1. Introduction

Freeform optical components have been shown to enable dramatic advances in both optical performance and system packaging with applications spanning both imaging and non-imaging regimes [1–8]. While developments in deterministic manufacturing enable the fabrication of complex optical surfaces, form metrology for these surfaces remain a challenge, making it difficult to complete the loop from design to as-built quality of the components. It is highly desired that the form metrology technique provides sufficiently low measurement uncertainty. For instance, on the order of tens of nanometers up to a hundred nanometers in order to be sufficiently lower than typical optical tolerances for form, which may be on the order of λ/2 to λ/4 [9] (∼300 nm to ∼150 nm at the He-Ne wavelength). It is also desired that this level of measurement uncertainty is achieved within a reasonable measurement time and that the technique is non-part-specific, i.e. with little to no downtime between measuring freeform surfaces of varying prescriptions. These characteristics enable the metrology technique to be used not only as final verification but also as feedback along the manufacturing process chain. Moreover, being in non-tactile contact with the optical surface is preferred to minimize the risk of damage.

The optical coherence tomography (OCT) technique is highly relevant in the context of these metrology requirements. OCT utilizes the narrow coherence lengths of broadband sources to axially section the surface under test. It is inherently non-contact, non-part-specific, and has been applied in the investigation of surface topography for biological samples [10]. Depending on the spectral bandwidth of the system and the data analysis algorithm employed, the axial localization capability may be on the order of tens of nanometers or better. By performing telecentric scanning across the surface under test, OCT is highly flexible across different freeform prescriptions [11]. Once the OCT system has been aligned and well characterized, a range of optics of different prescriptions may be measured with minimal downtime as the surface under test is de-coupled from the system overall.

At the same time, within the conventional OCT architectures, there are various limitations in the context of metrology for optical components. The two main categories of OCT are time domain (TD-) and Fourier domain (FD-) OCT, the latter of which includes spectral domain (SD-) OCT that utilizes a broadband laser with a dispersive spectrometer and swept source (SS-) OCT that utilizes a laser that sweeps in wavelengths with a photodetector [12]. In TD-OCT, the reference arm scans axially for every lateral point across the surface under test, which not only greatly reduces the measurement speed but also, more importantly, makes it so that the measurement uncertainty is limited by the uncertainty of the encoder of the actuating motor in the reference arm. Errors in the reference arm’s mechanical motion directly determine the sag measurement error. FD-OCT overcomes this limitation by eliminating the reference arm translation and measuring a modulated spectrum instead. Typically, a fast Fourier transform (FFT) is performed in software on the measured spectrum to recover the depth information encoded in the modulation. In SS-OCT, the measurement uncertainty is limited by the swept source laser in terms of the inherent trigger jitter noise as well as the limited spectral bandwidth in comparison with the supercontinuum laser often used in SD-OCT [13,14]. In SD-OCT, the dispersive spectrometer largely determines the metrology capability of the system. On one hand, to obtain sub-micron axial sectioning capability, a spectral bandwidth of several hundreds of nanometers is required. On the other hand, to obtain a measurable sag range on the order of millimeters to be relevant for optical surfaces, a spectral resolution on the order of 0.01 nm is required. The wide spectral bandwidth and narrow spectral resolution combined make the dispersive spectrometer one of the most challenging and expensive components for an SD-OCT system [15,16]. Moreover, due to the need for FFT in data processing, the measured signal of conventional FD-OCT requires linearization in wavenumber space, either via software methods [17] or hardware methods [5,18], both of which have their own associated challenges.

We developed a cascade OCT (C-OCT) technique that is customized towards the task of freeform optics metrology. In C-OCT, a secondary interferometer is connected in cascade to a primary interferometer where the freeform sample is placed. There is no translation of the reference arm in the primary interferometer, thus eliminating this source of error in comparison with TD-OCT. A broadband laser source such as a supercontinuum laser is used, which is free from trigger jitter noise. The secondary interferometer is based on the principle of Fourier transform spectroscopy and replaces the dispersive spectrometer of a typical SD-OCT system. This configuration importantly enables larger spectral bandwidth simultaneously with finer spectral resolution with no trade-off between these two specifications. Moreover, the secondary interferometer performs the necessary Fourier transform in hardware, eliminating the need for wavenumber linearization of the signal.

For the secondary interferometer, a novel configuration with a rotating optical cube is developed to enable high-speed measurements. In general, OCT (apart from full field OCT) takes measurements point by point to form a point-cloud representation of the sample under test. To sufficiently sample a freeform optical surface that may be tens of millimeters in diameter, a grid of 100 by 100 points to 1000 by 1000 points may be required. To constrain the total measurement time within practical limits, it is necessary that the time spent on each point is on the order of milliseconds. The rotational nature of the configuration developed as well as the symmetry of the cube geometry are both highly leveraged for this speed requirement.

A prototype setup was built to experimentally validate both the C-OCT technique as well as the rotating cube interferometer. Given that C-OCT is a point-cloud technique, it is worthwhile to understand the performance of the technique at a given point on a sample surface, after which mechanisms to scan and fill the three-dimensional point cloud may then be introduced.

In this paper, the mathematical framework of C-OCT is derived in Section 2 in relation to conventional OCT methods. The optical design and working principle of the rotating cube interferometer is developed in Section 3. Experimental results with the prototype C-OCT setup is presented and discussed in Section 4.

2. C-OCT and its mathematical framework

The mathematical framework describing the working principle of C-OCT is derived in relation to conventional TD-OCT and FD-OCT. As will be shown, C-OCT may be understood as a hybrid of these two conventional methods. Given that the C-OCT technique is developed for the metrology of optical components, which are in many cases mirrors or the surfaces of lenses with a single air/material interface as the measurand, the framework presented here assumes a single surface under test. The parts of the derivation concerning TD-OCT and FD-OCT will be familiar to those who are in the OCT field; however, they are derived in a variant fashion to facilitate intuitive understanding of C-OCT.

At the core of OCT is the Michelson interferometer, which is shown schematically in Fig. 1. Let the field at the source be written as ${E_0}(k)$, where k is the wavenumber. ${l_r}$ and ${l_s}$ denote the single pass optical path length of the reference and sample arms, respectively. The combined field at the output of this interferometer is expressed as

 

Fig. 1. Schematic diagram of a Michelson interferometer, a common core component of OCT systems.

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Placing a photodetector at the output of the interferometer shown in Fig. 1 captures a photocurrent. Over the integration time $\Delta t$ of the detector, the time-averaged photocurrent detected is expressed as

In the context of OCT, the measurand of interest is the optical path length of the sample arm, i.e. ${l_s}$. As may be seen from Eq. (3), one method to obtain this measurand is to vary ${l_r}$ until ${l_r} - {l_s} = 0$, where Eq. (3) achieves maximum value. In other words, the left-hand side of Eq. (3) becomes ${\overline P _1}({l_r})$. The form of the expected signal is a constant floor with a burst of fringes whose envelope peaks at the reference mirror axial position that satisfies ${l_r} = {l_s}$. Combined with mechanisms to laterally displace the optical beam with respect to the sample surface to scan point by point, this framework describes the principle of conventional TD-OCT.

Another method to obtain the measurand ${l_s}$ is to circumvent the integration over k in Eq. (3), for example by detecting ${E_1}(k)$ with a dispersive spectrometer as shown in Fig. 2(a). In this case, the detected photocurrent may be re-written as

 

Fig. 2. Schematic diagram of (a) conventional SD-OCT in comparison with (b) C-OCT.

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To compare the signal from FD-OCT to that of TD-OCT and C-OCT, we examine the form of the signal given by Eq. (5), post Fourier transform. For simplicity and without loss of generality, let the coefficients ${\alpha _r}(k) = {\alpha _s}(k) = 1/2$ (i.e. only accounting for the beamsplitter that is assumed to give the theoretical 50:50 split ratio). We rewrite Eq. (5) as

The C-OCT technique we developed is shown schematically in Fig. 2(b), where instead of detecting ${E_1}$ with a dispersive spectrometer, the field is sent into a secondary interferometer that forms a Fourier transform spectrometer. In the simplest implementation, the secondary interferometer may comprise two planar mirrors where one is stationary while the other is moving to introduce OPD between the two arms. For the secondary interferometer in C-OCT where light is in collimated space, a large spectral bandwidth is readily obtained as the only elements limiting the bandwidth are the optical materials, coatings, and the detector wavelength range. Its spectral resolution is inversely proportional to the OPD scanned by the translating mirror, which means that finer spectral resolution may be achieved by enlarging the OPD scan range and is independent of the spectral bandwidth requirements.

Let the OPD introduced by the secondary interferometer be $l \equiv 2({l_1} - {l_2})$, where ${l_1}$ and ${l_2}$ denote the single pass optical path length of the two arms, respectively. Note that we have defined l as the total resultant OPD from the secondary interferometer; this notation facilitates our understanding, especially in the case of a non-traditional interferometer configuration such as the one shown in Section 3. The field reaching the photodetector is expressed as

As may be seen from Eq. (12), the signal from a C-OCT system is expected to have three components. The first is a constant floor given by the first two terms within the curly brackets of Eq. (12). The second component is a fringe burst whose envelope peaks at $l = 0$; this is the DC term. The third component consists of two fringe bursts that are conjugate to each other and whose envelopes peak at $l ={\pm} 2\Delta $, respectively. The measurand ${l_s}$ may then be obtained via the distances between the conjugate fringe bursts and the DC term, similar to FD-OCT.

To visually compare the analytical forms of the signals from TD-OCT, FD-OCT, and C-OCT, a simulation was performed following the mathematical framework developed here. The same ${l_s}$ measurand at a single point on the sample was used, which in this case is axially 10 µm longer than the optical path length of the reference arm. A broadband Gaussian source spectrum was used, and the detector sensitivity was assumed to be constant and the same across the three techniques. The results of this simulation were normalized and shown in Fig. 3. In the FD-OCT case, an absolute value was taken after inverse Fourier transforming the obtained signal, which corresponds to obtaining the envelope of the fringe bursts. The measurements from both FD-OCT and C-OCT show 20 µm which is twice the measurand due to double-pass as expected. This factor of two also scales the abscissa, which is why the FWHM of the envelopes of FD-OCT and C-OCT as shown are twice that of TD-OCT, which may be seen visually in Fig. 3.

 

Fig. 3. Comparison of the analytical form of the signal with the measurand at ${l_s} = {l_r} + 10$(µm). (a) Normalized TD-OCT signal versus reference arm location ${l_r}$. (b) Normalized absolute value of the signal post inverse Fourier transform for FD-OCT versus z, which is the conjugate of k in the Fourier transform kernel. (c) Normalized C-OCT signal versus l, which is the total OPD introduced by the secondary interferometer.

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With the C-OCT mathematical framework derived in relation to conventional TD-OCT and FD-OCT, we note the following insights. Firstly, the Fourier transform secondary interferometer may be interpreted in a time-domain sense as matching the OPD of the primary interferometer. This time-domain perspective is reminiscent of tandem interferometry, though the latter only measures a single point with no field of view scanning over the sample [20,21]. Following this interpretation, the C-OCT technique effectively transforms a difficult-to-measure OPD, in this case the sag of a freeform surface measured against a reference surface, into the OPD of the secondary interferometer that is more readily measured. There are many more degrees of freedom in terms of interferometry configuration and design for the secondary interferometer compared to that for the primary interferometer. For instance, various approaches from the field of displacement measuring interferometry [22] may now be applied to optimize for the desired level of measurement uncertainty.

Secondly, in conventional FD-OCT, it may be difficult to understand physically the conjugate signal, as there is no sample physically at −20 µm (double-pass) for instance in the case of the Fig. 3 simulation. However, this conjugate becomes intuitive after examining the secondary interferometer in C-OCT. If we take on the time-domain interpretation of C-OCT where the secondary interferometer scans through OPD to match the OPD of the primary interferometer, it may be seen that there are two instances where the OPDs are matched, since the two arms in the secondary interferometer are mutually interchangeable.

3. High-speed interferometer with a rotating optical cube

In Fig. 2, the secondary interferometer is shown as another Michelson architecture with a simple axially-scanning mirror. While this configuration helps to explain the working principle of C-OCT, the oscillatory linear motion of the scanning mirror does not lend itself to high-speed actuation with travel ranges of several millimeters to be relevant for freeform optics metrology.

To enable high-speed measurements with a sufficiently large OPD scan range, we developed an interferometer where a rotating optical cube is used to generate OPD as a function of the rotation angle. A schematic of this design is shown in Fig. 4. The optical cube undergoes continuous rotation with no acceleration or deceleration during measurements, which is much more stable and robust compared to a translating mirror. This rotational motion is also much more scalable in speed depending on the motor used. In addition to the broadband laser source for C-OCT, a frequency stabilized helium-neon (He-Ne) laser is co-located for alignment and calibration.

 

Fig. 4. Schematic diagram of a C-OCT setup with a secondary interferometer that utilizes a rotating optical cube to generate OPD as a function of the rotation angle.

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As the cube rotates, OPD is introduced as a function of the rotation angle. The geometry used to derive the analytical expression for this OPD scan is shown in Fig. 5, where the OPD generated is the difference between the optical path length from point A to $A^{\prime}$ and that from point B to $B^{\prime}$. It was found that there exist regions of constant zero OPD and regions of varying non-zero OPD. The critical rotation angles that bound the non-zero OPD regions, which are used for measuring the surface under test, are related to the aspect ratio of the cross section of the rotating optic and the refractive index of the optic material. One instance of the cube just past a critical angle is shown in Fig. 5(c). It may be found through geometry that the angular range to obtain non-zero OPD for an N-BK7 cube (i.e. an aspect ratio of 1:1 for the cross section) is approximately ${\pm} 13.6^\circ $ about the nominal position shown in Fig. 5(a) for an infinitely thin beam; in reality this angular range will be correspondingly shortened depending on the beam diameter. Due to the symmetry of the cube, the non-zero OPD region occurs four times per revolution, making it highly leveraged for measurement speed as the rate of measurement is four times faster than the cube rotation speed.

 

Fig. 5. Geometry and definitions of the rotating optical cube in top-down view with the ray-paths shown in red going through the cube. The nominal position is defined in (a), an instance during rotation within a non-zero OPD region is shown in (b), and an instance during rotation within a zero OPD region is shown in (c) that is just past a critical angle. The cube is shown rotating counter-clockwise from (a) to (b) to (c).

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Over the non-zero OPD regions, the analytical expression for OPD is derived in Appendix B to be

 

Fig. 6. (a) Plot of OPD versus rotation angle from 0° to 90° with an optical cube, exhibiting constant zero OPD regions and a varying non-zero OPD region that is highly linear. This behavior is repeated 4 times during one full revolution. (b) Plot of the negligible difference between the OPD found using the analytical equation and that with the ray-trace simulation within the non-zero OPD region, validating Eq. (13).

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According to the literature, a high-speed interferometer configuration based on a rotating rectangular prism has been reported [23]. For comparison, with the same optical ray-trace model that was used for the rotating cube, we modelled a rotating prism with a rectangular cross section of aspect ratio 2:1 as a test case. The double-pass OPD for the two cases are plotted together in Fig. 7.

 

Fig. 7. Plot of OPD versus rotation angle over one full revolution for a rotating optical element with a square cross section of 20 mm by 20 mm versus that with a rectangular cross section of 20 mm by 10 mm. The regions with missing OPD data for the rectangular prism are from rays not reaching the return mirrors due to total internal reflection within the prism.

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As may be seen from Fig. 7, the rectangular prism is able to offer almost twice as large an OPD scan range with half of the material mass compared with the cube. However, in the context of freeform optics metrology, the cube configuration is more advantageous for the following reasons. Firstly, during one full revolution, there exist angular regions over which total internal reflection occurs within the rectangular prism, resulting in the rays not reaching the detector and rendering such regions unusable; this condition does not exist for the cube. In addition, while both designs offer four regions of quasi-linear OPD scan per revolution, those with the cube are equal in OPD scan range while those with the rectangular prism are highly unequal; in fact, two of the four OPD scan regions with the rectangular prism are barely visible in Fig. 7 at the ±90° marks. This characteristic combined with the lost signal due to total internal reflection make the rectangular prism have effectively only 2 usable scan regions versus 4 with the cube, meaning that when driven at the same speed, an interferometer with the cube will be able to make measurements two times faster than that with the rectangular prism. Secondly, outside of the quasi-linear OPD regions, the cube gives constant zero OPD whereas the rectangular prism gives a significant non-zero OPD that varies with angle. This varying OPD behavior outside of the quasi-linear OPD regions makes it more difficult in experiment to tell apart which are the regions containing useful information about the sample under test. Considering these differences and the observation that most standard size (1 in. to 2 in. in diameter, 1 in. = 2.54 cm) optical components rarely have surface sags beyond several millimeters, using a rotating cube in this case is preferred.

As may be seen from Eq. (13), $OPD(\theta )$ is also dependent on the refractive index n and thus the wavelength. The OPD scan over the wavelength range of interest for our prototype C-OCT setup is plotted in Fig. 8(a) for given $\theta $ angles at the edge of the scan range. It was found that the OPD dependence on wavelength is non-linear with up to a cubic term and the non-linearity increases with the cube rotation angle away from the nominal position. The analytical model across wavelength was validated using the same numerical ray-trace model with the negligible difference between the two shown in Fig. 8(b), enabling the use of the analytical equation to next simulate the form of the C-OCT signal with this rotating cube interferometer.

 

Fig. 8. (a) Plot of OPD versus wavelength for cube angle of 13° and 12.9° away from the nominal position. (b) The negligible difference between the OPD obtained using Eq. (13) and that with the ray-trace simulation, validating Eq. (13) across wavelength.

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As derived in Section 2, the analytical form of the C-OCT signal is given by Eq. (11) and Eq. (12). Equation (13) is substituted into Eq. (12) using $l = OPD(\theta ;n(k))$ to obtain the form of the signal that accounts for the dispersion of the rotating cube. The simulated signal at an example sag of 1 mm is shown in Fig. 9, which has been zoomed in to show the three fringe bursts. As may be seen, the cube dispersion leads to asymmetric broadening of the left and right fringe bursts and their maximum intensities are correspondingly lowered. These results show that the signal-to-noise ratio in experiment is expected to decrease with increasing sag, which may in turn bound the total measurement volume of the system. It may also be seen that the DC fringe burst remains symmetric due to the symmetry of the secondary interferometer design. It was found that when this design symmetry is broken, e.g. when the two arms are not nominally equal in OPD, the DC fringe burst asymmetrically broadens as well. As such, this characteristic may be utilized in experiment to fine align the secondary interferometer using the shape of the DC fringe burst as feedback.

 

Fig. 9. The simulated C-OCT signal obtained with the rotating cube secondary interferometer for a sample sag of 1 mm, zoomed in to show the left, DC, and right fringe bursts. The abscissa range is kept the same across the three plots for ease of comparison.

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4. Experimental results and discussions

A prototype setup was built to validate the C-OCT technique at a given point on a sample surface. As each point in the measurement point-cloud array undergoes the same physical process and software processing, experimental data at a single point is crucial for the validation of this technique on a system level. We outline the experimental setup and the data processing methods used, followed by the experimental results and discussions.

The prototype setup follows the schematic shown in Fig. 4. The broadband source used is a supercontinuum laser (SuperK Versa, Koheras) that has been spectrally filtered using a shortpass filter (10CGA-715, Newport) and a longpass filter (10SWF-1100-B, Newport) to span a spectral range from approximately 720 nm to 1080 nm, which enables an axial PSF FWHM of approximately 1 µm. The signal obtained with this broadband laser is hereinafter referred to as the near-infrared (NIR) signal. The pump source for this supercontinuum laser is at 1064 nm and therefore a notch filter (#86-128, Edmund Optics) was used to enable a more uniform spectrum. Two mirrors (PF10-03-P01, ThorLabs) were used as the reference and sample surface in the primary interferometer. A motorized rotation stage (CR1-Z7, ThorLabs) was used to actuate the optical cube in the secondary interferometer. The optical cube was made in-house from N-BK7 and is 20 mm by 20 mm, enabling the measurement of up to 3 mm of sample surface sag departure from the reference surface. A frequency stabilized red He-Ne source (HRS015B, ThorLabs) was aligned to be co-linear with the NIR beam within the secondary interferometer. Two photodetectors were used, one for the NIR signal (PDA100A, ThorLabs) and the other for the He-Ne signal (PDA100A2, ThorLabs). The two detectors were connected to one digitizer board (ATS9350, AlazarTech) for synchronous detection. Custom data acquisition scripts were developed in MATLAB to control the cube rotation and digitize the detected signals.

The first step in processing the obtained raw data is to calibrate for speed variations of the rotating optical cube. The steps involved are shown in Fig. 10 and are as follows. To better illustrate this process, we show a zoomed-in area about the central DC fringe burst for Figs. 10(a)–10(c); the process described utilizes the entire signal. First, the 50% reference level of the He-Ne signal is determined by detecting the high and low state levels of the signal with the state level tolerance set to ±5% (see MATLAB midcross function for more details). This reference level is then removed which is similar to removing the mean of the signal. The zero-crossing locations of the He-Ne is then detected. As shown in Fig. 10(a), the crossing points with positive slopes are denoted on top of the He-Ne with “x” markers. Note that either positive slope or negative slope crossing points may be used but not both at the same time in order to maintain one wave of OPD between the crossing points used. Detecting the zero-crossing points gives the mapping of number of He-Ne waves versus pixel, shown in Fig. 10(b). Given that the NIR and He-Ne signals are simultaneously detected, we use this mapping with spline interpolation to calibrate the NIR signal such that it is linear with respect to the number of waves of He-Ne. This process is shown schematically with the dashed lines in Fig. 10(a) and the result is shown in Fig. 10(c). This calibration effectively mitigates for speed variations in the optical cube rotation, which may be seen visually by comparing the NIR signal frequency irregularities between Fig. 10(a) and Fig. 10(c). Note that we have centered the coordinate origin at the peak of the DC fringe burst after calibration; this is because in this technique it is the relative locations of the side fringe bursts with respect to the DC fringe burst that give us the measurement, rather than the absolute locations. The zoomed-out view in Fig. 10(d) shows the three fringe bursts as expected from the mathematical framework derived in Section 2.

 

Fig. 10. One instance of experimentally obtained data with the prototype C-OCT setup. (a) Simultaneously detected NIR signal and He-Ne signals. The shown He-Ne signal is after removing its 50% reference level and detecting its positive zero crossing points. (b) The mapping of number of waves of He-Ne versus the signal data points, which is used to calibrate the NIR signal. (c) The NIR signal after calibration so that it is linear with respect to the number of He-Ne waves. (d) The zoomed-out view of the entire NIR signal, showing the three fringe bursts, which validates the mathematical framework of Section 2.

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With the NIR signal calibrated to be linear with respect to the number of He-Ne waves, the next step is to convert the abscissa to length units. This conversion is not simply performed by multiplying the number of He-Ne waves by the He-Ne wavelength. With the rotating optical cube, the paths of the rays traversed by the He-Ne beam are different from those traversed by the NIR beam due to the cube index of refraction resulting in different refraction angles at different wavelengths. An analytical calibration function is built to correlate the number of He-Ne waves to the sag indicated by the NIR beam. The process to build this function is as follows. First, Eq. (13) is substituted into Eq. (12) using $l = OPD(\theta )$ to form the simulation model used to generate the expected NIR signal. It is important to take into account $n(k)$ within these equations, for which the Sellmeier equation is used. Next, a series of sag departures that we refer to as sag ground truths are input into this simulation model and the expected NIR signal for each sag input is generated. The He-Ne signal is generated using Eq. (13) and is independent of the sag ground truths. Finally, for each sag input, the peaks of the envelopes of the fringe bursts in the NIR signal are found and the number of He-Ne waves between the DC fringe burst and the peaks of the side envelopes are obtained. Sufficient sampling was used in wavenumber k to ensure no aliasing and in rotation angle $\theta $ to ensure smooth fringes as well as equivalent distances from the DC fringe burst to the side fringe bursts. This calibration function is shown in Fig. 11(a).

 

Fig. 11. (a) The analytical calibration function to correlate the number of He-Ne waves to the sag experienced by the NIR beam. (b) The residual of the analytical calibration function after removing its linear fit.

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As may be seen from Fig. 11(a), the analytical calibration function is highly linear with the linear fit showing ${R^2} = 99.9999982\%$. However, within the context of optical metrology, it may not be sufficient to extract the slope of the linear fit to use as a calibration factor across sag range. The analytical calibration function after removing its linear fit is shown in Fig. 11(b) with up to approximately one wave peak-to-valley of residual; this residual nonlinearity is attributed to the asymmetric broadening of the axial PSF as shown in Section 3. For the desired level of measurement uncertainty that is on the order of tens of nanometers up to a hundred nanometers as discussed in Section 1, the full calibration function of Fig. 11(a) should be used. This calibration function is applied to the experimental data shown in Fig. 10(d) and the final result is shown in Fig. 12, which may now be used to obtain the sag measurement in length units.

 

Fig. 12. The final result in length units of the experimental data shown in Fig. 10 after processing with the analytical calibration function of Fig. 11(a).

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Consecutive measurements were carried out at a given point on the sample surface to quantify precision. The processing method described here is applied across these experimental data. The final measurement for each data is taken to be the mean value between the sag measured using the left fringe burst and that using the right fringe burst. Twenty consecutive measurements are shown in Fig. 13, the RMS of which is 26.1 nm, or approximately λ/24 at the He-Ne wavelength.

 

Fig. 13. Consecutive single-point measurements with the prototype C-OCT setup showing 26.1 nm RMS, which is approximately λ/24 at the He-Ne wavelength.

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From these experimental results, there are several aspects worth discussing. First, as may be seen from Fig. 10(a) and Fig. 10(c), there are quasi-monochromatic oscillations outside of the axial PSF region. This is attributed to the strong 1064 nm pump source of the supercontinuum laser source, some power of which remains even after the applied notch filter. The effect of these oscillations is negligible as their peak-to-valley magnitudes are more than five times smaller than that at the center of the DC fringe burst and they extinguish rapidly outside of the region shown in Fig. 10(a) and Fig. 10(c).

Secondly, it was observed that the sag measurement using the left fringe burst is slightly different from that using the right fringe burst. In the case of Fig. 12, this difference is 67.5 nm; over the twenty consecutive measurement shown in Fig. 13, this difference is found to be 76.9 ± 58.9 nm. These differences are attributed to residual misalignments in the secondary interferometer that disrupts its symmetric design as well as slight imperfections in the dimensions of the optical cube. Currently with the prototype setup, the motorized rotation stage used is not straightforward to program for continuous rotation and is slow with less than 1 rotation per minute. A high-speed configuration is planned for the next development of this system with continuous rotation capabilities to enable rapid feedback, which will guide fine alignment and isolate residual errors due to alignment from that due to the cube quality. An experimental sag calibration function may then be built in addition to the analytical one shown in Fig. 11.

Lastly, the ∼176.7 µm measured sag depth position of the sample surface (Fig. 13) was chosen arbitrarily as an example point for experimental testing. As C-OCT is a point-cloud technique, the results obtained validate both the working principle of C-OCT as well as that of the rotating cube interferometer.

5. Summary

In this work, we developed and investigated a technique, C-OCT, towards the task of freeform optics metrology. A rotating cube interferometer design was developed for the C-OCT secondary interferometer to target high-speed measurements. A prototype C-OCT setup was built and demonstrated precision of ±26.1 nm (∼λ/24 at the He-Ne wavelength) at a given point on the sample surface. The experimental results obtained validate both the mathematical framework presented for the C-OCT technique as well as the working principle of the rotating cube interferometer. With the technique validated, a high-speed rotating cube together with lateral scanning mechanisms are being implemented in the next generation of the system to rapidly scan the surface of a sample, leading to surface measurements of freeform optical components.

Appendix A

We derive here Eq. (2) that describes the time-averaged photocurrent generated by an impinging field on a photodetector. The instantaneous intensity of the field as a function of time is given by

(14)$$I(t) = c{\mathrm{\epsilon}_0}|E(t){|^2}, $$

where c is the speed of light in vacuum and ${\mathrm{\epsilon}_0}$ is the permittivity in vacuum [24]. Over the integration time of the detector, the time-averaged intensity may be written as

(15)$$\overline I = \frac{{c{\mathrm{\epsilon}_0}}}{{\Delta t}}\int\limits_{{t_1}}^{{t_2}} {|E(t){|^2}dt}, $$

where $\Delta t$ is the integration time. Note that $\overline I $ is implicitly a function of ${t_1}$ which is the time stamp at which the intensity is interrogated. For our OCT system, we use a broadband, pulsed laser source. Under the assumption of stationarity, the time average of the pulse intensity over an infinite time window is the same as the time average of the intensity within a finite time window containing a sufficient number of pulses. Therefore, we obtain

(16)$$\begin{aligned} \overline I &= \frac{{c{\mathrm{\epsilon}_0}}}{{\Delta t}}\int\limits_{ - \infty }^\infty {|E(t){|^2}dt} \\ &= \frac{{c{\mathrm{\epsilon}_0}}}{{\Delta t}}\int\limits_{ - \infty }^\infty {|\widetilde E(\omega ){|^2}d\omega } \end{aligned}, $$

where $\widetilde E(\omega )$ is the Fourier transform of $E(t)$ and the Parseval’s theorem was used. Using the relationship $\omega = ck$, Eq. (16) may be re-written as

(17)$$\overline I = \frac{{{c^2}{\mathrm{\epsilon}_0}}}{{\Delta t}}\int\limits_{ - \infty }^\infty {|\widetilde E(ck){|^2}dk} = \frac{{{c^2}{\mathrm{\epsilon}_0}}}{{\Delta t}}\int\limits_{ - \infty }^\infty {|\widetilde {\widetilde E}(k){|^2}dk}, $$

using the fact that c is a constant and may be absorbed into the functional form of $\widetilde {\widetilde E}(k)$. A photodetector has a spectrally dependent responsivity $\Re (k)$ and an active area $A$. In this framework, we keep the spatial dependence implicit for simplicity of the equations; in other words, $\widetilde {\widetilde E}(k)$ is implicitly $\widetilde {\widetilde E}(k;x,y)$. Therefore, the time averaged photocurrent from a detector may be expressed as

(18)$$\overline P = \frac{{{c^2}{\mathrm{\epsilon}_0}}}{{\Delta t}}\int\limits_{ - \infty }^\infty {\int\limits_A {\Re (k)|\widetilde {\widetilde E}(k){|^2}} dAdk}. $$

Appendix B

We derive here Eq. (13) that describes the OPD generated by the rotating cube interferometer. The geometry used for this derivation is shown in Fig. 14 and is consistent with that of Fig. 5. All angles and lengths are taken to be positive and the angles are in degrees. For parameters of the two rays, all unprimed symbols correspond to Ray 1, while the primed symbols correspond to Ray 2.

 

Fig. 14. Geometry used to derive the analytical expression Eq. (13) for the rotating cube interferometer. In this diagram, boldface symbols refer to lengths while italicized symbols are angles.

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In addition to the variables defined in Fig. 5, D denotes the lengths of ray segments in glass, X is the projection of the Ray 1 glass segment ($D$) onto the horizontal axis. $Y^{\prime}$ is the projection of the Ray 2 glass segment ($D^{\prime}$) onto the vertical axis. i denotes the incidence angle measured from the surface normal, t denotes the transmitted angle measured from the surface normal, a is the elevation angle of Ray 1 measured from the horizontal axis, and $a^{\prime}$ is the elevation angle of Ray 2 measured from the vertical axis.

The following relations and identities are derived using geometrical principles.

(19)$$i = \theta$$

(20)$$a = i - t$$

(21)$$a^{\prime} = i^{\prime} - t^{\prime}$$

(22)$$i^{\prime} = 90 - \theta$$

(23)$$X = D\cos (a)$$

(24)$$Y^{\prime} = D^{\prime}\cos (a^{\prime})$$

In the current case the cube is surrounded by air, the refractive index of which is assumed to be 1. Using Snell’s Law, we have

(25)$$\sin (i) = n\sin (t). $$

Using the blue-shaded right triangle of Fig. 14, we obtain

(26)$$\sin (t) = \frac{{\sqrt {{D^2} - {L^2}} }}{D}. $$

Combining Eq. (25) and Eq. (26), we obtain

(27)$$D\sin (i) = n\sqrt {{D^2} - {L^2}}. $$

Using Eq. (27) and Eq. (19), we solve for $D$ to obtain the geometrical path length of Ray 1 within the glass,

(28)$$D = \frac{{nL}}{{\sqrt {{n^2} - {{\sin }^2}(\theta )} }}. $$

Similarly, the geometrical path length of Ray 2 within the glass may be derived. In this case, Eq. (22) instead of Eq. (19) is used to relate the incidence angle to the cube rotation angle. As such,

(29)$$D^{\prime} = \frac{{nL}}{{\sqrt {{n^2} - {{\sin }^2}(90 - \theta )} }}. $$

The total optical path length (OPL) for either ray may be broken down into two components, i.e. the distance traveled through air multiplied by the refractive index of air (assumed to be 1) and the distance traveled through glass multiplied by its refractive index n. As may be seen from Fig. 14, the distance traveled through air is the total length of the bounding box minus the projection of the ray through the glass along the horizontal axis $X$. Therefore, the OPLs for Ray 1 and Ray 2 are given respectively by

(30)$$OP{L_1} = (\sqrt 2 L - X) + nD, $$

(31)$$OP{L_2} = (\sqrt 2 L - Y^{\prime}) + nD^{\prime}. $$

Therefore, the OPD between the two rays after they traverse the cube once is

(32)$$OPD = OP{L_1} - OP{L_2} ={-} X + Y^{\prime} + nD - nD^{\prime}. $$

From Eq. (32), it may be seen that the dependence on the bounding box dimension ($\sqrt 2 L$) has vanished, which suggests that one may use as reference a bounding box of arbitrary dimension as long as it is symmetric about the center of rotation of the cube and is aligned to the axis we have chosen here. Using Eq. (23) and Eq. (24), we rewrite Eq. (32) as

(33)$$OPD = D[n - \cos (a)] + D^{\prime}[\cos (a^{\prime}) - n]. $$

Using Eq. (19), Eq. (20), and Eq. (25), we derive the expression for the angle a in terms of the rotation angle $\theta $ and obtain

(34)$$a = \theta - \arcsin (\frac{{\sin (\theta )}}{n}). $$

Similarly, we use the relationship between i and $i^{\prime}$ to derive the expression for the angle $a^{\prime}$ in terms of the rotation angle and obtain

(35)$$a^{\prime} = 90 - \theta - \arcsin (\frac{{\sin (90 - \theta )}}{n}). $$

Finally, we substitute Eq. (28), Eq. (29), Eq. (34), and Eq. (35) into Eq. (33) to obtain

(36)$$\begin{aligned} OPD &= \frac{{nL}}{{\sqrt {{n^2} - {{\sin }^2}(\theta )} }}\{ n - \cos [\theta - \arcsin (\frac{{\sin (\theta )}}{n})]\} \\ &\quad+ \frac{{nL}}{{\sqrt {{n^2} - {{\sin }^2}(90 - \theta )} }}\{ \cos [90 - \theta - \arcsin (\frac{{\sin (90 - \theta )}}{n})] - n\} \end{aligned}. $$

Funding

National Science Foundation (IIP-1338877, IIP-1338898, IIP-1822026, IIP-1822049); Corning Incorporated Office of STEM Graduate Research Fellowship.

Acknowledgments

We thank Synopsys, Inc. for the education license of LightTools. We thank Gustavo A. Gandara-Montano, Changsik Yoon, and Romita Chaudhuri for stimulating discussions.

Disclosures

The authors declare no conflicts of interest.

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