1.

INTRODUCTION

With the success of the laser ranging interferometer in the GRACE Follow on mission, future satellite to satellite tracking mission is going to employ heterodyne laser interferometry for tracking the miniature range variation between two satellites. The present work aims to propose a new off-axis design of the laser ranging interferometer for the next generation gravity and geodesy mission, with a view that this will be one of the viable options to be considered for the laser metrology in a prospective, future satellite to satellite tracking gravity mission in China. In place of the triple mirror assembly adopted in the GRACE Follow on mission, an optical system comprising of lens and roof top mirrors is employed to generate laser retro-reflection between two satellites. The receiving (RX) beam and transmitting (TX) beam are enforced to be anti-parallel by a control loop with the differential wavefront signal (DWS) readout from the quadrant photodiodes (QPD) as controller and a fast steering mirror is employed to actuate the pointing of the local beam. The optical system is also designed in such a way to reduce the tilt to length coupling noise.

The outline of the paper may be given as follows. In Section 2, detailed design of the off-axis optical bench layout is presented. Section 3 describes simulation results of the new layout, including the preliminary consideration of the tilt-to-length coupling noise. In addition, thermal stability of the optical bench is also considered. Section 4 provides the concluding remarks and future outlook of our work.

2.

OPTICAL BENCH DESIGN

The schematic diagram of the off-axis optical bench design is shown in Fig. 1. The RX beam enters the optical bench through the receiving aperture. Upon passing through the lens L1 and beamsplitter BS1, about 90% of the initial RX beam couples with the beamsplitter BS2. After being equally split by BS2, the RX beam is transmitted through the lenses L4 and L5 respectively. Both the LO beam and TX beam originate from a single beam that propagates to the optical bench through the fiber coupler, and the direction of the injecting beam is then controlled by the fast steering mirror FSM. After being split by BS1, 10% of the injecting beam is reflected to the BS2 and then interferes with the RX beam, while 90% of the injecting beam is transmitted to the lens L3, then reflected by mirrors M1 and M2, and then propagates to the distant spacecraft. The beat note signal is received by the quadrant photodiodes QPD1 and QPD2 respectively. The phase signal derived from the beat note not only gives the pathlength variation information but is also used to calculate the horizontal and vertical differential wavefront sensing (DWS) signal. A DWS feedback loop concept, first proposed by Vitali Müller [1], is then employed here. The DWS signal is utilized to measure the misalignment of the receiving beam relative to the local beam, and the result will be fed back to steer the FSM mirror. In this way, a feedback control loop is established. In addition, both beamsplitters have a wedge of 0.5°, which serve to reduce the influence of back scattered stray light. The angle between two roof top mirrors (M1-M2) is chosen to be 135°. The mirrors plays the role of guiding an optical beam out of the optical bench and propagate to the distant spacecraft. An advantage of roof top mirrors is that a slight rotation of an optical beam will not alter the reflection angle of the beam.

Figure 1:

The schematic drawing of the off-axis optical bench design. The RX beam and TX beam enter and exit the optical bench through different light paths.

In the proposed optical layout, five keplerian telescope lens systems are employed. These lens systems serve to minimize the beam walk on the surface of quadrant photodiode as well as the TX beam pointing jitter. At the same time, the design also attempts to optimize the heterodyne efficiency and carrier to signal ratio.

As Fig. 2 shows, for a lens system, if the rotation center of the incident beam is placed at the front focal point of the first lens, when the input beam rotates, the output beam that passes through the lens system will take the back focus of the second lens as the rotation center. The optical path length between these two focal points will stay unchanged. The relationship between the absolute value of the rotation angle of the incident light relative to the optical axis and the absolute value of the rotation angle of the output light relative to the optical axis is:

Figure 2:

Rotation center of the incident beam placed at the front focal point of the first lens implies that the back focus of the second lens will be the rotation center for the output beam.

and

where θ_i represents the rotation angle of the incident light relative to the optical axis, θ_o represents the rotation angle of the output light relative to the optical axis, m_a represents the angular magnification of the lens system, and , represent the back focal length of the lens, , .

As Fig. 1 shows, the L1-L4 and L1-L5 lens systems are designated as RX lens systems; L2-L4 and L2-L5 lens systems as LO lens systems; L2-L3 lens system as TX lens system. In addition to compressing and expanding the beam size, the RX lens system images the receiver reference point (RX-RP) located at the center of receiving aperture to the center of QPD; the TX lens system images the rotation point of the beam that is reflected by the FSM to the transmitter reference point (TX RP), while the waist of the beam is put on the FSM surface; the LO lens system images the rotation point on FSM surface to the center of quadrant photodiode surface. The relationship of the angular magnifications among the three lens systems may be worked out to be[2]:

m_a,rx, m_a,lo, m_a,tx denote respectively the angular magnification of the RX, LO and TX lens system. As the beam rotates along its reference point, the phase of the wavefront received by the photodiode remains stable. The reference point (RP) of the optical bench is located at the center of the line joining TX-RP and RX-RP (see figure 1), and coincides with the center of mass of the satellite. Such optical design is designed to suppress the tilt-to-length coupling noise caused by the angular jitter of the satellites. At the same time, lens systems also serve to adjust the beam parameters of the RX, LO and TX beams so as to achieve a higher heterodyne efficiency and improve the power density received by the photodiodes.

In our setup, a combination of the lens system and mirrors function as a laser retro-reflector in such a way that RX and TX beam are enforced to be antiparallel, as depicted in Fig. 3. The relationship between angles and magnifications may be expressed as as[2]:

Figure 3:

Assembly of lens systems and mirrors.

where θ_rx,out, θ_lo,out, θ_tx,out are the output angles of the RX, LO and TX beams. θ_rx,in, θ_lo,in, θ_tx,in are the incident angles of RX beam, LO beam and TX beam respectively. With the help of closed feedback loop with the DWS signal as controller and leave aside for the time being noise source like the piston noise, we have

and it follows that

Therefore

According to the optical bench design,

Together with (4), we then have

By using the DWS feedback loop to compensate for the spacecraft attitude jitter, the RX beam and TX beam will remain antiparallel according to our design.

Based on the off-axis optical design, theoretical calculations indicate that such configuration may reach a heterodyne efficiency of 0.92 and a carrier to noise ratio of 98.1dB-Hz. Though in practice, with noises and assembly errors of optical elements taken into account, it is expected that the numbers are lower than those given by the theoretical values.

3.

IFOCAD SIMULATION

To verify the feasibility of the optical design, a simulation model was built using IfoCAD[3]. Parameters input into the simulation are based on those of commercial optical components. A schematic diagram of the simulation optical bench is given in Fig. 4. All the parameters and the results are output in double precision. As Fig. 1 shows, the RX beam, originated from a distant satellite of 200km apart, is received and clipped by the receiving aperture. The received beam behind the aperture has an approximately flat wavefront and a constant intensity[1, 5], known as a flat-top beam. In order to simulate the propagation of a flat-top beam within the optical bench, a mode expansion method (MEM) is applied[2, 5]. In the IfoCAD model, the origin of the RX flat-top beam is put at the center of the receiving aperture, where the front focal point of L1 is located. Also, a gaussian beam is generated in the simulation as the source of the LO beam and TX beam, whose waist is put on the surface of the FSM.

Figure 4:

The schematic drawing of the off-axis optical bench design, output by Optocad[4]. The RX beam is in red, while the local beam and TX beam are shown in blue color. The triangle markers give the position of the beam waist, the green marker is the tangential waist indicator and the red marker is the sagittal waist indicators.

In the proposed optical layout, five lens systems are employed. Take the L1-L4 RX lens system as a typical lens system in the IfoCAD simulation, as illustrated in Fig. 5. In order to find out the equivalent back focal point of L1 that is located behind the beamsplitter BS2, auxiliary beam b1 and beam b2 are employed. The origin of auxiliary beam b1 is put on the front focal point of the lens L1, and b1 propagates through the optical axis of L1. By trials and errors, the positions of beamsplitter BS1 and BS2 are then worked out with the location of L1 being fixed so as to minimise the tilt to length coupling noises. The origin of beam b2 is slightly separated from the origin of the auxiliary beam b1, and the offset is set up to 1μm. After passing through the L1, BS1 and BS2, the b1 beam couples with the b2 beam at the equivalent back focal point of L1. The equivalent back focal point of L1 is then obtained, where the back focal point of L4 is located. In this way, the L1-L4 RX lens system is built. The rest of the lens systems may be built up in a similiar way in the simulation model by using the method above.

Figure 5:

Illustration of the L1-L4 RX lens system. Auxiliary beams are used to build up the L1-L4 RX lens system. Beam b1 is shown in blue; beam b2 is in red.

Detailed information about the optical components used in the off-axis design is shown in Tab. 1. The ideal performance of the off-axis layout using design A could reach a heterodyne efficiency of 0.92 and a carrier to noise ratio of 98.1dB-Hz. Compared to design A, design B used a LO beam with a smaller waist radius, which is easier to realize in the experiment. However, due to the smaller LO beam size, the heterodyne efficiency of Configuration B is reduced to 0.31, but a carrier to noise ratio of 90.0 dB-HZ that is higher than that of the GRACE-FO optical bench. In what follows, parameters of Design B will be used in the simulation, because the waist size is easier to realize with a commercial fiber injector and the CNR value is still acceptable even with a smaller waist radius of 1mm. Further optimisation of this optical design is still ongoing.

Table 1:

Parameters of the optical components used in simulation. rRX/TX,AP is the radius of both the receiving and transmitting aperture; ω0 is the waist radius of the beam coming out from the fiber coupler; η is the heterodyne efficiency; CNR is the carrier to signal ratio.

After establishing the off-axis simulation model and optimizing the relative position of the optical components, the local tilt-to-length coupling noise is investigated. We rotate the RX beam along the RX-RP in yaw and pitch degrees of freedom for ±2mrad, with the DWS feedback loop active, as shown in Fig. 6. After passing through the optical components, the RX beam interferes with the LO beam. Beatnote signal is received by the QPD and the variation of the LPS signal may then be read out, with this the local TTL coupling noise is worked out. This gives a preliminary evaluation of the feasibility of the optical design as far as local TTL is concerned.

Figure 6:

Illustration of the local TTL coupling noise simulation. The RX beam is rotated along the RX-RP point in yaw and pitch degrees of freedom for ±2mrad, and the beatnote signal is captured by the QPD.

As a preliminary investigation of the TTL noise generated by the relative attitude jitter between two spacecrafts, a remote photodiode (PD) is introduced into the simulation model, as depicted in Fig. 7. Such a photodiode, with a sufficient active area, represents the remote satellite located at 200km away from the TX-RP of the local optical bench. The TX beam transmitting from the local optical bench propagates to the remote PD and interferes with a gaussian beam placed on the surface of the remote photodiode, the beat note is then captured by the remote PD. The gaussian beam on the remote PD represents the LO beam of the remote optical bench and the waist radius should be large enough to reduce the effect caused by the wavefront of the beams. The local optical bench then rotates along the RP point in yaw and pitch degrees of freedom, and LPS from remote and local photodiodes is obtained. In this way, the TTL noise generated by the pitch and yaw of the attitude variation is then worked out.

Figure 7:

Illustration of the global TTL coupling noise in the simulation. The local optical bench is rotated along the RP point in yaw and pitch degrees of freedom for ±2mrad. A photodiode is placed 200km away from the TX-RP point, acting as a simplified model of the remote satellite.

The local TTL coupling noise results are shown in Fig. 8 and the global TTL coupling noise results are displayed in Fig. 9. The figures depict the horizontal and vertical DWS signal of QPD1 and QPD2 with the control of the DWS feedback loop; The TTL coupling noise caused by rotating the RX beam and rotating the local optical bench. All the results are given separately in the yaw and pitch degrees of freedom. The output data is based on double precision.

Figure 8:

Results of local TTL simulation.

Figure 9:

Results of global TTL simulation.

The GRACE Follow on 1B IMU data is employed in the simulation of attitude jitter of spacecrafts[6], with only the yaw and pitch degrees of freedom considered, as illustrated in Fig. 10. With the attitude data of the GRACE Follow on mission input into the simulation, the local optical bench is rotated along the RP point first in yaw and then in pitch. The DWS feedback loop is activated to compensate for the misalignment of the RX and LO beam. LPS signal variation is then derived and the power spectral density of the LPS variation generated by TTL coupling is then computed, see Fig. 11. It may be seen from the PSD diagram that the noise of LPS at 1mHz is higher than that at higher frequencies. This suggests that there is still room for optimisation of the optical design considered here.

Figure 10:

Attitude variation in yaw and pitch degrees of freedom from the 1B IMU data of the GRACE Follow on mission.

Figure 11:

(a) LPS signal variation and (b) power spectral density of the LPS variation generated by spacecraft attitude jitter with input from the 1B IMU GRACE Follow on data.

Temperature fluctuation also leads to optical pathlength variation and such thermal instability is also analysed, again in a preliminary way in the simulation. The temperature variation considered is between ±3K [7]. In the simulation, the baseplate of the optical bench and the components are assumed to expand uniformly. For the off-axis optical layout, components are made of N-BK7 and fused-silica material, the baseplate is made of titanium. The coefficient of thermal expansion of N-BK7 is α_BK7 = 7.1 × 10^-6/°C, and the change in index of refraction with temperature is δ_BK7 = 2.4 × 10^-6/°C; for fused-silica, α_FS = 0.55 × 10^-6/°C, δ_FS = 11.9 × 10^-6/°C; for titanium, α_Ti = 8.6 × 10^-6/°C. Considering the thermal expansion of the baseplate and the components, and the change of the index of refraction of beamsplitters and lenses, the optical pathlength variation due to temperature variation can be evaluated. The results are depicted in Fig. 12, the maximum value of total TTL is 10.3μm/rad, and the thermal drift of the optical bench is within the noise budget. Further investigation of the thermal effect will be based on the β angle subtended between the orbital plane and the solar vector.

Figure 12:

Thermal noise results of off-axis optical bench. Temperature variation is divided into several steps, and angular motion is introduced while activating the DWS feedback loop. Optical bench is rotated in yaw(a) and pitch(b) degrees of freedom.

4.

CONCLUDING REMARKS

Presented in this work is a preliminary off-axis optical design for a prospective future NGGM mission, with a view that this will be a feasible optical design to be considered for future NGGM mission in China. The basic structure of the optical design is briefly sketched and only a few dominant noise sources are considered in a simplified setting. Experimental work is ongoing to further understand the practical aspects of the design and the build up of a prototype. At the same time, indepth noise analysis and further optimisation of the optical design are being carried out. We hope to report on the progress in the near future.