Advances in reconfigurable optical design, metrology, characterization, and data analysis

During the iterative sequence of design, fabrication and metrology, mathematical tools are required for describing surfaces and vector fields, such as distortion. Modal processing is one useful method for reducing a surface or vector distribution into a list of fit coefficients which can be accurately modeled by a computer [5–8]. Ideally the basis functions that make up these modal distributions are orthogonal, computationally robust, and efficient. Efficiency is critical for reconfigurable optical systems because the optical prescription is expected to substantially change and fast calculations permit rapid response. For dynamic metrology (example, section 4.1), numerical calculation speed may be the limiting factor in system performance.

One approach that meets these goals is employing a two-dimensional (2D) extension of Chebyshev polynomials to describe surfaces. Chebyshev polynomials have broad use in the optical community for applications ranging from phase unwrapping algorithms [9] to Gaussian beam diffraction [10].

Vector distributions are described by Chebyshev gradient polynomials [11], which can describe slope data, and Chebyshev curl polynomials [12] that describe rotational data. Both of these sets are orthonormal across rectangular apertures. If the common Laplacian terms are only included once, these gradient and curl polynomials can be used to fit any vector data in the rectangular domain.

Mathematically, the one-dimensional Chebyshev polynomials of the first kind (T) are defined as:

$\begin{equation}\begin{array}{*{20}{l}} {{T_{m + 1}}\left( x \right) = 2x{T_m}\left( x \right) - {T_{m - 1}}\left( x \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }\\ {{\textrm{where}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {T_0}\left( x \right) = 1,{T_1}\left( x \right) = x,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \;{\textrm{for}}{\mkern 1mu} {\mkern 1mu} - 1 \le x \le 1} \end{array}.\end{equation} \tag{ 1 }$

Two-dimensional Chebyshev polynomials (F) are expressed as:

$\begin{equation}{F_j}(x,y) = F_n^m(x,y) = {T_m}(x){T_n}(y).\end{equation} \tag{ 2 }$

Finally, the gradient and curl polynomials (G and C polynomials) [11, 12] are defined by the equations:

$\begin{equation}\begin{array}{*{20}{l}} {{{\vec G}_j}\left( {x,y} \right) = \vec G_n^m\left( {x,y} \right) = \nabla F_n^m\left( {x,y} \right) = \frac{\partial }{{\partial x}}F_n^m\left( {x,y} \right)\hat i + \frac{\partial }{{\partial y}}F_n^m\left( {x,y} \right)\hat j,} \\ {{{\vec G}_j}\left( {x,y} \right) = {T_n}\left( y \right){{T{^{^{\prime}}}}_m}\left( x \right)\hat i + {T_m}\left( x \right){{T{^{^{\prime}}}}_n}\left( y \right)\hat j} \end{array}\end{equation} \tag{ 3 }$

$\begin{equation}\begin{array}{*{20}{l}} {{{\vec C}_j}\left( {x,y} \right) = \vec C_n^m(x,y) = \frac{\partial }{{\partial y}}F_n^m\left( {x,y} \right)\hat i - \frac{\partial }{{\partial x}}F_n^m\left( {x,y} \right)\hat j{ },} \\ {\vec C_n^m(x,y) = {T_m}\left( x \right){{T{^{^{\prime}}}}_n}\left( y \right){ }\hat i - {T_n}\left( y \right){{T{^{^{\prime}}}}_m}\left( x \right){ }\hat j{ }} \end{array}.\end{equation} \tag{ 4 }$

In general, the polynomial sets are defined in the Cartesian coordinate system, with (x, y) axes. Terms m and n are indices of polynomial functions in x and y, respectively. They are both positive integer values in the range 0 to infinity. They are used to denote the order of the polynomial. For 2D polynomials, an additional variable, j, is introduced that denotes the order of the polynomial, in the same way that m and n denote polynomial order for 1D polynomials. Since the 2D polynomials are a function of 1D polynomials, j can be written in terms of m and n and this relation is described in [11]. The terms $\hat i$ and $\hat j$ represent unit vectors in the x and y directions, respectively. The terms with the prime symbol (') denote gradients, e.g. T' (x) is the gradient of T in the x dimension.

As an example of the vector polynomials, consider modeling imaging distortion with a quiver plot from a three mirror anastigmat system with rectangular aperture as shown in figure 1 [12]. The model is taken from the standard Zemax OpticStudio [13] library.

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The distortion map is fit once with the G and C polynomials and then for comparison, it is fit another time using Zernike-based S and T polynomials [14, 15]. For both sets of fittings, the first 12 (6 gradient and 6 curl) polynomials were chosen. However, the common terms in both sets (i.e. for which the Laplacian is zero) was only counted once.

To show the performance of the Chebyshev-based vector polynomial, residual error from both polynomial fits was calculated as the RMS (root mean squared) error in x and y position as well as percentage errors in both positions, calculated as follows:

% Error = 100% × [rms (real error)/RMS (reference position)] (6) .

Here, the reference position means the ideal (simulated) distortion, i.e. predicted—real position from the distortion map.

As can be seen from table 1, the G & C polynomial based fitting provides a more accurate representation of the ideal distortion map, with lower error in both x and y positions.

These Chebyshev-based polynomials maintain several advantages [11, 12] over other sets of polynomials. To start, the vector sets can be derived straightforwardly from the scalar sets without needing any orthogonalization techniques, such as Gram–Schmidt orthogonalization. Additionally, Chebyshev polynomials have well-known recursion relations that can be computed very quickly and with minimal loss of numerical precision. For the design and analysis of reconfigurable optical systems, these polynomials offer significant advantages.

For some reconfigurable optical systems utilizing multiple beam paths sharing a common optical component (e.g. Dynamic K-mirror system in section 2.3) freeform optics are often used in order to correct configuration-dependent wavefront aberrations. Traditional optical design approaches require human's intuition in order to select the optimal location and numbers of the freeform optics, which significantly impact the overall manufacturing cost and alignment tolerancing. The modal mathematical framework (section 2.1) combined with an objective freeform surface selection process provides an efficient optical system design and optimization capability.

When evaluating a design to determine if breaking rotational symmetry and employing freeform surfaces [16] is worth the effort, a data-driven methodology should be employed. The decision to increase the number of degrees of freedom through freeform surfaces can yield reduced volume, increased field of view, or increased imaging performance [17, 18], but typically comes at a high cost. Therefore, careful choice of which surfaces to apply freeform terms is necessary in order to maximize their impact.

To guide this selection process, a parametric fitness function is developed using modal wavefront fitting [19, 20]. The fitness function combines multiple metrics from aberration control to manufacturability to provide a single data-driven metric. The detailed prescription of the fitness function and its actual optical design application is available [19, 20].

Equation (5) is one form of the fitness function proposed,

$\begin{equation}f = {w_1}{U_{{\text{fr}}}} - {w_2}\sqrt {{{\left( {\Delta S_{\text{f}}^{{\text{RMS}}}} \right)}^2} + {{\left( {\Delta S_{\text{r}}^{{\text{RMS}}}} \right)}^2}} + {w_3}\Delta S_x^{{\text{RMS}}} + {w_4}\Delta S_y^{{\text{RMS}}} - {w_5}F_\Delta ^{{\text{PSD}}},\end{equation} \tag{ 5 }$

where ${U_{{\text{fr}}}}$ is the minimum fractional overlap between the forward $\left( \,\,f\,\,\right)$ and the reverse $\left( r \right)$ data, which captures the input data quality. $\Delta S_{\text{f}}^{{\text{RMS}}}$ and $\Delta S_{\text{r}}^{{\text{RMS}}}$ are parameters capturing how well a single surface represents the ensemble of all input data at the surface. $\Delta S_x^{{\text{RMS}}}$ and $\Delta S_y^{{\text{RMS}}}$ are parameters that capture the amount of potential aberration correction over the entire field of view at the surface. $F_\Delta ^{{\text{PSD}}}$is the parameter to represent the difficulty manufacturing the surface. Each parameter set has its own unique weight and sums together to provide a single metric to evaluate the surface, where a large positive value of the function is optimal.

The data used to inform the selection comes from forward and reverse ray trace data at every surface of interest within the design. An aberrated optical system will have discrepancies between these data, while a non-aberrated system will have identical data, as shown in figure 2.

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The typical goal of optimization is to take an aberrated system and turn it into a non-aberrated system. Therefore, throughout the process of optimization, as more freeform terms are employed, the data input into the fitness function will converge.

To validate efficiency of the computational tools, the parametric fitness function is employed on a real-world project, the tomographic ionized-carbon mapping experiment (TIME) [21, 22] including a reconfigurable dynamic K-mirror configuration. The dynamic system is needed to de-rotate the field on the sky.

Shown in figure 3 is an optical design optimized in Zemax that utilized freeform surfaces on K2, P1, P2, and C1. This system enables a linear field of view mm-wave experiment on the 12 m Atacama large millimeter array (ALMA) [23] prototype radio telescope.

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The mirrors for TIME and their mechanical structure have been fabricated out of aluminum, as shown in figure 4. The mechanical position of the mirrors and sled was set by a laser-tracker, first in a laboratory setting (a) and (b), and then again within the telescope cabin (c) and (d).

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The mirror surfaces designed using local optimization guided by human experience and intuition were compared to a local optimization process guided by the parametric fitness function defined in equation (5). (Note: The parametric merit function was defined using the standard optical design merit function editor communicating with customized Matlab function.) The fitness function was used to guide the local optimization in an optimal manner, and a worst-case manner to demonstrate its ability to objectively guide. In all three cases, the same merit function was used for the local optimization process. A smaller merit function means that the design has better satisfied all constraints. The final merit function value in all three cases is shown in table 2, where the optimal fitness function was able to find a lower local minimum than the human designer experience.

Near-flat or convex optical surface metrology using traditional deflectometry often requires an impractical display or light source size, which has been one of the most challenging dynamic range issues in precision deflectometry approaches. In this section, we discuss a method to simplify the metrology setup.

Initially, deflectometry was utilized for testing concave surfaces, or surfaces which are only very slightly convex. This is due to the nature of deflectometry, which requires that a source area is large enough to satisfy a ray path from the source to the full optical aperture area under test, and then into a camera after reflection. Even for extremely large optical surfaces, such as the 8.4 m Giant Magellan Telescope primary mirror segments [31], the required source area can be as small as 220 mm × 40 mm for a full aperture test [32] if the source and camera are placed near the center of curvature of the optic.

For optics with convex surfaces or convex regions, the required source area can range from very large to infinitely large depending on the surface curvature. For reflective optics, it is impossible to place the source and camera at the center of curvature of the optic. To solve the issue of testing challenging optics such as convex optics using deflectometry, a methodology known as infinite deflectometry (ID) achieves a 2π steradian measurement volume [33]. Other systems have tackled this problem but usually rely on a large number of screens [29] which increases the chance for calibration errors and minimizes measurement flexibility.

ID achieves its wide testing range by creating a virtual source enclosure around the test optic. To do this, a source is tilted to create a ramp over the test optic, and a camera is mounted facing down towards the optic. In this configuration, a portion of the optic under test is measurable. To close the loop and measure the full aperture, the test optic is mounted on a precision rotation stage and clocked a fixed interval, which creates a new 'virtual' deflectometry system at each angular position.

By sweeping through the full 2π rotation, screens of the virtual systems create a 'tipi' source enclosure around the optic under test and allow for testing an extreme range of surface slopes, from concave to convex. The basic principle of the ID system, from the physical hardware to the virtual source enclosure, is shown in figure 5.

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The ID system has been used to successfully test both fast convex optics and highly freeform optical surfaces. In one test scenario a fast F/1.26 large convex optic with a 50 mm diameter clear aperture was measured using both an ID system and a commercial interferometer as shown in figure 6.

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The interferometer was limited to measuring a 45.29 mm subaperture of the optic. Therefore the reconstructed surface maps of the 45.29 mm diameter common aperture were used for comparison, with the standard Noll-ordered Zernike terms 1:4, 1:6, 1:21, and 1:37 removed. The results demonstrated a strong match between both test methods across all spatial frequencies. After removing Zernike terms 1:37, the interferometer reported 18.48 nm RMS surface height while the ID reconstruction reported 16.26 nm RMS, demonstrating the system can achieve optical quality testing for large, fast, convex optics [33].

It is worth to note that the interferometric and deflectometric approach come with their own advantages and limits supplementing each other. The scanning ID hardware setup costs only a few thousand dollars comparing to the around tens of thousands dollar interferometric system for a similar accuracy. However, the rotational scanning process of the deflectometry takes much longer time (e.g. minutes to hours depending on the unit under test) compared to a single measurement time of interferometry (with limited measurement area). For the stitching interferometry system, the overall measurement time and cost will greatly depend on the overall system approach and architecture.

For a more challenging test, a highly freeform Alvarez lens was measured using the ID system. Because no null optic was available for this surface, and the fringe density with a spherical null exceeded the range of the available commercial interferometers, the surface was cross checked using a KLA-Tencor Alpha-Step D-500 profilometer. The profilometer profile measurement, SP, and the height of the same profile taken from the Alvarez ID map, are reported. The results of the ID reconstruction, the designed Alvarez lens, and the profilometer measurements are shown in figure 7.

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With these tests, the ID method has proven to be a viable test solution and expands deflectometry to providing optical quality testing for convex optics and highly freeform optics which are traditionally challenging to test.

A challenging area of optical fabrication process is the initial shaping process (e.g. generating or grinding). Aggressive processing can quickly remove material, but finding suitable metrology tools to measure the performance is difficult. Most optical metrology systems rely on visible wavelength reflections, but this light is strongly scattered with little useful signal.

Early in the fabrication process, deflectometry with infrared radiation enables optical figuring of a ground surface [34, 35]. This greatly reduces the cost and time for fabricating freeform optics since later polishing stages remove material up to a thousand times slower than during grinding.

Demonstrations of optical figure convergence highlights the usefulness of infrared deflectometry, but understanding the uncertainty in the measurement is important. To do this it is necessary to compare deflectometry measurements to another established metrology system, such as was done with a 6.5 m diameter, rotationally symmetrical conic mirror using long-wave infrared (LWIR,7–12 µm) light [36].

The mirror was clocked in 30° steps to average out systematic errors in the deflectometry system. The computed shape deviation from the ideal conic surface is shown in the left image in figure 8. In this data, power is the largest shape error. The mirror substrate is borosilicate glass, and temperature gradients are known to change the surface figure. The temperature gradient in the mirror was measured simultaneously with the deflectometry data acquisition, and a thermal deformation model was used to compensate for the these effects. The surface error after thermal compensation is shown in the right side of figure 8.

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Figure 9 shows the difference between the deflectometry and interferometric data. The RMS difference is 0.73 μm, and most of this is power. When power is removed, the total error is 0.33 µm RMS. For comparison, the overall surface shape measured by interferometry shows 0.45 µm RMS of mostly astigmatism. The infrared deflectometry is able to detect that level of astigmatism [37]. To consider high order error the first 37 Zernike terms (Noll order) are subtracted from the discrepancy between deflectometry and interferometry. This data has an RMS value of 0.04 μm which is in good agreement with the 0.02 µm RMS of high spatial frequency features measured by interferometry. While there are some artifacts with 30° spacing due to the clocking of the mirror during testing, the residual is quite small. This shows that infrared deflectometry can accurately measure mid spatial frequencies.

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This data shows infrared deflectometry can achieve sub-micron surface accuracy and could be used to polish an optic to a few hundred nanometers RMS error. However, its primary benefit is early in the manufacturing process where material can be removed much faster but other visible-light metrology techniques fail to operate.

On a subsequent project, the infrared deflectometry hardware was reconfigured to guide the grinding process of a different mirror that is 1.5 m diameter. This new mirror was received with spherical surface shape and the grinding process reshapes the surface to be parabolic. Figure 10 shows two images of the surface error during fabrication. On the left is the surface after generation, and before grinding. The mid-spatial frequency structure left by the generating tool is clearly visible. On the right is the surface error during fabrication.

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These two data sets in figure 10 demonstrate two key aspects of infrared deflectometry. The first is that the figure error can be measured very early in the fabrication process. The other is the wide dynamic range and easy reconfigurability of the instrument. The data on the left shows the error of the surface with respect to the sphere that was generated. The data on the right shows the error of the surface with respect to the desired parabola. The only change that was made when switching between the two ideal shapes was in the processing software.

Null solutions, such as LWIR interferometers, are commercially available. However, a large number of custom null optics would be required to measure and guide fabrication of a continuously evolving workpiece. LWIR deflectometry is advantageous for its flexibility during the optical shaping process.

Finding a good thermal source for LWIR light has historically been a limiting factor. The size of a deflectometry source fundamentally limits the dynamic range of measurable slopes and the uniformity directly limits measurement precision, while the signal to noise ratio (SNR) limits the testable range of surfaces. The implications of a low-quality source are particularly insidious for IR deflectometry because the canonical source, a hot scanning tungsten ribbon, is prone to flexing at higher temperatures and longer ribbon lengths, and cannot efficiently scale LWIR spectral flux with higher applied input power.

To extend the reach of infrared non-null measurement to high-sag optics and improve the source SNR, a scalable source architecture called LITMIS, or long-wave infrared time-modulated integrating-cavity source shown in figure 11 was developed [38, 39]. The design utilizes small, self-contained, input nodes which act as pseudo-blackbody radiators when an electric current is applied. LITMIS flexibly places these input nodes into an aluminum integrating cavity, which diffusely emits light from a custom exit aperture.

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An advantage of this architecture is that the input sources have relatively low thermal mass and can therefore be quickly time modulated. Digital control enables binary modulation of the source so background noise can be subtracted in-situ. Because of the large amount of background thermal radiation present in most test scenarios, up to date background removal is critical for accurate and high SNR testing.

The LITMIS source has been tested against the traditional tungsten ribbon IR deflectometry source, in which a ground glass optic (Glass¹⁵⁰⁰), a machined aluminum disk (Al_Room), and a machined aluminum disk heated to 150 °C (Al₁₅₀) were all tested using both sources.

The reconstructed surface topology maps of all three test cases are shown in figure 12. The LITMIS source was able to fully test all of the surfaces, achieving 2–5 times larger SNR for the tested optics as compared to the tungsten ribbon. The tungsten ribbon however did not have a sufficient SNR to fully measure the Glass¹⁵⁰⁰ optic. Notably the Al₁₅₀ sample could not be measured using the ribbon source due to high thermal noise generated, while the LITMIS source provided sufficient data due to the in-situ background removal of the time modulated source.

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These demonstrations of the LITMIS architecture show that the new source provides a reliable, high SNR source for infrared deflectometry. The ability to create custom emission apertures further adds to the reconfigurability and flexibility of long-wave infrared deflectometry systems.

When you see an image through a mirror, the angle of the mirror determines the area where you can see. If you change the tip-tilt angle of the mirror, the image moves. As a slope-measuring device, configuring deflectometry for orientation measurement is straightforward.

Computationally, a sinusoidal image is created on the source. As the mirror moves, the image motion produces a phase shift in the sinusoidal pattern, and the tip-tilt angle can be calculated from the phase shift. The parameters necessary for the alignment calculation are the distance between the image pattern and the mirror, and the amount of phase shift due to the misalignment.

A proof-of-concept test system comprised of seven hexagonal mirrors is illustrated in figure 13 [42]. Each hexagonal segment can be independently tilted. This type of scenario can occur for active optics systems in large, segmented-mirror telescopes such as the James Webb Space Telescope [43] or the Thirty Meter Telescope [44].

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As a gold-standard reference to compare performance, an autocollimator was aligned with a flat mirror M4 in figure 13, and then the system was oriented to see the reflected pattern through the camera. While scanning the M4 mirror angle, the measured angle from both deflectometry and autocollimator were monitored concurrently and compared. As presented in figure 14 (bottom), the difference between the autocollimator and the deflectometry system is about 0.8 µrad RMS, which is smaller than the noise (error bar) of the data.

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To highlight system adaptability, tip-tilt values were measured in real-time at ∼15 Hz. As shown in figure 15, an actual change in tilt angle on M4 was applied after 5 s.

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Meanwhile, a global perturbation was introduced around 12 s and 15 s into the measurement (two red boxes in figure 15). By monitoring the entire seven segments altogether, the environmental perturbation is clearly recognized as a common motion and compensated for the real M4 orientation change relative to the other segments. The simultaneous measurement capability allows relative orientation data to be distinguished from environmental perturbation [42].

Presently the real-time deflectometry system is limited to measuring tip/tilt alignment only. However the presence of many pixels on each mirror segment, and efficient parameterization methods such as the Chebyshev polynomial functions (section 2.1) open the possibility for higher order shape errors to be measured in real time.

Dynamic deflectometry methods have found several applications in industrial and scientific environments, enabling new processes with potential to enhance time to market as well as production quality and speed. An example of an industrial application under development is the so-called IEMTF or 'induction forming', in which magnetic fields work to heat and shape a workpiece into a compound curve formed in an computer controlled adjustable mold [45, 46]. In this particular application, infrared deflectometry show in figure 16 is proposed to measure the shape quality of the produced panels in real time, being able to adjust the mold and accommodate for incremental steps until achieving the desired final shape.

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In this infrared metrology system, the LITMIS (section 3.3) light source can be applied and the illuminating light can be temporarily modulated. The metrology signal (i.e. deflectometry pattern) can be distinguished under varying temperature workpiece conditions during the heating process. This closed loop manufacturing method can produce several different shapes in the same production line with minimal setup changing, and because the shape of the produced pieces is being monitored while it is being thermoformed, real time changes can be performed reducing mold wear, spring back, and other types of effects that may introduce variations in the desired shape quality.

The induction forming has been proposed to produce a variety of complex shaped metallic panels with applications in several industries. Reflector panels for high frequency radio telescopes and satellite downlinks are currently under development in the University of Arizona [47]. Other applications include compound curves for architectural cladding systems, solar power metallic reflectors and different pieces for automotive, marine and aerospace applications.

As a final example of dynamic measurement, the OASIS (Orbiting Astronomical Satellite for Investigating Stellar Systems) concept is presented. This system utilizes a 20 m inflatable aperture for next-generation terahertz space telescopes [48, 49]. A schematic of the concept is shown in figure 17, and builds on heritage from the 14 m inflatable aperture experiment (IAE) mission.

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As a technology demonstrator, a 1 m class OASIS Mylar spherical reflector has been successfully manufactured, inflated, and tested at the University of Arizona as shown in figure 18 [49]. The Hencky surface shape and its stability must be calibrated as a function of time, balloon pressure, and thermal environment. Reconfigurable, non-null deflectometry is an effective tool for characterizing the surface variations because the hardware is low cost, provides high spatial resolution and does not require a customized nulling configuration such as a computer generated hologram is needed for interferometry.

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The future of very large, space-based telescopes will enable terahertz observations to study the origins of stars, planets, molecular clouds, and galaxies. The OASIS demonstrator is a direct application of a dynamic metrology system's continuous monitoring capability.