one has

where the armlength $L_{\alpha}(t)$ is a function in time, $\alpha,\beta=1,2,3,1^{\prime},2^{\prime},3^{\prime}$ that enumerates all the links that constitute the diagram, the superscripts “$\rightarrow$” and “$\leftarrow$” indicate the two forward and backward evolutions in time, and the summations are carried out for all the links on the optical paths.

The detector armlengths ${L}_{i}(t)$, as a function of time, can be rewritten using Taylor expansion about the instant of measurement as

One substitutes the above equation into Eq. (7), groups up the terms proportional to the armlengths and their time derivatives, and divides both sides by $\frac{L}{c}$. The resulting delay-time residual along an closed optical path reads

where ${L_{i},L_{i^{\prime}}}$ represent the armlengths, $\dot{L}_{i},\dot{L}_{i^{\prime}}$ are their first-order time derivative, $\ddot{L}_{i}$ correspond to the second order terms, all evaluated at a given instant. One ignores the difference in the second order terms, so that $\ddot{L}_{i}\approx\ddot{L}_{i^{\prime}}$. The dimensionless coefficients $b_{i},b_{i^{\prime}}$, $d_{i},d_{i^{\prime}}$, and $f_{i}$ are governed by the specific diagram. The explicit forms of these coefficients are given, for instance, in Tab. 5 for the second-generation fourteen-link diagrams and Tab. 7 for the modified second-generation sixteen-link ones in the following section. In deriving Eq. (9), one also makes use of the approximation $L=cT$ where $L\sim L_{i}$. Therefore, $\delta t$ essentially contains the contributions of second-order terms, which are not considered in the first and second TDI combinations. The remaining higher-order contributions from the Taylor expansion and those from the above approximations are insignificant and thus attributed to “$\cdots$”.

The criterion for the first generation TDI is derived by only taking into account the leading terms in the expansion Eq. (9). In particular, the first generation TDI is obtained by requiring

On the other hand, the modified first generation TDI demands that

Apparently, the first generation TDI gives a rigorous solution if the detector constellation is static, namely, $L_{i}(t)=L_{i^{\prime}}(t)\equiv L_{i}$. The modified first-generation TDI serves as a solution if the constellation is rigid and rotates at a constant speed. This is because, for the latter case, one effectively has constant armlengths $L_{i}(t)=L_{i}\neq L_{i^{\prime}}=L_{i^{\prime}}(t)$ due to the symmetry in time translation.

For the second generation TDI, one further includes the first-order derivative terms. The criterion for the second generation reads

However, generally one has $\dot{L}_{i}\neq\dot{L}_{i^{\prime}}$, and some discrepancy from $\dot{L}_{i}=\dot{L}_{i^{\prime}}$ is expected (also see the explicit form given by Eq. (IV) below). In other words, the most dominant contribution on the r.h.s. of Eq. (9) gives

Therefore, following the line of thought of the modified first-generation TDI, one may distinguish the rates of change of arm lengths in the two opposite directions. This gives rise to the modified second generation TDI, which satisfies

In this case, one is free of the problem caused by the speed mismatch $\dot{L}_{i}\neq\dot{L}_{i^{\prime}}$. Now, Eq. (9) gives

which is essentially of second order in derivatives, and is expected to be of minor magnitude. Eqs. (13) and (15) will be utilized to estimate the optical path descrepancies and explicitly evaluated in Sec. IV.

We now turn to discuss the enumeration process of the geometric TDI. In Ref. [23], the search was performed by enumerating a total of ${4^{n}}$ possible diagrams with $n$ links. However, the computationally expensive task can be somewhat relieved by the following considerations. It is noted that both the laser propagation and the time evolution directions can be denoted by a binary digit, and subsequently, $2\times 2=4$, there are four possibilities. To be specific, one adopts the convention that the clockwise propagations $\rm{S/C~{}3}\to S/C~{}1\to S/C~{}2\to S/C~{}3$ are denoted by 1, and the counterclockwise propagations $\rm{S/C~{}3}\to S/C~{}2\to S/C~{}1\to S/C~{}3$ by 0. Also, the laser beam that propagates forward in time is denoted by “1” and backward in time by “0”. As explicitly shown in Fig. 3, as long as one considers the preceding spacecraft where the laser beam was deflected, among the above four possibilities, one of them (indicated by the wavy line) is always a null bigram and can readily be excluded. As a result, apart from the first link, there are only three relevant possibilities for laser propagation. Therefore, a ternary search algorithm is adopted to look for the feasible TDI combinations, with a total of $3^{n}$ possible diagrams with $n$ links, which significantly reduces the computational load. The three possible optical paths are denoted by “0”, “1”, and “2”. In particular, the two choices with the same time evolution direction are denoted by “0” and “1” according to the cyclic order for the spatial armlength. The remaining one is denoted by “2”. Also, without loss of generality, one assumes that the index of the start spacecraft $i=1$ and the first link is “1”. For instance, the eight-link Michelson combination shown in Fig. 2 is thus written as $\left\{10012001\right\}$.

Consider the case of $n$-link combinations as an example. The enumeration process start from the diagram $\left\{1,\underbrace{0,...0}_{n-1}\right\}$, which corresponds to a ternary number with the value $n_{\mathrm{min}}=3^{n-1}$. It ceases at the diagram $\left\{1,\underbrace{2,...2}_{n-1}\right\}$ with the the maximum value $n_{\mathrm{max}}=2\times 3^{n-1}-1$. The difference $n_{\mathrm{max}}-n_{\mathrm{min}}+1=3^{n-1}$ gives the number of possible diagrams. By exhausting all these diagrams using the criterion given by Eqs. (10)-(II.2), it is necessary to check whether the obtained solutions are redundant.

For the completeness of the discussion, we briefly mention why the geometric TDI solution satisfying Eq. (7) furnishes a valid TDI solution of Eq. (6). According to Ref. [23], one assigns the corresponding observable Eq. (5) to each node of a geometric TDI solution and then sums up individual contributions from all the grids with appropriate time delays. A valid solution thus implies that all the terms regarding the three independent laser frequency noise cancel out entirely. It is apparent that Eq. (7) dictates that the laser frequency noise $p_{i}$ of the starting S/C $i$ cancel out. This is because the two optical paths through the two virtual routes are identical when they terminate at S/C $j$, and subsequently, the time delays along the two routes are the same. However, due to the specific form of Eq. (5), the first links of both routes also inevitably brings in an additional noise term originating from the receiving spacecraft of the link multiplied by a minus sign. It is noted that this additional term is eliminated by the corresponding observable $\eta_{i}$ assigned to this grid since it is readily verified that both terms experience the same amount of time delay and thus are identical up to a minus sign. One may verify that such cancellations can be carried out iteratively until the grid where the two routes meet. Neither of the two laser frequency noise terms $p_{j}$ from the two corresponding links is subjected to any time delay, and therefore all the noise terms are annihilated.

For the first-generation TDI combinations, one assumes that the six armlengths satisfy $L_{i}=L_{i^{\prime}}={\rm{const}}.$, but $L_{1}\neq L_{2}\neq L_{3}$. The modified first-generation TDI combinations further assumes that the six armlengths are distinct, namely, $L_{i}(t)=L_{i}\neq L_{i^{\prime}}=L_{i^{\prime}}(t)$. There are only four modified first-generation TDI combinations, and their space-time diagrams of the geometric TDI solutions are illustrated in Fig. 4. The red-solid and blue-dashed line segments correspond to the two synthesized routes, either of which might contain both forward and backward propagations in time. The initial and terminal grids of the two routes are indicated by the black squares and black dots, respectively. Also, without loss of generality, the reference time $t=0$ is assigned to the terminal S/C 1. In this study, the above convention is adopted by the remaining plots.

To be specific, following the convention developed in [31], the time series of Figs. 4 (a), (b), (c) and (d) are illustrated in Tab. 1.

Accordingly, the resulting TDI combinations are found to be consistent with the existing literature [15]. For Fig. 4(a),

It corresponds to the Michelson combination. The first four terms of Eq. (16) correspond to the virtual optical paths via the links $1-2-3-4$, and the last four terms correspond to the one in terms of the links $5-6-7-8$.

Fig. 4(b) gives the Relay TDI combination, which reads

The first four terms of Eq. (17) correspond to the virtual optical paths via the links $1-2-3-4$, and the last four terms correspond to the one in terms of the links $5-6-7-8$. Based on the cycle rules between the three spacecraft, the Relay TDI combination contains $U(t),V(t),W(t)$, where $U(t)$ was denoted as $W(t)$ in some literatures [15, 34].

Fig. 4(c) gives the Monitor TDI combination, which reads

where ${\cal D}_{\bar{3}}$ is an advance operator which is the inverse of the corresponding delay operator, satisfying ${\cal D}_{\bar{3}}{\cal D}_{3}={\cal D}_{3}{\cal D}_{\bar{3}}=1$. The first four terms of Eq. (18) correspond to the virtual optical paths via the links $1-2-3-4$, and the last four terms correspond to the one in terms of the links $5-6-7-8$. Because the modified first-generation TDI combination ignores the commutator of time-delay operators, the Eq. (18) can be further simplified to

Fig. 4(d) gives the Beacon TDI combination. The combination $P(t)$ is different from the other three given above. It consists of two independent virtual optical loops. For the space-time diagram, we choose $t=0$ and S/C 1 as the final grid for both loops, and one finds:

The expression in literatures [15, 34] is

The time delay operator ${\cal D}_{1}$ acts on the Eq. (20) given by the geometric space-time diagram, which is the same as the mathematical Eq. (21).

It is noted that the four TDI combinations given above do not form a complete generating set for an arbitrary modified first-generation TDI combination. By adding two cyclic combinations to these four combinations, one furnishes a complete basis [19, 24].

Using the criterion given by Eq. (II.2) to study the twelve-link diagrams, one finds three geometric TDI combinations, as shown in Fig. 5. The corresponding time series are illustrated in Tab. 2.

It is noted that $\left[\alpha\right]_{1}$ is a known form of the second-generation TDI combination, which reads

The first six terms of the Eq. (22) correspond to the virtual optical paths via the links $1-2-3-4-5-6$, and the last six terms correspond to the one in terms of the links $7-8-9-10-11-12$. Comparing with the optical path with $\left[\alpha\right]_{1}$, $\left[\alpha\right]_{2}$, and $\left[\alpha\right]_{3}$, it can be found that the $\left[\alpha\right]_{2}$, $\left[\alpha\right]_{3}$ combinations of alternative forms spans shorter time in the time domain. Therefore, the alternative TDI combinations are not easily affected by instrumental gaps or glitches. On the other hand, the alternative TDI combination $[\alpha]_{2}$ has better sensitivity in the high-frequency region. This is because the averaged response function and noise PSD have fewer zeros, as shown by Figs. 19 and 20 of Sec. V.

We give in the second and third columns of Tab. 3, the coefficients $d_{i},d_{i^{\prime}}$ and $f_{i}$ defined earlier by Eq. (9) of the twelve-link 2nd-generation geometric TDI combinations. These quantities are then utilized to eventually evaluate the residual $\frac{c\delta t}{L}$ and shown by the fourth column of the table (see the derivations given in Sec. IV).

Similarly, by employing the criterion given by Eq. (II.2) to study the fourteen-link diagrams, one finds four geometric TDI combinations, as shown in Fig. 6. The corresponding time series are given in Tab. 4.

One observes that the resulting geometric TDI solutions can be expressed in the modified first-generation TDI combination. In particular, $[U]_{1}^{14},[U]_{2}^{14}$ are the summation of two modified first-generation Relay combinations. $[EP]_{1}^{14},[EP]_{2}^{14}$ are the summation of a modified first-generation Monitor combination and a modified first-generation Beacon combination. In order to distinguish it from the sixteen-link TDI combinations introduced later, these combinations are denoted by the superscript “14”.

We enumerate in the second and third columns of Tab. 5, the coefficients $d_{i},d_{i^{\prime}}$ and $f_{i}$ defined earlier by Eq. (9) of the fourteen-link 2nd-generation geometric TDI combinations. These quantities are then utilized to eventually evaluate the residual $\frac{c\delta t}{L}$ and shown by the fourth column of the table (see the derivations given in Sec. IV).

Using the criterion given by Eq. (II.2) to study the sixteen-link diagrams, one finds fourty geometric TDI combinations as shown in Tab 6, 8, 10. Using the criterion given by Eq. (II.2) to study the sixteen-link diagrams, we found nine solutions as the modified second-generation TDI combinations. They are given in Tab 6.

When compared with the existing standard TDI combinations in literature [15], two of the solutions, $[X]_{1}^{16},[X]_{2}^{16}$ are of Michelson-type. The corresponding space-time diagram is illustrated in Fig. 7. It is observed that $[X]_{1}^{16}$ is identical to the traditional form Michelson combination in the literature [15, 34], which reads

The first row of Eq. (17) corresponds to the virtual optical paths via the links $1-2-3-4-5-6-7-8$, and the second row corresponds to the one in terms of the links $9-10-11-12-13-14-15-16$. Similarly, the space-time diagram of the combination $[X]_{2}^{16}$ is also shown in Fig. 7(b).

To determine whether an obtained geometric TDI solution is equivalent to an existing one in the literature, one may repeat the process to construct the solution based on their modified first-generation counterpart. Accordingly, it can be shown that the combinations $[U]_{1}^{16},[U]_{2}^{16},[U]_{3}^{16}$ are Relay-type TDI, and their space-time diagrams are illustrated in Fig. 8. To be specific, by using Fig. 8(a), the solution $[U]_{1}^{16}$ reads

One rewrites the relevant terms of the modified first-generation $U(t)$ combination Eq. (17), we have

By multiplying Eq. (25a) by $({{\cal D}_{2^{\prime}1^{\prime}3^{\prime}}}-1)$ and Eq. (25b) by $({\cal D}_{33^{\prime}2^{\prime}1^{\prime}\bar{3}}-1)$, one rewrites $[U]_{stan}^{16}$ as

and therefore $[U]_{1}^{16}$ is indeed the second-generation Relay combination in the literature [34].

Similarly, the combinations $[E]_{1}^{16}$ and $[E]_{2}^{16}$ shown in Fig. 9 are the Monitor-type TDI solutions. The geometric TDI solution given in Fig. 9(a) implies

To compare with the combination $[E]_{stan}^{16}$ of the Monitor-type in the literature [27], one simplies individual terms of Eq. (18) as follows

By multiplying Eq. (28a)by $(1-{\cal D}_{2^{\prime}1^{\prime}1\bar{2}^{\prime}})$ and Eq. (28b) by $(1-{\cal D}_{311^{\prime}\bar{3}})$, one rewrites $[E]_{stan}^{16}$ as