Tilt-to-length noise coupled by wavefront errors in the interfering beams for the space measurement of gravitational waves

Ya Zhao; Ya Zhao; Jia Shen; Jia Shen; Chao Fang; Heshan Liu; Zhi Wang; Zhi Wang; Ziren Luo; Ziren Luo

doi:10.1364/OE.397097

1. Introduction

In the spaceborne and long-baseline laser interferometers such as the Laser Interferometer Space Antenna (LISA) and the Taiji program, there are three spacecraft forming an equilateral triangle with arm lengths of millions of kilometres [1–3]. In order to successfully detect the gravitational waves, the measurement noise of the detectors needs to meet $1 pm/\sqrt Hz$ within the frequency band from $0.1 mHz$ to $1 Hz$ [4,5]. This ultra-low noise level demands tightly on all aspects of these gravitational wave detectors. The science interferometer shown in Fig. 1, one of the subsystems of the interferometric system, is responsible for measuring the optical path change between the local optical bench and the optical bench on a remote spacecraft [6].

Fig. 1. The simplified diagram of the science interferometer shows the process that the far-field wavefront is clipped and narrowed through a telescope to the optical platform (lenses used to fold the optical path are omitted), and then hitting the detector through an imaging system. A pupil imaging conjugate system consists of the telescope and the imaging system. It overlaps the exit pupil (EXP) of the telescope with the entrance pupil (ENP) of the imaging system, and placing the detector at the EXP of the imaging system. The green beam of an off-axis field of view is worked by the pupil imaging conjugate system to hit the same position of the detector with the same optical path (Fermat principle) compared to the red beam of the on-axis field of view.

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As shown in Fig. 1, the local telescope is a 4-mirror Schiefspiegler system with 80 magnification and a diameter of the primary mirror of 400 mm, and the imaging system is a Kepler system with 2.5 magnification [7]. As a result, the received flat-top beam has a diameter of $2mm$ on the detector. Furthermore, due to the spacecraft jitter and the breathing of the constellation, the received beam jitters on the detector and interferes with the static local beam at varying angles. In principle, the pupil imaging conjugate system can make the received beam imaged on the detector without beam walk and eliminate the effect of beam tilt on the change of optical path. However, it can not eliminate the TTL noise coupled with the wavefront error of the interfering beams, which are currently induced by the imperfections of the optical systems because of the manufacture and adjustment errors. The coupling coefficient is currently required to be below $25pm/\mu rad$ within $\pm 300\mu rad$ of the wavefront misalignment between the interfering beams [8].

C P Sasso et al. analytically investigated the TTL coupling between the wavefront misalignment and the low-order aberrations of the interfering beams by extracting the phase information from the complex amplitude of a single element photodiode (SEPD) [9]. However, the quadrant photodiode (QPD) consisting of four active segments, is mostly used as the detector of the interferometric systems to obtain the DWS (differential wavefront sensing) signal for measuring small angular changes [10–12]. The phases of the four quadrants can be combined in different ways to get the overall phase of the entire QPD, and the phases obtained may be different [13].

In this paper, we analytically calculate the TTL noise coupled by aberrations including the third to the seventh order, by applying the two most common QPD path length definitions (the $AP$ signal and the $LPF$ signal). For the $AP$ signal, we also analyze the effect of the lateral shift between the interfering beams and the detector on TTL coupling and derive an expression which theoretically explains a compensating relationship between the lateral shifts of the detector and optical lens in the experiments [14]. Meanwhile, the expressions for calculating the minimum sensitivity tilt for the TTL noise are given. Finally, we carry out Monte Carlo simulations for the $LPF$ signal and the three kinds of $AP$ signals with three different settings of the lateral shift by trying arbitrary wavefront errors. The results intuitively show that the $AP$ signal can more greatly reduce the TTL noise coupled with aberrations compared to the $LPF$ signal, and introducing the appropriate lateral shift of the interfering beams on the detector can further reduce the TTL coupling. The AP signal with $\lambda /10$ wavefront error of the interfering beams is better than the LPF signal with $\lambda /20$ wavefront error when other states are the same. Therefore, the analyses C P Sasso et al. performed in [9,15] may exaggerate the influence of aberrations on the TTL noise to put forward too high demand for wavefront quality.

2. Optical model

The far-field wavefront is narrowed by the off-axis telescope as a flat-top beam, which then superimposing with the local Gaussian beam pass the imaging system together to interfere on the QPD [16]. By proper adjustment, the relative position between the two interfering beams can go below microns level. For the sake of simplicity, we assume that the centers of the two beams coincide and set the center as the coordinate origin shown in Fig. 2. Consequently, the local Gaussian beam on the detector can be written as (in polar coordinates):

(1)$$E_{1}(r,\theta,z,t)=e^{\frac{-r^{2}}{\omega(z)^{2}}} e^{-i k\left [\frac{r^{2}}{2R(z)}+w_{1}(r,\theta)\right ]+i\omega_{1}t},$$

where $k=2\pi /\lambda$ is the wavenumber, $\lambda$ is the wavelength. $(r,\theta )$ is the polar coordinate on the detector plane, $\omega (z)=\omega _{0} \sqrt {1+(\frac {z}{z_{0}})^2}$ is the spot size on the detector, where $\omega _{0}$ is the waist size, $z$ is the distance away from the waist position, $z_{0}=\pi \omega _{0}^{2}/\lambda$ is the Rayleigh distance. $R(z)=z\big [1+(z_{0}/z)^2\big ]$ is the radius of the wavefront curvature. The phase term $\frac {-r^2} {2R(z)}$ is similar to defocus, and whether it is equal to 0 depends on whether $z$ is equal to 0. $w_{1}(r,\theta )$ represents the wavefront error of the local beam created by defective optical elements. $\omega _{1}$ is the angular frequency.

Fig. 2. The interfering beams hit centrally (left) and eccentrically (right) on the QPD. In the pictures, $D1$, $D2$, $D3$ and $D4$ represent the valid integral regions in the four Cartesian coordinate quadrants. Blue numbers 1,2,3,4 indicate the four segments in the QPD, respectively. $\epsilon _{x}$ and $\epsilon _{y}$ are the eccentricities in the x-direction and y-direction between the interfering beam and the QPD. $S1$, $S2$, $S3$, $S4$ and $S5$ stand for the changed effective integral regions of the four segments induced by the eccentricity.

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The flat-top beam on the detector can be derived as (in polar coordinates):

(2)$$E_{2}(r,\theta,t)= \left\{ \begin{array}{ll} e^{-ik\left [r\alpha\cos(\theta-\theta_{\alpha})+w_{2}(r,\theta)\right ]+i\omega_2 t} & r \leq r_{0}\\ 0 & \textrm{else}\\ \end{array}, \right.$$

where $\alpha$ is the small tilt angle of the beam about an axis that is orthogonal to a line making an angle of $\theta _{\alpha }$ with the $x$ axis on the detector plane, $r_{0}$ is the radial dimension of the flat-top beam. $w_{2}(r,\theta )$ also represents the wavefront error of the flat-top beam caused by defective optical elements including the off-axis telescope. $\omega _{2}$ is the angular frequency.

Provide that the interfering beams can be detected without clipping, it is easy to extract the phase from an integration of the complex overlap term over the detection area. Since the phase is time-independent, we set $t = 0$ and the integrand becomes:

(3)$$O_{ovi}(r,\theta)=\int_S E_{1} E_{2}^{\ast} dr^{2}=\int_S e^{\frac{-r^{2}}{\omega(z)^{2}}}e^{ikW(r,\theta)}dr^{2},$$

where $W(r,\theta )=r\alpha \cos (\theta -\theta _{\alpha })+\Delta w(r,\theta )$, represents the total wavefront error including tilt, and $\Delta w(r,\theta )=w_{2}(r,\theta )-w_{1}(r,\theta )-\frac {r^2}{2R(z)}$, denotes the static wavefront error of the interfering beams. Then the phase information is extracted from the argument of (3). In this paper, the range of the static wavefront error $\Delta w(r,\theta )$ discussed is less than $0.1064\mu m$, and the range of the tilt angle $\alpha$ discussed is within $\pm 300 \mu rad$. So based on computational accuracy, the exponential term $e^{ikW(r,\theta )}$ is approximate to:

(4)$$e^{ikW(r,\theta)}{\approx}1+ikW(r,\theta)-\frac{1}{2}k^2 W(r,\theta)^{2}-\frac{i}{6}k^{3} W(r,\theta)^{3}.$$

As shown in Fig. 2, the complex amplitude for every segment of such a QPD can be calculated. And there are the two most common ways to combine the complex amplitudes of the four segments in the current projects. The first phase definition selected by LISA Pathfinder is the $LPF$ signal, which first sums the complex amplitudes of each segment of the QPD and then calculates the argument from the sum [13]:

(5a)$$\phi_{LPF}=\arg \left( \sum_{n=1}^{4} O_{ovi}^{n} \right).$$

Here, the slit of the QPD is not considered in the following calculations because the effect of the ten microns scale slit gap of the universal QPD on the total complex amplitude is negligible. As a result, we can directly integrate the entire circle to calculate $\phi _{LPF}$ (And $\arg x$ is approximated as $Im(x)/Re(x)$ in the calculation):

(5b)$$\begin{aligned}\phi_{LPF}&=\arg \left[ \int_{0}^{1}\rho e^{-\rho^{2}/\omega_{r}^{2}}\bigg(\int_{0}^{2\pi} e^{ikW(\rho,\theta)}d\theta\bigg)d\rho \right] \\ &{\approx}\arg (a_{0}+ia_{1}-a_{2}-ia_{3}) \\ &{\approx}\frac{(a_{1}-a_{3})(a_{0}+a_{2})}{a_{0}^2}, \end{aligned}$$

where $\omega _{r}=\omega (z)/r_{0}$, $\rho =r/r_{0}$ and

(5c)$$a_{n}=\frac{1}{n!}\int_{0}^{1} \rho e^{-\rho^{2}/\omega_{r}^{2}}\bigg(\int_{0}^{2\pi} [kW(\rho,\theta)]^{n}d\theta\bigg)d\rho.$$

The second phase definition is the $AP$ signal, which first takes the resulting phase of each segment of the QPD and then averages it. By assuming $\sqrt {\epsilon _{x}^{2}+\epsilon _{y}^{2}} \ll r_{0}$, the shape of $Si$ areas can be regarded as a rectangle, as shown in Fig. 2. Therefore, it is more suitable to calculate the complex amplitudes of the $Si$ areas with Cartesian coordinates. And the exponential term $e^{ikW(x,y)}$ is approximated up to the first order because the $Si$ areas are very small:

(6)$$e^{ikW(x,y)}{\approx}1+ikW(x,y).$$

The effective integral regions of the four segments in the QPD are shown in the Fig. 2. Accordingly, the calculations of the complex amplitudes of the four segments are given in (7a$\sim$7d):

(7a)$$O_{ovi}^{1} =\int_{D1} O_{ovi}(r,\theta)+\int_{S1+S4+S5} O_{ovi}(x,y),$$

(7b)$$O_{ovi}^{2} = \int_{D2}O_{ovi}(r,\theta)-\int_{S1}O_{ovi}(x,y)+\int_{S2}O_{ovi}(x,y),$$

(7c)$$O_{ovi}^{3} =\int_{D3}O_{ovi}(r,\theta)-\int_{S2+S3+S5}O_{ovi}(x,y),$$

(7d)$$O_{ovi}^{4} =\int_{D4}O_{ovi}(r,\theta)-\int_{S4}O_{ovi}(x,y)+\int_{S3}O_{ovi}(x,y).$$

Here we take (7a) as an example to show the phase calculation process of the first segment of the QPD. The calculation of the complex amplitudes of the $D1$ area is shown in (8a) and (8b).

(8a)$$O_{ovi}^{D1}{\approx}d_{0}+id_{1}-d_{2}-id_{3},$$

(8b)$$d_{n}=\frac{1}{n!}\int_{0}^{1}\rho e^{-\rho^2/\omega_{r}^{2}}\bigg(\int_{0}^{\pi/2} [kW(\rho,\theta)]^{n}d\theta\bigg)d\rho,$$

When calculating the complex amplitudes of the $Si$ area, $\epsilon _{x}$ and $\epsilon _{y}$ are expressed via the polar vector $\epsilon ,\theta _{\epsilon }$ ($\epsilon _{x}=\epsilon *\cos \theta _{\epsilon }$, $\epsilon _{y}=\epsilon *\sin \theta _{\epsilon }$, $\theta _{\epsilon } \in [0,2\pi ]$):

(8c)$$O_{ovi}^{S1+S4+S5}{\approx}s_{0}+is_{1},$$

(8d)$$\begin{aligned}s_{n}&=\int_{0}^{1}\int_{\epsilon_{r} \cos \theta_{\epsilon}}^{0}e^{-(x_{r}^{2}+y_{r}^{2})/\omega_{r}^{2}}[kW(x_{r},y_{r})]^{n}dx_{r}dy_{r} \\ &+\int_{\epsilon_{r} \sin \theta_{\epsilon}}^{0}\int_{0}^{1}e^{-(x_{r}^{2}+y_{r}^{2})/\omega_{r}^{2}}[kW(x_{r},y_{r})]^{n}dx_{r}dy_{r} \\ & +\int_{\epsilon_{r} \sin \theta_{\epsilon}}^{0}\int_{\epsilon_{r} \cos \theta_{\epsilon}}^{0}e^{-(x_{r}^{2}+y_{r}^{2})/\omega_{r}^{2}}[kW(x_{r},y_{r})]^{n}dx_{r}dy_{r}, \end{aligned}$$

where $x_{r}=x/r_{0}$, $y_{r}=y/r_{0}$, $\epsilon _{r}=\epsilon /r_{0}$ are the normalized parameters. Combining (8a) and (8c), we can get the complex amplitudes of the first segment of the QPD:

(8e)$$O_{ovi}^{1}{\approx}d_{0}+s_{0}-d_{2}+i(d_{1}+s_{1}-d_{3})).$$

For the complex amplitudes of the other three segments, we just change the integral regions of (8a$\sim$8d) according to (7b$\sim$7d) to calculate the expressions similar to (8e). Eventually, the $AP$ signal can be expressed as:

(8f)$$\phi_{AP}{\approx}\frac{1}{4} \frac{Im(\prod_{n=1}^4 O_{ovi}^{n})}{Re(\prod_{n=1}^4 O_{ovi}^{n})}.$$

It is easy to notice that the $LPF$ signal is insensitive to the static lateral displacement $\epsilon$. In contrast, the lateral position offset of the interfering beams on the QPD will change the complex amplitude of each segment on the QPD and consequently alter the $AP$ signal. However, it does not mean that the $AP$ signal is a bad phase definition. On the contrary, this character can provide the $AP$ signal a certain degree of freedom to offset the phase noises induced by aberrations, which will be verified theoretically in the next section.

3. Coupling of TTL noise with aberrations

3.1 Simple analysis with the spherical aberration and the coma of Seidel aberrations

First, simple analytical expressions are derived to visually show the coupling relationship between the phase noises and the aberrations for the $LPF$ signal and the $AP$ signal. Here we set $R(z)=\infty$, and choose the radial symmetric spherical aberration and the asymmetric coma in Seidel aberrations to form $\Delta w(r,\theta )$. As a result, $W(r,\theta )$ and $W(x,y)$ become:

(9a)$$W(r,\theta)=r\alpha \cos (\theta-\theta_{\alpha})+A_{c} r^3\cos (\theta-\theta_{c})+A_{s} r^4,$$

(9b)$$W(x,y)=\alpha (x\cos \theta_{\alpha}+ y\sin \theta_{\alpha})+A_{c} (x^2+y^2) (x\cos\theta_{c}+y\sin\theta_{c})+A_{s} (x^2+y^2)^2,$$

where $A_{c}$ and $A_{s}$ represent the coefficients of the coma and the spherical aberration, respectively, and the coma is symmetric about a line making angle of $\theta _{c}$ with the $x$ axis. Next, we carry out the symbolic calculations of the integration (5c) and (5b) by utilizing Mathematica (the Mathematica code is available in the supplementary material as we shown in Code 1 [17]). Because we focus on the coupling between the small rotation angle $\alpha$ and the phase, the terms unrelated and higher than second order of $\alpha$ are omitted during the simplification. Therefore, the longitudinal pathlength signal (LPS) $LPS_{LPF}$ of the $LPF$ signal is ($\phi _{LPF}/k$):

(10a)$$LPS_{LPF}{\approx}b_{1}*A_{c} A_{s}\alpha+b_{2}*A_{s} \alpha^{2},$$

(10b)$$\small b_{1}=-\frac{k^2 \left(20 e^{\frac{2}{\omega _r^2}} \omega _r^8+4 \left(5 \omega _r^8+4 \omega _r^6+\omega _r^4\right)-e^{\frac{1}{\omega _r^2}} \left(40 \omega _r^8+16 \omega _r^6+8 \omega _r^4+4 \omega _r^2+1\right)\right) \cos \left(\theta _c-\theta _{\alpha }\right)}{2 \left(e^{\frac{1}{\omega _r^2}}-1\right){}^2},$$

(10c)$$\small b_{2}=-\frac{k^2 \left(4 e^{\frac{2}{\omega _r^2}} \omega_r^6+2 \left(2 \omega _r^6+\omega _r^4\right)-e^{\frac{1}{\omega _r^2}} \left(8 \omega _r^6+2 \omega _r^4+2 \omega _r^2+1\right)\right)}{4 \left(e^{\frac{1}{\omega _r^2}}-1\right){}^2}.$$

Meanwhile, the $LPS_{AP}$ of the AP signal is:

(11a)$$LPS_{AP}{\approx}\left[p_{1}*A_{s} A_{c} +(p_{1}^{\epsilon 1}+p_{1}^{\epsilon 2}*A_{s}^{2})*\epsilon_{r} \right]\alpha+p_{2}* A_{s}\alpha^{2},$$

where the expressions of $p_{1}$ and $p_{2}$ are too long, thus being denoted by the power series of $\omega _{r}$ with the coefficients $p1_{n}$ and $p2_{n}$, as shown in (11b) and (11c), respectively. (the full expressions can be shown by the supplementary material as we shown in Code 1 (Ref. [17])).

(11b)$$p_{1}=k^{2} \cos(\theta_{\alpha}-\theta_{c})\sum_{n=0}^{8} p1_{n} \omega_{r}^{n},$$

(11c)$$p_{2}=\sum_{n=0}^{6} p2_{n} k^{2} \omega_{r}^{n},$$

The coefficient of $\alpha ^{2}$ contains one term concerning the lateral shift $\epsilon _{r}$, and this term is so negligible that can be omitted when $\epsilon _{r}$ is within $\pm 0.02$. $p_{1}^{\epsilon 1}$ (11d) is independent of aberrations. $p_{1}^{\epsilon 2}$ (11e) is also denoted by the power series of $\omega _{r}$ with the coefficients $p1\epsilon 2_{n}$ (the full expressions are provided in the supplementary material as we shown in Code 1 (Ref. [17])).

(11d)$$p_{1}^{\epsilon 1}=\frac{2 \cos (\theta_{\alpha} -\theta_{\epsilon} ) e^{\frac{1}{\omega_{r}^2}} \textrm{erf}\left(\frac{1}{\omega_{r}}\right) \left(\sqrt{\pi } e^{\frac{1}{\omega_{r}}} \omega_{r} \textrm{erf}\left(\frac{1}{\omega_{r}}\right)-2\right)}{\pi ^{3/2} \left(e^{\frac{1}{\omega_{r}^2}}-1\right){}^2 \omega_{r}},$$

(11e)$$p_{1}^{\epsilon 2}=k^{2}\cos(\theta_{\alpha}-\theta_{\epsilon})\sum_{n=-1}^{8}p1\epsilon 2_{n} \omega_{r}^{n}.$$

The Eqs. (10b) and (11b) indicate that the coupling degree of the coma $A_{c}$ with TTL noise depends on the cosine of the included angle between $\theta _{\alpha }$ and $\theta _{c}$ whatever for $LPS_{LPF}$ or $LPS_{AP}$. When we set $A_{s}=A_{c}=0$ to eliminate the wavefront mismatch of the interfering beams, $LPS_{LPF}$ is free for tilting, but $LPS_{AP}$ has a linear coupling relation with the angle $\alpha$ because of the lateral shift $\epsilon _{r}$. Similarly, the coupling degree of the lateral shift $\epsilon _{r}$ with TTL noise for $LPS_{AP}$ depends on the cosine of the included angle between $\theta _{\alpha }$ and $\theta _{\epsilon }$.

The coupling coefficients $\delta _{LPS}^{LPF}$ and $\delta _{LPS}^{AP}$, calculated by taking the derivatives of (10a) and (11a) with respect to $\alpha$, characterize the sensitivity of $LPS$ to the tilting angle.

(12a)$$\delta_{LPS}^{LPF} =\frac{d(LPS_{LPF})}{d(\alpha)}=b_{1}*A_{c} A_{s}+2 b_{2}*A_{s}\alpha,$$

(12b)$$\delta_{LPS}^{AP} =\frac{d(LPS_{AP})}{d(\alpha)}=p_{1}*A_{s} A_{c}+(p_{1}^{\epsilon 1}+p_{1}^{\epsilon 2}*A_{s}^{2}) *\epsilon_{r}+2 p_{2}*A_{s} \alpha.$$

It is worth noting that the constant term of $\delta _{LPS}^{AP}$ can be eliminated if $\epsilon _{r}$ satisfies:

(13)$$\epsilon_{r}^{opt}=-\frac{p_{1}*A_{s}A_{c}}{(p_{1}^{\epsilon 1}+p_{1}^{\epsilon 2} A_{s}^{2})},$$

where $\epsilon _{r}^{opt}=\epsilon _{opt}/r_{0}$ is the normalized optimal lateral shift. Equation (13) indicates that the component of the lateral shift in the direction which is orthogonal to the jittering axis of the beam, works for reducing the TTL noise coupled with the aberrations. Meanwhile, we can obtain the two optimal angles $\alpha _{LPF}^{0}$ and $\alpha _{AP}^{0}$, which can minimize $\delta _{LPS}^{LPF}$ and $\delta _{LPS}^{AP}$ to 0, respectively:

(14a)$$\alpha_{LPF}^{0}=-\frac{b_{1}*A_{c}A_{s}}{2b_{2}*A_{s}},$$

(14b)$$\alpha_{AP}^{0}=-\frac{p_{1}*A_{s} A_{c}+(p_{1}^{\epsilon 1}+p_{1}^{\epsilon 2} A_{s}^{2})*\epsilon}{2 p_{2}*A_{s}}.$$

Assuming that $\theta _{\alpha }=\theta _{\epsilon }=\theta _{c}=0$ and $\omega (z)=r_{0}=1mm$ to consider one-dimensional situation, the above results are intuitively shown on Fig. 3. By adjusting $\epsilon$ to $\epsilon _{opt}$, the TTL noise coupled with the coma can be surely eliminated for the AP signal, and $\alpha _{AP}^{0}$ simultaneously becomes 0. And if $\epsilon$ is not too large, it is obvious that the $AP$ signal is less sensitive than the $LPF$ signal for the TTL noise coupled with aberrations.

Fig. 3. Left: $\epsilon _{opt}$(13) changing with $A_{s}$ and $A_{c}$. Right: $LPS_{LPF}$, $LPS_{AP}$ and their coupling coefficients $\delta _{LPS}^{LPF}$, $\delta _{LPS}^{AP}$ with different values of $A_{s}$ and $A_{c}$, versus the wavefront tilt $\alpha$.

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3.2 Comprehensive theoretical analysis with Zernike polynomials

An excellent choice as basis functions for representing functions such as wavefronts is Zernike circle polynomials, which are orthogonal over a unit circle in the form [18]:

(15)$$Z_{i}(\rho,\theta)= \left\{ \begin{array}{ll} \sqrt{2(n+1)} R_{n}^{m}(\rho)\cos(m\theta) & \textrm{i is even and} \ m \not = 0\\ \sqrt{2(n+1)} R_{n}^{m}(\rho)\sin(m\theta) & \textrm{i is odd and} \ m \not = 0\\ \sqrt{n+1} R_{n}^{0}(\rho) & m = 0 \end{array}, \right.$$

where $R_{n}^{m}(\rho )$ (16) is the radial polynomial, $n$ and $m$ are positive integers, and $n-m \geq 0$. $\rho =r/r_{0}$ is the normalized radial coordinate. The index $i$ is a polynomial-ordering number and a function of $n$ and $m$.

(16)$$R_{n}^{m}(\rho)=\sum_{s=0}^{(n-m)/2} \frac{(-1)^{s}(n-s)!}{s!(\frac{n+m}{2}-s)! (\frac{n+m}{2}-s)!} \rho^{n-2s}.$$

There are several ways of representing the Zernike polynomials, with one difference being the ordering of terms. Here, the Fringe Zernike polynomials, currently used by several vendors of interferometers and interferometric data analysis software, are chosen in the analysis. In order to fully characterise the manufacturing and adjustment errors of the optical systems, the first 25 terms of the Fringe Zernike polynomials are used to fit the wavefront error of the interfering beams (the piston term being omitted). Furthermore, except the defocus term and three spherical aberration terms, the cosine and sine terms of other Zernike aberrations are combined to represent the magnitudes and orientations of these aberrations. For example, when both x and y Zernike tilts are used to present the wavefront tilt, the aberration may be written as the form:

(17)$$a_{2}Z_{2}(\rho,\theta)+a_{3}Z_{3}(\rho,\theta)=A_{1}^{Ti}\rho\cos(\theta-\theta_{Ti}),$$

showing a wavefront tilt of magnitude $A_{1}^{Ti}=r_{0}\alpha =2\sqrt {a_{2}^{2}+a_{3}^{2}}$ about an axis that is orthogonal to a line making an angle of $\theta _{Ti}=\theta _{\alpha }=\tan ^{-1}(a_{3}/a_{2})$ with the $x$ axis. Where $a_{i}$ is the expansion coefficient before combination and $A_{j}^{aber}$ is the magnitude of the aberration after combination (its subscript is the serial number $j$ and superscript is the abbreviation of the corresponding aberration). The complete orthonormal Zernike polynomials $Z_{j}(\rho ,\theta )$ in polar coordinates and $Z_{j}(x,y)$ in Cartesian coordinates are given in Table 1, where $\theta _{aber}$ presents an orientation angle of the corresponding aberration inclined with the $x$ axis and its subscript is the abbreviation of the corresponding aberration. The effect of the phase term $\frac {-r^2} {2R(z)}$ can be represented by the defocus Zernike term. As a result, the wavefront error can be presented as:

(18a)$$W(\rho,\theta)=\sum_{j}^{14} A_{j}^{aber} Z_{j}(\rho,\theta),$$

(18b)$$W(x,y)=\sum_{j}^{14} A_{j}^{aber} Z_{j}(x,y).$$

The expansion coefficients $A_{j}^{aber}$ are given by:

(19)$$A_{j}^{aber}=\frac{1}{\pi} \int_{0}^{1} \int_{0}^{2\pi} W(\rho,\theta) Z_{j}(\rho,\theta) \rho d\rho d\theta$$

The aberration variance is given by:

(20)$$\delta_{W}^{2}=\sum_{j}(A_{j}^{aber})^{2},$$

where $\delta _{W}$ can be regarded as the root-mean-square (RMS) value of the wavefront error because the piston term $a_{0}$ is not included here.

Table 1. The fourteen Zernike aberrations consist of The first 25 terms of Fringe Zernike polynomials.

View Table | View all tables in this article

After (9a) and (9b) are replaced by (18a) and (18b), $\delta _{LPS}^{LPF}$ and the corresponding optimal tilt angle $\alpha _{LPF}^{0}$ can be expressed as:

(21a)$$\delta_{LPS}^{LPF}=B_{1} +2 B_{2} \alpha,$$

(21b)$$\alpha_{LPF}^{0}=-\frac{B_{1}}{2 B_{2}},$$

(21c)$$B_{1}=\sum_{n=1}^{29} B1_{n}*A1_{n}^{LPF},$$

(21d)$$B_{2}=\sum_{n=1}^{7} B2_{n}*A2_{n}^{LPF},$$

where $B_{1}$ (21c) is the sum of 29 factors $A1_{n}^{LPF}$ (21g) with the coefficients $B1_{n}$, the 21 factors of which are the pairwise combinations between the three coma terms and the group $AA_{1}^{LPF}$ (21e) including the four spherical terms (including defocus) and the three astigmatism terms, and the other 8 factors are the pairwise combinations between the two trefoil terms and the group $AA_{2}^{LPF}$ (21f) including the three astigmatism terms and the primary tetrafoil term. $B_{2}$ (21d) is the sum of 7 factors $A2_{n}^{LPF}$ (21h) including the 7 terms in the group $AA_{1}^{LPF}$ with the coefficients $B2_{n}$ (21h).

(21e)$$AA_{1}^{LPF} \Longrightarrow \left\{ A_{2}^{DE},A_{5}^{PS},A_{9}^{SS},A_{14}^{TS},A_{3}^{PA},A_{7}^{SA},A_{12}^{TA}\right\},$$

(21f)$$AA_{2}^{LPF} \Longrightarrow \left\{ A_{3}^{PA},A_{7}^{SA},A_{12}^{T^A},A_{10}^{PTE}\right\},$$

(21g)$$A1_{n}^{LPF} \Longrightarrow \left\{\{A_{4}^{PC},A_{8}^{SC},A_{13}^{TC}\}*AA_{1}^{LPF}, \{A_{6}^{PTR},A_{11}^{STR}\}*AA_{2}^{LPF}\right\},$$

(21h)$$A2_{n}^{LPF}\Longrightarrow AA_{1}^{LPF}.$$

Because of too many factors in (18a) and (18b), it is very hard to directly calculate $\delta _{LPS}^{AP}$ by using the Eq. (8a). Therefore, we expand $\prod _{n=1}^4 O_{ovi}^{n}$ and neglect some factors to reduce the amount of calculation through only considering the terms that are not higher than the second order of $\alpha$ and the first order of $\epsilon _{r}$ (the Mathematica codes of the specific simplification process are shown in the supplementary material as we shown in Code 1 (Ref. [17])). As a result, $\delta _{LPS}^{AP}$ and the corresponding optimal tilt angle $\alpha _{AP}^{0}$ can be expressed as:

(22a)$$\delta_{LPS}^{AP}=\left[P_{1}+ (P_{1}^{\epsilon 1}+P_{1}^{\epsilon 2})*\epsilon_{r} \right] +2 \left(P_{2}+ P_{2}^{\epsilon}*\epsilon_{r} \right) \alpha,$$

(22b)$$\alpha_{AP}^{0}=-\frac{\left[P_{1}+ (P_{1}^{\epsilon 1}+P_{1}^{\epsilon 2})*\epsilon_{r} \right]}{\left(P_{2}+ P_{2}^{\epsilon}*\epsilon_{r} \right)},$$

(22c)$$P_{1}=\sum_{n=1}^{40} P1_{n}*A1_{n}^{AP},$$

(22d)$$P_{2}=\sum_{n=1}^{8} P2_{n}*A2_{n}^{AP},$$

(22e)$$P_{1}^{\epsilon 1}=p1\epsilon 1,$$

(22f)$$P_{1}^{\epsilon 2}=\sum_{n=1}^{51} P1\epsilon 2_{n}*A1\epsilon_{n},$$

(22g)$$P_{2}^{\epsilon}=\sum_{n=1}^{5} P2\epsilon_{n}*A2\epsilon_{n},$$

where $P_{1}$ (22c) is the sum of 40 factors $A1_{n}^{AP}$ (22j) with the coefficients $P1_{n}$, which are the pairwise combinations between the group $AA_{1}^{AP}$ (22h) including the four spherical terms, the three astigmatism terms and the primary tetrafoil term and the group $AA_{2}^{AP}$ (22i) including the three coma terms and the two trefoil terms. $P_{2}$ (22d) is the sum of 8 factors $A2_{n}^{AP}$ (22k) including the 8 terms in the group $AA_{1}^{AP}$ with the coefficients $P2_{n}$. $P_{1}^{\epsilon 1}$ (22e) is equal to $p_{1}^{\epsilon }1$ (11d), which is independent of aberrations. $P_{1}^{\epsilon 2}$ (22f) is the sum of 51 factors $A1\epsilon _{n}$ (22l) with the coefficients $P1\epsilon 2_{n}$, the 36 factors of which are the pairwise combinations (where any two different terms are combined in pairs and every term can also be combined with itself in pairs to form the pairwise combinations) of the group $AA_{1}^{AP}$, and the other 15 factors are formed in the same way of pairwise combinations of the group $AA_{2}^{AP}$. $P_{2}^{\epsilon }$ (22g) is the sum of 5 factors $A2\epsilon _{n}$ (22m) including the 5 terms in the group $AA_{2}^{AP}$ with the coefficients $P2\epsilon _{n}$.

(22h)$$AA_{1}^{AP}\Longrightarrow \left\{A_{2}^{DE},A_{5}^{PS},A_{9}^{SS},A_{14}^{TS},A_{3}^{PA},A_{7}^{SA},A_{12}^{TA}, A_{10}^{PTE}\right\},$$

(22i)$$AA_{2}^{AP}\Longrightarrow \left\{A_{4}^{PC},A_{8}^{SC},A_{13}^{TC}, A_{6}^{PTR}, A_{11}^{STR}\right\},$$

(22j)$$A1_{n}^{AP} \Longrightarrow \left\{AA_{1}^{AP}*AA_{2}^{AP}\right\},$$

(22k)$$A2_{n}^{AP}\Longrightarrow AA_{1}^{AP},$$

(22l)$$A1\epsilon_{n}\Longrightarrow \left\{AA_{1}^{AP}*AA_{1}^{AP}, AA_{2}^{AP}*AA_{2}^{AP}\right\},$$

(22m)$$A2\epsilon_{n}\Longrightarrow AA_{2}^{AP}.$$

Like the expression (10b) or (11b), the coefficients $B1_{n}$, $B2_{n}$, $P1_{n}$ and $P2_{n}$ are the functions of the normalized radius $\omega _{r}$ and the cosine of the add and subtract combination between $\theta _{\alpha }$ and $\theta _{aber}$ (corresponding to the aberration terms of $A1_{n}$ and $A2_{n}$). Like the expression (11d) or (11e), $P1\epsilon 2_{n}$, $P2\epsilon _{n}$ are the functions of the normalized radius $\omega _{r}$, the cosine of the add and subtract combination among $\theta _{\alpha }$, $\theta _{\epsilon }$ and $\theta _{aber}$ (corresponding to the aberration terms of $A1\epsilon _{n}$ and $A2\epsilon _{n}$). The texts of these coefficients are not shown here due to limited space. But providing that the cosine terms are equal to 1 by setting $\theta _{\alpha }$, $\theta _{\epsilon }$, and other $\theta _{aber}$ to be 0, and all the magnitudes $A_{j}^{aber}$ are equal to $\lambda /10$, Fig. 4 shows every sub-term of $B_{1}$, $P_{1}$ (left) and $2*B_{2}*\alpha _{max}$, $2*P_{2}*\alpha _{max}$ (middle) versus the normalized radius $\omega _{r}$. And the right of Fig. 4 shows $P_{1}^{\epsilon 1}$, $P_{1}^{\epsilon 2}$, and $2*P_{2}^{\epsilon }* \alpha _{max}$ (right) versus the lateral shift $\epsilon _{r}$ ($\omega _{r}=1$) (for consistency, $B_{2}$, $P_{2}$ and $P_{2}^{\epsilon }$ multiplied by $2\alpha _{max}$, where $\alpha _{max}=300\mu rad$).

Fig. 4. Left: every sub-term of $B_{1}$, $P_{1}$ versus $\omega _{r}$. Middle: every sub-term of $B_{2}*2\alpha _{max}$, $P_{2}*2\alpha _{max}$ versus $\omega _{r}$. Right: every sub-term of $P_{1}^{\epsilon 1}$, $P_{1}^{\epsilon 2}$ and $P_{2}^{\epsilon }$ versus $\epsilon _{r}$.

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In general, we can notice that the sub-term values of $B_{1}$, $2*B_{2}*\alpha _{max}$ are larger than $P_{1}$, $2*P_{2}*\alpha _{max}$, respectively. And it is also worth noting that in the coefficients of $\epsilon _{r}$, the three largest sub-terms are $P_{1}^{\epsilon 1}$ and the two terms about $(A_{2}^{DE})^{2}$, $A_{2}^{DE}*A_{5}^{PS}$ of $P_{1}^{\epsilon 2}$, whose cosine terms are all $\cos (\theta _{\alpha }-\theta _{\epsilon })$. Consequently, $P_{2}^{\epsilon }$ can be ignored for its smaller values and we set $\theta _{\epsilon }=\theta _{\alpha }$ when calculating the normalized optimal lateral shift $\epsilon _{r}^{opt}$ shown as:

(23)$$\epsilon_{r}^{opt}=-\frac{P_{1}}{P_{1}^{\epsilon 1}+P_{1}^{\epsilon 2}}.$$

4. Results

Because the actual $W(\rho ,\theta )$ spectrum (18a) is not available, it is necessary to predict the TTL noise coupled with aberrations under the different constraints of $\delta _{W}$(20) by adopting the Monte Carlo simulation for the analytical results of the previous section.

To give an example, we set a situation that the beam wavelength $\lambda =1064nm$, the radius of the flat-top beam and the spot size of the Gaussian beam on the detector $\omega (z)=r_{0}=1mm$. Meanwhile, the coefficients $A_{j}^{aber}$ (except $A_{1}^{Ti}$) are randomly generated according to (20) in the constraint of $\delta _{W}=\lambda /20=53.2nm$ and the orientation angles $\theta _{aber}$ (except $\theta _{Ti}$) are randomly generated in the interval of $[0,2\pi ]$. Figure 5 (left) shows a randomly generated wavefront, and the right of Fig. 5 indicates that the value of $\epsilon _{opt}$ depends on the orientation of the wavefront tilt due to the random orientation angles $\theta _{aber}$ of some aberrations in this wavefront sample. Figure 6 and 7 show the absolute values of two kinds of $\delta _{LPS}^{AP}$ and $\delta _{LPS}^{LPF}$ versus both the horizontal tilt angle $\alpha _{x}=\alpha \cos (\theta _{\alpha })$ and the vertical tilt angle $\alpha _{y}=\alpha \sin (\theta _{\alpha })$, respectively. As expected, the optimal lateral shift can slightly reduce $\delta _{LPS}^{AP}$. Meanwhile, the red dotted lines in the three pictures indicate that the distributions of the optimal wavefront tilts are one or two continuous curves and are different for three kinds of situations.

Fig. 5. Left: a randomly generated wavefront. Right: the optimal lateral shift $\epsilon _{opt}$ versus the orientation angle of the wavefront tilt $\theta _{\alpha }$.

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Fig. 6. The coupling coefficient $|\delta _{LPS}^{AP}|$ of the $AP$ signal. The red dotted lines indicate the positions with the zero value according to the Eq. (22b). Left: using the Eq. (22a) only. Right: introducing $\epsilon _{opt}$ calculated by (23) to (22a).

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Fig. 7. The coupling coefficient $|\delta _{LPS}^{LPF}|$ of the $LPF$ signal according to (21a). The red dotted line indicates the position with the zero value according to (21b).

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It is noticed that the premises of the parabolic approximation used in the previous theoretical analysis are that $\left | kW(\rho , \theta ) \right | \ll 1$ and $\epsilon _{r} \ll 1$. In order to verify the validity of the approximations for the large misaligment, the wavefront error and the lateral shift, we calculate the residual error $\delta _{approx}$ (24) between the approximate analytical solutions $LPS_{approx}$ and the corresponding numerical simulations $LPS_{num}$ of the $LPF$ signal and the $AP$ signal with $10^{4}$ arbitrary wavefront samples (the analytical solutions of the LPS signals are the primitive functions of (21a) and (22a) without the constant term). Figure 8 shows that, the mean values of $\delta _{approx}^{LPF}$ and $\delta _{approx}^{AP}$ are $-5.7\%$ and $2.4\%$, with the standard deviations of $13\%$ and $10\%$, respectively.

(24)$$\delta_{approx}= (LPS_{approx}-LPS_{num})/LPS_{num}.$$

Fig. 8. Histogram of $10^4$ Monte Carlo calculation of the residual errors $\delta _{approx}^{LPF}$ and $\delta _{approx}^{AP}$ when the radius of the flat-top beam and the spot size of the Gaussian beam on the detector $\omega (z)=r_{0}=1mm$. And we set the $RMS$ wavefront error $\delta _{W}=\lambda /10$, the wavefront misalignment $\alpha =300 \mu rad$ and the lateral shift $\epsilon =60\mu m$ to consider the maximum error.

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In the following analysis, we first carry out Monte Carlo algorithm to generate $10^{4}$ groups of $A_{j}^{aber}$ (expect $A_{1}^{Ti}$) and $\theta _{aber}$ (including $\theta _{\alpha }=\theta _{Ti}$), respectively, in the cases of $\lambda /40$, $\lambda /20$ and $\lambda /10$ of $\delta _{W}$. Next, we use the wavefront tilt $\alpha =A_{1}^{Ti}$ to maximize $|\delta _{LPS}^{AP}|$ and $|\delta _{LPS}^{LPF}|$ in the range of $\pm 300\mu rad$ for considering the worst situation. Meanwhile, the corresponding optimal lateral shifts $\epsilon _{opt}$ are calculated according to (23). Specifically, as for the $AP$ signal, we carry out three cases for different lateral shift vector $\vec {\epsilon }$. Among them, case 1 is that $\epsilon =\epsilon _{opt}$ and $\theta _{\epsilon }=\theta _{\alpha }$ to consider the compensation mechanism of the lateral shift, case 2 is that $\epsilon =0$ to assume the perfect centering, and case 3 is that $\epsilon$ and $\theta _{\epsilon }$ are randomly generated in the range of $\pm 20\mu m$ and $0 \sim 2\pi$, respectively, to consider the position error. Meanwhile, we use case 4 to represent the $LPF$ signal.

Figure 9 illustrates the simulation results and Table 2 shows the mean values and the proportions of $|\delta _{LPS}|$ less than the required $25pm/ \mu rad$ in the samples of the four cases. Providing that $\delta _{W}=\lambda /40$, we can notice that most of all samples of all the cases meet the required $25pm/ \mu rad$. When $\delta _{W}=\lambda /20$, although the mean values of all the cases are less than the required $25pm/ \mu rad$ and the samples of the three cases of $AP$ signal nearly fulfill the requirement, the proportion of samples satisfying the requirement for the $LPF$ signal is reduced to $55\%$. If $\delta _{W}$ rises to $\lambda /10$, only the samples of the case 1 can nearly meet the requirement. And in general, we can also notice that $\epsilon _{opt}$ increases with the deterioration of the wavefront quality.

Fig. 9. In the wavefront error $\delta _{W}$ of $\lambda /40$, $\lambda /20$ and $\lambda /10$. Left: the sample distributions of the coupling coefficient $|\delta _{LPS}|$ of the four cases. Right: the sample distributions of the optimal lateral shift $\epsilon _{opt}$ in the case 1.

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Table 2. The specific mean values and the proportions of $|\delta _{LPS}|$ less than the required $25pm/ \mu rad$ in the samples of the four cases when $\delta _{W}$ is $\lambda /40$, $\lambda /20$ and $\lambda /10$, respectively.

View Table | View all tables in this article

Finally, in the range of $0 \sim \lambda /10$ and the step of $\lambda /100$ for $\delta _{W}$, we repeat the above calculation process. As shown in Fig. 10, the data points are linked into lines, respectively. The results indicate that provided that $\delta _{W}$ is less than $\lambda /50$, the mean value of $\delta _{LPS}$ for the case 3 is slightly larger than other cases because of the term $P_{1}^{\epsilon 1}$ concerning the lateral shift $\epsilon _{r}$. But with the increase of $\delta _{W}$, the difference between the case 2 and the case 3 is getting smaller, which indicates that the random deviation within $\pm 20$ microns for the lateral eccentricity of the interfering beams at the detector has a limited effect on the $AP$ signal. Meanwhile, the mean value of $|\delta _{LPS}|$ and the number of samples that do not meet the requirement for the $LPF$ signal, increase faster than the other $AP$ signals. Among them, case 1 performs best for reducing the TTL noise coupled with aberrations. Correspondingly, the mean value and the maximum value of the optimal lateral shift, increase from 0 to $12.8\mu m$ and from 0 to $59.5\mu m$, respectively, while $\delta _{w}$ increases from 0 to 0.1$\lambda$.

Fig. 10. In the range of $0 \sim \lambda /10$ and the step of $\lambda /100$ for $\delta _{W}$. Left: the specific mean values and the proportions of $|\delta _{LPS}|$ less than the required $25pm/ \mu rad$ in the samples of the four cases versus $\delta _{W}$. Right: the specific mean values and maximum values of the optimal lateral shift $\epsilon _{opt}$ versus $\delta _{W}$ in the case 1.

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5. Conclusion

Owing to the tilts and jitters of the spacecraft and angular amplification of the telescope, the received beam is misaligned and jittered to interference with the local beam. As a result, the wavefront aberrations of the interfering beams can couple with the jitter and generate the TTL noise.

In this paper, we first construct the optical model of the phase noise under the two phase definitions (the $AP$ signal and the $LPF$ signal), when the flat-top beam and the Gaussian beam interfere in the case of the wavefront tilt, aberrations and eccentricity on the QPD. Then taking into account the spherical aberration and the coma of Seidel aberrations, we carry out the relevant analytical computations to obtain the slightly simpler expressions of the phase noises for the $AP$ signal and the $LPF$ signal, respectively. Go further, we take advantage of the first 25 terms of the Fringe Zernike polynomials for constituting the wavefront aberrations to obtain the expressions (21a) and (22a) of the coupling coefficients between the TTL noise and aberrations. Meanwhile, we analytically derive the expression (23) of the optimal lateral shift which can reduce the coupling degree between the TTL noise and aberrations for the $AP$ signal.

On the above bases, we implement Monte Carlo algorithm to generate arbitrary $10^4$ samples for different $RMS$ wavefront aberrations in the range of $0 \sim \lambda /10$. Finally, we set $\omega (z)=r_{0}=1mm$ to estimate the coupling coefficients for the $LPF$ signal, and the $AP$ signals with three different lateral shifts (zero, random in the range of $\pm 20\mu m$, and the optimal) of the interfering beams on the detector. And considering Murphy’s law, we optimize the wavefront tilt in the range of $\pm 300\mu rad$ to take the maximum values of the coupling coefficients.

The results show that only if the $RMS$ wavefront error is less than $\lambda /20$, the mean coupling coefficients of the $LPF$ signal can be less than the required $25 pm/ \mu rad$. However, even if the $RMS$ wavefront error equals $\lambda /10$, the mean coupling coefficients of the three kinds of $AP$ signals are all less than the required $25 pm/ \mu rad$, and further for the $AP$ signal introducing the optimal lateral shift, the proportion of the samples fulfilling the requirement can reach $95.4\%$. Therefore, we can conclude that the $AP$ signal is a better method of the phase definition than the $LPF$ signal to suppress the influence of aberration on the TTL noise.

Correspondingly, the Eqs. (22a),(22c) and (22h $\sim$ 22j) suggest that the TTL noise coupled with the coma and the trefoil aberrations can be mostly compensated through introducing the optimal lateral shift. Normally, these aberrations are caused by the lateral alignment errors of the lens or mirrors in the practical optical systems [19,20]. Definitely, it may be difficult to directly adopt the above method for the practical space environment because of the random orientation of pointing jitter, but easy for the ground experiments due to the available orientation of rotating angle, e.g. the phenomena in the experiment [14].

If the fitted Fringe Zernike coefficients of the wavefront error between the interfering beams are available, the analytical results in this paper can be used to predict the TTL coupling noise and calculate the optimal lateral shift between the interfering beams and the QPD to reduce the TTL noise coupled by wavefront aberrations.

Funding

Chinese Academy of Sciences (XDB23030000); the Projects of Science Technology Development Plan of Jilin Province under Grant (20190302102GX).

Disclosures

The author declare no conflicts of interest.

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$i$	$j$	$Z_{j} (ρ, θ)$	$Z_{j} (x, y)$	Aberration name
2,3	$1$	$ρ \cos (θ - θ_{α})$	$x \cos (θ_{α}) + y \sin (θ_{α})$	Tilt
4	$2$	$\sqrt{3} (2 ρ^{2} - 1)$	$\sqrt{3} [2 (x^{2} + y^{2}) - 1]$	Defocus
5,6	$3$	$\sqrt{6} ρ^{2} \cos (2 θ - (θ_{P A}))$	$\sqrt{6} [2 x y \sin (θ_{P A}) + (x^{2} - y^{2}) \cos (θ_{P A})]$	Primary astigmatism
7,8	$4$	$\sqrt{8} (3 ρ^{3} - 2 ρ) \cos (θ - (θ_{P C}))$	$\sqrt{8} (x \cos (θ_{P C}) + y \sin (θ_{P C})) (3 ρ^{2} - 2)$	Primary coma
9	$5$	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	Primary spherical
10,11	$6$	$\sqrt{8} ρ^{3} \cos (3 θ - θ_{P T R})$	$\sqrt{8} [x \cos (θ_{P T R}) (x^{2} - 3 y^{2}) + y \sin (θ_{P T R}) (3 x^{2} - y^{2})]$	Primary trefoil
12,13	$7$	$\sqrt{10} (4 ρ^{4} - 3 ρ^{2}) \cos (θ - θ_{S A})$	$\sqrt{10} (2 x y \sin (θ_{P A}) + (x^{2} - y^{2}) \cos (θ_{S A})) (4 ρ^{2} - 3)$	Secondary astigmatism
14,15	$8$	$\sqrt{5} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos (2 θ - θ_{S C})$	$\sqrt{8} (x \cos (θ_{S C}) + y \sin (θ_{S C})) (10 ρ^{4} - 12 ρ^{2} + 3)$	Secondary coma
16	$9$	$\sqrt{5} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	$\sqrt{5} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	Secondary spherical
17,18	$10$	$\sqrt{5} ρ^{4} \cos (4 θ - θ_{P T E})$	$(ρ^{4} - 8 x^{2} y^{2}) \cos (θ_{P T E}) + 4 x y (x^{2} - y^{2}) \sin (θ_{P T E})$	Primary tetrafoil
19,20	$11$	$\sqrt{8} (5 ρ^{5} - 4 ρ^{3}) \cos (3 θ - θ_{S T R})$	$\sqrt{8} (x \cos (θ_{P T R}) (x^{2} - 3 y^{2}) + y \sin (θ_{P T R}) (3 x^{2} - y^{2})) (5 ρ^{2} - 4)$	Secondary trefoil
21,22	$12$	$\sqrt{5} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) \cos (2 θ - θ_{T A})$	$\sqrt{10} (2 x y \sin (θ_{P A}) + (x^{2} - y^{2}) \cos (θ_{S A})) (15 ρ^{4} - 20 ρ^{2} + 6 ρ)$	Tertiary astigmatism
23,24	$13$	$\sqrt{5} (35 ρ^{7} - 60 ρ^{5} + 30 ρ^{3} - 4 ρ) \cos (2 θ - θ_{T C})$	$\sqrt{8} (x \cos (θ_{S C}) + y \sin (θ_{S C})) (35 ρ^{7} - 60 ρ^{4} + 30 ρ^{2} - 4)$	Tertiary coma
25	$14$	$\sqrt{5} (70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1)$	$\sqrt{5} (70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1)$	Tertiary spherical

Case $δ_{L P S} [p m / μ r a d]$ $δ_{W}$ [ $λ$ ]	1/40		1/20		1/10
Case $δ_{L P S} [p m / μ r a d]$ $δ_{W}$ [ $λ$ ]	Mean	Proportion ( $< 25$ )	Mean	Proportion ( $< 25$ )	Mean	Proportion ( $< 25$ )
Case 1	2.61	100 $%$	5.26	100 $%$	10.61	95.4 $%$
Case 2	3.11	100 $%$	7.42	100 $%$	18.71	72.3 $%$
Case 3	8.61	100 $%$	11.65	98.7 $%$	20.53	65.4 $%$
Case 4	9.84	96.7 $%$	22.64	55.4 $%$	53.85	15.7 $%$

$i$	$j$	$Z_{j} (ρ, θ)$	$Z_{j} (x, y)$	Aberration name
2,3	$1$	$ρ \cos (θ - θ_{α})$	$x \cos (θ_{α}) + y \sin (θ_{α})$	Tilt
4	$2$	$\sqrt{3} (2 ρ^{2} - 1)$	$\sqrt{3} [2 (x^{2} + y^{2}) - 1]$	Defocus
5,6	$3$	$\sqrt{6} ρ^{2} \cos (2 θ - (θ_{P A}))$	$\sqrt{6} [2 x y \sin (θ_{P A}) + (x^{2} - y^{2}) \cos (θ_{P A})]$	Primary astigmatism
7,8	$4$	$\sqrt{8} (3 ρ^{3} - 2 ρ) \cos (θ - (θ_{P C}))$	$\sqrt{8} (x \cos (θ_{P C}) + y \sin (θ_{P C})) (3 ρ^{2} - 2)$	Primary coma
9	$5$	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	Primary spherical
10,11	$6$	$\sqrt{8} ρ^{3} \cos (3 θ - θ_{P T R})$	$\sqrt{8} [x \cos (θ_{P T R}) (x^{2} - 3 y^{2}) + y \sin (θ_{P T R}) (3 x^{2} - y^{2})]$	Primary trefoil
12,13	$7$	$\sqrt{10} (4 ρ^{4} - 3 ρ^{2}) \cos (θ - θ_{S A})$	$\sqrt{10} (2 x y \sin (θ_{P A}) + (x^{2} - y^{2}) \cos (θ_{S A})) (4 ρ^{2} - 3)$	Secondary astigmatism
14,15	$8$	$\sqrt{5} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos (2 θ - θ_{S C})$	$\sqrt{8} (x \cos (θ_{S C}) + y \sin (θ_{S C})) (10 ρ^{4} - 12 ρ^{2} + 3)$	Secondary coma
16	$9$	$\sqrt{5} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	$\sqrt{5} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	Secondary spherical
17,18	$10$	$\sqrt{5} ρ^{4} \cos (4 θ - θ_{P T E})$	$(ρ^{4} - 8 x^{2} y^{2}) \cos (θ_{P T E}) + 4 x y (x^{2} - y^{2}) \sin (θ_{P T E})$	Primary tetrafoil
19,20	$11$	$\sqrt{8} (5 ρ^{5} - 4 ρ^{3}) \cos (3 θ - θ_{S T R})$	$\sqrt{8} (x \cos (θ_{P T R}) (x^{2} - 3 y^{2}) + y \sin (θ_{P T R}) (3 x^{2} - y^{2})) (5 ρ^{2} - 4)$	Secondary trefoil
21,22	$12$	$\sqrt{5} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) \cos (2 θ - θ_{T A})$	$\sqrt{10} (2 x y \sin (θ_{P A}) + (x^{2} - y^{2}) \cos (θ_{S A})) (15 ρ^{4} - 20 ρ^{2} + 6 ρ)$	Tertiary astigmatism
23,24	$13$	$\sqrt{5} (35 ρ^{7} - 60 ρ^{5} + 30 ρ^{3} - 4 ρ) \cos (2 θ - θ_{T C})$	$\sqrt{8} (x \cos (θ_{S C}) + y \sin (θ_{S C})) (35 ρ^{7} - 60 ρ^{4} + 30 ρ^{2} - 4)$	Tertiary coma
25	$14$	$\sqrt{5} (70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1)$	$\sqrt{5} (70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1)$	Tertiary spherical

Case $δ_{L P S} [p m / μ r a d]$ $δ_{W}$ [ $λ$ ]	1/40		1/20		1/10
Case $δ_{L P S} [p m / μ r a d]$ $δ_{W}$ [ $λ$ ]	Mean	Proportion ( $< 25$ )	Mean	Proportion ( $< 25$ )	Mean	Proportion ( $< 25$ )
Case 1	2.61	100 $%$	5.26	100 $%$	10.61	95.4 $%$
Case 2	3.11	100 $%$	7.42	100 $%$	18.71	72.3 $%$
Case 3	8.61	100 $%$	11.65	98.7 $%$	20.53	65.4 $%$
Case 4	9.84	96.7 $%$	22.64	55.4 $%$	53.85	15.7 $%$

Tilt-to-length noise coupled by wavefront errors in the interfering beams for the space measurement of gravitational waves

Abstract

1. Introduction

2. Optical model

3. Coupling of TTL noise with aberrations

3.1 Simple analysis with the spherical aberration and the coma of Seidel aberrations

3.2 Comprehensive theoretical analysis with Zernike polynomials

4. Results

5. Conclusion

Funding

Disclosures

References

Supplementary Material (1)

Cited By

Figures (10)

Tables (2)

Equations (63)

Optics Express