1. Introduction

The Earth gravity field measurements, especially the time-variable gravity field measurements, are able to give a wealth of knowledge about the hydrologic cycle, sea-level changes, glacier mass changes, as well as mass movements inside Earth from seasonal to decadal and provide important help for solving the increasingly serious problems of resources, environment and disasters facing mankind. The global and homogeneous gravity field can be obtained by means of the inter-satellite ranging. In 2002, the Gravity Recovery and Climate Experiment (GRACE) mission [1], which tracked the distance between two satellites and its rate of change by using inter-satellite microwave system in K/Ka band to derive the model of the Earth gravity field, was launched and worked until 2017. Its successor, GRACE Follow-On mission [2], launched successfully in 2018, was loaded with an inter-satellite laser ranging interferometer(LRI) as a technology demonstrator for the first time and achieved the ranging precision of $10 \ nm/\sqrt {Hz}$ at a frequency of $40\ mHz$.

With the success of the GRACE and GRACE Follow-On missions, the researches on future gravity missions with higher ranging precision and longer lifetime are ongoing. Different to the off-axis optical bench design of GRACE Follow-On LRI project, which produces the lateral offset between the transmitting(TX) beam and receiving(RX) beam and keeps the beams anti-parallel with the method of triple mirrors assembly(TMA), several kinds of on-axis optical bench designs that use the polarized light and multiple sets of lens system have been proposed in recent years [3–5]. By combining advantages of less optical components of GRACE Follow-On LRI project and higher carrier-to-noise ratio(CNR) of the on-axis designs with the help of telescopes, the new off-axis topology of the optical bench for future gravity missions is put forward in this paper. In this design, five sets of lens system are used to restrain the tilt-to-length(TTL) coupling noise and the differential wavefront sensing(DWS) feedback loop is applied to make sure the TX beam and RX beam anti-parallel to each other. Furthermore, the higher CNR for a single channel of the photoreceiver will be realized, as the following content will demonstrate. Hence, the off-axis optical bench design will be a competitive choice for China’s future gravity missions and serves as a technology demonstrator for the ESA-NASA Laser Interferometer Space Antenna (LISA) mission [6], as well as the Chinese space-borne gravitational wave observatories Taiji [7] and TianQin [8] missions.

The structure of this paper is organized as follows. In Section 2, we introduce the design principle of the off-axis optical bench. In Section 3, the calculating method of the CNR for one segment of the photodiode is discussed in detail. The key parameters of the optical bench design are determined and results of the CNR are given in Section 4. Finally, we draw our conclusions in Section 5.

2. Design concept of the off-axis optical bench

As its name implies, the off-axis optical bench design means that the TX beam and RX beam are anti-parallel to each other but not on the same axis, as shown in Fig. 1. Clipped by the receiver aperture at first, the RX beam from the distant satellite passes through the lens L1, then the most part of the RX beam transmits through the beamsplitter BS1. Next, about one half of the beam penetrates the beamsplitter BS2, then passes through the lens L4 and finally arrives at the quadrant photodiode QPD1. The other half is reflected by the BS2, then passes through the lens L5 and is detected by the QPD2. On the other hand, the local(LO) Gaussian beam, whose waist is set to be at the center point of the reflecting surface of the fast steering mirror(FSM), is injected from the fiber coupler to the optical bench and is reflected by the FSM, then transmits through the lens L2. The majority of the LO beam penetrates the BS1, lens L3, then is reflected by the mirrors M1, M2 in turn and is transmitted to the remote satellite. It needs to be pointed that the use of two mirrors instead of a single mirror is for maintaining the direction of the TX beam relative to the optical axis. At the same time, the small part of the LO beam is reflected by the BS1 and goes along with the RX beam passing through the BS1. At the end of the light path, the heterodyne interference happens between the LO and RX beam at the QPDs which convert an optical power into an electric current by exploiting the inner photoelectric effect. The photocurrent is then converted to a voltage by a transimpedance amplifier in the photoreceiver. At last, the phasemeter track the phase of voltage signals provided by the photoreceiver and this phase carries information about changes in distance between the satellites.

 

Fig. 1. The diagrammatic drawing of the optical bench design for future gravity missions.

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In this design, there are five sets of Keplerian telescopes: L1-L4(L1-L5) as the receiving telescopes, L2-L4(L2-L5) as the local telescopes and L2-L3 as the transmitting telescope. For the receiving telescopes, the front focus of L1 coincides with the center of the receiver aperture and the back focus of L4(L5) is placed at the center of the active area of the QPD1(QPD2), in order to keep the optical path of the RX beam from the aperture to photodiodes unchanged when the RX beam rotates around the center point of the aperture, which is called the RX reference point(RX RP). Similarly, the front focus of the L2 is positioned at the center point of the reflecting surface of the FSM, and the back focus of the L3 is named as the TX reference point(TX RP). Therefore, the optical paths of both the LO beam from the FSM to photodiodes and the TX beam from the FSM to the TX RP remain constant even though the FSM rotates around the center point of its reflecting surface. The midpoint of the TX RP and RX RP is defined as the reference point(RP) of the optical bench, which coincides with the center of the test mass in the accelerometer and the mass center of the whole satellite. While the local satellite rotates around the RP, the DWS signal of the QPD1(or QPD2) is used to control the FSM attitude. As the DWS signal goes close to 0, the LO beam overlaps gradually with the RX beam and the TX and RX beam are nearly anti-parallel to each other in the meantime. At the same time, the contributions of displacements of the TX RP and RX RP to the ranging between two satellites cancel out so that the TTL coupling of the total measurement system is restricted significantly. Besides, the arrangement of double QPDs is for hot/cold redundancy as needed.

Next, the relations between transversal magnifications of the receiving, transmitting and local telescopes will be derived. $m_{RX}, m_{LO}$ and $m_{TX}$ denote the absolute value of transversal magnifications of the receiving, local and transmitting telescopes respectively. It needs to be noted that the transmitting telescope expands the TX beam while the receiving and local telescopes shrink the RX beam and LO beam respectively. The incident angles into telescopes and the emergent angles out of telescopes of the RX, TX and LO beam satisfy the following equations:

In order to make the TX beam anti-parallel to the RX beam, the equation

Combining all above equations, we have

The relation will be helpful for the derivation of the CNR in the next section.

3. Calculations of carrier-to-noise ratio

A major function of the optical bench is to provide the heterodyne interference signal with the high CNR because the sensitivity of the phase readout of the phasemeter can be expressed as the inverse of the CNR $C/N_0$,

Here the unit of the CNR is Hz and it’s usually expressed by dB-Hz. The criteria is generally used for the phase tracking within GPS receivers [9], and has been applied in GRACE and GRACE Follow-On missions to evaluate system noise in the microwave system [10]. In order to ensure that the phasemeter continuously tracks the phase of the input signal without the loss of the phase lock, the minimum requirement for the CNR in GRACE Follow-On LRI project is 70 dB-Hz corresponding to $10^7$ Hz, which means a phase readout accuracy of $0.3\ mrad/\sqrt {Hz}$ according to eq. (7). This requirement will be applied in the following discussions about the CNR of the optical design. The calculations of the CNR below follow the approach presented in the reference [3], whereby the corresponding equations have been modified to account for our optical bench design.

The CNR is generally defined as the ratio of carrier power to noise power spectral density(PSD), which is used to estimate the quality of the beatnote signal. Therefore the CNR of a photoreceiver can be expressed as

Here $I_{PR}$ is the equivalent input current noise of the photoreceiver [11] and its typical value of

The $PSD[\delta P_{LO}(f)/P_{LO}]$, which denotes relative fluctuations of the local laser power, is required to be within $1.0\times 10^{-15}/Hz(f>4MHz)$ for gravity field measurements [12].

Consequently, the $C/N_0$ for a single channel of the photoreceiver can be expressed approximately as

3.1 Optimal local beam power

In order to calculate the $C/N_0$ in eq. (14), we need to know these parameters: the LO beam power $P_{LO}$, RX beam power $P_{RX}$ and heterodyne efficiency $\eta$. For the sake of convenience, eq. (14) can be rewritten in the form of

It is evident that $f(P_{LO})$ is just the function of the LO beam power and it has a maximum at some value of $P_{LO}$. The function $f(P_{LO})$ is displayed in Fig. 2. After simple calculations, it comes to a conclusion that $f(P_{LO})$ reaches its maximum value 63.5 dB-Hz when $P_{LO}=0.198\ mW$. Therefore, when the LO beam power is set to be $0.198\ mW$, eq. (15) can also be changed to the form of

For meeting the preceding requirement of 70 dB-Hz, the second term on the right-hand side of the equal sign in the above equation needs to excess 6.5 dB-Hz. Next, the expression of the RX beam power $P_{RX}$ and the heterodyne efficiency $\eta$ is going to be determined.

 

Fig. 2. $C/N_0$ in dB-Hz as a function of the LO beam power for a single channel of the photoreceiver.

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3.2 Receiving beam power

It is obvious that the $C/N_0$ increases with the RX beam power from eq. (15). The laser beam intensity and the aperture size decide the power within the receiver aperture of the local satellite. The far-field intensity of the Gaussian beam for small misalignments of the TX beam relative to the line-of-sight is

Here $r_{TAP}$ is the radius of the transmitting aperture and $\omega _{LO}$ represents the waist radius of the LO beam. The $\psi _1(x)$ function, which accounts for the increased divergence angle due to the clipping, is a polynomial of 6th order with coefficients given in the Table 1 and is only valid in the range $0.2<x<2.0$. Considering the transmitting efficiency $\eta _{RX}$ for the RX beam from the receiver aperture to one segment of the photodiode and the relation

Table 1. The coefficients of the polynomial functions $\psi _1(x)=\sum _{i=0}^{6} p_i\cdot x^i$ and $\psi _2(x)=\sum _{i=0}^{6} q_i\cdot x^i$. [3]

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3.3 Heterodyne efficiency of interference

The heterodyne efficiency $\eta$ of the interference is another important factor in eq. (15) and raising the heterodyne efficiency means improving the overlap of RX and LO electric field. The expression of the heterodyne efficiency $\eta _{CPD}$ of the interference between the LO Gaussian beam and the RX beam, which has a constant intensity and phase within the aperture and zero magnitude outside, on the active area of a circular photodiode with radius $r_{PD}$ can be written as [3]

Figure 3 shows that the heterodyne efficiency $\eta _{CPD}$ increases with the ratio of $\omega _{LO,PD}$ and $r_{PD}$. Moreover, $\eta _{CPD}$ is extremely close to 1 when $\omega _{LO,PD}\gg r_{PD}$.

 

Fig. 3. The graph of the heterodyne efficiency $\eta _{CPD}$ versus the ratio $\omega _{LO,PD}/r_{PD}$.

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If $\alpha _{RX,LO}\neq 0$, the heterodyne efficiency for a single segment can be derived approximately by the model [3]

According to the optical design, there are the following relations:

Thereupon, the full approximate expression of the heterodyne efficiency for a single segment of the photodiode is written as

So far, the complete expression of the CNR for one channel of the photoreceiver has been derived by substituting eq. (28) and eq. (23) into eq. (17). Qualitatively, in addition to setting $P_{LO}$ for a single segment of the photodiode to be $0.198\ mW$, it’s vital for improving the CNR to increase the RX beam power and the heterodyne efficiency of interference between the RX and LO beams. In the following section, studying the CNR quantitatively will help determine the critical parameters of the optical bench design for future gravity missions.

4. Results and discussions

Before calculating results of the CNR, the values or ranges of the following parameters need to be ascertained:

First of all, $1064\ nm$ wavelength laser with $100\ mW$ power and the mean distance $L=100\ km$ between two satellites are chosen in the preliminary plan for China’s future gravity missions. In the power budget of the optical design, $3\ mW$ laser is allocated for the frequency stabilization and around $1.6\ mW$ laser is assigned to be detected by eight segments of two quadrant photodiodes. A portion of the remaining laser is used as the TX beam without the clipping by the transmitting telescope and $P_{TX}$ can be described by

In the first step, we concentrate on the high case: $\alpha _{TX}=\alpha _{RX}=0$, which means that the directions of local and remote satellites coincide perfectly with the line-of-sight. Apart from that, four typical values {1.0, 1.5, 2.0, 2.5 mm} are chosen for the waist radius $\omega _{LO}$ of the LO beam. The two parameters $m_{TX}$ and $r_{RAP}$ are scanned in the ranges $[1,14]$ and $[1,14]mm$ respectively. Fig. 4 displays the distribution of the $C/N_0$ on the plane of $m_{TX}$ and $r_{RAP}$. It can be seen that the $C/N_0$ of almost all points in the diagram excesses 70 dB-Hz and the $C/N_0$ of partial regions is even above 115 dB-Hz when $\omega _{LO}$ is equal to $1.0\ mm$. As the waist radius of the LO beam increases, the maximum of the $C/N_0$ drops to about 105 dB-Hz. Besides, $m_{TX}$ that corresponds to the maximums of the $C/N_0$ decreases with $\omega _{LO}$ because of the clipping by the transmitting telescope with the fixed aperture radius of $12.7\ mm$.

 

Fig. 4. The distribution diagram of the $C/N_0$ on the plane of $m_{TX}$ and $r_{RAP}$. The four panels correspond to four discrete values of $\omega _{LO}$: 1.0, 1.5, 2.0 and 2.5 mm. Different colors(from blue to red) of points in the diagram denote the different amounts of the CNR.

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It’s time to discuss about the effect of misalignment angles of the local and remote satellites with respect to the line-of-sight on the $C/N_0$. Since the $1\ mm$ waist radius of the LO beam corresponds to the higher CNR and it’s convenient to use commercially available fiber injectors for future experiments, $\omega _{LO}$ is set to be $1.0\ mm$. Similarly, there are four choices for $\alpha _{RX}(=\alpha _{TX})$: $20,40,60,80\ \mu rad$ and the scanning ranges of $m_{TX}$ and $r_{RAP}$ are $[1,14]$ and $[1,14]mm$ as well. For convenience, we define

As mentioned earlier, the range of x and y should be $[0.2,2.0]$, where the fitting functions $\psi _1(x)$ and $\psi _2(y)$ are valid. In addition, the parameter space with $G(\alpha _{RX}) <0.13$ or $F(\alpha _{TX}) <0.13$ is also excluded. The scanning results of the $C/N_0$ are shown in Fig. 5. To make it look clearer, the ranges of coordinates for the X-axis of the two lower panels are changed to $[1,8]$. Obviously, the surviving regions are getting smaller due to the restriction of above-mentioned conditions and the maximum of the $C/N_0$ also decreases by about 20 dB-Hz as the angles $\alpha _{RX}$ and $\alpha _{TX}$ become larger. Based on this diagram, we can select $m_{TX}=4$ and $r_{RAP}=10\ mm$ for our optical bench design, meanwhile, the corresponding CNR for a single segment of the photoreceiver is close to 92 dB-Hz. Same as GRACE Follow-On mission, the radius of the photodiodes $r_{PD}$ is chosen to be $0.5\ mm$. Therefore, the transversal magnifications of the receiving and local telescopes are also determined: $m_{RX}=0.05,\ m_{LO}=0.2$.

 

Fig. 5. The distribution diagram of the $C/N_0$ on the plane of $m_{TX}$ and $r_{RAP}$. The four panels correspond to four discrete values of $\alpha _{TX}$: 20, 40, 60 and 80 $\mu rad$. Different colors(from blue to red) of points in the diagram denote the different amounts of the CNR. White areas with $x,y<0.2,\ x,y>2.0,\ F(\alpha _{TX})<0.13$ or $G(\alpha _{RX})<0.13$ have been excluded. The black point at (4,10) in the diagram denotes the parameter that is selected for the final design. Note that the ranges of coordinates for the X-axis of the two lower panels are changed to $[1,8]$ for clarity.

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The Table 2 shows the comparison of key parameters and the CNR of our optical design and the optical design of GRACE Follow-On mission [2,3]. It is obvious that the $C/N_0$ of our design is considerably larger even for the worst case mainly because of applications of a series of telescopes, shorter satellite separation and larger laser power. Finally, it comes to the conclusion that the optical bench design in this paper has more significant advantages than the optical design of GRACE Follow-On mission and it is a hopeful candidate for China’s future gravity missions.

Table 2. The comparison of key parameters and the CNR of the optical bench design in this paper and the optical bench design in GRACE Follow-On mission.

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5. Conclusion

In this work, we come up with a novel off-axis optical bench design which absorbs the virtues of the off-axis optical bench design of GRACE Follow-On mission and other on-axis optical designs. The detailed constructions of the design and the principles that the TX and RX beam are kept anti-parallel with DWS feedback loop and the TTL coupling noise is suppressed by introducing multiple sets of lens systems are described in detail. The expression of the carrier-to-noise ratio is derived and key parameters of the optical bench design are also ascertained. Finally, we confirm that the carrier-to-noise ratio for a single channel of the photoreceiver is larger than that of the optical bench design of GRACE Follow-On mission. The numerical simulations and experimental demonstrations of the optical bench design is still ongoing. The China’s future gravity missions can choose it as the optical bench with high CNR allowing for a robust phase tracking.