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Abstract
In this paper, we propose an extended-aperture Hartmann wavefront sensor (HWFS) based on raster scanning. Unlike traditional HWFS, where there is a trade-off between the dynamic range and spatial resolution of wavefront measurement, our extended-aperture HWFS breaks the trade-off and thus could achieve a large dynamic range and high spatial resolution simultaneously. By applying a narrow-beam raster-scanning scheme, the detection aperture of our HWFS is extended to 40 × 40 mm2 without using the enlarging 4f relay system. The spatial resolution of our setup depends on the scanning step, the pinhole size, and the wavelength. The sensitivity and dynamic range can be adjusted flexibly by varying the axial distance between the pinhole plane and the imaging sensor plane, because our decoupled large dynamic range could be reasonable traded-off to achieve better sensitivity. Furthermore, compared with tradition HWFS, our method does not need to compute the positions of a two-dimensional spots array where complicated spots tracking algorithms are necessary to achieve high dynamic range, thus remarkably reduces the spots aliasing issue and the computational cost. It should be noted that this scheme is not only applicable for HWFS but also for Shack-Hartmann wavefront sensor (SHWFS) with microlens array to achieve higher accuracy and better power efficiency. Experiments were performed to demonstrate the capability of our method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Compared with interferometer methods in wavefront sensing, Hartmann wavefront sensors (HWFS) perceive the wavefront slope instead of the optical path length difference (OPD) and have advantages of relatively simple, low cost, compact, robust and insensitive to vibration [1,2]. The HWFS have been used in a variety of applications including ophthalmology, laser beam quality measurement, optics testing, and optical system alignment [3–5]. In conventional HWFS there exists a well-known trade-off between the dynamic range with the measuring sensitivity and with the spatial resolution, where the increase of dynamic range results in a loss in measuring sensitivity or spatial resolution, and vice versa [6,7]. Specifically, in conventional HWFS the dynamic range mainly depends on the pitch size on the imaging sensor which is the separations of the micro-apertures of the aperture array, and the axial distance from the plane of aperture array to the plane of imaging sensor. A larger pitch size increases the sampling distance and thus reduce the spatial resolution. A longer axial distance between the aperture array and the imaging sensor increases the measurement sensitivity while sacrifices the available pitch size and thus decreases the dynamic range. Dynamic range, sensitivity, and resolution are important factors for the wavefront sensor, and the coupling among them prevent the traditional HWFS from being applied in situations where optimal performance in all the factors are important. [8]. Many of the former works had been devoted to increasing the dynamic range [9–12], yet they did not solve the entanglement issue radically and their capacities of extending the dynamic range are limited. Moreover, in conventional HWFS, spots tracking is an important process determining whether the wavefront slope under test is out of the measuring range of the HWFS and greatly influences the correctness or even success of the subsequent wavefront reconstruction procedure. In order to exploit all their potential on enlarging dynamic range for their formerly proposed schemes, the corresponding spots tracking algorithm has to be developed simultaneously and seemingly tends to increase the risk of aliasing in spots tracking and also increase the computation cost eventually [13,14]. On the other hand, to extend another significant factor, the detection area or the aperture, the common way is to use a 4f relay system to scale the incident beam size before the entrance of the HWFS or use a larger size imaging sensor [15]. However, using the 4f relay system not only perplexes the measuring light path but also introduces some unwanted aberrations, from the optics alignment and compensation operations of the 4f system [16,17]. It should be noted that a larger size imaging sensor is hard to obtain and expensive [18].
In this paper, we propose an extended-aperture HWFS based on raster scanning RS-HWFS. The large dynamic range and high resolution could be achieved simultaneously that breaks the trade-off in traditional HWFS or SHWFS. Our approach is theoretically different from the stitching methods which raster scan the whole aperture with a HWFS and then stitch all the fractionally reconstructed small wavefront maps together to form a large wavefront map [19,20]. By applying a narrow-beam raster-scanning scheme, the detection aperture of our HWFS is extended to $40 \times 40 mm^2$ without using the enlarging 4f relay system. The spatial resolution of our setup depends on the scanning step distance, the pinhole size and the wavelength. The sensitivity and dynamic range can be adjusted flexibly by varying the axial distance between the pinhole plane and the imaging sensor plane, because our decoupled large dynamic range could be reasonable traded-off for a better sensitivity. Furthermore, compared with tradition HWFS, our method does not need to compute the positions of a two-dimensional spots array where complicated spots tracking algorithms are necessary to achieve high dynamic range, thus remarkably reduces the spots aliasing issue and the computational cost. It should be noted that this scheme is not only applicable for HWFS but also for Shack-Hartmann wavefront sensor SHWFS with microlens array to achieve higher accuracy and energy saving purposes. Experiments had been performed to demonstrate its feasibility and practicability. The targeting application of our method is for optical testing where large field of view, dynamic range, and good spatial resolution are desired simultaneously. As an alternative method, RS-HWFS does not need a compensator for free-form lens wavefront measurement and is more robust and adaptable than interferometer. We expect that it could be applicable for examining free-form lenses, of which the wavefront profile may range from mild to extreme that urges to be dealt with both relatively high resolution and large dynamic range, which is still a challenge for present commercial HWFS [21].
2. Principles
The setup for the conventional HWFS is shown in Fig. 1. As we can see, the collimator flattens the spherical wave emitted from the point-shape light source after a pinhole, generating the incident reference wavefront. The collimated incident wavefront will then be reshaped after passing through the transparent sample and carry its profile information. The reshaped wavefront will be discretely sampled by the pinhole array and divided into many small spots. The x-component and y-component of the wavefront slopes can be calculated as:
Fig. 1. The illustration of the conventional HWFS setup that mainly includes a light source, a pin-hole, a collimator, a pin-hole array and an imaging sensor.
Our proposed RS-HWFS is illustrated in Fig. 2. Its main difference from conventional HWFS is that it replaces the pinhole array with a single pinhole, and the sample is driven by a two-dimensional translation stage(not shown in the figure) and could laterally travel along the two orthogonal x and y directions. To complete a wavefront detection with RS-HWFS, raster scanning over the sample should be performed, through which the measuring aperture size is extendable and controlled by the two-dimensional translation stage mounted under the sample. In addition, the spatial resolution could be appropriately chosen by adjusting the scanning step size. Note that the beam size of the RS-HWFS is smaller than that of conventional HWFS, which enables easy alignment measurement and thus easier to keep the flatness of the reference wavefront. Using a single pinhole, the RS-HWFS needs to retrieve only one spot position which is more robust than measuring a set of spots positions produced by the pinhole array in conventional HWFS. However, to measure the whole wavefront, the RS-HWFS has to scan the sample instead of acquiring the complete wavefront data in only one shot. Although this strategy loses the capability for real-time acquisition, the RS-HWFS has many advantages including extendable measuring aperture, breaking the trade-off between dynamic range and resolution, and avoiding complicated spots tracking algorithm.
Fig. 2. The illustration of the RS-HWFS setup. Its main difference from the conventional HWFS is that the pin-hole array is replaced by a single pinhole, and the sample is driven by a two-dimensional translation stage(did not draw in the figure) and can laterally travel along the two orthogonal x and y directions.
The imaging sensor plane is usually separated into dozens of equivalent small sections allocated to every pinhole for tracking the corresponding spots in conventional HWFS. When the dynamic range of the HWFS is enough for the slopes of the wavefront under test, all spots on imaging sensor could be positioned inside their corresponding pitches, which ensures the success of the spots tracking process. As we can see in Fig. 3(1), all the nine spots appear inside their pitches so that we can successfully calculate all their centroid positions within every pitch denoted by $(x_n,y_n)$. When the slopes of the wavefront under test reach or are larger than the dynamic range of the HWFS, some spots representing these local slope areas may escape to their neighbor pitch from their own one. On the imaging sensor plane, this situation could result in some errors for the spots tracking process called the spots aliasing, which perplexes the spots centroid calculation. As we can see in Fig. 3(2), the top left spot totally escapes to a neighbor pitch and the centroid calculation is null for that pitch because there is no spot could be detected; another situation shown in Fig. 3(2) is that the middle left pitch contains two spots inside the boundary and may cause an error of centroid calculation because of the spots aliasing.
Fig. 3. Demonstration of the Spots tracking aliasing in conventional HWFS. (1) is the situation of successful spots tracking and (2) is the situation of unsuccessful spots tracking or the spots aliasing
From the above demonstration we can see that the dynamic range is majorly relevant to the half pitch length allocated on the imaging sensor plane. In conventional HWFS the imaging sensor plane has to be evenly divided into many small pitches in order to sense the wavefront under test, which is discretely sampled by the pinhole array so that the pitch size is predefined and expected to be as large as possible. However, on the other hand the pitch size could not be arbitrarily assigned because it also determines the space between two adjacent sampling spots or, the sampling step size, which defines the spatial resolution. As we can see in the Fig. 4(1), $\Delta x$ denotes as the allowable maximum displacement of a spot inside a pitch, and obviously is the half length of a pitch, which is not only relevant to the dynamic range, but also relevant to the resolution in conventional HWFS. On the contrary, the RS-HWFS depends on raster scanning to fill up the sampling resolution so that the whole imaging sensor plane does not have to be divided and could achieve large pitch for recording spots from each scanning step, as we can see in Fig. 4(2). This modification significantly increases the dynamic range of the RS-HWFS, compared to conventional HWFS, leading to a breakthrough over the well-known trade-off between dynamic range and resolution in conventional HWFS, and achieving extended aperture only with an inexpensive translation stage cooperate with a normal size imaging sensor, of which large size could be very expensive especially in some situations when using the infrared or UV light, and moreover, reducing the computation cost for spots tracking and avoiding the complicated spots tracking algorithms.
Fig. 4. Dynamic range comparison between traditional HWFS and RS-HWFS. The reason for why the RS-HWFS can extend the dynamic range is that the imaging sensor plane does not need to be divided into small pitches and instead could utilize the whole sensor plane to record the only spot in every step of scanning
By means of the raster scanning the dynamic range no longer has an effect on spatial resolution but multiplies the range because the imaging sensor plane does not have to be divided into blocks. The raster scanning scheme is depicted in Fig. 5(1). Evidently the size of the measuring aperture is extendable without implementing a 4f telescope system that commonly used in conventional HWFS and depends on the total traveling length of the translation stage, and the spatial resolution depends on the scanning step size, the size of the sampling pinhole, and the wavelength. Fig. 5(2) shows the flow of data acquisition and spots centriod relocation process in RS-HWFS. Different from other large aperture and high resolution methods that have to record a large size image of spots array which is inevitably memory consuming and processing complicated, our RS-HWFS stores only $x$ and $y$ centroid coordinates from each scanning step. After finishing the scanning, we will get a stack of centroid coordinates of the ordered spots. All these coordinates from each scanning step should firstly subtract the reference origin (denoted as the reference one in red) and then be relocated to their corresponding positions according to the sample plane with the order of the scanning steps as shown in Fig. 5 (1).
Fig. 5. Description of the process of raster scanning and data recording of RS-HWFS. (1) is the raster scanning scheme over an extendable aperture and (2) is rding and pre-processing, with which the size of the measuring aperture depends on the total traveling length of the sample carried by translation stage, and the spatial resolution depends on scanning step size. Different from other large aperture and high resolution methods that have to record a large size image of spots array, which is inevitably memory consuming, our scheme compactly stores only one pair of $x$ and $y$ centroid coordinate on the imaging sensor plane in each scanning step. After finishing the scanning, all these local coordinates from each scanning step will firstly subtract the reference origin and then be relocated to their corresponding global positions on the sample plane.
Another conflicting relationship between dynamic range and sensitivity in HWFS is illustrated in Fig. 6. The distorted wavefront under test is examined by the pinhole aperture and then diffracted at a distance $f$ along the axial direction, causing a diffraction spot centroid displacement $\Delta x$, from its original reference position with a planar wavefront incident in red. Obviously, with the same slope angle of the distorted wavefront under test, when the axial distance $f$ from the pinhole plane to the imaging sensor plane is getting longer, the narrow beam with a local slope of the distorted wavefront diffracts at a longer distance and causes a larger displacement on the imaging sensor plane, from its original reference position. A larger displacement benefits smaller wavefront slope detection, yet the available or effective dynamic range within a pitch becomes smaller, as we can see in Fig. 6. Moreover, considering the wave property, the beam could not be as narrow when it has to diffract at a longer distance $f$ and therefore dissipates more energy before hitting the imaging sensor. Especially in conventional HWFS the longer the diffraction distance the larger the diffraction spot, which raises the risk of crosstalk between any two adjacent spots on the grid divided sensor plane. The dynamic range ($\alpha _{max}$), sensitivity ($\alpha _{min}$) and spatial resolution ($R$) in conventional HWFS are defined with the characteristic parameters of the setup [8], represented as:
where the spatial resolution $R$ is defined as the pitch size of the pinhole array $\rho$, which determines the sampling distance; $\alpha _{min}$ is defined as 1-pixel displacement ($\delta x$) of spot centroid on the observation plane levered by the wavefront distortion to be detected; $\alpha _{max}$ is the maximum allowable displacement of the spot centroid; and $f$ is the space between the pinhole array and the observation plane. In RS-HWFS, there are some differences including the pitch of the pinhole array $\rho$ for the dynamic range $\alpha _{max}$ is re-defined as the half entire length of the imaging sensor; and the spatial resolution $R$ related parameter $\rho$ is redetermined as the length of scanning step $L$ and obviously decoupled with the dynamic range, represented as:Therefore, the spatial resolution of our setup depends on how finely the stage moving, the pinhole size, and the wavelength. The step size of scanning and its positioning accuracy determine the spatial resolution, and obviously, the pinhole size should be smaller than the scanning step, while the pinhole size also could not be too small to get enough light impinged on the sensor, because of the diffraction effect related to the wavelength. On the other hand, because the dynamic range of our RS-HWFS is decoupled with the spatial resolution, it could be optionally traded-off for increasing the sensitivity by conveniently adjusting the axial distance $f$ from the pinhole plane to the imaging sensor plane.
Fig. 6. The relation between sensitivity and dynamic range in HWFS. A longer axial distance $f$ from the pinhole plane to the imaging sensor plane results in a higher sensitivity but shorter dynamic range because the shorter available displacement $\Delta x$ for the spot in its pitch. Because our RS-HWFS releases the space of the imaging sensor to build a large dynamic range, so it could be optionally traded-off for increasing the sensitivity by conveniently adjusting the axial distance $f$ from the pinhole plane to the imaging sensor plane.
Fig. 7 is the brief illustration of data acquisition and processing flow comparison between conventional HWFS and RS-HWFS. In conventional HWFS, before measuring the spots array, the reference spots array should be recorded in advance in order to track the spots displacements. Then the wavefront map could be reconstructed from the quiver plot of spot’s centroid difference by performing the integration processes [22]. For RS-HWFS, the measuring spots array is replaced by measuring a stack of spots with the same reference spot. With no need for spots tracking, we could calculate the spots centroid difference directly, and then relocate all these centroid differences to their real positions based on the scanning order on the sample plane, to form the quiver plot and wavefront map. This improvement could make the RS-HWFS robuster and easier for alignment.
Fig. 7. Illustration of data acquisition and processing procedure comparison between the conventional HWFS and RS-HWFS.
Since we focus on static optics measurement, in our opinion, the time cost restriction of RS-HWFS is looser than some of the WFS used for real-time measurement. Actually, the time cost largely depends on the configuration, for example the speed of translation stage, how many frames taken and the delay you set in each scanning step, the size of measurement aperture and the sampling density you choose. Therefore, in different configuration the time cost should be different.
3. Experiments
In our setup, the RS-HWFS with an extended aperture of $40 \times 40mm^2$, we chose $f = 10mm$, $\Delta x = 2.2\mu m$, $\rho = 1000 \times 2.2\mu m$, and pinhole diameter of $50\mu m$. Step size $L = R = 800\mu m$ is chosen to shorten the acquisition time over this extended aperture. According to Eq. (3), the dynamic range $\alpha _{max} = 0.11rad$ and the sensitivity $\alpha _{min} = 0.22mrad$, which are generally better than that of the commercial SHWFS, with the THORLABS WFS30-14AR SHWFS as an example. The comparison could be seen in Table 1. Obviously the dynamic range, sensitivity and aperture size of RS-HWFS are about 11 times, almost 2 times and 4 times better than that of the THORLABS-WFS30-14AR, respectively.
Table 1. Important features comparison of RS-HWFS, with the common use SHWFS of THORLABS-WFS30-14AR.
Because of the these advantages of RS-HWFS, we could not find a conventional or state-of-the-art WFS, which could achieve similar performances simultaneously, to compare directly with our test result using the same sample. In order to validate the feasibility of RS-HWFS, as the refractive power measurement does not have so much restriction and it is also of interest for us, experiments have been done with two spherical lenses as the standard samples, one has optical refractive power of +2.73$m^{-1}$ (focal length of +366mm) and the other has optical refractive power of −0.56$m^{-1}$ (focal length of −1786mm), which are already known by measuring them with a commercial lensmeter. The sampling matrix has $50 \times 50$ elements when the step size is $800\mu m$ over the extended aperture $40 \times 40mm^2$, and the time cost in this configuration is 17 minutes. The imaging sensor is DMK 72AUC02 monochrome industrial camera of the Imaging Source, a $1/2.5$ inch Micron CMOS sensor with $2.2\mu m$ pixel size. The wavelength of the LED light source is $546nm$. The verification experiment result, tested by our protocol, is shown in Fig. 8. Figure 8(a) is the slope map, Fig. 8(c) is the wavefront map, of the spherical lens has 366mm focal length; Fig. 8(b) is the slope map, and Fig. 8(d) is the wavefront map, of the spherical lens has −1786mm focal length. We tested the optical refractive power of each standard sample with their slope data in our configuration thirty times by manually rotating or shifting the sample slightly. Results are shown in Fig. 9, from which we get their Root Mean Square (RMS) values of +2.773$m^{-1}$ and −0.5548$m^{-1}$, denoted by horizontal solid lines. The differences of the RMS values from their typical values, denoted by horizontal dash lines, are about $1.57\%$ and $1\%$, respectively, possibly caused by sample tilt while mounted on the holder, could be acceptable for the detection of the refractive power.
Fig. 8. The slope map and wavefront map of the sample spherical lens, tested with RS-HWFS. (a) is the slope map acquired with raster scanning, (c) is the reconstructed wavefront map, of the spherical lens has 366mm focal length; so as to (b) and (d), of the spherical lens has −1786mm focal length.
Fig. 9. The observed optical refractive powers of the spherical lens has 366mm focal length (a), the spherical lens has −1786mm focal length (b), with RS-HWFS prototype.
The maximum wave gradients in reconstruction, which mainly occurs at the margin of the extended aperture $40 \times 40mm^2$ over a spherical wavefront, could be estimated by resorting to trigonometry conduct, are about $0.055 rad$ for the convex sample and $0.011 rad$ for the concave sample, with assumption that spherical aberration of the samples are not pronounced. In our RS-HWFS setup these observed values are $0.058 rad$ and $0.012 rad$ in $x$ component, and $0.058 rad$ and $0.013 rad$ in $y$ component, equal to $52.5\%$ and $11\%$ in $x$ component, and $52.7\%$ and $11.6\%$ in $y$ component of the dynamic range $\alpha _{max} = 0.11 rad$ in our RS-HWFS setup, respectively. The minimum wave gradients in reconstruction, are $0.67 mrad$ and $0.5 mrad$ in $x$ component, and $0.45 mrad$ and $0.3 mrad$ in $y$ component, respectively, which are slightly higher and close to the sensitivity $\alpha _{min} = 0.22 mrad$ in our setup. The results are summarized in Table 2.
Table 2. Standard samples tested with RS-HWFS. SP: standard power; TP: Testing power. The percentage in the TP represents its difference from the SP. The percentages in the $X \alpha _{max}$ and $Y \alpha _{max}$ represent their proportion compared to the dynamic range $\alpha _{max}=0.11rad$.
Due to the raster scanning strategy, the trade-off between dynamic range and spatial resolution no longer exists in RS-HWFS and therefore we could enhance the spatial resolution by increasing the number of scanning step while maintaining the same large dynamic range of each sampling spot, which could not be done by conventional HWFS. We have tested this property and expected that it could be applicable for examining of free-form lenses, because the wavefront profile of them may range from mild to extreme that needs both high spatial resolution and large dynamic range simultaneously. Following the former setup configuration, by changing the scanning density from $50 \times 50$, to $100 \times 100$, $200 \times 200$, and $400 \times 400$ over the same $40 \times 40mm^2$ field-of-view (FOV) on a free-form lens sample, the applicability and the flexibility of the RS-HWFS were demonstrated. The sampling density and their displacements of centroid in experiment were tested by our protocol, as shown in Fig. 10. Figure 10(a)-(d) are the slope maps with $50 \times 50$, $100 \times 100$ $200 \times 200$, and $400 \times 400$ sampling density, corresponding to $800\mu m$, $400\mu m$, $200\mu m$, and $100\mu m$ in spatial resolution, respectively. Obviously, the cases of $200 \times 200$ and $400 \times 400$ sampling densities are both superior in spatial resolution compared with the commercial SHWFS of THORLABS WFS30-14AR, where the spatial resolution in sampling length is $300\mu m$, as we can see in Table 1.
Fig. 10. The sampling density and slope map of the sample free-form lens, tested by RS-HWFS. (a) is the slope map with $50 \times 50$ sampling density, (b) is the result with $100 \times 100$ sampling density, (c) is the result with $200 \times 200$ sampling density, and (d) is the result with $400 \times 400$ sampling density.
The experiment result with different sampling densities mentioned above, and their wavefront map, profile and contour of the sample free-form lens, tested by our protocol, are shown in Fig. 11. Figure 11(a), (b) are the wavefront map, profile and contour of the sample free-form lens with $50 \times 50$ sampling density. Fig. 11(c), (d) are the result with $100 \times 100$ sampling density. Fig. 11(e), (f) are the result with $200 \times 200$ sampling density. And Fig. 11(g), (h) are the result with $400 \times 400$ sampling density. Their RMS and Peak-to-Valley (PV) values are also shown on top of the wavefront maps, respectively. It is normal for RMS and PV values varied from measurements of different densities in a certain extent when the differences are less than $0.001\lambda$, or about $1\%$ deviation. As we can see from Fig. 11(a), (b) the wavefront profile is grainy so that the sampling grid is obvious, and apparently the margin of profile circle has heavy jagged fashion that could be detrimental for measuring the surrounding region of optics. In Fig. 11(c), (d) the grain structure of the profile is smoother with increasing sampling density but at the margin the jagged shape is still obvious. When the spatial resolution is decreased to $200\mu m$ in Fig. 11(e), (f) and $100\mu m$ in Fig. 11(g), (h), the wavefront profiles are much smoother. The value of each sampling grid represents the mean result of low, middle, and high-frequency wavefront of the corresponding sampling point on the specimen so that a dense sampling should be beneficial for wavefront measurement. In other words, higher spatial resolution brings about a larger amount of information of the input wavefront and benefits for pursuing the fidelity in reconstruction process of the HWFS. This also could be quantitatively observed by analyzing their spatial frequency spectrum, which is shown in Fig. 12. When increasing the sampling density, there emerged some new higher-frequency components in reconstruction and at the same time the amplitudes of all the frequency components are also intensified.
Fig. 11. The experiment result with different sampling density and their wavefront map, profile and contour of the sample free-form lens, tested by our protocol. (a), (b) are the wavefront map, profile and contour of the sample free-form lens with $50 \times 50$ sampling density. (c), (d) are the result with $100 \times 100$ sampling density. (e), (f) are the result with $200 \times 200$ sampling density. And (g), (h) are the result with $400 \times 400$ sampling density.
Fig. 12. The spatial frequency spectrum of the reconstructed wavefront with different sampling densities.
4. Discussion and conclusion
Since we are interested in static optical testing inside house, the influence of air could hardly affect the test result like turbulence in the space. The positioning accuracy of the translation stage is within 10 microns, and our maximum sampling density in our experiment is 400$\times$400 for an aperture of 40$\times$40 millimeters, in other word, 100 microns in spatial resolution. Actually, conventional HWFS only takes a small portion of the wavefront slopes through each sub-aperture to represent that of the respective pitch, which is widely acceptable for lens measurement. Thus one-tenth positioning error of the translation stage should not affect the testing of the first derivative of the wavefront profile noticeably. To improve the positioning accuracy of translation stage, one way is to use a servo motor to build up a close-loop control system, which could feedback the real-time position of the stage; or directly replacing it with a higher accuracy translation stage. It is not much more expensive to get a positioning accuracy below 1 micron. Another way to improve accuracy is to calibrate this positioning error by using an optical wedge with a small refractive angle as the sample. The measuring spot displacement should increase linearly along the wedge direction if there is no positioning error, thus we can repeat this measurement to get a set of data to quantify the positioning error of translation stage and use them to calibrate the stage in each scanning step. Since this is a raster scanning scheme, it is important to note that the translation stage along x and y direction should be both calibrated. For the vibration insensitive performance during the scanning procedure, we have tested the method of setting a delay in each scanning step and taking an average acquisition from several frames to mitigate the possible error from the stage’s vibration. After we compare the results using this method with the one with normal raster scanning, there is no evident difference in wavefront profile between them. Obviously, this difference depends on the stability of the translation stage and in our case this vibration problem could be reasonably neglected.
Breaking the trade-off between dynamic range and spatial resolution is significant for HWFS in optical detection because it liberates the sampling distance to achieve the preferred accuracy of reconstruction for the wavefront under test, while maintaining the dynamic range adequate to the application scenarios where the sharp slopes of local wavefront of interest could be measured. So far we achieved about 11 times larger dynamic range than that of the typical commercial Hartmann type wavefront sensor. Actually, we can easily enlarge the dynamic range further by using an imaging sensor with more pixel number. Another reward of large dynamic range is that we can use it to trade-off for better sensitivity in situations where the subtle changes of slopes need to be measured. The measuring aperture on the sample, is extendable in our scheme and only limited by the maximum traveling range of the translation stage. The customary spots tracking process in our scheme is robuster and simpler than commercial HWFS, and also could significantly reduce the memory cost towards large aperture detection or dense sampling circumstances. It should be noted that the spatial resolution in HWFS depends on the step size in scanning, and could be chosen for different purposes. The main drawbacks of the HWFS is the slow measurement speed because of the scanning process. We expect that our HWFS are useful for measuring static samples where high spatial resolution, high sensitivity, large dynamic range and low cost are preferred simultaneously.
Funding
National Natural Science Foundation of China (61205192); National High-tech Research and Development Program (2015AA020944).
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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