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Abstract
We report an Optical Differentiation Wavefront Sensor based on a telephoto lens system and binary pixelated filters. It provides a five-fold reduction in the system length compared to a 4f system with identical effective focal length. Measurements of phase plates with this system are compared to measurements performed with a commercial low-coherence interferometer. The telephoto-lens-based system can measure wavefronts with accuracy better than λ/10 Root Mean Squared (RMS) at λ=633 nm. Experimental investigation shows that the system has a high tolerance to components alignment errors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Wavefront measurement is one of the key requirements in many areas of optics such as astronomy, metrology, and laser engineering. In metrology, an optical component can be characterized by measuring the wavefront change it induces. Iterative wavefront metrology is needed during manufacturing. Freeform optics are based on surfaces that have no rotational or translational symmetry [1]. They enable enhanced system performance and packaging due to the high degrees of design freedom they provide [1–3]. The manufacturing quality of freeform optics depends on the capabilities of the available metrology tools, but characterizing these optics remains challenging [4,5].
Shack-Hartman Wavefront Sensors (SHWS) are limited by their dynamic range and resolution for use in freeform metrology [6]. To overcome these limitations, modified SHWSs using additional components and advanced processing have been proposed [7–12]. Wavefront measurements based on phase retrieval have been applied for high-precision freeform optics characterization [13]. Interferometers aided with adaptive nulling components have freeform metrology capability [14–17]. Other emerging techniques are deflectometry [18,19], Optical Coherence Tomography [20] and point-cloud optical profilometers [21].
The Optical Differentiation Wavefront Sensor (ODWS) measures an input wavefront by modulating the far-field of the input field and processing the resulting near-field fluence [22–28]. The far-field is modified by a linear amplitude-transmission filter, and the wavefront slope in the direction of the filter’s gradient can be calculated from the resulting fluence. Two orthogonal directions of the filter can be used to obtain two sets of wavefront slopes that can be integrated using algorithms such as the Southwell procedure [29]. Advantages of the ODWS include dynamic range, spatial resolution, achromaticity, and relatively high signal-to-noise ratio, which are desirable features in domains such as freeform metrology and adaptive optics in astronomy [26,27,30,31]. A Shack-Hartman wavefront sensor’s resolution is limited by the pitch of the lenslets, and its dynamic range is limited by the ability to correctly identify the spots arising from wavefronts with high slopes. An ODWS has high resolution because it depends on the pixel pitch of the camera and has adjustable high dynamic range since it is proportional to the filter width, which can be practically very large. Whereas the standard ODWS relies on a filter with a spatially linear transmission for the field amplitude, a generalized Optical Differentiation Wavefront Sensor with nonlinear filters has recently been proposed to optimize the trade-off between dynamic range and sensitivity [32–34]. These demonstrations used a liquid-crystal-based light modulator, which is wavelength and polarization dependent, to introduce amplitude modulation in the Fourier plane of the object under test. They are based on a 4f configuration, in which two lenses of identical focal length are used to perform a Fourier transform from the input pupil to the Fourier plane, in which the filter is located, then from the Fourier plane to an image plane of the input pupil, where the fluence is detected.
An ODWS based on a 4f configuration and binary pixelated filters has been previously demonstrated [30,35]. In such system, the linear transmission filter is implemented using a spatially dithered distribution of small transparent or opaque pixels. This filter, implemented by lithography of a metal layer deposited on a glass substrate, is achromatic and polarization independent [36]. Tradeoffs between far-field size and size of the binary pixels can lead to impractically large system lengths. For practical applications, a system configuration enabling significant size reduction is needed. In this article, we demonstrate an ODWS based on a telephoto system that accurately characterizes freeform optics and significantly reduces the system length. We study its robustness to the misalignment of system components. Section 2 describes a model of the telephoto-lens-based ODWS. Section 3 presents the experimental design, evaluates the metrology of freeform phase plates, and analyzes the system tolerance to alignment errors. Finally, Section 4 concludes this article.
2. Telephoto-lens-based ODWS
Figure 1 shows an ODWS system, where the first two lenses form a telephoto system with effective focal length (EFL) f. The length from the vertex of the first lens to the focal plane is written as kf, where k (the telephoto ratio) is by design smaller than 1. A gradient transmission filter is placed at the focal plane of the telephoto system. The wavefront to be tested is at the input plane at a distance d0 from the vertex of the first lens. The third lens having a focal length f3 is positioned at a distance a from the filter to image the input plane on the detection plane at distance b.
Let (x1, y1), (x2, y2) and (x3, y3) represent the coordinate planes of the test object, filter, and detector, respectively. We assume the input field to have a constant amplitude A0 and a phase $\varphi $, i.e., $u({{x_1},{y_1}} )= {A_0}{e^{j\varphi ({{x_1},{y_1}} )}}$. This field propagates through the telephoto system and is focused on the amplitude filter whose transmission varies linearly according to $\frac{{{x_2}}}{W} + \frac{1}{2}$, where W is the filter’s width. The field after modulation by the filter propagates through the third lens and is detected by the camera. Fresnel propagation from input to output planes using the matrix method leads to the simplified relation between detector fluence and input wavefront slope
The accuracy of an ODWS implemented with a spatially dithered distribution of transparent and opaque pixels depends on the ratio of the far-field size and pixel size. Smaller pixels generally provide better accuracy because they provide a transmission that is closer to the ideal linear profile [30,35]. However, the pixel size is in general limited by fabrication technology and physical effects that can occur in components with features size of the order of the optical wavelength. The relative error on the pixel size, and therefore on the synthesized transmission profile, increases for smaller pixels because of fabrication tolerance. Increasing the focal length of the optical system in an ODWS allows for an increase in precision with spatially dithered pixels, but such increase can be impractical when using a standard 4f arrangement. The telephoto system solves this issue by offering a large EFL in a relatively small footprint. While the larger EFL yields a smaller dynamic range, this can be compensated in practice by using larger filters, which are allowed by conventional lithography techniques. The use of a telephoto lens with a large EFL could also benefit other implementations of the ODWS, in particular those based on pixelated liquid-crystal spatial light modulators.
3. Experimental demonstration
3.1 Experimental implementation
The experimental setup (Fig. 2) closely resembles the system layout presented in Fig. 1. Lenses 1, 2 and 3 are commercial off-the-shelf lenses (Thorlabs), chosen as plano-convex, plano-concave and plano-convex, respectively. The collimated input laser beam ($\lambda \; \,$ 633 nm) with a diameter of 20 mm propagates through a freeform phase plate which introduces a wavefront distortion. The beam then propagates through the telephoto system and is focused onto the transmission filter. The modulated beam propagates through the third lens and is captured on the CMOS camera (The Imaging Source) located at the image plane of the phase plate. The system design parameters are shown in Table 1 referring to the layout in Fig. 1.
Table 1. Design parameters for an ODWS implemented with a telephoto system.
The transmission filter is based on a spatially dithered distribution of binary pixels to synthesize the linear amplitude transmission. It is custom-designed using the four-weight error-diffusion algorithm [37] and fabricated as a metal-on-glass component via commercial lithographic techniques. Filters with pixel size ranging from 2.5 $\mu \textrm{m\; }$ to 10 $\mu \textrm{m}$ have been designed, fabricated, and tested. The 2.5 $\mu $m pixel size is the optimal choice in presence of detection noise and filter transmission non-linearity caused by fabrication uncertainty [35]. Therefore, the experiments in this article were performed using the pixelated filter with pixel size of 2.5 $\mu $m and width W equal to 1 cm. The range of slopes that can be measured around the mean slope is [-2.3, 2.3] waves/mm, and the dynamic range is 4.5 waves/mm.
Accurate wavefront reconstruction requires optimum far-field spot size that is proportional to the EFL so that the sampling from the binary pixelated filter is sufficient [30]. A telephoto EFL ∼3.5 m was chosen based on different experimental trials to reconstruct a plane wavefront accurately, i.e., when no aberration is induced on the input wave with a phase plate.
Referring to the layout in Fig. 1, for convenience in determining the experimental telephoto EFL, the system was aligned to be afocal, i.e., the distance between the filter and Lens 3 is $a = {f_3}.$ In this case, the magnification is $m = \frac{{{f_3}}}{{EFL}}$, and the experimental EFL can be determined from the measured magnification without the need to determine a or b accurately. Both ${f_3}$ and EFL determine the magnification of the system and hence the measurement resolution. The parameters of the optical system determined experimentally are m = 1/11.4044, EFL $(f )$ = 3421.3 mm, yielding a spatial resolution in the input plane equal to 50.2 $\mu $m based on the 4.4-$\mu $m camera pixel size.
The total length of this telephoto-ODWS system, from object to detection plane is ∼1.5 m. For a similar EFL and measurement resolution using the same camera, the length of an ODWS without a telephoto (${f_1} = 3.5\; \textrm{m},\; {f_3} = 300\; \textrm{m}$) would be 7.6 m. In this demonstration, the use of a telephoto system reduces the footprint by 5 compared to a 4f- ODWS system. More compact implementations with longer EFL are possible following the same concept.
3.2 Measurement of freeform phase plates
In order to assess the performance of the telephoto-lens-based ODWS, we have measured the wavefront induced by two phase plates. The phase plates primarily have coma aberration along with relatively small power and other aberrations. The aberration content of these two phase plates are 2.7 and 4.2 waves in terms of peak-to-valley (PV) wavefront, 0.4 and 0.67 waves in terms of root-mean-square (RMS) wavefront at 633 nm. These phase plates were fabricated with an UltraForm machine [38] and characterized with an UltraSurf machine (a non-contact coordinate measuring machine using a Lumetrics OptiGauge probe which uses low-coherence interferometry) [21]. Each phase plate was measured once with a spatial resolution of 250 $\mu $m. The UltraSurf data is interpolated to match the ODWS data resolution. A reference wavefront without any phase plate is measured with the ODWS to account for static aberrations present in the system. This reference wavefront is subtracted from each test wavefront. The difference in the wavefronts reconstructed by the ODWS and the UltraSurf is taken as a metric for accuracy validation, after numerical removal of the piston and tip/tilt.
The measurements of phase plate #1 by UltraSurf, ODWS and their comparison are shown in Figs. 3(a), 3(b) and 3(c), respectively. Similarly, the measurements of phase plate #2 are shown in Fig. 4. Each ODWS measurement is an average of 10 measurements. For lateral registration of the wavefronts, the centroid of the peak at center of the phase plates is used as fiducial. For clocking alignment, each wavefront is rotated in a range of angles and for each angle, Zernike polynomials are fit to the wavefront. The wavefronts primarily have coma aberrations, and we choose coma in the horizontal direction as reference. The angle that gives the minimum Zernike coefficient for horizontal coma is chosen for numerical rotation before calculation of the wavefront difference. The wavefront alignments were done separately for UltraSurf and ODWS measurements in MATLAB. In future, fiducials can be machined on the parts for wavefront alignment and easier comparison.
Fig. 3. Wavefront of phase plate #1: (a) UltraSurf measurement [PV: 2.66λ, RMS: 0.4λ]; (b) ODWS measurement [PV: 2.32λ, RMS: 0.37λ]. (c) Difference (RMS 0.09λ).
Fig. 4. Wavefront of phase plate #2: (a) UltraSurf measurement [PV: 4.25λ, RMS: 0.67λ,]; (b) ODWS measurement [PV: 3.8λ, RMS: 0.7λ]; (c) Difference (RMS: 0.09λ).
The telephoto-lens-based ODWS wavefront measurements are comparable with UltraSurf measurements; their RMS difference is 0.09$\lambda $ for both phase plates. This difference may be caused by imperfect registration and may include inherent error in either measurement technique. The ODWS measurements for each phase plate show an average precision of ∼λ/50, quantified as the RMS of difference between each wavefront map and the average of all the measurements.
3.3 ODWS system tolerance analysis
In this section, we analyze the error in wavefront measurement due to misalignment of the ODWS components along the optical axis. First, we give a description of the alignment of the telephoto ODWS system. Referring to Fig. 2, the three lenses are initially put at their respective positions according to the design in Table 1 and are optically aligned for tilting and centering. An object with known dimension is placed in front of Lens-1 at the design object distance and is imaged to a camera, from which the object location and magnification of the system are determined. The amplitude filter, mounted on a three-axis translation stage, is placed at the focal plane of the telephoto system. Its longitudinal position is determined by a knife edge test, using the sharp transition from opaque to transparent transmission at the edge of the filter. The filter is centered with respect to the beam using fiducials added at the four edges of the filter. During the alignment, a Shack-Hartman wavefront sensor is used after Lens-3 to ensure that the system is afocal. The exact EFL of telephoto is determined by dividing the focal length of Lens-3 by the magnification.
Referring to Fig. 1, the alignment uncertainties are primarily from the position of the object, filter, and detector. Hence for tolerance analysis, each of these components was separately moved from its ideal position over a range of [-8 mm, +8 mm], in steps of 2 mm (only one component was moved at a time while the other components were kept at their ideal positions). These longitudinal translations are facilitated by motorized translation stages.
Figures 5(a) and 5(b) show the wavefront measurement error in relation to variation of the object and filter position respectively. The measurement error is quantified as the RMS of the difference between wavefront measured at misaligned position and wavefront at ideal position. These figures show that the ODWS is tolerant to object and filter position misalignments. The maximum RMS error with respect to ideal position is within λ/10 for both phase plates. Considering that the range of misalignments ([-8 mm, +8 mm]) is significantly larger than the typical uncertainty during alignment, we conclude that this system can tolerate typical misalignments of the object and filter positions. Figure 6 shows that the wavefront RMS error also increases with the misalignment of the detector. It is within λ/12 and λ/20 for a misalignment range of [-8 mm, 8 mm] and [-2 mm, 2 mm], respectively.
Fig. 5. RMS difference of wavefront measurements from ideal position when varying (a) the object position, and (b) the filter position.
Fig. 6. RMS difference of wavefront measurements from ideal position when varying the detector position.
Figures 5 and 6 also show that the measurement of wavefronts with steeper slopes is more sensitive to misalignments of the object, filter and detector. Phase plate #2 has higher wavefront amplitude (1.6 times) and steeper slope (3.2 times for the maximum slope) at edge compared to phase plate #1. It therefore has higher error than phase plate #1. It is noted that the detector misalignment data were obtained with a reduced pupil diameter of 17 mm to compare the results for the same misalignment range. For a full 20-mm beam, the wavefront measurements were not valid when misalignment is larger than +4 mm as the much higher wavefront slope at the edge diffracts the beam away.
The presented tolerance analysis was performed for component misalignments along the optical axis. Object and detector position misalignments in the x and y directions result in transverse shifts of the measured fluence profiles. This can be handled during the registration process after wavefront reconstruction from the measured fluences. Transverse translation of the filter in the direction orthogonal to its gradient has no impact on the ODWS operation because this does not modify the modulation induced on the field. Translation of the filter in the direction of the gradient introduces a global tilt on the reconstructed wavefront. Such tilt is not a concern for components metrology.
4. Discussion and conclusion
The main advantages of the ODWS are its scalable dynamic range and high resolution compared to other wavefront sensors. As an example, a commercial Shack Hartman sensor WFS20-5C from Thorlabs has microlenses of diameter 150 $\mu $m, focal length of 4.1 mm and wavefront measurement aperture diameter of 5.4 mm. When measuring a beam/object with 20 mm diameter, considering the magnification factor of 3.7, the expected spatial resolution and dynamic range for wavefront slope will be 555.6 $\mu $m and ±7.6 waves/mm, respectively. For comparison with the implemented telephoto ODWS system for measuring the same 20-mm beam, the expected ODWS spatial resolution and dynamic range are 50.2 $\mu $m and ±11.5 waves/mm, respectively when the filter width is increased to 5 cm. The spatial resolution is less than one tenth and the dynamic range is 1.5 times that of the expected values from the Shack Hartman sensor. If the beam size is reduced, the ODWS dynamic range will be further increased proportionally because the effective focal length can be reduced by the same factor. Furthermore, commercial lithography allows for scaling the filter width to the level of tens of centimeters, if needed. For a filter size of 10 cm, the expected spatial resolution and dynamic range are 50.2 $\mu $m and ±23 waves/mm, which is significantly higher than what Shack Hartman sensor can offer.
We have formulated the concept of a compact ODWS based on a telephoto system and experimentally demonstrated wavefront retrieval with such diagnostic based on spatially dithered binary pixelated filters. The ODWS measurements of phase plates are consistent with those from a commercial scanning low-coherence interferometer. The total system length is five times shorter than a 4f-ODWS system of identical effective focal length, yet allowing for accurate metrology of free-form optics. We have shown experimentally that the demonstrated ODWS has an alignment tolerance for accurate measurements that can easily be met in practical conditions. This novel implementation of an ODWS with a telephoto-lens expands the domain of applicability for this technique. In particular, compact systems allowing for the metrology of components with large wavefront slopes as well as wavefront characterization for real-time adaptive optics in astronomy.
Funding
National Science Foundation (1711669).
Acknowledgments
We thank Scott DeFisher at OptiPro systems (Rochester, NY, USA) for manufacturing and measuring the phase plates. Portions of this work were presented at the Optical Design and Fabrication Congress in 2021, #OM3B.4.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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