Topology optimization of multicomponent optomechanical systems for improved optical performance
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Abstract
The stringent and conflicting requirements imposed on optomechanical instruments and the everincreasing need for higher resolution and quality imagery demands a tightly integrated system design approach. Our aim is to drive the thermomechanical design of multiple components through the optical performance of the complete system. To this end, we propose a new method combining structuralthermaloptical performance analysis and topology optimization while taking into account both component and system level constraints. A 2D twomirror example demonstrates that the proposed approach is able to improve the system’s spot size error by 95% compared to uncoupled system optimization while satisfying equivalent constraints.
Keywords
Topology optimization Multidisciplinary design optimization System optimization Optical instrumentation Optomechanics Thermoelasticity Structuralthermalopticalperformance analysis1 Introduction
1.1 Optomechanical instruments
The optical performance generally encompasses the systems’ image quality, optical resolution, and image position accuracy. Both image quality and optical resolution are typically quantified by the spot size or image blur diameter (e.g. Welsh 1991; Doyle et al. 2012). The spot size of an aberrationfree system is limited by the wavelength of light. This diffraction limit determines the minimum blur diameter that is achievable by an optical system and hence provides a reference for image quality. The spot size mainly depends on the wavefront quality due to geometric aberrations, whereas the beam position accuracy depends on how accurate the optical components are positioned/oriented. These performance metrics can be determined using geometric ray tracing, which traces the propagation of light rays through an optical system (e.g. Spencer and Murty 1962). For flat or spherical singlemirror systems the wavefront error scales linearly with the Surface Form Error (SFE) of the deformed surface and the pointing error is directly related to the tilt of the surface (e.g. Genberg 1984, 1999). The optical performance metric of interest thus depends on application and system composition.
Many factors may contribute to the inability of an optical system to produce a perfect image, including chromatic and geometrical aberrations, fabrication and alignment errors, (lack of) selfweight and environmental effects such as temperature fluctuations. This work focuses on the reduction of optical performance errors of reflective optical systems induced by (quasi)static thermal loads. Therefore, structural deformations and temperature differences of the frame should neither excessively distort the mount nor the optical surface. This implies that a mechanically disconnected frame and optical surface combination would be optimal. However, the presence of thermal loads requires material to abduct the heat from the optical surfaces to the frame. In addition, the optical components also require a stiff design, as the structure must constrain the components such that they are not damaged or irreversibly moved after exposure to external conditions such as vibrations, thermal shocks and gravity. To limit excitation from external vibrations, the fundamental elastic eigenfrequency must be higher than a critical lower limit. Adding more practical constraints such as maximum mass and material usage, often linked to costs, the optomechanical mirror support design clearly involves multiple conflicting structural requirements. To make wellfounded and justified design tradeoffs, careful consideration of the thermomechanical and dynamic performance of optical mounts is required. Optimization techniques can aid in this process.
1.2 Problem definition
The current typical design approach of optomechanical instruments is characterized by the optical discipline creating a performance error budget that defines deformation limits for each optical component to the structural discipline. In an iterative design process, the thermal discipline provides the temperature fields and gradients, after which the structural discipline aims to realize a design which meets the deformation limits during thermal and other environmental load cases. From this point of view, an optical system functions as long as the components remain within allowed tolerances of their nominal locations, orientations and deformations. Thus, in the existing thermomechanical design process, the optical performance is not considered directly. Instead, each component is designed and optimized separately to meet a priori defined deformation limits.
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systemlevel optical performance metrics that drive the thermomechanical design and optimization process,
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simultaneous optimization of all components for the combined metrics, and
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systemlevel constraints that, where possible and applicable, replace componentlevel constraints.
1.3 Approach: simultaneous multicomponent multidisciplinary topology optimization
Modelbased structural optimization techniques can aid in further improving the performance of optical instruments. Topology optimization is a systematic, bottomup structural optimization approach that provides maximum design freedom without any prior knowledge of the design. The procedure optimizes the material layout within given design domains in order to maximize a performance measure, while subjected to a given set of loads, boundary conditions, and constraints (e.g. Bendsøe and Sigmund 2003). The method, in combination with mathematical programming, has shown to be able to solve complex multidisciplinary problems with multiple active nonlinear response functions and is capable of producing innovative solutions.
The mentioned design requirements have previously been investigated in topology optimization. This includes thermal loads and prescribed temperature differences on stiffness problems (e.g. Rodrigues and Fernandes 1995; Li et al. 2001a; Gao and Zhang 2010; Deaton and Grandhi 2013) and the application to the design of thermomechanical compliant mechanisms (e.g. Sigmund 2001; Ansola et al. 2012). Recently, Zhu et al. (2016) presented a shape preserving design method for topology optimization. By suppressing the elastic strain energy in the local domain, the shape of the concerned domain can be effectively maintained to satisfy the requirements. This is extended by Li et al. (2017) to achieve desired deformation behavior within local structural domains by distinguishing and suppressing specific deformation in a certain direction. The fundamental eigenfrequency as a response function has been thoroughly investigated (e.g. Ma et al. 1995; Pedersen 2000; Du and Olhoff 2007; Tsai and Cheng 2013).
Previously, topology optimization has shown its benefits in the design of mirror mounts for optical performance. For example the minimization of deformations of optical surfaces under static (Park et al. 2005; Sahu et al. 2017) and thermal loads (Kim et al. 2005) and mass minimization while constraining deformations as well as the fundamental eigenfrequency (Hu et al. 2017a). Furthermore, semikinematic flexible mirror mounts are topologically optimized by, Hu et al. (2017b) who minimize surface form errors subjected to both static and thermal loads constrained by a minimum natural eigenfrequency, and Van der Kolk et al. (2017), who achieve optimal damping characteristics at source frequencies. System optical performance metrics have not been included in the topology optimization framework yet.
An integrated Multidisciplinary Design Optimization (MDO) approach is often applied to couple all involved physics and profit from the interactions between different disciplines, resulting in superior designs. We apply an integrated StructuralThermalOptical Performance (STOP) analysis and optimization procedure for this purpose to utilize simultaneous optimization of the optical, structural and thermal design aspects (e.g. Johnston et al. 2004; Doyle et al. 2012; Kuisl et al. 2016). Prior work using STOP optimization couples various analysis tools to obtain the optical performance. This shows great improvement, although the optimization often includes only a limited number of variables such as facesheet thickness and strut diameters (e.g. Williams et al. 1999; Michels et al. 2005; Bonin and McMaster 2007). The results indicate that device performance can profit from integration of optical knowledge, and that further improvement is accessible when all design parameters are considered. The STOP optimization procedure includes optical knowledge at the thermomechanical design level, but the design freedom is not yet fully exploited.
Coupled multicomponent topology optimization can aid in exploiting the component interactions. Simultaneous topology optimization of multiple components has mainly focused on layout design and combined topology and joint location optimization (e.g. Chickermane and Gea 1997; Li et al. 2001b; Zhu et al. 2009, 2015). Topology optimization involving component interactions to improve a system performance has previously been investigated by Jin et al. (2016, 2017), with the focus on simultaneous optimization of multiple coupled actuator mechanisms to minimize the coupling interaction. However, this study was restricted to a single physical discipline.
In this work, we combine the foregoing and additionally apply multidisciplinary optimization to a component assembly and optimize for a systemlevel optical performance metric. This approach will be referred to as integrated System Design Optimization (SDO). To determine the system optical performance, this study uses a simplified version of geometric ray tracing, the ray transfer matrix analysis. The method uses the paraxial approximation to construct a linear operator that describes the behavior of an optical system (e.g. Nazarathy et al. 1986; Smith 2007; Fischer et al. 2008).
The innovation point of the SDO method compared to existing methods is threefold: it combines topology optimization and a full STOP analysis, uses systemlevel optical performance metrics to drive the thermomechanical design of multiple components simultaneously, and uses systemlevel constraints to replace multiple equivalent componentlevel constraints.
We hypothesize that an integrated structuralthermaloptical thermomechanical design optimization procedure taking into account all system components improves the system optical performance compared to individual component optimization, while subjected to equivalent design constraints.
The formulation of the coupled thermomechanical discretized equilibrium equations and modal analysis, topology optimization formulation and sensitivity analysis of a generalized response function are described in Section 2. The extension to a full STOP topology optimization approach is given in Section 3, which focuses on the optical performance prediction from finite element analysis results. The method is first validated on a singlecomponent system, after which the hypothesis is tested by numerical optimization of a twomirror example as discussed in Section 5. The results are followed by a discussion, recommendations and conclusions presented in Sections 6 and 7.
2 Formulation of coupled thermomechanical analysis framework for topology optimization
The design variables following from density based topology optimization, one belonging to every finite element in the domain, are bounded by a lower and upper bound, i.e. \(0 < \underline {s} \leq s_{i} \leq 1\) with \( i = 1,2,...,\hat {n}\), where \(\hat {n}\) is the number of variables. The underbound \(\underline {s}\) has a very small value (to avoid numerical issues) denoting the absence of material and, \(\bar {s} = 1\) providing the element with the assigned initial material properties. Each design variable can have any intermediate value within the given bounds. The material properties are interpolated using a penalization function as discussed in Section 2.1. Sections 2.2 to 2.4 discuss the thermomechanical and modal analysis, followed by the respective sensitivity analyses.
2.1 Penalization scheme
Intermediate density values are implicitly penalized to gradually force the design variables to approach their bounds and facilitate interpretation and improve manufacturability. Most material properties are a function of the design variables, which varies depending on the applied penalization scheme. Common interpolation functions for the Young’s modulus R_{E}(s) and conductivity R_{k}(s), are the SIMP and RAMP functions (Bendsøe and Sigmund 1999).
2.2 Thermomechanical equilibrium equations
2.3 Modal analysis and mean eigenvalue
2.4 Sensitivity analysis
3 Optical performance metrics and sensitivities
This section describes various optical performance metrics relevant for the analysis of reflective optical systems. First, the SFE response will be discussed, after which the analysis of average positional accuracy and spot size are explained. Most commonly the SFE is expressed by the Root Mean Square Error (RMSE) of the deformed configuration compared to the undeformed or another predefined configuration (Genberg 1984). For 3D unit disk surfaces the surface errors are often expressed in Zernike polynomials, which are directly related to typical optical aberrations. For diffraction limited flat or spherical singlemirror systems there exists a simple relation between the RMSE and the Strehl ratio. This is the peak aberrated image intensity compared to the maximum attainable intensity using an unaberrated system. The wavefront is proportional to surface front error. Though, for complex mirrors or multimirror systems the WFE is not directly related to the SFEs and the image quality can only be determined by ray tracing techniques.
To analyze a multicomponent system without using numerical ray tracing, the deformed surfaces can be approximated by a fit to directly obtain contributions to the optical surface misalignments, i.e. rigid body movements and SFEs. Next, the ray transfer matrix analysis can be used to track the position and angle of a paraxial ray though a multicomponent system, leading to a measure for the averaged positional accuracy. In order to track the rays, all system properties, i.e. optical paths lengths, incident angles and specific properties of the optical components, as well as component transfer functions and misalignments should be known. Finally, the spot size is quantified by the mean average deviation of all rays with respect to the averaged spot position.
3.1 Surface form error
3.2 Positional accuracy
Depending on the application, it is often essential that the image is kept within certain bounds (e.g. within the boundaries of a sensor). To determine the location of a light ray on the image plane we use paraxial ray tracing of multiple rays. Considering a situation where all optical components are symmetric around the optical axis, the positional error of ray j (i.e. the distance of ray j with respect to the optical axis on the image plane), here denoted by ε_{ j }, depends on the radial distance and angle of the ray with respect to the optical axis when entering the system, which will be denoted by vector r_{0}. Furthermore, it depends on all misalignments δ_{1},...,δ_{ N } of all reflective optics (N is the number of components) and system specific constant parameters p (initial optical path lengths d and angles of incidence ϕ), thus ε_{ j }(r_{0,j},p,δ_{1},...,δ_{ N }).
The lower order misalignments of a surface i in 2D, are the change in curvature κ_{ i }, the axial displacement δ_{z,i} (despace), the rotational misalignment 𝜃_{y,i} (tip/tilt), and the radial displacement δ_{x,i} (decenter). The decenter is directly calculated from the tangential displacement of the surface vertex. Other misalignments can be derived from the coefficients of the surface fit b_{ i }, which is calculated by the linear least square regression in (16) and shown in Fig. 3.
The radius of curvature \({R_{i} = \frac {1}{\kappa _{i}}}\) is assumed to be constant (parabolic) over the surface for small angular misalignments, that is \({\left (\frac {\mathrm {d}z}{\mathrm {d}x}\right )_{i} \ll 1}\), and defined as the reciprocal of the curvature κ_{ i }, which equals \({\kappa _{i} \approx \left (\frac {\mathrm {d}^{2}z}{\mathrm {d}x^{2}}\right )_{i}}\). The misalignments are stored as δ_{ i } = [R_{ i } 𝜃_{y,i} δ_{z,i} δ_{x,i}]^{ T }. Determination of the despace, tip/tilt and radius of curvature from the surface fit do not take into account the radial displacement distribution nor the average radial displacement (decenter) of the mirror surface.
3.2.1 Ray transfer matrix analysis
General misaligned ray transfer matrices for generalized misaligned space propagation and mirror surfaces (Yuan et al. 2011). Here R_{ i } is the radius of curvature, δ_{z,i} and δ_{x,i} are despace and decenter misalignments, 𝜃_{y,i} the tip/tilt contribution and d_{ i } the optical path lengths of a principal ray i with angle of incidence ϕ_{ i } on component i
3.2.2 Averaged positional error and sensitivities
3.3 Spot size and sensitivities
4 Numerical implementation
This section describes practical considerations of the implementation. The initial conditions are set such that designs are initialized with a uniform density field that exactly satisfies the volume constraint. The optimization problem is solved using the Method of Moving Asymptotes (MMA) (Svanberg 1987). The optical performance measures are relatively sensitive to design changes. Therefore, the algorithm is set more conservative (the move limit move is set to 0.1) in order to avoid large jumps in the design space.
5 Results
This section discusses two case studies applying the foregoing theory and demonstrating the validity of the proposed method. First, the focus is on the optimization of a single mirror mount to minimize SFEs. Next, the proposed method including the STOP analysis is tested on a twomirror case.
5.1 Singlecomponent surface form error minimization
The focus of this study is on the optimization of a flat mirror mount design subjected to a uniform temperature increase. The goal is to minimize the resulting spotsize due do both boundary and loading conditions. For a singlemirror system the spotsize is directly related to the SFE of the mirror surface, hence minimizing the SFE will suffice. The aim is to show that the proposed method works well for single component problems under various conditions. Both the boundary conditions and eigenfrequency constraint play a dominant role in the possibility to improve the optical performance. The study verifies the singlecomponent topology optimization procedure and investigates the influence of the eigenfrequency constraint on the resulting topology and performance.
5.1.1 Problem definition
The objective is to minimize the MSE of the optical surface Γ due to the specified thermal environment, while constrained by a minimum mean eigenfrequency and a maximum volume. Therefore, the mean eigenvalue (8) is adopted as a constraint such that the mean eigenvalue must be higher or equal than the minimum elastic mean eigenfrequency \(\underline {\omega }_{\mathrm {n}}^{2}\). A parameter sweep is performed over a range of minimum eigenfrequency constraints to investigate the influence of the eigenfrequency constraint. The range spans from minimum eigenfrequency constraints where the constraint is inactive up to values where the structure is unable to satisfy the constraint. Additionally, a constant volume constraint is added, in order to ensure a fair comparison.
5.1.2 Optimization results
5.2 Twomirror system spotsize minimization
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Uncoupled System (US) optimization, and
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Coupled System (CS) optimization, where the integrated SDO approach is applied.
Both optimization procedures make use of a full STOP analysis, though only the integrated approach considers the component interactions and applies both system and component level constraints. Note that the uncoupled optimization problem is not a typical design approach used in practice, since it does consider system optical performances. The separate components are however artificially decoupled with relation to the optical performance, in order to investigate the difference with respect to the coupled SDO approach.
The optimization aims to minimize the spot size error due to prescribed boundary conditions of the frame and thermal loads from the propagating beam, while bounded by a the position accuracy, mass, and eigenfrequency constraints. For multicomponent systems both position accuracy and spotsize depend on the rigid body motions and SFEs of all components involved. The main target is to investigate whether, and how, the components make use of the capability to interact and compensate for each others optical aberrations and how this may benefit the optical performance.
5.2.1 Problem definition
Consider the schematic structuralthermaloptical system consisting of two flat reflecting surfaces supported by optical mounts with design domains as shown in Fig. 2. An incoming converging beam with a perfect wavefront is reflected by two mirrors before it is focused onto a sensor with a theoretical spot size of zero (if it were unconstrained by the diffraction limit).
The mesh is structured and consists of 10000 Quad 4 isoparametric elements (4 Gauss points and bilinear shape functions) per domain. Each domain consists of Aluminium, with Young’s modulus E = 70 GPa, Poisson ratio 0.35, density ρ = 2700 kg/m^{3}, coefficient of thermal conductivity k = 250 W m^{− 1} K and CTE α = 25μm/mK^{− 1}.
Both mirrors are subjected to known rigid body movements from the housing, which is modelled as a rigid interface. The interfaces of the first mirror mount are considered constant at 1 K difference with respect to ambient conditions, whereas the interfaces of the second mirror are constant at 0 K difference. The first mirror is subjected to a decenter rigid body effect of δ_{x,1} = 200 μm and a the same amount of despace misalignment. The left side of the second mirror is moved out by 200 μm and down the same magnitude. The right side of the second mirror interface is also moved out by 200 μm. This causes the second mirror to initially have a despace and tip/tilt error.
Performance of the obtained optomechanical systems, and properties of their individual mirror mounts: RMS spot size, RMS positional error, volume, RMS SFE, mean eigenfrequency rigid body movements and mirror curvature. The values between parentheses indicate the value at the first iteration
The lower bound on the design variables is \({\underline {\mathbf {s}} = 10^{3}}\). The density filter radius equals the length of two finite elements and the Heaviside projection parameters are set to β = 1.5 and η = 0.45. The termination criteria are ε_{Δs} = 0.0015, ε_{Δf(s)} = 0.002, and ε_{g(s)} = 0.02.
5.2.2 Optimization results
The two approaches result in some significant differences, see Table 2 and Fig. 7. Whereas the curvature of the first mirror in both approaches is brought as close as possible to zero, the second mirror in the coupled approach is made even more concave than its original shape. Note that the spot size of US is not simply the average of both mirrors. The positional error constraint of UM2 is active, although the combined positional error of US is far below the allowable error. Both systems have satisfied the allowable mass, however, the integrated SDO has transferred mass from CM2 to CM1.
The optimization process has lowered the RMS spot size diameter from 0.4 mm in the first iteration to 106.5 μm and 4.7 μm for the uncoupled and coupled optimization approaches, respectively. Thus, the integrated SDO approach used for the coupled case proves a ratio of increase in performance of 22.6 times compared to the uncoupled method, equivalent to an improvement of 95.6%. Note that the integrated SDO approach is able to lower the RMS spot size to below the diffraction limit of the system, that means it is not possible to further improve the optical performance measure with a geometrybased performance metric.
6 Discussion and recommendations
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topology optimization of mirror mounts for the systems’ optical performance expressed in terms of spot size and position accuracy using a full structuralthermaloptical performance analysis procedure, and
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simultaneous topology optimization of multiple mirror mounts to exploit the interaction of aberrations between different components due to thermal loads and frame movements and minimize a system optical performance while constrained by dynamic stiffness, weight and optical performance measures, both on a component and system level.
The results of the first case study, shown in Fig. 5, indicate there is a certain bandwidth of minimum eigenfrequency constraint values that influence the ability of the optimizer to minimize the surface deformations. Thus, conflicting requirement tolerance values should be thoroughly investigated, as their limits can highly influence the topological layout and performance.
The results of the second study indicates that the uncoupled optimization aims to design two perfectly flat mirrors, whereas the layout of the second mirror in the coupled optimization is such that its misalignments (mainly curvature) effectively compensate for the misalignments of the first mirror, resulting in a spotsize improvement of over 95%, without reduction of any other performance aspect considered in the optimization.
The activity of the positional error constraint in the optimal solution of the uncoupled optimization (while the system easily remains within the accuracy limits) indicates the system is unnecessarily overconstrained, see Table 2. The SDO approach makes use of the enlarged feasible domain, which is apparent from the mass transfer between the mirrors in the optimal solution and the significant differences in topologies. Thus, the typical design approach unnecessarily overconstrains problems, whereas the SDO approach enlarges the feasible domain and gives the optimizer more freedom to minimize the objective while still satisfying all constraints.
During optimization the SFE constraints are not always satisfied, which means that optical analysis is based on optical properties that do not accurately describe the deformed surface leading to inaccurate results. However, the converged optimal solutions do satisfy the SFE constraints and hence accurately describe the system’s optical performance metrics. A more enhanced surface error determination method may result in faster convergence, and additionally a different optimal solution.
The RMS spot size diameter of the uncoupled system reaches the diffraction limit twice, halfway the iteration history, but the optimization continuous because component responses do not satisfy all constraints and termination criteria. On top of that, when the optimizer is able to decrease the objective function of the second mirror, the system spot size increases again. This indicates that the mirrors exactly counteract each others error during optimization, although the optimizer is unaware of this information. Thus, component interaction should be included to obtain a more optimal solution in terms of system performance.
Thus, the case study demonstrates that an integrated structuralthermaloptical design optimization procedure taking into account all system components improves the system optical performance compared to individual component optimization, while subjected to the same (or equivalent) design constraints.
The system designed using the SDO method keeps a considerable margin above the competitor and hence the loads may increase considerably before the system’s spot size diameter reaches that of the uncoupled variant. Note that the individual components designed using the SDO method are only applicable to this specific system configuration with these specific loads and boundary conditions. In general, the SDO method will result in designs that are more tailored to a specific case and generally are less robust with respect to other loading conditions not considered in the optimization.

application towards various different load cases to verify and validate the general applicability of the method,

simultaneous optimization of the housing and optical components and simultaneous optimization when their domains are merged into a single mesh, in order to give the optimizer the freedom to relocate the boundary locations,

extension to multiple, and different types of components, e.g lenses, gratings, prisms and initially curved mirrors,

extension to 3D structures and consideration of manufacturability,

including the uncertainty in both thermal and mechanical loading, i.e. robust design,

extension to multimaterial topology optimization to achieve higher performances as there are more possibilities to counter effect thermal expansion, create conductive isolation, as well as damp out external vibrations (Van der Kolk et al. 2017), and

enhancing thermal modeling and control by e.g. considering designdependent heat loads affected by the misalignments, including radiation influences, and simultaneously optimizing locations and input of active thermal components (heaters/coolers).
It is expected that the more components the system consist of and the stronger their interaction is, the greater the benefit of the SDO method will be. Allowing the optimizer to distribute unavoidable errors over multiple components in the system, instead of letting the designer impose the error budgets on each component, enlarges the feasible domain and the potential for superior system designs.
7 Conclusions
The key to satisfy next generation optomechanical system requirements is to not distribute error budgets over components a priori, but to consider and optimize the system as a whole. This allows for focus where it matters, without overconstraining the system unnecessarily. A structuralthermaloptical performance analysis is able to expose the performance metrics that matter for optomechanical systems without relying on intermediate derived performance indicators. For a single component system or multicomponent uncoupled optimized system, only minimizing deformations (and nothing else) leads to better optical systems. However, there is additional room for improvement when multicomponent systems are optimized in a coupled fashion as this allows for error compensation between components. Since the feasible design space of the system level optimization completely encapsulates that of the individual component optimization, the globally optimal performance of the coupled system is always better or equal to the uncoupled optimization approach. This is also shown by the results of the numerical example; coupled optimization based on the full structuralthermaloptical performance analysis is able to reduce the spot size of a twomirror system with 95% compared to uncoupled component optimization to below the system’s diffraction limit. The coupled analysis allowed the two mirrors to compensate for each others errors, which is a mechanism that would be otherwise invisible to the optimizer. Despite the fact that real systems are more complex than the simplified example considered in this study, it shows that optomechanical designers should aim for considering and integrating multiple components and physics simultaneously in the design loop, and thus apply the SDO approach, when requirements seem irreconcilable.
Footnotes
 1.
The number of optical elements in a system is generally larger than the number of optical components, since space propagation is also an optical element, i.e. \({\tilde {N} \geq N}\). In most optomechanical applications \({\tilde {N} = 2N + 1}\).
Notes
Acknowledgements
The authors thank R. Saathof and J. Day for their recommendations during the period of this research and K. Svanberg for the use of the MATLAB implementation of the MMA optimizer (Svanberg 1987).
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