1. Introduction

Mobile camera has been widely used in daily life and professional fields [1–5], which usually employ sophisticated lens designs to achieve pro-level performances with low cost and small size [6–10]. The sophisticatedly designed lens assembly is quite sensitive to manufacturing tolerances and varying environments (e.g., temperature, stress, pH, and humidity) [11–14]. Extreme environments may induce lens deformation, performance degradation, and even non-functionality. Thus, the sustainability of the lens in extreme environments must be proved by conducting the reliability test; and a comprehensive knowledge of the lens deformation during the reliability test is critical for optimizing the lens design and fabrication processes. To study the lens deformation in this case, a non-destructive measurement for multi-surface lens assembly is particularly desired.

Currently, a wide variety of excellent contact and non-contact profilometry techniques [15–22] are commercially available. However, they can only measure a single lens piece once a time, but cannot analyze the multi-surface lens assembly as a whole. The popular modulation transfer function (MTF) test [23,24] can directly assess the imaging performance degradation, but cannot resolve the specific deformation in the lens structure accounting for the degradation. Available measuring techniques for lens assembly only aim at measuring the central distances between surfaces [25,26] and element eccentricity [27], but cannot provide full-size deformograms (2D maps showing deformation) of each surface. X-ray micron computed tomography seems an effective non-destructive measuring technique for the lens assembly; however, its resolution is limited in the micrometer scale, and the imaging and reconstruction incur an excessive computational time [28,29]. Thus, a fast and non-destructive method is in desperate need for measuring the nanoscale changes in the multi-surface lens assembly.

Three-dimensional (3D) optical coherence tomography (OCT) [30] is particularly suitable for non-destructively imaging the multi-surface lens assembly by measuring the echoes of each surface. Using the 3D OCT image, an inspection of defective regions of lenses has been reported by Lee et al. [31]. Ranging the echo delay of the intensity envelop, a lens surface profile can be measured. Though the axial resolution (FWHM of the envelope) of the OCT intensity is in micron scale (typically 1-20 µm), the accuracy of measuring intensity center can reach sub-micron scale using digital center correction [32] or point cloud transform [33]. With adequate averaging, the accuracy can reach as high as 55 nm [34]. Integrating the phase information of the echoes, the axial motion sensitivity can be further improved. Recent researches regarding surface profile or thickness measurements have manifested sensitivities down to sub-nanometer [35,36], but the dynamic range was confined to half center wavelength due to phase wrapping. Combining both intensity and phase, the dynamic range can be extended to the level of intensity measurement, and the accuracy can be refined to the level of phase measurement [37]. Despite the high accuracy, in measuring a multi-surface lens assembly, phase sensitive OCT cannot correctly get the surface profile because the surfaces are distorted due to the refraction of the probing light when passing through the lens-air interfaces [38]. Existing methods [38–40] correcting the distortion can only interpolate intensity pixels but cannot retain phase information, which means the phase-level distortion correction is almost impossible. Moreover, the lens needs to be dismounted from the measuring equipment for testing and remounted after the test. The pre- and post-test mounting mismatch would greatly deviate the measured deformation. In brief, combining the intensity and phase information, OCT is promising for revealing the test-induced deformation in the multi-refractive-surface lens in the nanometer scale, but the potential is challenged by the refractive distortion, as well as the phase wrapping and mounting mismatch problems.

In this paper, we proposed an optical coherence digital-null deformography (ODD) for measuring the test-induced deformation in multi-refractive-surface lens assemblies with nanometer accuracy. To circumvent the refractive distortion, a digital null alignment (DNA) method was developed to physically align the post-test lens to the digital null (i.e., the lens data prior to the test). The changes between the aligned lens and its digital null were measured with an intensity centroid shift (ICS) at micron scale and a joint wavenumber (k)-depth (z) domain phase shift (kz-PhS) at nanoscale. Experimental validations of the accuracy of ICS, kz-PhS and DNA were conducted sequentially with pre-known exerted changes, and the feasibility of the proposed ODD method was demonstrated in a drop test.

2. Methodology

2.1 Data volume acquisition

The proposed ODD system was developed based on a swept-source OCT configuration with a line scan rate of 100 kHz. The swept source (SL131090, Thorlabs, Inc.) was operated at a center wavelength of 1300 nm with a spectral bandwidth of 100 nm, thus providing a measured axial resolution of 14 µm in air. As shown in Fig. 1, the output light was split into two parts by Coupler 1 and routed into the sample and reference arms, respectively, which were recombined and interfered in Coupler 2, and then recorded using a balanced detector. In the sample arm (photograph shown in the inset of Fig. 1), the probing beam was focused onto the lens sample with a spot size of 12.96 µm using a telecentric scan lens (f = 54 mm). The beam was scanned using a set of galvanometers on a 2D mesh $({x,\; y} )$ for OCT volume acquisition, where x and y are the coordinates on the scan lens principal plane. The lens sample was mounted on a self-designed tooling fixture, whose empirical positioning repeatability was: x, y, z < 10 µm, θ_x, θ_y< 0.1°, and θ_z < 1° based on experimental observations. The fixture was fixed onto a hexapod (H-811. I2, Physik Instrumente), which could exert a desired position and orientation on the lens sample via six-axis movement ($x,\; y,\; z,{\theta _x},\; {\theta _y},{\theta _z}$). The resolution of the hexapod was $x,\; y < 0.2\; \mathrm{\mu m}$, $z < 0.08\; \mathrm{\mu m}$, and ${\theta _x},\; {\theta _y},{\theta _z} < 0.0002^\circ $. Detailed information of the system setup was provided in Supplement 1, Note 1.

 

Fig. 1. Schematic of the ODD setup. The left inset (green box) shows an image of the sample arm. PC, polarization controller; DC, dispersion compensators; BD, balanced detector. $I({x,y,k} )$ represents the acquired data volume.

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Repeated volumetric scans were conducted before and after the reliability tests. Each OCT volume consisted of 512 frames (y direction); 512 A-scans was in each frame (x direction); and 2048 pixels was in each A-scan (k direction). By Fourier transforming along the k-direction we can get the depth (z-direction) resolved image with an effective field of view (FoV) of 11 × 14 × 14 mm³ (z × x × y). Before the reliability test, a pre-test raw volume ${I_1}({x,y,k} )$ of the lens was acquired as a digital null. The lens was then dismounted for testing, and remounted for post-test acquisition. Thereafter, a post-test raw volume ${I_2}({x,y,k} )$ of the lens was acquired for null measurement with the digital null. Note that the acquisition of ${I_2}({x,y,k} )$ may be repeated until the DNA was completed (see Section 2.4).

2.2 Data preprocessing

Raw data volume of lens. The lens contained a series of surfaces. The light backscattered from each surface interferes with the reference light, and the spectral interferogram in each A-scan can be expressed as a summation of each surface:

Fourier Transform. As shown in Fig. 2, the pre-test volume ${I_1}({x,y,k} )$ in the wavenumber domain was Fourier transformed into the depth domain to reconstruct the 3D depth-resolved complex raw volume $\widetilde {{I_1}}({x,y,z} )$, which could be decomposed into intensity and phase volumes. The same operations were conducted on the post-test volume ${I_2}({x,y,k} )$. A subsequent null measurement would resolve the inter-volume shifts.

 

Fig. 2. Overview of the ODD framework. Pre-test volume of the lens was acquired as the digital null, and the post-test volume was acquired for null measurement with the digital null. In the preprocessing module, both two spectral volumes were converted to the surface OPD maps and masked phase volumes. Thereafter, the null measurement module compares the OPD maps and phase volumes, calculating the inter-volume changes. The mounting mismatch was iteratively minimized in the DNA module. When DNA was completed, the deformation on all surfaces were finally output as deformograms. FT, Fourier transform; ICS, intensity centroid shift; kz-PhS, joint wavenumber(k)-depth(z)-domain phase shift; k-PhS, wavenumber-domain phase shift; z-PhS, depth-domain phase shift; DNA, digital-null alignment.

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Lens mask. The light backscattered from the lens surfaces contained both single- and multi-scattering components. The single-scattering component was scattered only once at the surface being detected but transmitted at all other lens surfaces. In contrast, the multi-scattering component was scattered multiple times between lens surfaces, forming a complex and uninterpretable optical path. The former is useful signal, whereas the latter generates artifacts and causes troubles to the surface searching. To address this problem, a lens mask was generated by tracing the probing beam on a lens model created from its nominal shape (lens design document). Details about computing the lens mask was presented in Supplement 1, Note 2. The lens mask was multiplied to the raw intensity and phase volumes. The masked volumes eventually preserve only the information of the actual surfaces with a clean background. Thereafter, the intensity volume would go through intensity centroid searching, while the phase volume was directly sent to the null measurement.

Searching intensity centroid. Positions of lens surfaces were determined in this step by searching intensity centroid in the masked intensity volume. First, the discrete voxelated positions of each lens surface were located using a graph shortest path algorithm [41], which iterated through all frames, and in each single frame, the algorithm identified and connected the voxels with maximal intensity from the leftmost A-scan to the rightmost A-scan. The accuracy of the discrete positioning of surfaces is equal to the voxel size. Thereafter, the accuracy along axial direction (z direction) was improved using a center correction method [42] which calculated the axial intensity centroid (i.e., the intensity weighted average of axial voxelated positions) in each A-scan. In this study, the calculation of the weighted average involved two adjacent axial voxels on both sides of the surface. The improved accuracy along axial direction is about 1/5 of the voxel height, which will be proved later in the paper. The axial positions of the intensity centroid show the OPD between lens surfaces and the reference arm, so they are called the surface OPD map. They would be used in the subsequent null measurement.

2.3 Null measurement

As shown in Fig. 2, the null measurement module took in the pre- and post-test surface OPD maps as well as the masked phase volumes to evaluate inter-volume changes. Two types of changes can be computed: the mounting mismatch and deformation. The mounting mismatch is a combined six-axis displacement ($\mathrm{\Delta }x,\; \mathrm{\Delta }y,\; \mathrm{\Delta }z,\; \mathrm{\Delta }{\theta _x},\mathrm{\Delta }{\theta _y},\mathrm{\Delta }{\theta _z}$) of a whole lens assembly, which is used for alignment purpose; and the deformation is the regional OPD change on each lens surface, which is computed only after all mounting mismatches were adequately suppressed. To measure those inter-volume changes, we offered two methods based on intensity and phase signals, respectively, with different accuracy and range. In practical cases, they should be chosen appropriately based on the magnitude of the changes (detailed usage was in the next section). In this section, we only introduce their algorithms.

Micron-scale intensity centroid shift (ICS) measurement. To measure the mounting mismatch based on intensity signal, we should pick a surface-of-benchmark to conduct point cloud registration. First, the OPD map of this surface was constructed as a point cloud. And a rigid point cloud registration function [43] was used to stitch the post-test point cloud to the pre-test point cloud. The output of the stitching is a transformation matrix, $\mathrm{\mathbb{T}} = {({{t_{ij}}} )_{4 \times 4}}$, and the six-axis displacements ($\mathrm{\Delta }x,\; \mathrm{\Delta }y,\; \mathrm{\Delta }z,\; \mathrm{\Delta }{\theta _x},\mathrm{\Delta }{\theta _y},\mathrm{\Delta }{\theta _z}$) can be decoupled from $\mathrm{\mathbb{T}}$ by:

To measure the deformation, the pre-test OPD was subtracted from the post-test OPD for each surface. The subtraction was conducted on every A-scan. Once completed, we obtained the OPD-change map showing the deformation. The accuracy of ICS outputs is comparable to the accuracy of the surface OPD map, which is in micron scale.

Nanoscale joint wavenumber(k)-depth(z) domain phase shift (kz-PhS) measurement. By exploiting the phase shift, the accuracy of the measurement of inter-volume changes can be further improved to the nanometer scale. By taking a difference operation between the pre- and post-test phase volumes, one obtained the depth-domain phase shift (z-PhS) $\mathrm{\Delta }{\varphi _{mz}}$ on surface S_m, which was $\mathrm{\Delta }{\varphi _{mz}} = 2{k_0}\mathrm{\Delta }{d_{mz}}$ at the voxel positions of the surface, where $\mathrm{\Delta }{d_{mz}}$ is the OPD change of S_m calculated from z-PhS. The $\mathrm{\Delta }{d_{mz}}$ could be measured with high accuracy from $\mathrm{\Delta }{\varphi _{mz}}$ using $\mathrm{\Delta }{d_{mz}} = \mathrm{\Delta }{\varphi _{mz}}/2{k_0}$. However, the measuring range of $\mathrm{\Delta }{d_{mz}}$ was within ${\pm} {\lambda _0}/4$, given that the valid range of $\mathrm{\Delta }{\varphi _{mz}}$ was $- \pi \sim \pi $ due to the $2\pi $ ambiguity, where ${\lambda _0}$ is the center wavelength.

To circumvent the ambiguity in measuring the OPD change, the wavenumber-domain phase shift (k-PhS) was studied. An illustration was presented in Fig. 3. According to the phase term in Eq. (1) (i.e., the content in the cosine function), the k-PhS $\mathrm{\Delta }{\varphi _{mk}}$ can be expressed as a linear function of sweeping wavenumber: $\mathrm{\Delta }{\varphi _{mk}} = 2k\mathrm{\Delta }{d_{mk}} + \mathrm{\Delta }\theta $, where $\mathrm{\Delta }{d_{mk}}$ (the OPD change of S_m calculated from the k-PhS) is half of its slope. To extract the $\mathrm{\Delta }{\varphi _{mk}}$ of the surface S_m, a bandpass filter (the hollowed blue bar in Fig. 3(a)) was applied to the depth domain complex volume $\tilde{I}({x,y,z} )$ at the voxelated position of surface S_m along the axial direction for each A-scan, which preserved the positive-delay signal of S_m a surface while rejected its conjugate and any other signals. A 15-pixel one-dimensional Gaussian window was used as the bandpass filter in this study. Thereafter, the filtered A-scan was inverse Fourier transformed to the wavenumber domain to obtain the complex spectrum of S_m. After obtaining the pre- and post-test spectra (Fig. 3(b)), the conjugation of the post-test spectrum was multiplied to the pre-test spectrum. From the angle of the conjugation product, k-PhS $\mathrm{\Delta }{\varphi _{mk}}$ can be extracted (Fig. 3(d)). Thereafter, linear fitting was performed on the $\mathrm{\Delta }{\varphi _{mk}}$ to calculate the slope, and half of the slope was the OPD change $\mathrm{\Delta }{d_{mk}}$.

 

Fig. 3. An illustration of the k-PhS method. (a) One frame of the pre-test lens depth domain complex volume (only real value was visualized here), in which the hollowed blue bar indicates the bandpass filter for S3 applied on one arbitrary A-scan. (b) The real part of the pre-test and post-test spectra from the filtered A-scan. (c) The phase of the spectra in (b) around the start wavenumber (left) and the end wavenumber (right). The initial phase was dropped, and the end points were marked with arrows. (d) The phase difference of the pre- and post-test spectra (the k-PhS).

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Using k-PhS, an OPD change of ${\pm} \; 5\; \mathrm{\mu m}$ corresponded to a maximal $\mathrm{\Delta }{\varphi _{mk}}$ span of ${\pm} \; 2.86\; \textrm{rad}$, which is without any risk of phase wrapping in the wavenumber domain. Thus, $\mathrm{\Delta }{d_{mk}}$ could be readily obtained from the slope of $\mathrm{\Delta }{\varphi _{mk}}$ and was not influenced by ambiguity. However, although the k-PhS approach is capable of OPD change determination with larger range, it is subject to a larger displacement error than z-PhS [36]. To overcome this shortcoming, a joint phase shift in both wavenumber and depth domain (the kz-PhS) was developed by combining the advantages of the z-PhS (accurate) and k-PhS (large range). In the kz-PhS method, $\mathrm{\Delta }{d_{mz}}$ and $\mathrm{\Delta }{d_{mk}}$ are first calculated separately and then joint together using the following equation:

To measure the mounting mismatch, a plane was fitted to $\mathrm{\Delta }{d_m}({x,y} )$ using least square error method. The tilt angle $\mathrm{\Delta }{\theta _x}$ and $\mathrm{\Delta }{\theta _y}$ were calculated from the arctangents of the x and y slopes (Details were shown in Supplement 1. Note 3). The axial displacement $\mathrm{\Delta }z$ was the $\mathrm{\Delta }{d_m}$ value at the center of the surface. However, the transverse displacements $\mathrm{\Delta }x,\mathrm{\;\ \Delta }y$, and the rotation $\mathrm{\Delta }{\theta _z}$ had no contribution to $\mathrm{\Delta }{d_m}$; thus, they cannot be measured with kz-PhS.

To measure the deformation, we simply adopt the $\mathrm{\Delta }{d_m}({x,y} )$ after mounting mismatches were adequately suppressed. Using kz-PhS, we could measure inter-volume changes with nano scale accuracy.

2.4 Digital-null alignment (DNA)

As shown in Fig. 2, after acquiring the digital null pre-test, the lens was dismounted for testing, and then re-mounted for post-test acquisition. To match the refractive distortion of the post-test lens to the digital null, we designed a digital-null alignment (DNA) module to suppress the mounting mismatch, where a hexapod was actuated to conduct six-axis motion for compensation. The amount of compensation was calculated in the null measurement module. In this work, the surface S1 was selected as the alignment benchmark for mounting mismatch estimation, as it was the only surface that was free of refractive distortion. Moreover, it receives the strongest laser power without being attenuated, so that it had the highest SNR.

Considering the relatively poor positioning repeatability of the lens fixture, DNA adopted ICS to estimate a wide-range coarse mounting mismatch in the first few loops. Every time after actuating the hexapod, the raw OCT volume of lens was acquired again to replace the previous ${I_2}({x,y,k} )$, and then the next loop started. The DNA module would continue using ICS for mismatch estimation until the amount in the last loop converged below the resolution limit of ICS. Once it happened, kz-PhS was adopted in substitution. By using kz-PhS for mismatch estimation, the compensation for the z translation and θ_x, θ_y tilts could be refined. After the shifts were under the resolution limit of kz-PhS, the mounting mismatch is considered adequately suppressed, and the loop was stopped. It should be noted that the movements of tilt/rotation and translation were conducted separately in this study, given that the origin of the hexapod coordinate was not concentric with the center of S₁, and the rotational movements were coupled with translational movements.

After the termination of DNA, the deformograms of all surfaces were calculated. If the maximal deformation of a surface was under 5 µm, we adopted kz-PhS for higher accuracy; if the maximal deformation was beyond 5 µm, we adopted ICS instead. With the integration of DNA module, the ODD would run in a closed loop form as summarized in Table 1.

Table 1. The closed-loop form of ODD with DNA implemented

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3. Experiment and results

3.1 Demonstration of lens imaging and preprocessing

The preprocessing module was demonstrated with a representative lens sample (4-piece aspheric plastic assembly), a photograph of which is shown in Fig. 4(a). Figure 4(b) was an OCT cross-section acquired along the red dashed line, in which all the lens surfaces (indicated by yellow arrows) were observed despite the presence of multi-scattering artifacts (red arrows). Figure 4(c) shows the corresponding physical cross-section of Fig. 4(b) that was modeled according to its nominal shape. It is observed that the probing beams (the red lines in Fig. 4(c)) refracted at each surface due to the changes of refraction index, and the OCT image did not resemble the lens in real world. With reference to the ray tracing results, the distorted surfaces in the OCT image were predicted and the lens mask was generated, as shown in Fig. 4(d), which was in good agreement with the acquired OCT cross-section in Fig. 4(b). After applying the lens mask, all the single-scattering OCT signals were well preserved in the masked lens cross-section, and all the lens surfaces (S₁-S₈ in Fig. 4(e)) were extracted successfully. By searching surfaces in all frames in the volume, the lens surface OPD maps were constructed, as shown in Fig. 4(f).

 

Fig. 4. Lens mask and intensity centroid searching procedures in the preprocessing module. (a) Top view photograph of a representative four-piece aspheric plastic assembly. (b) Raw OCT intensity cross-section at the red dashed line in (a); yellow arrows indicate the lens surfaces contributed by single-scattering signals; red arrows indicate the multi-scattering artifacts; blue arrow indicates the low SNR region. (c) Nominal shape of the physical lens entity in the same cross-section. Red lines indicate the simulated beam passing through lens surfaces. (d) Distortion-predicted OCT image of the lens in the same cross-section, where the surfaces were colored black, and the lens mask regions were encircled by gray lines. (e) Refined OCT intensity cross-section by multiplying lens mask in (d) to the raw cross section (b) and the searched lens surfaces (S₁-S₈, color curves). (f) Volumetric rendering of the lens surface OPD map. Scale bars, 0.5 mm.

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3.2 Accuracy and sensitivity evaluation of the null measurement

To evaluate the accuracy of the null measurement, ICS or kz-PhS based null measurement was used to measure the translational and rotational movements exerted on a sample lens by a moving stage. By comparing the measured value to the exerted value (the ground truth), we can get the measurement error, and thus calculate the sensitivity and accuracy, where the sensitivity is quantified using standard deviation (STD) and the accuracy was quantified using root-mean-square error (RMSE) (for definition and calculation, see Supplement 1, Note 4).

Deformation. The deformation was simulated by exerting axial movements to induce OPD change. For a more accurate exertion, a piezo-electric stage (P-733.3CD, Physik Instrumente, repeatability < 2 nm) was inserted between the fixture and the hexapod to conduct movement with nanoscale accuracy. The micro-scale ICS was evaluated by measuring a 1-um axial movement. During data acquisition, X-Y scanning was conducted. The OPD changes in all A-scans and all surfaces were counted as independent OPD-change measurements. The STD and RMSE of all measurements were both 3.13 µm, thus indicating that the ICS had the same sensitivity and accuracy of 3.13 µm.

The nanoscale kz-PhS was evaluated by measuring an incremental series of axial movements (ranging from 0-5 µm) on the lens sample. During data acquisition, the X-Y scanning spans were set to zero to conduct repeated A-scan. The OPD changes in all A-scans and all surfaces were counted as independent measurements. The results are plotted in Fig. 5(a). As can be seen from the magnified red box in Fig. 5(a), phase $2\pi $ ambiguity occurred when the movement was larger than ${\lambda _0}/4$ in the conventional z-PhS measurement. In contrast, the proposed kz-PhS demonstrated a measuring range of 0-5 µm free of ambiguity. The STD remained stable within the entire range (see the STD line in Fig. 5(b)); and the average STD was 4.15 nm, indicating the deformation sensitivity. The accuracy of the kz-PhS is indicated by the RMSE line in Fig. 5(b). As observed, lower movement measurements correlated to lower RMSE values. In the small movement region, kz-PhS has an accuracy of 6.19 nm in measuring a 50-nm movement, which were in good agreement with the sensitivity. By scanning in the X and Y directions, the cross-sectional (Fig. 5(b)) and volumetric (Fig. 5(c)) deformograms were acquired accordingly with sensitivities of 22.11 nm (Fig. 5(d)) and 57.38 nm (Fig. 5(e)), respectively. The RMSE values were only slightly greater than the STD, which indicated that the measurement accuracy was in good agreement with the sensitivity. However, the sensitivity decreased as the scanning dimensions increased. It is highly probable that this was due to the following: 1) the limited spatial repeatability of the X-Y scanner caused the OPD fluctuations; and 2) the surface region with large gradient (see the blue arrow in Fig. 4(b)) caused in a limited energy backscattered to the probe and increased the phase noise. The averaging of repeated measurements could effectively smooth out the noises to increase the sensitivity at the cost of measuring time.

 

Fig. 5. Sensitivity and accuracy evaluation of kz-PhS in measuring deformation. (a) Plots of the measured axial movements with respect to the exerted movements. The upper left insert is the magnified view of the lower left red box. The error bar indicates the 99.7% confidence interval. (b) The statistics for every exerted movement in (a). (c) Cross-sectional deformogram and (d) volumetric deformogram under an exerted movement of 100 nm. (e) Corresponding cross-sectional histogram, and (f) volumetric histogram, respectively. Note that (a) and (b) share the same x ticks; (c) and (d) share the same color bar, which is encoded with the displacement. A lens mask has been applied in (c) and (d).

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Mounting mismatch. The mounting mismatch was simulated by exerting translational or rotational movements ($\mathrm{\Delta }x,\mathrm{\;\ \Delta }y,\mathrm{\;\ \Delta }z,\mathrm{\;\ \Delta }{\theta _x},\mathrm{\Delta }{\theta _y},\mathrm{\Delta }{\theta _z}$). The exertion was conducted by a hexapod (H-811. I2, Physik Instrumente), whose resolution has been presented in 2.1. To evaluate the accuracy and sensitivity, a series of single-axis displacements (1, 1/10, and 1/100 of the empirical positioning repeatability of the tooling fixture) were exerted by the hexapod, which was then measured by ICS or kz-PhS. During data acquisition, X-Y scanning was conducted. The mismatch of S₁ measured at each exertion was counted as an independent measurement. For each value, measurements were repeated for six times. The results of measurements are listed in Table 2. As reported in Table 2, the measured mismatch was in good agreement with the exerted displacement. As the displacement decreased, the STD as well as the difference between the RMSE and STD also decreased, indicating an increase in the sensitivity and accuracy. Regarding the resolution limit of a measurement as the case wherein the mean and STD reach the same order of magnitude, we concluded that the resolution limits of ICS were: translation 1 µm, tilt 0.01°, and rotation 0.1°. Also, the resolution limits of kz-PhS were: axial translation 0.1 µm and tilt 0.001°. The highest sensitivity of the mounting mismatch was 0.04 µm for z translation (kz-PhS), 0.24 µm for x or y translation (ICS), 0.0003° for tilt (kz-PhS), and 0.03° for rotation (ICS).

Table 2. Measurements of mounting mismatch using ICS and kz-PhS.^a

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3.3 Quantified visualization of DNA

To test the performance of the DNA module, a six-axis combined motion (θ_x, θ_y = 0.2°; θ_z = 2°; x, y = 10 µm, z = 5 µm) greater than the positioning repeatability of the fixture was exerted on the lens sample using a hexapod. The mounting mismatch was estimated and fed back to the hexapod as six-axis movements, and the hexapod moved to restore the pre-exertion lens location iteratively. The whole process was visualized with the deformogram of S₁. As shown in Fig. 6, the magnitude on S₁ decreased gradually with the loop, indicating a converging process. Loop 0 (Fig. 6(a)) means the initial condition after exertion but before alignment, with a corresponding deformogram that indicates a noticeable red-to-blue symmetric pattern, which was due to the bulk horizontal translation and tilt; and the region marked by the orange box displayed a notch pattern that resulted from the position change of the lens element trim point, which could provide important clues for aligning θ_z. Typical deformograms under different kinds of mismatches were presented in Fig. 6(a1)-(a4) for readers’ reference. The initial alignment (the first four loops) was conducted using ICS. After the first four loops, the deformogram converged from ∼5 µm (Fig. 6(a)) to ∼0.5 µm (Fig. 6(c)), and further converged to ∼100 nm (Fig. 6(d)) after switching to kz-PhS in loops 5 and 6. The loop stopped when the residual mounting mismatch was $\mathrm{\Delta z\;\ < \;\ 0}\textrm{.1}$ µm and θ_x, θ_y< 0.001° (Loop 6 in Fig. 6(d), central $\mathrm{\Delta z}$ was ∼86 nm, and θ_x and θ_y were ∼0.0005°). The deformograms of the deeper surfaces (Fig. 6(d1)-(d4)) also manifest that the success of alignment was consistent on every surface. The accumulated six-axis motion of θ_x= 0.1977°, θ_y = 0.1936°; θ_z = 1.929°; x = 11.3 µm, y = 8.2 µm, z = 5.4 µm was compensated by the hexapod, which was in good agreement with the exerted value. Thus, the proposed DNA module, when using the residual OPD as the criterium of the alignment, demonstrated a satisfactory performance.

 

Fig. 6. The converging process of mounting mismatches during the DNA of a six-axis combined movement (θ_x, θ_y = 0.2°; θ_z = 2°; x, y = 10 µm, z = 5 µm), visualized by the deformogram of S₁. The color bar on the first row is used to display the deformogram after (a) Loop 0 (the initial condition) and (b) Loop 2. The color bar on the second row is used to display the deformogram after (c) Loop 4 and (d) Loop 6. The conducted movements between loops were displayed. (a1)-(a4) Typical deformograms of different kinds of mismatches. (d1)-(d4) Deformograms on all lens surfaces after Loop 6.

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3.4 ODD in a drop test

To demonstrate the proposed ODD in practice, a lens drop test was performed, and the deformogram of each lens surface was measured. The converging process of mounting mismatches during DNA were posted in Supplement 1, Note 5; and the deformograms of all surfaces after DNA were posted in Fig. 7. As shown in Fig. 7(a), the deformogram of S₁ was flat (80–120 nm) in most regions, thus indicating that the post-test remounted lens was successfully aligned to the pre-test digital null via the DNA. The abrupt change (from 0 nm to -800 nm) in the bottom left may correspond to a locally deformed region. A similar deformogram pattern was observed in S₂ (Fig. 7(b)), which can be attributed to that they belong to the same lens piece. Similarly, S₃ (Fig. 7(c)) and S₄ (Fig. 7(d)) presented a similar centrosymmetric pattern. In particular, it was concave in the lower part, convex in the upper part, and changed smoothly from a minimum of -1.1 µm to a maximum of 1.3 µm. Such a centrosymmetric pattern can be attributed to both the deformation and the relative shift and tilt between Pieces 1 (S₁ and S₂) and 2 (S₃ and S₄) arising from the drop test. Moreover, Pieces 1 and 2 indicated a correlated deformed region in the lower left corner, which can be attributed to the same cause, such as a severe collision force in a drop.

 

Fig. 7. Lens deformograms measured after the drop test. (a)-(d) Projectional deformograms of Surfaces (a) 1, (b) 2, (c) 3, and (d) 4. (e) The 3D rendered surface OPD maps color encoded with deformation. A representative 2-piece aspheric plastic assembly was used.

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4. Discussion and conclusion

The main advantage of ODD over current lens measuring techniques is the ability to resolve all optical elements in a lens assembly. The maximal number of detectable pieces in an assembly is theoretically decided by the axial range of the OCT system. In our work, the swept source OCT offered a 11-mm range, which is enough for measuring most mobile lenses; and an increased sampling frequency can further boost the imaging range. ODD has wide adaptability in terms of the lens size. It is best for medium-sized lenses, which is comparable in size to the mobile lens with apertures of a few millimeters or tens of millimeters. When the size gets bigger, ODD still applies, at the cost of a larger data size, which means more time on acquisition and processing. Modern lenses designed with aspherical, freeform, and other special surfaces always have a complex internal structure. The refraction of the probing light in the assembly would induce low SNR regions (see blue arrows in Fig. 4(b) for example) and even detection blind spots. In the future work, we would overcome this drawback by changing current telecentric scanning to a customized scanning pattern with different incident angle of each A-scan to cover the blind spots. Moreover, the ability of detecting multiple pieces was also challenged by the multi-scattering artifacts. Taking the advantage that the lens shape is known a priori, ray tracing was conducted on the nominal lens structure to predict the location of every true optical surface in the distorted 3D OCT image, based on which a lens mask was formed to select every single surface while rejecting the artifacts and all other reflections. As a result, we achieved an automated and error-resistant surface detection. Note that the mask width in axial direction is 2.5 pixels, so the minimum requirement of clearance between lens elements is the 2.5 pixels, which is equivalent to 27 µm.

The main novelty of ODD is the concept of null measurement, which circumvented the challenge of refractive distortion that was characterized but unsolved for the multi-refractive-surface optics. In our method, the pre-test lens interferogram was recorded as a digital null, and was compared with its post-test interferogram, which is a self-referred style. There are three major advantages of using the digital null: (1) the test-induced lens deformation was directly measured without the requirement of correcting the refractive distortion, because an undeformed lens should experience the same refractive distortion as its digital null; (2) the phase-sensitive measurement can be conducted to achieve high accuracy; (3) the simultaneous measurement of all surfaces in a lens assembly was enabled. This method was inspired by null interferometry, such as the null lens measurement using reference surface or computer-generated hologram [18,44], which compared the test lens with a reference wavefront created by customized null optics or digital hologram. Different from the conventional null interferometry, we expanded the scope to multi-surface measurement and achieved nanoscale accuracy using phase-sensitive OCT.

Mounting mismatch was a main challenge for ODD, which was analogous to the bulk motion in the phase-sensitive OCT imaging of live subjects. And it was inevitable as the lens sample must be remounted to go through the test. Our solution was the digital-null alignment (DNA) based on hexapod six-axis motion. Given that we had reliable mounting mismatches calculated by ICS and kz-PhS, they can be fed back to direct the motion of the hexapod. Combining the lateral scanning and axial (OPD) ranging, the axial translation (z), lateral translation (x, y), tilt (θ_x, θ_y), and rotation θ_z can all be computed and compensated accordingly. The DNA technique was a combination of hardware and software, which can restore the mounting of the pre-test digital null physically and precisely (phase scale). In contrast, the prevalent alignment techniques [45,46] in OCT are purely based on post-processing, and they cannot preserve the phase information of a complicated multi-refractive-surface optics.

The limited range of phase-sensitive measurement was another challenge for ODD, which was typically confined to half the center wavelength due to phase wrapping. In phase-sensitive OCT, a numerical unwrapping algorithm based on global minimal phase variance optimization is widely used [47,48]. However, it only calculates the relative shift to eliminate abrupt phase jumps, but does not take into consideration that the unwrapped phase may deviate from the true value by an integer multiples of 2π. In contrast, a wavenumber domain phase microscopy technique [36] can measure profiles without phase wrapping and the potential absolute 2π deviation. This method requires the signal to locate sparsely and have distinct sidelobes in the depth domain, which is highly suitable for samples with discrete interfaces like a lens assembly. Our concept of kz-PhS is originated from the wavenumber domain phase microscopy, but we use the phase difference instead of absolute phase. Our kz-PhS can effectively extend the range of phase measurement to ± 5 µm and meanwhile does not introduce an extra phase wrapping problem within this range. Hence, the proposed kz-PhS has a superior performance in comparing OPD change over Ref. [36] which commonly requires an extra unwrapping step in the wavenumber domain.

One limitation of this study is the accuracy of DNA. Currently, the resolution limit of DNA cannot fully utilize the sensitivity of kz-PhS due to the inferior resolution of hexapod movement. Luckily, the residual mismatch is prospective to be digitally compensated on the ground that the residual z values on S₁ are almost consistent throughout all the surfaces (see Fig. 6(d)-(g)). In the future, the residual values on S₁ would be deducted from other deeper surfaces to further cancel out the residual. Moreover, the relatively poor alignment accuracy of ${\theta _z}{\; }$ rotation also needs attention. Lens is a centrosymmetric sample, which means ${\theta _z}{\; }$ rotation will induce very limited OPD change. In this study, the feature employed to align ${\theta _z}$ is the position of trim points of the lens elements (see the orange box in Fig. 6(a)), and it achieved a resolution of approximately 0.1°, which is not as high as the tilt resolution. Although the accuracy in ${\theta _z}$ is not strictly required in the detection of centrosymmetric samples, improvement of ${\theta _z}$ alignment accuracy would be beneficial to reduce the phase decorrelation. Viable methods for improvement include the addition of bump features on the surface extension to increase ${\theta _z}{\; }$ asymmetry or the use of a camera module to register the featured shapes on the lens barrel based on a computer vision method.

Another limitation is that the current ODD only uses a single probe; thus, it is only sensitive to the deformation component along the probing beam direction (analogous to the Doppler-angle limitation). To restore the lateral deformation, a total of three probing beams from different directions were theoretically desired [49]. Future research would focus on implementing measurement of lateral deformation. We would consider using multiple probes or changing the azimuth of the lens to multiple values by hexapod to get the OPD changes along different probing directions and thus restore both the axial and lateral deformation.

Currently, the MTF test is the most widely used technique for measuring test-induced imaging quality degradation. However, the MTF test result provides no strategy for design optimization, given that the lens assembly is a black box in which the designer has limited information regarding its internal deformation. The lens tolerance to extreme conditions can only be evaluated by simulations instead of measurements. The ODD is a revolutionary tool in the lens reliability test, as it can provide the full-size deformogram of each lens surface, which can assist in identifying regions susceptible to extreme environments. Many commercial applications can benefit from ODD. For example, the drop and strain resistance test of mobile phone lenses, the thermal resistance test of surveillance lenses and automotive lenses, the reliability test of free-form optics, etc. The ODD can also benefit high-end applications where there is a necessity for high accuracy and non-invasive detection. It should be noted that although our research is motivated by reliability test and aimed to get test-induced deformation, the scope of this technique is promising to extend widely because the selection of digital null is very flexible. We are measuring deformation if selecting the pre-test lens interferogram as the null. We foresee that the absolute profile of each surface can be measured if the null is replaced by a physical or even computer-generated standard lens interferogram. In this case, the exact 3D position of every surface inside a lens assembly can be obtained by adding the deformograms to the positions of the standard surfaces.

In conclusion, we proposed ODD to non-destructively measure the reliability-test-induced changes in lens assemblies of multiple refractive surfaces. ODD relies on the DNA to suppress the mounting mismatch and circumvent the effect of refractive distortion. And the kz-PhS provide accurate deformation measurement down to nanoscale (4.15 nm) and up to 5 µm without any phase-wrapping concerns. Considering the affordable cost of OCT and the experimental results which demonstrate a high accuracy, ODD is a cost-effective technique to evaluate lens reliability tolerance, and is prospective to conduct absolute profile measurement. Moreover, it can accelerate lens optimization and iteration in a wide range of applications.