1. Introduction

The Laser Interferometer Space Antenna (LISA) – an equilateral triangle of three spacecraft with a side length of 2.5 × 10⁶ km – aims to measure picometer changes in the distance between free-falling masses placed inside the spacecraft [1, 2]. As shown in Fig. 1, the measurement is split into two steps: i) the measurement of the test-mass motions with respect to onboard optical-benches and ii) the measurement of the spacecraft distance. To this end, each satellite is equipped with two telescopes (afocal, off-axis, beam expanders having provisional 134× magnification) that simultaneously transmit and receive 1064 nm laser beams and heterodyne the received beams with local references [3,4]. A critical aspect is the total displacement noise in a one way test-mass to test-mass link [3],

 

Fig. 1 Schematic of the LISA measurement. The time-of-flight of the light separates the snapshots of the two spacecraft. Each telescope transmits and receives light to and from the other. The transmitted light is phase locked to the received one, setting up a transponder-like scheme. The grey boxes are the test masses; the dots indicate where the test-mass motion (blue) and spacecraft distance (yellow) are measured. The dashed lines are the axes of the transmitted beams. The green arrows are the z axes; the blue arrows are the line-of-sight to the far spacecrafts; α and β are the transmission and reception angles; we assume that the local beam is parallel to the transmitted one. Adapted from [2]

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For the transmission, each telescope takes a collimated beam, with a diameter of 2.24 mm, and transforms it into a beam having a 300 mm diameter. For the reception, each telescope collects, with a 300 mm diameter aperture, the light sent by the far spacecraft and reduces it to a 2.24 mm diameter beam. Each telescope operates between a pair of conjugate pupils and maps angular motions in the sky into angular movements in an optical bench without any transverse displacement, and ideally without length instabilities.

Wavefront aberrations and telescope jitters interact and cause a measurement noise. Firstly, because of aberrations, the received wavefront deviates from sphericity. Therefore, the transmitter jitter leads to changes of received phase and, consequently, to apparent variations of the spacecraft distance [6,7]. Secondly, the receiver jitter misaligns the interfering beams and leads to phase changes of the heterodyne signal [8,9].

The wavefronts quality, both in the transmission and reception, must be good enough that the expected $10\hspace{0.17em}\text{nrad}/\sqrt{\text{Hz}}$ jitters do not cause phase noise in the heterodyne signal [3, 4]. In previous papers, we investigated separately the noise originating from the transmitter and receiver jitter [10,11].

In this paper, we build on those results and report about an end-to-end study of the measurement noise from the beam launch to the far field and from the reception to the heterodyne signal. Since the relationship between the measured and actual distances is computationally expensive, we used an analytic parametric surrogate for the phase of the heterodyne signal and its sensitivity to jitter. It takes the form of a polynomial expansion and allows for fast sampling from the possible wavefront aberrations. When studying the phase sensitivity to the jitter, it is possible to single out the root-mean-square deviation from flatness of the wavefronts as a global parameter predicting the mean noise.

Sections 2 and 3 model the far-field propagation of the transmitted wavefront and interference. Sections 4.2 and 4.3 give the polynomial approximations of the far-field and heterodyne-signal phases, which are the main results of our previous studies. Section 4.4 works on these approximations and provides the noise and sensitivity to the jitters of the heterodyne signal. In section 5 we carry out a Monte Carlo simulation of the measurement noise by sampling from random wavefront aberrations, examine how the sensitivity to the jitters depends on the design parameters and aberrations, and develop criteria for the noise assessment.

2. Far-field propagation

By using the scalar and paraxial approximations, the optical field propagating between the spacecraft is

The complex amplitude of the transmitted beam,

The paraxial propagation in free space is given by the Rayleigh-Sommerfeld integral

We approximate the received field by the spherical wave

3. Heterodyne interferometry

We describe the local,

In (5), due to the negligible size of the telescope aperture relative to the size of the received beam, we approximated the received wavefront by a plane one having a phase delay (or advance) ϕ_far that depends on the transmission angle and aberrations. The propagation through the receiver optics introduce anew wavefront aberrations, which are summarized by S₁(r).

Though they matter in terms of contrast and shot noise, the interfering-beam amplitudes do not affect the phase of the interference signal. Therefore, we set both to unity. By neglecting the piston terms of S₁(r) and S₂(r), we do not consider the phase retardation related to the optical lengths of the beam paths through the receiving telescope and optical bench.

The phase of the heterodyne signal is kz + ϕ, where

The ϕ_far phase depends on the transmission angle α through the tilt component of S_TX. The transmission is on-axis and α is null when the tilt aberration of S_TX is zero and the transmitted beam propagates along the z axis (see Figs. 1 and 2). The ϕ_Ξ phase depends on the reception angle β through the tilt component of S_RX. The reception is on-axis and β is null when the tilt aberration of S_RX is zero and the received and local beams propagate in parallel (see Figs. 1 and 2).

 

Fig. 2 Hypothetical angular map of the beams (green: local and transmitted, blue: received) contributing to the measurement of the spacecraft distance. The two scales, one magnified 134 times, provide the angles in the sky and optical bench. For the sake of simplicity, we assumed the local and transmitted beams parallel. The far spacecraft moves along the dashed line; the circle is the z axis (the far spacecraft after a light round trip). The telescope jitter makes the line-of-sight to the far spacecraft and the z axis noisy (red random walks); α and β are the (istantaneous) transmission and reception angles; α₀ and β₀ are their mean values.

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In (10), we assumed a Gaussian intensity profile. However, if the receiver is modelled as an ideal beam-expander, the interference is between a local Gaussian beam and a received top-hat beam. Both Gaussian and (ideal) top-hat beams are limit cases. Furthermore, ϕ_far and ϕ_Ξ require the evaluation of the same integrals (6) and (12). In the Gaussian beam limit case here considered, they differ only by the replacement of $w_{\text{RX}}^2$ with $2w_{\text{TX}}^2$. In the top-hat limit case, they are identical.

4. Phase noise

4.1. Zernike modes

The phase profiles S_TX(r) and S_RX(r) are expressed in terms of Zernike modes. Therefore,

To ensure that S_TX,RX(r) are real, the relationship $z_n^{-m}=z_n^{m*}$ holds. Therefore, $z_n^0$ are real and, if m ≠ 0, $z_n^{\pm m}=\left|z_n^m\right|{\text{e}}^{\pm \text{i}{\theta }_n^m}$, where $\left|z_n^{-m}\right|=\left|z_n^m\right|$. Eventually, the magnitudes $\left|z_n^m\right|$ are expressed in radians. The n = 1 term,

We will also use the in-plane and out-of-plane components ${\zeta }_x=\left|z_1^1\right|\text{cos}\left({\theta }_1^1\right)$, ${\zeta }_y=\left|z_1^1\right|\text{sin}\left({\theta }_1^1\right)$, ${\alpha }_x=\alpha \text{cos}\left({\theta }_1^1\right)$, ${\alpha }_y=-\alpha \text{sin}\left({\theta }_1^1\right)$. Similar equations hold for the η_x,y and β_x,y components.

4.2. Far-field phase

By expressing S_TX(r) in term of Zernike modes and limiting the analysis to the lowest order couplings between tilt and higher-order aberrations, the phase of the received wavefront in advance or delay to the kz retardation is

The a_ij coefficients – which we derived in [10] – depend on the normalized radius of the transmitted beam, w′_TX = w_TX/r_TX, and the modal amplitudes $z_n^m$ of the transmitted wavefront. By limiting the aberrations to defocus, astigmatism, coma, trefoil, and spherical, they are given in the appendix.

As shown in [7, 10], for each specific wavefront aberration S_TX, there exists an optimal transmission angle α_opt – which can be found by solving ∇_αϕ_far = 0 – that nullifies the ϕ_far sensitivity to the transmission angle.

4.3. Heterodyne-signal phase

To study the phase of the heterodyne signal, we express the misalignment of the interfering beams in term of the reception angle β, where β = 0 rad means that the interfering beams are parallel. Therefore, the angle β means that the tilt aberration $z_1^1$ of the phase profile S_RX(r) is associated with the Mβ misalignment of the interfering beams, where M ≈ 134 is the telescope magnification.

By expressing the phase profile S_RX(r) in terms of Zernike modes and limiting the analysis to the lowest order couplings between tilt and higher-order aberrations, we obtain

The b_ij coefficients – which we derived in [11] – depend on the normalized mean-radius of the interfering beams w′_RX = w_RX/r_D and the amplitudes of the phase-profile aberrations. By limiting the aberrations to defocus, astigmatism, coma, trefoil, and spherical, they are given in the appendix. Owing to the identity of the (6) and (12) integrals, the a_ij and b_ij coefficients in (18) and (21) are linked by $a_{ij}(w^{\prime })=b_{ij}\left(\sqrt{2}{w}^{\prime }\right)$.

As shown in [11], for each specific aberration S_TX of the interference phase-profile there exists an optimal reception angle β_opt – which can be found by solving ∇_βϕ_Ξ = 0 – that nullifies the ϕ_Ξ sensitivity to the reception angle.

4.4. Phase noise

To compensate for the disturbances, the telescope pointings are continuously corrected, which corrections jitter both the transmission and reception angles and, according to (18) and (21), interfere with the phase of the heterodyne signal (see Fig. 2). The LISA error budgeting is done in the frequency domain, with noises expressed as power spectral densities. However, since no dynamics is involved in the propagation of the wavefronts, the transfer function that maps jitters into phase noise is independent of frequency. Therefore, we considered white uncorrelated jitters and computed how the jitter variance propagates into the phase-noise variance.

By linearization of (18) and (21), the variance of the ϕ = ϕ_far + ϕ_Ξ phase is

The transfer function is ℌ = σ_ϕ/σ_jitter, where

The a_ij and b_ij coefficients depend on the amplitudes of the Zernike modes. To find a criterion for the noise assessment, we need an expression determined by the deviations of the wavefront from flatnsses alone. The sought expression can be obtained by the averaging ℌ² over the $z_n^m$ amplitudes constrained to a predetermined root-mean-square amplitude σ_S of the wavefront aberrations. Therefore, the amplitudes of the Zernike modes of S_TX and S_RX are constrained by

By averaging over the azimuths ${\theta }_n^m$ of the $z_n^m$ amplitudes – which are assumed identical, independent, and uniformly distributed – we average over all the orientations of the Zernike modes. Since the ${\theta }_n^m$ origin is arbitrary, we can set the tilt azimuth to zero, so that the average, ${〈{ℌ}^2〉}_{{\theta }_n^m}$, cannot depend on the directional tilt-components. Hence, we obtain

The average over the magnitude $\left|z_n^m\right|$ of the Zernike modes requires to integrate the polynomials (29-b) over the hyperellipsoid (26). A derivation of these integrals is in [14]. The result is

5. Results

The jitter and aberrations of the transmitter and receiver contribute to the noise of the interferometric measurement. The received wavefront was approximated by a plane one whose phase ϕ_far advances or delays with respect the dynamical one (encoding the sought distance) because of the transmitter jitter and aberrations. An ideal receiver – that is, a perfect compressor – does not aberrate the received wavefront, also if we take the truncation into account. The additional advance or delay ϕ_Ξ of the signal phase takes the receiver non-ideality and jitter into account. Eventually, the average of the transfer function (25) over aberrations having the same root-mean-square amplitude summarizes the noise contributions of both the transmitter and receiver.

The asymptotic values,

 

Fig. 3 On-axis (left) and off-axis (right, the angles α₀ and β₀ are set to 1 μrad) sections of ${〈{ℌ}^2〉}_{z_n^m}$ vs. the normalized radii of the transmitted and detected beams, w′_TX and w′_RX. The filled areas indicate the standard-deviations. Blue: transmission; red: reception. The root-mean-square amplitude of S_TX and S_RX is 74 mrad.

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As shown in Fig. 3, when the normalized radii of the transmitted and detected beams, w′_TX = w_TX/r_TX and w′_RX = w_RX/r_D tend to zero (Gaussian-beam approximation) and infinity (plane-wave approximation) the heterodyne signal is insensitive to the jitter. Also, as shown in Fig. 3 (left), the on-axis terms $\overline{g_0\left(w_{\text{TX}}^{\prime }\right)}{\sigma }_S^4$ and $\overline{g_0\left(w_{\text{RX}}^{\prime }\right)}{\sigma }_S^4$ of (31) do not contribute to ${〈{ℌ}^2〉}_{z_n^m}$. Figure 3 (right) shows that the off-axis terms are maximum when the normalized radii are w′_TX ≈ 0.4 and w′_RX ≈ 0.6, with $\overline{g_2\left(w_{\text{TX}}^{\prime }\right)}{\sigma }_S^2$ (blue line) predominates over $\overline{g_2\left(w_{\text{RX}}^{\prime }\right)}{\sigma }_S^2$ (red line).

The parabolic approximations (18) and (21) hold if S_TX,RX < 1 rad over the effective domains of the integrations (4) and (12). However, while the wavefront deviations from flatness satisfy this constraint for all practical purposes, we would like (18) and (21) to be satisfactory approximations for tilt aberrations as significant as possible. Heuristically, S_TX,RX < 1 rad implies $2\left|z_1^1\right|w_{\text{TX},\text{RX}}^{\prime }<1\hspace{0.17em}\text{rad}$ or, equivalently, α < 1/(kw_TX) and β < 1/(kMw_RX) = 1/(kw_TX). By using w_TX = 150 cm, we obtain α, β ≲ 1 μrad.

To test the accuracy of the approximations made, Fig. 4 compares (18) against the numerical integration of (6). Since (6) cannot depend on the azimuth origin, we can always set ${\theta }_1^1=0$ and express it in term of the transmission angle α. The approximation is good up to about 1 μrad and acceptable up to about 3 μrad. It is worth noting that – since the square coefficients are of the first order, while those of the linear terms are of the second – ϕ_far and ϕ_Ξ are minimum or maximum near to the on-axis transmission/reception.

 

Fig. 4 Left: comparison of the numerical (dots) and approximate (lines) calculations of the far-field phase; α₀ is the transmission angle. Right: histogram of 10⁴ calculations of the fractional errors when α₀ = 2 μrad. The Zernike spectra were randomly generated, but constrained to a root-mean-square amplitude of 74 mrad. Colors indicate different runs.

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To quantify the approximation error, we calculated the fractional errors of 10⁴ approximated ϕ_far values when α₀ = 2 μrad and the Zernike spectra of S_TX were randomly generated, but constrained to a root-mean-square amplitude σ_S = 74 mrad. Figure 4 shows that, on the average, (18) underestimates ϕ_far by about 8%, with a standard deviation of 10%. Since it implies the same integration, the same is true for (21).

As regards the approximation (25) of ℌ², since the calculations of ∇_αϕ_far and ∇_βϕ_Ξ are the same, but the width parameter of ∇_βϕ_Ξ is scaled down by $\sqrt{2}$, the leading ℌ² term is ∇_αϕ_far (see Fig. 3, red lines vs. blue lines, and Fig. 11, w′_RX axis vs. w′_TX axis). Also in this case, the Zernike spectra of S_TX were randomly generated, but constrained to a root-mean-square amplitude σ_S = 74 mrad. Figure 5 compares the values of ∇_αϕ_far obtained from (18) against the differentiation of the numerical integration of (6). The figure shows that, when considering large transmission angles, (18) and (21) neglect significant higher-order contributions. Figure 6 shows the histogram of 10⁴ calculations of the leading ∂_αϕ_far term, where ∂_α is the directional derivative along α and α₀ = 2 μrad. Hence, on the average, (25) underestimates ℌ by about 30%, with a standard deviation of 13%.

 

Fig. 5 Comparison of the numerical (dots) and approximate (lines) calculations of ∇_αϕ_far. Left: the derivative is parallel to α₀. Right: the derivative is orthogonal to α₀. The Zernike spectra were randomly generated, but constrained by a root-mean-square amplitude of 74 mrad. Colors indicate different runs

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Fig. 6 Histogram of 10⁴ calculations of the fractional errors of the |∂_αϕ_far| approximation when α₀ = 2 μrad. The Zernike spectra were randomly generated, but constrained to a root-mean-square amplitude of 74 mrad.

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In the αβ plane, the contour lines of ${〈{ℌ}^2〉}_{z_n^m}^{1/2}$ are ellipses having the axes parallel to constant transmission and reception angles. Figure 7 shows the sections β₀ = 0 rad and α₀ = 0 rad for varying values of the normalized radius of the transmitted and detected beams. If the normalized radius is null (Gaussian beam approximation), the heterodyne signal is insensitive to jitter. As the normalized radii grow, ${〈{ℌ}^2〉}_{z_n^m}^{1/2}$ increases but returns to zero when the normalized radius tend to infinity (plane waves approximation). When w′_TX = w′_RX = 1, Fig. 7 also shows upper and lower bounds calculated as plus/minus one approximate standard deviation

 

Fig. 7 Sections β₀ = 0 rad and α₀ = 0 rad of ${〈{ℌ}^2〉}_{z_n^m}^{1/2}$ for varying values of the normalized radii w′_TX and w′_RX. In the w′_TX = w′_RX = 1 case, the bounds (standard deviations) of the ℌ’s values are also given (filled areas). The root-mean-square amplitude of S_TX and S_RX is σ_S = 74 mrad. The radius of the transmitted beam is w_TX = 150 mm.

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To trade off between the jitter, wavefront quality, and transmission/reception angles, Fig. 8 shows the mean jitter transfer function for varying root-mean-square deviations from flatness of the wavefronts. It also proves the agreement between the analytical and Monte Carlo averaging of ℌ². The top axis ticks give the mean peak-to-valley deviation from flatness of each Monte Carlo set.

 

Fig. 8 Mean jitter transfer function for different deviations from flatness of the transmitted and received wavefronts. Dots: numerical (Monte Carlo) average. Solid lines: analytical calculation (31). The top axis ticks are the mean peak-to-valley deviations from flatness of the wavefronts. In the α₀ = 0 mrad and β₀ = 2 mrad case, the gray area indicates the bounds (standard deviations) of the Monte Carlo set of ℌ’s values. The normalized radii of the transmitted and detected beams are w_TX/r_TX = w_RX/r_D = 1. The radius of the transmitted beam is w_TX = 150 mm. The horizontal (dashed) line is the upper bound to ${〈{ℌ}^2〉}_{z_n^m}^{1/2}$ when the design parameters in [3,4] are taken into account.

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With about 2 μrad beam divergence and a conservative analysis of the achievable optical and alignment imperfections, to maximise the received power, the transmitter must point the receiver, to within 10 nrad [3]. Hence, the transmission angle α₀ is negligible. As regards the reception angle, owing to the point-ahead angle, it cannot be null, unless the local reference is tilted with respect to the transmitted beam. The noise requirement (1) is for interfering-beam misalignments up to 300 μrad, a statement that comes from a top-level breakdown and was adopted as a conservative and rather stringent one [8,9]. Since the telescope reduces the misalignment by the magnification, a 300 μrad angle means that, in the sky, the noise requirement (1) is for reception angles β₀ up to 2.2 μrad.

To cope with a $10\hspace{0.17em}\text{nrad}/\sqrt{\text{Hz}}$ jitter [3, 4], the jitter transfer function must be constrained to ${〈{ℌ}^2〉}_{z_n^m}^{1/2}\precsim 1$ pm/nrad, as indicated by the dashed line in Fig. 8. By considering (31), where α₀ = 0 mrad and β₀ = 2 μrad, this means that the root-mean-square aberrations of the wavefronts must be constrained to 97 mrad (equivalently, to λ/65). Furthermore, if we take the ${〈{ℌ}^2〉}_{z_n^m}^{1/2}$ dispersion (one standard deviation, see Fig. 8) into account, the the root-mean-square aberrations must be constrained to 70 mrad (equivalently, to λ/90).

To investigate how the heterodyne phase depends on the specific set of Zernike amplitudes, we calculated the |∇_αϕ_far|₀ derivative for 10⁴ random aberrations of the transmitted wavefront constrained by σ_S = 74 mrad. After sorting the results in ascending |∇_αϕ_far|₀ order, Fig. 9 shows the |∇_αϕ_far|₀ moving-average and Fig. 10 shows the moving average of the Zernike amplitudes and correlation of the defocus and spherical aberrations. The same results hold for the phase sensitivity to the receiver jitter, |∇_βϕ_Ξ|₀.

 

Fig. 9 Moving average (2500 element windows, shifted by 1250 elements) of 10⁴ sensitivities (sorted in ascending order) of the heterodyne phase to the transmitter jitter. The random wavefront aberrations are constrained by σ_S = 74 mrad.

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Fig. 10 First and second order statistics of the Zernike amplitudes of 10⁴ random aberrations of the transmitted wavefront sorted according increasing sensitivity to the jitter. The aberrations are constrained by σ_S = 74 mrad. Left: moving average of $\left|z_n^m\right|$ over 2500 element windows, shifted by 1250 elements. Right: correlation of the defocus and spherical aberrations over 2500 element windows, shifted by 1250 elements.

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Fig. 11 Coefficients of the (36-B6) polynomials. The horizontal lines are asymptotic values (plane-wave approximation). The limits of A₂ and A₄ when w_□/r_□ → 0 are ±1.

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Figure 9 shows that there is room for improvements by identifying how aberrations (having the same root-mean-square amplitude) combine constructively or destructively. Figure 10 (left) shows that a low sensitivity is correlated to small defocus and astigmatism. This is consistent with (27), because – on the average and with the w′_TX = 1 choice made, see Fig. 11 – the leading g₂(w′_TX) term of ${\left|\nabla \alpha {\varphi }_{\text{far}}\right|}_0^2$ is dominated by the coefficients of the ${\left|z_2^0\right|}^2$ and ${\left|z_2^2\right|}^2$ amplitudes. Also, a low sensitivity correlates to aberrations having the largest spherical aberration because the coefficient of the ${\left|z_4^0\right|}^2$ amplitude is the smallest. Figure 10 (right) shows that correlated defocus and spherical aberrations combine to keep sensitivity low. In fact, they contribute with opposite signs to (27).

6. Conclusion

Heterodyne interferometry, where laser beams are simultaneously transmitted and received by onboard telescopes, will sense the separation of the LISA spacecraft down to picometre sensitivity. Due to the interaction with the wavefront aberrations, telescope jitter affects the measured distances. In previous papers [10, 11], we developed surrogates of the far-field and heterodyne-signal phases, in the form of polynomial expansions, that allow for reduced computational loads and analytic evaluation of the measurement noise.

We build on these separate investigations by using them to map analytically the spacecraft jitter into the phase noise of the heterodyne signal. Here, we combined the wavefront aberrations and jitters in a global computation, including far-field propagation, non-ideal reception, and heterodyne detection of the accumulated phase. Also, we averaged analytically the jitter transfer function over transmission and reception aberrations constrained by given root-mean-square amplitudes and developed criteria for trading off between allocations for aberrations, jitters, designs, and operations.

After fixing the root-mean-square aberrations, the jitter-induced noise depends on the radii of the transmitted and detected beams, the telescope and detector apertures, and the transmission and reception angles. To achieve the targeted performance, using the present estimates of the radii, apertures, angles, and jitter [3,4], the root-mean-square aberrations must be less than 97 mrad (equivalently, λ/65). This figure is significantly more demanding than the present estimate [3,4].

Future calculations of the wavefront aberrations in the far field, the aberration impact on the interferometric signal, and the aberrations and jitter effect on the measurement of the spacecraft separation will require dedicated numerical tools. Our parametric surrogate is a test-bed to validate them.