1.1

Rotating Synthetic Aperture Technology

Whether ground-based or space-based, a telescope’s ability to resolve fine details in the imaged scene is a function of its light collecting ability. The root of this dependency can be traced back to the Rayleigh Criterion, which for a diffraction limited system with a circular aperture, relates angular resolution to aperture diameter and wavelength as shown in Equation 1.

The angular resolution of the imaging system is given by θ, the wavelength of light is λ, and the diameter of the imaging payload’s aperture is D. To achieve better imaging resolutions, the goal is to minimize θ, which ultimately translates into the need to maximize D. Traditional telescopes have light collecting areas that are circular in shape while some modern designs, including the James Webb Space Telescope, use segmented optics to approximate circular apertures.¹ Both traditional circular apertures and the more modern segmented designs are forms of filled aperture imaging systems, or imagers that sample the entire frequency space of the object scene in a single observation. Dilute aperture imaging systems sample the frequency space of the object scene through a sequence of observations. Typically, either the imaging system or target is reoriented between captures to introduce new portions of the target scene into the payload’s field of view (FOV). Application of image post processing algorithms to the sequence of captured images can reconstruct the complete image that would have been captured by the filled aperture system. At its core, rotating synthetic aperture (RSA) technology is an application of dilute aperture imagery to spacecraft design. Figure 1 shows a CAD rendering of a cubesat-scale RSA.

Figure 1.

CAD rendering of a cubesat-scale RSA.²

Given the requirement of dilute aperture systems to take a sequence of images with different relative perspectives to fully sample the object scene frequency space, the concept of operations (CONOPS) of an RSA space telescope is different from the CONOPS of filled aperture designs. RSA telescopes operate by pointing the principal optical axis at a target and tracking said target for the full duration of an observation. While tracking, the telescope rotates its rectangular aperture about the optical axis at a precisely controlled rate. Images are captured throughout the rotation. An observation maneuver is completed once the full spatial frequency content of the object scene is captured. In the case of a rectangular strip mirror, a 180° rotation is required to complete an observation.

RSAs have the potential to unlock the design space of extremely high spatial resolutions and high temporal frequencies without incurring significant mass and cost barriers. As discussed previously, RSAs are capable of producing complete images with angular resolutions equal to, and in some cases, better than those captured by circular aperture systems 2.1. This performance improvement is achieved while reducing the payload’s light collecting area by nearly a factor of eight. RSAs have the potential to become the next revolutionary design for space-based observing systems.

2.1

Factors impacting image resolution

To determine the theoretical best resolution of an imaging system, aberrations including defocus and distortion are overlooked to presume a perfect optical system. In doing so, the system’s limiting resolution becomes a function of just the physical behavior of light as it interacts with optical elements. Diffraction describes the deviation of light from rectilinear propagation. The laws of diffraction differ depending on the distance between the source and observation points from the diffracting element. When distances are small, near-field, or Fresnel diffraction takes place. However, since our focus is on space telescope applications, a special case of Fresnel diffraction called far-field, or Fraunhofer diffraction, is considered.

The conditional assumption of the Fraunhofer diffraction formula is that the Fraunhofer condition is met. This condition can be understood by considering figure 2 which shows the annotated geometry for a line of point sources and an observation point P. The Fraunhofer condition requires the lengths r_i to be linear with respect to the dimensions of the diffracting element. This requirement reduces the sensitivity of the diffracted wavefront to the phase component containing r_i. Writing r_i in terms of the y coordinate using a Maclaurin series expansion, we obtain equation 2.

Figure 2.

Annotated geometry for a line of length D of point sources and an observation point P.³

The nonlinear term in equation 2 can be considered negligible when R is very large. This implies that the distances from the source and observation points to the diffracting element must be large, hence the terminology “far-field” diffraction. While there is no particular distance for determining the necessary value of R for this assumption to hold, a practical rule of thumb is given in equation 3.

R represents the smaller of the two distances from source and observation point to the diffracting element, a is the greatest width of the diffracting element, and λ is the source wavelength. In the case of a space telescope, the distance from target source to aperture can be considered to be ∞. However, the distance from the telescope aperture to the observation plane of the instrument is often on the order of 1-10m and does not strictly meet the condition in equation 3. Nevertheless, in the context of the RSAs considered in this work, the Fraunhofer diffraction formula will still produce results that are accurate to the first order.³ An additional note is that strategic placement of lenses or shaped mirrors within the imaging instrument can collimate light from the other diffractive optical elements onto the detector causing R to appear to be ∞ as shown in figure 3.

Figure 3.

L1 is used to collimate light from source S so it appears to be coming from ∞. L2 is used to focus the various angular frequencies of diffracted light to points on the image plane at P and σ. In this manner, L2 acts as a Fourier transform of the diffracted light.³

The theoretical best imaging resolution of an optical system is defined in terms of the ability to resolve adjacent peaks in the irradiance distribution at the image plane. Most imaging systems have circularly-shaped apertures. Equation 4 provides the normalized irradiance distribution for a circular aperture as a function of angular separation, θ.

The irradiance distribution is normalized by dividing by the irradiance at the center of the pattern, I(0). The radius of the aperture diameter is a and the wavenumber is k = 2π/λ. The first order Bessel function, J₁, is evaluated at the quantity ka sin θ for a range of θ. Figure 4 plots the irradiance distribution for a circular aperture with radius of 35cm as a function of angular distance.

Figure 4.

Irradiance distribution for circular aperture.

For a rectangular-shaped aperture, the normalized irradiance distribution is described by equation 5.

With Y and X representing Cartesian coordinates in the image plane, a representing the length dimension of the rectangular aperture, b representing the width dimension of the rectangular aperture, and R representing the distance from the diffracting element to the image plane, α’ = kaX/2R and β’=kbY/2R. A symmetric irradiance distribution is produced when a = b. However, for a 10: 1 aspect ratio RSA mirror as the diffracting element, the irradiance distribution is elongated along the width (X) axis as shown in figure 5.

Figure 5.

Irradiance distribution for rectangular aperture.

For axially symmetric irradiance distributions, an Airy disk describes the central maximum and lower order ripple patterns. Each dark ring in the Airy disk corresponds to a zero in the first order Bessel function J₁. There are a variety of criterion that exist for quantifying the theoretical best imaging resolution of an optical system based on the irradiance distribution. Consider two point sources of light that are being imaged by an optical device onto an image plane. One of the most popular resolving criterion is the Rayleigh criterion, which states that the point sources are just resolved when the center of one Airy disk falls on the first minimum of the second Airy disk. The familiar expression for angular resolution for circular aperture optical systems given in equation 1 is the angular distance for which the Rayleigh criterion is satisfied. The lack of axial symmetry in the irradiance distribution for an RSA shown in figure 5 implies that equation 1 is not the appropriate definition for angular resolution. Instead, the length dimension of the mirror has a higher angular resolution than the width dimension. The higher resolution along the length dimension is considered as the maximum resolution of the corresponding RSA in this work. This assumption is made on the basis that the filled image that is obtained after multi frame image registration is the data product of interest with an angular resolution equal to that of the high resolution dimension of the RSA.

While the optical components of an imaging system limit the amount of information transferred from the scene to the focal plane array (FPA), the characteristics of the image sensor also play a role in defining the maximum image resolution. The Q factor quantifies the ratio of the maximum spatial frequency resolveable by the optics to the maximum sampling frequency set by the sensor as defined in equation 6.

The detector sampling frequency is ξ_N = 1/p and the cutoff frequency for a circular aperture imaging system is ξ_C = 1/[λ(f/D)].⁴ The maximum spatial frequency of the optics is called the optical cutoff frequency and corresponds to the y=0 point of the imaging system MTF curve as shown in figure 6. A typical rule of thumb is to sample the information presented to the FPA at the Nyquist frequency, or at Q = 2, where the best spatial frequency coverage is obtained between detector and optics without ambiguity. Q values less than 2 result in aliased Fourier domain components resulting in information loss. However, for RSAs, intentional clocking of the primary mirror relative to the FPA can avoid overlap of the image spectrum in the Fourier domain for designs with Q < 2. A detailed Fourier domain analysis for RSAs is out of the scope of this paper. However, non-overlaping spectra in the Fourier domain enables restoration of the filled image to the imaging system’s cutoff frequency. Additionally, low Q systems have fewer pixels across a target than high Q systems resulting in more relaxed integration time requirements.⁵

Figure 6.

MTF curves for an imaging system’s optics and detector. This system is detector-limited corresponding to a Q < 2 system.

Given that the final data product of interest is the filled image obtained after multi frame registration, an additional aspect that would effect RSA image resolution is the amount of overlap present in consecutive image frames captured during an observation. The regions of the target scene that are overlapped are sampled at slightly different object points enabling the use of multi frame super resolution algorithms. Super resolution algorithms reconstruct spatial frequency content beyond the cutoff frequency of the imaging system. The resolution in the overlapping regions scales with the number of images captured. However, the improvement in resolution plateaus as more images are captured.⁶ While many multi frame super resolution algorithms exist, literature focused on determining the upper bound for resolution improvement suggest an improvement of 1.6X the system cutoff frequency is possible.⁷

2.2

Cost modeling

Some of the earliest cost models for telescopes were produced by the prominent astronomer and optical scientist Aden Meinel.⁸ Meinel surveyed the costs and aperture sizes of a variety of optical and radio ground-based telescopes in the late 1970s and early 1980s. Even in these early works, Meinel appreciated the importance of relating the cost of a telescope to its primary aperture size, which is a principal design driver. Meinel made an important distinction in the specific cost referenced for each design. Instead of using total program cost, which depended on telescope location and various sources of overhead, the cost of the telescope hardware alone was related to aperture diameter. Meinel’s scaling law evolved from a D^2.0 to a D^2.7 relationship as more data points were incorporated into the data set and more accurate cost and aperture data were made available.^8–10 While Meinel’s models can successfully predict cost to the first order, its application should be limited to the scope of ground-based telescopes. This is due to the fact that an entirely unique set of engineering design challenges exist for space-based telescopes that are not present for ground-based designs. As a result, a new set of models have been created that include space-based telescope data that reference the relation between aperture diameter and cost in Meinel’s early cost model.

More recent works by NASA Marshall Spaceflight Center senior optical scientist Philip H Stahl are based on data collected on both ground and space-based telescopes. A multivariable parametric cost model for both ground and space-based telescopes was developed by considering a set of cost estimating relationships (CERs) that predict optical telescope assembly (OTA) cost and are not correlated with one another. Stahl’s most recent work establishes the most current version of the cost model.¹¹ Stahl’s cost model is given in equation 7.

The S/G parameter specifies whether the model is for ground-based or space-based telescopes by specifying a 0 or 1 respectively. The D parameter corresponds to the telescope primary aperture diameter, λ refers to the diffraction limited wavelength of the optical system, T is the OTA operating temperature, and Y is the current year. This model assume a circular shape for the telescope’s primary aperture. To put the OTA cost in context of the more general project cost, Stahl showed that the OTA accounts for just 10%-15% of the total cost of astronomical telescopes. Nevertheless, a nearly five-fold cost savings on a subsystem that accounts for 10%-15% of the total cost of a project has the potential to save millions of dollars.

2.3

Modeling RSA dynamics

The torque on a rigid body that is rotating with respect to an inertial frame and is undergoing no external disturbance torques is given in equation 8.

The spacecraft moment of inertia (MoI) tensor is denoted as I, the angular velocity vector in the body frame is denoted as ω^B, and the torque in the body frame is denoted as τ^B. The total required momentum exchange for a particular body with known mass and geometry properties is defined as the minimum amount of cumulative torque required to execute a given reference trajectory over time. Deviation from the reference trajectory will always occur regardless of the implemented control algorithm. Nevertheless, required momentum exchange can serve as minimum requirements for spacecraft ADCS. This work is focused on quantifying variations in required momentum exchange for RSA imaging maneuvers as a function of RSA aperture dimensions and spin rates. Therefore, the orbit and imaging target were kept constant for all test cases.

The inertia tensor of a rigid body is a 3x3 matrix that describes its mass and geometry properties. For rigid body dynamics, it describes the torque needed for a desired angular acceleration about a certain rotation axis. Unconstrained three rotational degrees of freedom about the body axes are considered in this work for modeling RSA dynamics. The inertia tensor is constructed relative to a set of principal axes about which torques act independently from one another. Figure 7 shows a simple mass model created in Solidworks of an RSA spacecraft with body axes annotated. The RSA body frame is defined with its origin at the center of mass (CoM), the z-axis pointing along the principal optical axis, the y-axis along the length of the strip mirror, and the x-axis completing a right-handed coordinate frame.

Figure 7.

RSA mass model used to obtain a moment of inertia tensor for various RSA aperture and AR dimensions.

Assuming the RSA rigid body can be modeled as three rectangular prisms, the inertia tensor of the RSA mass model can be computed analytically. Since the MoI tensor of interest is about the system’s CoM, the system CoM must be computed first. Equation 9 describes how the system CoM was computed relative to an inertial coordinate frame, I, with origin located on the back face of the RSA bus at the intersection point of the RSA’s YZ and XZ planes of symmetry.

Subscript 1 corresponds to the rectangular prism representing the RSA bus, 2 represents the right hand mirror, and 3 represents the left hand mirror. The CoM of the components are easily determined by inspection using symmetry. The mass of each component was computed by multiplying the assigned material density by the component volume. Berrylium with a density of 1, 850kg/m³ was assigned to the mirrors while T6061 aluminum with a density of 2, 700kg/m³ was assigned to the spacecraft bus.

The parallel axis theorem was used to calculate the MoI about the system CoM. Equation 10 describes the MoI tensor for each component about the respective component CoM with m_i corresponding to the component mass, W_i is the component width, H_i is the component height, and L_i is the component length.

Equation 11 describes the MoI tensor of each component about the system CoM with X, Y, and Z corresponding to the distance between component and system CoM.

The system MoI tensor was computed as the summation of the component MoIs about the system CoM. The inertia tensor for a circular aperture mass model was computed in a similar manner with the exception that the mirror component MoI was computed for a solid cylinder as shown in equation 12.

The radius of the cylinder is R_cyl. A central portion of the cylinder was removed by subtracting the MoI of a rectangular prism such that the cylindrical mirror fit around the spacecraft bus without any conflicting intersections.

3.1

Resolution objective function

The first objective function pertains to the resolution of the synthesized image produced by an RSA. Several assumptions were made when constructing the resolution objective function. It was assumed that image sequences are registered and synthesized perfectly without any artifacts, the optical system is free of aberrations, and the optical cutoff frequency of the synthesized data product is equivalent to that of a circular aperture system. With these assumptions in mind, the image resolution of an Earth-observing RSA is given in equation 13.

The CMV20000 focal plane array was used as the detector in this work. The CMV20000 is a global shutter CMOS detector that has been used in several imaging instruments on spacecraft including the Mars 2020 Perseverance Rover’s engineering cameras and the Orbiting Carbon Observatory 3 context cameras.^{12, 13} The CMV20000 has a square pixel pitch of Δ_x = Δ_y = 6.4μm/px and a focal length of 0.5m was used for all test cases. While the radius of the Earth, R_e, and the orbital altitude, h, remain constant, the elevation angle, θ_elv varies. It is common for science requirements to set a threshold for the minimum elevation angle required before image acquisition can begin. For this work, a minimum elevation angle of 40° was specified. The image resolution is minimized when the spacecraft is nadir pointing.¹⁴ While an RSA will capture a sequence of images over different elevation angles, the nadir pointing image resolution was used in this design optimization study. Figure 8 shows a plot of the RSA synthesized and circular aperture image resolution as a function of aperture diameter for a spacecraft in a 300km altitude circular orbit.

Figure 8.

Image resolution as a function of aperture diameter in a 300km altitude circular orbit.

3.2

Cost objective function

Stahl’s cost model given in equation 7 predicts the cost of an OTA using several parameters, one of which is the telescope diameter. The diameter term in Stahl’s model assumes a circular aperture shape and can therefore be related to aperture area. While the length dimension of an RSA can be thought of as the equivalent dimension to a circular aperture’s diameter, the area of an RSA’s mirror is significantly less than the area of a circular mirror. To account for this areal difference, the diameter of a circular aperture corresponding to the area of the RSA mirror under consideration was computed as shown in equation 14.

The RSA area for a given AR and diameter is computed as A_RSA = D²/AR. Aperture diameters from 0.2m to 2.0 meters along with ARs of 5 : 1, 10 : 1, and 20 : 1 were considered in this work. The remaining terms in Stahl’s model were defined as S/G = 1, λ = 0.5μm, T = 273K, and Y = 2017 and kept constant throughout all test cases. Figure 9 shows a plot of Stahl’s cost model for a traditional circular aperture and an RSA using the equivalent diameter in equation 14 as a function of aperture diameter.

Figure 9.

OTA cost as a function of aperture diameter.

3.3

Angular momentum objective function

The third objective function is the total angular momentum exchange required for an imaging maneuver. In order to compute this quantity, Systems Tool Kit (STK) from Analytical Graphics Inc. (AGI) was used to propagate a spacecraft in a 300km altitude low Earth orbit. A single observing target was defined and the orbit was defined such that the spacecraft passes directly overhead. The spacecraft’s attitude profile was defined such that its body frame z-axis pointed at and tracked the target point throughout the imaging maneuver over elevation angles from ±40°.¹⁴

In order to compute the total angular momentum exchange, the torque profile as a function of time must be computed. As shown in equation 8, the angular velocity and acceleration profiles of the spacecraft must be known. In order to obtain these profiles, the spacecraft attitude was recorded from the STK simulation and differentiated over the duration of the observation maneuver from elevation angles ±40°.¹⁵ Figure 10 shows the quaternion profile and angular velocity of the simulated spacecraft. The angular velocity was differentiated using the discrete definition of the derivative as shown in equation 15.

Figure 10.

Quaternion and angular velocity profiles for a spacecraft in a 300km altitude orbit performing a target tracking maneuver over elevation angles ±40°.

Torque profiles were computed according to equation 8. The magnitude of the torque was then computed. Given the relation between angular momentum and torque shown in equation 16, a trapezoidal sum was used to approximate the magnitude of the total angular momentum exchange over the imaging maneuver.

Figure 11 shows plots of total angular momentum exchange as a function of aperture diameter for the three ARs considered. RSA spin rates from 5°/s to 20°/s were considered. For a detailed discussion on spin rate ranges for Earth-orbiting RSAs, see.¹⁴

Figure 11.

Total angular momentum exchange for an imaging maneuver as a function of aperture diameter.