# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 3 Optical Simulations  
In order to realize the OB design, we built an optical model using the interferometer simulation software IFOCAD [24], a collection of C++ libraries for simulating laser interferometers. The OB components, such as the beamsplitters and mirrors, the lenses, and the photoreceivers, are parameterized and included in the model. IFOCAD provides methods for tracing general astigmatic Gaussian beams through three-dimensional space, as well as for computing the relevant two-beam interferometric signals, such as the heterodyne efficiency, the longitudinal pathlength signal (LPS) and differential wavefront sensing (DWS) signals [25]. The LPS sensed by a single element photodiode is computed as  
\[s_{\text{LPS}}\equiv\frac{\phi}{k}=s+\frac{1}{k}\arg(\iint E_{1}E_{2}^{*}dS_{ \text{pd}}), \tag{4}\]  
where \(\phi\) is the interferometric phase, \(k=2\pi/\lambda\) is the wave number, \(s\) is the macroscopic accumulated optical pathlength difference during beam propagation through the setup, \(E_{1}\) and \(E_{2}\) are the complex amplitudes of the interfering beams, and \(dS_{\text{pd}}\) is a surface element in the detector surface. There exist several variants of the LPS signal for a QPD [26], of which we use the \"average phase\" definition, where the LPS is computed as the average phase over the four detector segments \(s_{\text{LPS}}=(\phi_{A}+\phi_{B}+\phi_{C}+\phi_{D})/4k\), where \(\phi_{A\ldots D}\) are the interferometric phases measured by each of the QPD segments. Note that \(s_{\text{LPS}}\) takes into account any effects stemming from the transverse distributions of the two interfering beams in the detector plane, as well as Gouy phase shifts, which are known to be potential sources of error in precision interferometers [27]. Polarization and stray light effects are neglected in the simulation, and will be the subject of future investigations. Parasitic polarization states, polarization fluctuations, and ghost beams, can couple to the LPS and DWS signals, and could have an impact on the interferometer performance [28; 29]. As shown in Figure 2, the RX beam, which has propagated for the inter-satellite distance of \(\sim\)200 km, is clipped to a radius of 8 mm by the RX/TX RP aperture located on the OB, and the diffracted beam's amplitude and phase profile is nearly flat. To simulate this \"flat-top\" beam, the mode-expansion method (MEM) [30] is adopted. In the MEM, the electrical field of the incoming Gaussian beam \(E(x,y)\) is decomposed into Hermite-Gaussian modes \(u_{mn}(x,y;q)\),  
\[E_{\text{MEM}}(x,y)=\sum_{m=0}^{N_{\text{max}}}\sum_{n=0}^{N_{\text{max}}-m}a_ {mn}u_{mn}(x,y;q)\exp(-iks), \tag{5}\]  
where \(a_{mn}\) is the mode amplitude, \(k\) is the wavenumber, \(s\) is a propagation distance, and \(N_{\text{max}}\) is the maximum expansion order, which limits the decomposition performance. Once the relative mode amplitudes are derived at the aperture, they are invariant through the paraxial propagation of the modes to the detectors downstream. Hence, the electrical field at an arbitrary plane along the optical axis can be computed as the sum over the Hermite-Gaussian modes propagated up to that plane. Figure 3 shows the amplitude and phase of the RX beam at the receive aperture and at the surface of QPD1, as obtained via the MEM simulation. The IFOCAD model is built based on the \"B\" configuration from Table 1, and it is drawn using OPTOCAD, a Fortran 95 module for tracing Gaussian TEM00 beams through an optical setup [31] (Figure 4). The B configuration is chosen due to the smaller size of the beam injected into the OB, which is straightforward to realize using commercially available fiber injectors. Figure 4: LRI OB as modeled in IFOCAD and drawn in OPTOCAD. The RX beams starts at the receive aperture (red). The TX and LO beams stem from the beam injected into the OB by the fiber injector (blue). The baseplate assumed in the thermal analysis is drawn as a rectangle enclosing all of the OB components.