## Abstract

A hybrid confocal-scan swept-source optical coherence tomography metrology system was conceived for simultaneous measurements of the refractive index and thickness profiles of polymeric layered gradient refractive index (GRIN) optics. An uncertainty analysis predicts the metrology capability of the system and guides the selection of an optimum working numerical aperture. Experimental results on both a monolithic and a GRIN layered sheet are demonstrated to be in close agreement with theoretical predictions. Index measurement precision reached 0.0001 and 0.0008 for measuring 2.8 mm and ~300 µm thick layers, respectively. The thicknesses of these layers were simultaneously measured with a precision of 0.28 and 0.17 µm, respectively.

© 2015 Optical Society of America

## 1. Introduction

As a fundamental physical property of an optical material, the refractive index governs the light propagation within the optical medium and gives rise to a broad range of optical phenomena. Accurate knowledge of the refractive indices of materials along with the confidence bounds on the values is critical to optical design, tolerancing, and implementation. To determine the refractive index of a sample, interferometric metrology techniques are generally confronted with the challenge that the measurand obtained reflects the optical thickness of a sample, which involves the coupling of the index with the physical thickness. Using low-coherence interferometry (LCI), several techniques have been proposed to decouple the index and physical thickness, which may be categorized into two main methodologies [1]: 1) evaluating the optical distortion of a reference induced by the sample inserted in the path [2–4] and 2) combining LCI with a focus tracking modality such as confocal scans or multi-photon microscopy [5–7] to obtain another independent relation between index and thickness.

Recently, a class of bio-inspired nanolayered-polymer-film-based gradient-index (GRIN) optics invented at Case Western Reserve University has achieved unprecedented modulations of refractive index within small volumes [8,9]. This breakthrough in the fabrication of GRIN optics brings forward new perspectives to light-weight, compact optical systems. The manufacturing process of these layered polymeric GRIN components was detailed in [10]. To summarize, a library of 50 µm thick nanolayered polymer films are first manufactured whose refractive indices, as predicted by the effective medium theory [11], depend on the volumetric ratio of the constituent polymer raw materials with substantially different refractive indices prior to film co-extrusion. A number of 50 µm thick films of varying polymeric compositions and therefore refractive indices are then stacked based on a prescribed gradient index recipe and then thermo-compressed to create a consolidated axial GRIN sheet [12]. This GRIN sheet is further thermoformed into a curved spherical GRIN preform [13], which is subsequently diamond-turned into a final spherical GRIN lens.

The implementation of the GRIN components in optical systems requires accurate metrology of their refractive index profiles, which is crucial to the aberration correction performance of the parts. Optical coherence tomography (OCT) [14] as a high-resolution, high-sensitivity and high-speed imaging technique has been utilized for the metrology of refractive indices [1–7]. The applications to date have been mainly demonstrated for homogeneous samples. In terms of GRIN reconstruction, by evaluating the optical distortion of the posterior surface based on ray tracing and optimization algorithms, de Castro et al. estimated the GRIN profile of a crystalline lens to the precision of 4 × 10^{−3} [4]. On the other hand, the method of integrating OCT with confocal scans enables direct measurements of the indices of layered samples without the assumption of a mathematical model for the index profile. To measure fine gradient index (e.g., 0.005 step gradient), however, the method needs to be carefully assessed to establish the protocols to reach the required capability in measurement accuracy, precision, and sensitivity.

In this paper, we demonstrate the uncertainty analysis and experimental implementation of a hybrid confocal-scan swept-source OCT (SS-OCT) system we conceived and discuss its metrology results on layered samples. We first investigate the algorithms for simultaneous measurements of index and thickness in Section 2; a comprehensive uncertainty analysis is carried out in Section 3 to identify the key operating conditions and achievable accuracy and precision; the system layout is described in Section 4; in Section 5, we provide the metrology results of the index profiles of a monolithic and a GRIN layered sheet, respectively, followed by a discussion in Section 6; finally, we conclude the paper in Section 7.

## 2. Algorithms for simultaneous estimations of refractive index and thickness

The original focus tracking methodology was first demonstrated using time domain OCT [1,5,6]. Fourier domain OCT (FD-OCT) (i.e., swept-source or spectral-domain OCT) [15] enables higher sensitivity and higher speed together with the simultaneous capture of back-reflection signals from multiple interfaces within a single shot acquisition of an interference spectrum, including without translating the reference mirror. The depths where the reflections occur are encoded in the modulation frequencies of the spectrum, through a Fourier transform of which a depth profile with multiple axial point-spread-functions (PSFs) located at the depths of the interfaces is obtained. The amplitude of each peak is modulated with respect to its distance from the focal point of the imaging lens and the reflectivity of the interface.

To exemplify the method, measurements on a single layer (or a homogeneous block) are illustrated in Fig. 1. By measuring the distance between the peaks of the two intensity PSFs on an FD-OCT depth profile as shown in Fig. 1(b), the group optical thickness (Δ*D*) between the top and bottom surfaces of the layer can be obtained, which is expressed as

*n*is the group refractive index and

_{g}*t*is the physical layer thickness. The group index

*n*is used in this expression given the low-coherence interferometry nature of OCT. In FD-OCT, no mechanical scan of a reference mirror is necessary to accurately compute the group optical path between two surfaces.

_{g}In order to extract simultaneously the group index *n _{g}* and the physical layer thickness

*t*, an additional independent relationship between these two quantities is required. In our approach, we integrated a confocal scanning technique to FD-OCT as schematically illustrated in Fig. 1(a) to obtain the additional information of the focal shift distance (Δ

*z*) needed to focus the objective lens from the top surface of the layer to the bottom surface, which yields an additional relationship between, in this case, the phase index of refraction

*n*and the thickness

_{p}*t*given as

*NA*is the numerical aperture of the objective lens, and

*n*is the phase index of the ambient air. In practice, Δ

_{air}*z*is obtained from the confocal intensity profiles of both the top and bottom surfaces as shown in Fig. 1(c), which are reconstructed in post-processing by tracing the peak amplitude of the respective PSFs on a sequence of depth profiles acquired while the objective lens is translated to focus through the top and bottom surfaces.

Note that the refractive index involved in the confocal-scan Δ*z* measurement is the phase index, whereas the FD-OCT Δ*D* measurement concerns the group index. To obtain independently the phase index, group index, and thickness of a layer, the conversion between the group and phase indices is required as described by the dispersion relation [16]

Combining Eqs. (1)-(3), *n _{p}* can be computed by solving the following quartic equation:

*n*is computed,

_{p}*n*and

_{g}*t*can be solved consecutively based on Eqs. (3) and (1).

## 3. Uncertainty analysis

An analytic solution to the quartic equation Eq. (4) can be readily obtained, yet it is in a rather complex form and is thus omitted here. In brief, assuming Δ*n _{disp}* is a known constant, the phase index

*n*may be expressed as a function of the four variables, i.e., the group optical thickness Δ

_{p}*D*, the focal shift Δ

*z*, working

*NA*and air phase index

*n*, as

_{air}*n*is associated with those of the four variables and may be divided into systematic and random errors, i.e., accuracy and precision, which are separately assessed.

_{p}According to the uncertainty propagation theory [17], assuming the measurements of Δ*D*, Δ*z*, *NA* and *n _{air}* are independent, the accuracy associated with

*n*may be expressed as

_{p}*B*with a subscript denotes the bias limit in each variable. Equation (6) relates the accuracy of

*n*with those of all involved variables. The explicit expressions for each partial derivative are not necessary to be carried out considering their complex mathematical forms. Denoting${B}_{{n}_{p},\Delta D}=\left(\frac{\partial f}{\partial \Delta D}\right){B}_{\Delta D},$${B}_{{n}_{p},\Delta z}=\left(\frac{\partial f}{\partial \Delta z}\right){B}_{\Delta z},$${B}_{{n}_{p},NA}=\left(\frac{\partial f}{\partial NA}\right){B}_{NA},$and ${B}_{{n}_{p},{n}_{air}}=\left(\frac{\partial f}{\partial {n}_{air}}\right){B}_{{n}_{air}},$ Eq. (6) may be rewritten as

_{p}*n*in Eq. (4), the bias error in

_{p}*n*induced solely by each variable (i.e., ${B}_{{n}_{p},\Delta D}$, ${B}_{{n}_{p},\Delta z}$, ${B}_{{n}_{p},NA}$, and ${B}_{{n}_{p},{n}_{air}}$) may be obtained. Based on Eq. (7), the overall accuracy of

_{p}*n*may then be estimated. Note that the root-sum-squares method is used for the propagation of errors [17].

_{p}Similarly, the precision associated with *n _{p}* may be estimated by analyzing the potential precision errors of individual variables (i.e., ${S}_{\Delta D},$${S}_{\Delta z},$${S}_{NA},$and${S}_{{n}_{air}}$, where

*S*with a subscript denotes the standard deviation in each variable that is used as a metric of precision) and their impact on the

*n*precision. Denoting${S}_{{n}_{p},\Delta D}=\left(\frac{\partial f}{\partial \Delta D}\right){S}_{\Delta D},$${S}_{{n}_{p},\Delta z}=\left(\frac{\partial f}{\partial \Delta z}\right){S}_{\Delta z},$${S}_{{n}_{p},NA}=\left(\frac{\partial f}{\partial NA}\right){S}_{NA},$and ${S}_{{n}_{p},{n}_{air}}=\left(\frac{\partial f}{\partial {n}_{air}}\right){S}_{{n}_{air}},$ the overall

_{p}*n*precision may be expressed as

_{p}As will be seen in Sections 3.3 and 3.4, the working *NA* and air phase index *n _{air}* are experimentally quantified for a given system and environment, and therefore may only induce bias errors. Thus, the ${S}_{{n}_{p},NA}$ and ${S}_{{n}_{p},{n}_{air}}$terms may be further omitted in Eq. (8). Once the accuracy and precision of

*n*are estimated, the uncertainty associated with the physical thickness

_{p}*t*is readily obtained based on Eqs. (1) and (3).

In Fig. 2, the top flow chart outlines the protocol of the uncertainty analysis method dividing into bias and precision analysis, each consisting of four steps. The protocol may serve as a guideline for the uncertainty analysis of general optical metrological instruments. The bottom flow chart expands on the guideline according to color codes that match those of the top chart. The bottom chart elaborates the procedures for the specific task of estimating the *n _{p}* (similar for

*t*) measurement uncertainty by the confocal-scan SS-OCT, which will be discussed in detail in the following subsections.

#### 3.1 Impact of ΔD uncertainty on n_{p} measurement uncertainty

The uncertainty of the confocal-scan SS-OCT system in determining the group optical thickness Δ*D* of a sample layer is impacted by the signal-to-noise ratio (SNR) of the back-reflections from both surfaces. The factors affecting the SNR in SS-OCT are 1) the intrinsic sensitivity fall-off with the increasing depth that is defined as the optical path difference (OPD) between the sample and reference paths and 2) the signal power fall-off with the defocus distance of the sample from the focal plane of the objective lens.

The sensitivity roll-off with depth is attributable to the finite instantaneous linewidth of the swept source. The convolution of the lineshape function with the spectral interferogram leads to the multiplication of the axial PSF with a fall-off curve through depth in the Fourier-transformed z-space. We experimentally measured the system sensitivity decay with depth and plot in Fig. 3(a) the peak intensities of the axial PSFs through a depth range of 5 mm at every 100 µm intervals. The signal power was normalized by its maximum at the first measured depth of ~20 µm instead of the zero to avoid the proximity of the DC term. It can be seen from the figure that the signal drops by about 10 dB at 5 mm depth.

On the other hand, the increase in the distance of a sample surface away from the focus of the objective lens causes the peak amplitude of the axial PSF of the surface to follow a fall-off curve as shown in Fig. 3(b). The blue curve was experimentally sampled by measuring the top surface of a fixed BK7 flat and translating the objective lens (working NA of 0.1782) at every 10 µm axially to form defocus distances of 0 – 2500 µm, which fits the theoretical fall-off curve in black [18].

The two signal fall-off factors and associated SNR lead to an increase in the measurement uncertainty when determining the axial locations of back-reflections captured by SS-OCT. We established a model in simulation that accounted for both factors and the noise sources of the A-scan trigger jitter, the source intensity noise, and the detector noise to quantitatively predict the uncertainty in measuring the group optical thickness Δ*D* of a sample layer composed of a dispersive material. Specifically, to account for the A-scan trigger jitter, a random shift (standard deviation of 0.055 nm) of the nominal wavelength $\overline{\lambda}$ was generated in simulation based on a trigger noise distribution experimentally quantified by an oscilloscope on the SS-OCT system, which is expressed as

*λ*represents the standard deviation of the wavelength shift and

*λ*is the perturbed wavelength. One instance of the simulated output OCT interference spectra

_{sim}*N*(

_{g}*λ*) for a given

_{sim}*λ*is assumed to follow a normal distribution and can be generated using a normal random number generator as

_{sim}*N*(

_{g}*λ*)>>> and

_{sim}*K*(

_{Ng}*λ*) represent the mean and variance of the output reading of the detector after 12-bit digitization.

_{sim}*K*(

_{Ng}*λ*) may be expressed as a quadratic function of <<<

_{sim}*N*(

_{g}*λ*)>>> as [19]

_{sim}*C*,

_{1}*C*, and

_{2}*C*correspond to the laser relative intensity noise, Poisson noise, and dark noise, respectively. The values of the coefficients may be experimentally evaluated for an OCT system with specific hardware.

_{3}*C*is negligible in the confocal-scan SS-OCT system where the relative intensity noise is largely canceled out by balanced detection.

_{1}*C*and

_{2}*C*were evaluated to be 0.38 and 6, respectively.

_{3}We used the model to simulate SS-OCT measurements of a 10/90% PMMA/SAN17 layer, a material composition used for either a layered monolithic sample (Section 5.1) or within a GRIN sample, with the group optical thickness Δ*D* varying from 50 – 5000 µm at 50 µm increment to predict the error in measuring these optical thicknesses. The top surface was fixed at a depth of 50 µm whereas the bottom surface went through a depth range of 100 – 5050 µm. A set of 1000 spectra containing the interference signals from both surfaces of a layer was simulated for each scenario of the bottom surface position (and layer optical thickness Δ*D*) with the signal fall-off with depth and defocus being accounted for. Gaussian random noise was generated by Eqs. (9)-(11) in the simulated spectra in order for the later to represent a realistic distribution of 1000 measurements conducted repeatedly in experiment. Each spectrum was then zero-padded and Fourier-transformed (axial sampling resolution of ~2 nm) to yield a depth profile (see Fig. 1(b)), where peak detection was performed to extract the axial locations of the layer surfaces. The mean and standard deviation of the measured location difference of the two surfaces (i.e., the group OPD) across the 1000 simulated repeated measurements compared to the ground truth yielded the bias and the precision of the Δ*D* measurements, respectively. Note that the objective lens was simulated to focus at a range of positions (trials at 50 µm interval) within the layer and a best focus position on the bottom surface was found to consistently yield the best precision in Δ*D* measurements. A constraint in simulating the spectral interference signals was to assure that the simulated spectral power remained below the saturation threshold of the balanced photodetector (PDB460C, Thorlabs, NJ, USA) through the translation of the objective from focusing on the top to the bottom surface of the sample. In simulation, we also varied the NA of the objective lens (see Fig. 5(a) for the axial signal decay profile with the defocus distance at different NAs) to investigate its impact on the measurement of different optical thicknesses Δ*D*. The results of the Δ*D* measurement precision *S _{ΔD}* at different NAs are shown in Fig. 4(a). For layers up to 3 mm thick, with the associated amount of

*S*error introduced in Eq. (4), the corresponding

_{ΔD}*n*and

_{p}*t*were numerically solved. The results of the uncertainty in

*n*and

_{p}*t*induced by a Δ

*D*precision error are plotted in Figs. 4(c) and 4(d), respectively. It can be seen that lower NA (i.e., 0.1 and 0.2) yields less uncertainty in measuring Δ

*D*,

*n*and

_{p}*t*as a result of the extended depth of focus allowing capturing both surfaces with sufficient SNR. At 0.2 NA, for instance, we consider 50, 300 and 3000 μm thick samples that represent thicknesses on the same order of magnitude as a GRIN film layer, a film stack sharing the same index, and an entire GRIN sheet (see Section 5.2), respectively. For these three test cases, Δ

*D*precision is ~12, 45 and 430 nm, respectively, which results in

*n*precision of 1.2 × 10

_{p}^{−4}, 7 × 10

^{−5}and 7 × 10

^{−5}, and

*t*precision of 3, 14 and 140 nm, respectively.

It is worth noting that the simulation also indicated that the Δ*D* measured for a dispersive sample may be biased (i.e., systematic deviation from the product of the *n _{g}* at the center wavelength and the geometrical thickness of the sample). This is caused by the higher order dispersion property of the broadband light that was omitted in the group index expression given by Eq. (3); however it is non-negligible during the transmission within a dispersive medium [20]. The bias

*B*is shown by the dashed black curve in Fig. 4(b), with a magnitude of ~0.28 μm for Δ

_{ΔD}*D*= 5 mm. To address the bias, the axial scale of the SS-OCT system was experimentally calibrated using a 2.0109 mm thick BK7 window (thickness known to the accuracy of within 0.1 µm). This calibration was regarded to have eliminated the bias at ~3 mm Δ

*D*and removed a linear term of the entire bias curve as shown by the solid magenta curve in Fig. 4(b). The residual bias after calibration is well within the uncertainty bounds defined by the system Δ

*D*measurement precision.

#### 3.2 Impact of Δz uncertainty on n_{p} measurement uncertainty

The uncertainty in measuring the focal shift distance Δ*z* (see Fig. 1(c)) is dominated by the uncertainty of the linear motorized stage (VP-25XL, Newport, CA, USA) used to translate the objective lens. In a laboratory environment with temperature control within 1°C (e.g. 22 ± 0.5°C) and controlled atmospheric pressure and humidity, the performance of the stage was regarded as conforming to the manufacturer’s test report conducted with a Renishaw laser interferometer. The repeatability of the stage (a source of *S _{∆z}* uncertainty) was guaranteed to be within 0.14 μm. The stage accuracy (a source of

*B*uncertainty) was estimated to be in the range of 0.15 – 0.95 μm (i.e., an approximate linearly increased error of ~0.4 μm per 1 mm travel) within a 2 mm travel of the stage, which is of practical interest to the confocal-scan SS-OCT experiments on GRIN samples.

_{∆z}Besides the stage motion uncertainty, the intensity noise associated with the reconstructed confocal intensity profiles also brings about another source of uncertainty in determining the locations of the peaks corresponding to the focus positions of the objective lens. Figure 5(a) shows the theoretical confocal intensity profiles from a surface of a 10/90% PMMA/SAN17 layer at different NAs. It is evident and expected that a larger NA leads to a sharper confocal PSF. With the intensity noise added to the ideal confocal intensity profiles, Lorentzian fitting to the peaks was investigated as a method of identifying the peak location that is potentially more robust to noise. Figure 5(b) shows an example of a simulated raw noisy confocal intensity profile at 0.2 NA in black with its Lorentzian fit shown in magenta. To test the performance of Lorentzian fitting, a fixed level of Gaussian random intensity noise with a standard deviation of 0.5% was added to all the theoretical confocal intensity profiles to estimate the error in identifying the peak locations in the presence of noise. Figure 5(c) shows the means and standard deviations of the differences of the measured peak positions from the ground truth based on 1000 trials each at different NAs. The error in locating the peaks significantly decreases with increasing NAs, trend shown for measuring both noisy raw data and the Lorentzian-fitted profiles. The Lorentzian fit demonstrates unbiased peak detection with 10x (NA = 0.8) to 25x (NA = 0.1) precision improvement over processing the raw data.

Note that the confocal intensity profiles are actually reconstructed by tracing the peak amplitude of the respective axial PSFs on a series of OCT depth profiles acquired through the translation of the objective focus at a 100 nm increment. The noise associated with the OCT spectra (detailed in Section 3.1) leads to noise on the confocal intensity profiles whose variance ${K}_{{I}_{c}}\left(z\right)$is found to be a quadratic function of the intensity${I}_{c}(z)$as

*C*,

_{1}*C*and

_{2}*C*were evaluated to be 6.4 × 10

_{3}^{−7}, 3.1 × 10

^{−5}and 0.059, respectively.

Considering this noise distribution, confocal intensity profiles with Gaussian random intensity noise defined by Eq. (12) were simulated for both the top and bottom surfaces of a 10/90% PMMA/SAN17 layer varying from 50 – 3000 µm thick at a 50 µm interval, each scenario with 1000 trials. The Lorentzian fitting method was used to locate the peaks of the confocal PSFs. The standard deviation of the difference of the measured objective Z positions to focus on the top and bottom surfaces across the 1000 trials yields another source of *S _{∆z}* uncertainty which, in combination with the stage precision error, leads to the overall

*S*based on the error propagation rule. Since the Lorentzian peak detection of the confocal PSFs does not introduce additional bias errors,

_{∆z}*B*is considered to be solely attributable to the stage inaccuracy.

_{∆z}*S*and

_{∆z}*B*as a function of

_{∆z}*∆z*are plotted in Figs. 6(a) and 6(d), respectively. A 0.2 NA is identified as most suitable for confocal-scan SS-OCT experiments considering the trade-off that higher NA may lead to less Δ

*z*measurement uncertainty whereas lower NA results in higher Δ

*D*precision. At 0.2 NA, the estimated Δ

*z*precision falls in the range of 200 – 314 nm for 30 μm – 2 mm Δ

*z*. The impact of the

*S*uncertainty on the precision of the

_{∆z}*n*and

_{p}*t*measurements are plotted as a function of the layer thickness

*t*in Figs. 6(b) and 6(c), and the inaccuracy of

*n*and

_{p}*t*induced by

*B*are shown in Figs. 6(e) and 6(f). Despite the increase in Δ

_{∆z}*z*uncertainty with the layer thickness, a thicker layer causes less uncertainty in the estimation of

*n*. With approximate Δ

_{p}*z*precision of 200, 202 and 300 nm for measuring 50, 300, and 3000 µm thick 10/90% PMMA/SAN17 samples at 0.2 NA, respectively, the induced

*n*precision errors are estimated to be 0.0048, 0.0008 and 0.0001, respectively, and the corresponding

_{p}*t*precisions are 155, 156 and 241 nm, respectively.

#### 3.3 NA calibration and impact of NA accuracy on n_{p} measurement accuracy

An optimum NA of ~0.2 was identified as discussed in Sections 3.1 and 3.2. In the actual experimental implementation, the working NA was selected to closely match the optimum condition and theoretically predicted to be 0.1788 based on an entrance pupil diameter of 3.27 mm, estimated from the nominal 1/e^{2} diameter of the collimated beam incident on the objective, a nominal objective focal length of 9 mm and an estimated ambient air phase index of 1.00027 based on the modified Edlén equation [21].

We next experimentally validated the working NA using a BK7 flat window standard with accurately known thickness and phase index. The thickness of the window (*t _{cal}*) was measured by a commercial frequency scanning interferometer (Tropel FlatMaster MSP, Corning, NY, USA) to be 2.0109 mm ± 0.1 μm based on 25 measurements across the 2 × 2 mm

^{2}lateral dimensions of the part. The estimated phase index (

*n*) of BK7 was 1.50349 ± 10

_{p,cal}^{−5}at 1318 nm in a stable laboratory environmental condition (22 ± 0.5°C) based on Sellmeier formula. By measuring the displacement

*Δz*of the objective lens to focus from the top to the bottom surface of the window, it can be derived from Eq. (2) that NA may be expressed as

During the experiment aimed at calibrating NA, the objective lens was translated by a motorized linear stage axially at 0.1 μm resolution for an overall 1.5 mm travel range to cover the range of focusing from the top to the bottom surfaces of the BK7 window. Each set of 100 repeated spectra were acquired at every axial position of the objective; a total of 1500 sets of interference spectra were collected to reconstruct the confocal intensity profiles of both window surfaces. From the 100 repeated measurements, a focal shift distance of 1325.79 ± 0.17 μm was measured to translate the objective focus from the top to the bottom surface of the window. Based on Eq. (13), the NA was evaluated to be 0.1782 ± 0.0014, whose potential bias magnitude was estimated to be 6 × 10^{−4}, as compared to the theoretical value of 0.1788.

Figure 7(a) plots the induced sample *n _{p}* measurement inaccuracy solved by Eq. (4) as a function of the NA bias introduced in the formula. The second vertical axis plots the corresponding relative

*t*inaccuracy. The dot on the curve shows that an estimated NA bias of 6 × 10

^{−4}leads to the

*n*and relative

_{p}*t*inaccuracies of 5 × 10

^{−5}and 3 × 10

^{−5}, respectively.

#### 3.4 Impact of n_{air} accuracy on n_{p} measurement accuracy

A last parameter in Eq. (7) that contributes to the uncertainty of *n _{p}* is the uncertainty of

*n*. A modified Edlén equation that takes into account the impact of thermal effect, relative humidity and pressure variations is recommended by NIST to evaluate the air index to an accuracy of better than 10

_{air}^{−6}[21]. Considering the performance of commercial data loggers that typically measure the temperature, relative humidity and pressure to the accuracies of ± 0.5 °C, ± 3% and ± 1%, respectively,

*n*may be determined with accuracy better than 10

_{air}^{−5}.

To investigate the amount of *n _{p}* measurement inaccuracy induced by the bias of

*n*,

_{air}*n*was varied in Eq. (4) and then

_{air}*n*and

_{p}*t*were numerically solved, assuming a 30 μm thick 10/90% PMMA/SAN17 sample. The resulting

*n*and relative

_{p}*t*inaccuracies induced by a

*n*bias at the working NA of the system (i.e., 0.1782) are shown in Fig. 7(b). Results indicate that the

_{air}*n*bias is linear to and nearly of the same order of magnitude as the

_{p}*n*bias. Specifically, the

_{air}*n*and relative

_{p}*t*biases induced by a

*n*bias limit of 10

_{air}^{−5}as a conservative estimate are 8 × 10

^{−6}and 5 × 10

^{−6}, respectively, as indicated by the dot on the curve. The results are consistent for samples of different thicknesses.

## 4. Confocal-scan SS-OCT system layout

The layout of the confocal-scan SS-OCT system is shown schematically in Fig. 8. The system is built on a fiber-based Mach-Zehnder interferometer (MZI) configuration. The source is a 20 kHz frequency swept laser (HSL-2100-WR, Santec, Japan) centered at 1318 nm with a full width at half maximum (FWHM) bandwidth of 125 nm, yielding an axial PSF of approximately 10 µm FWHM. Light entering the main OCT interferometer is split by a fiber coupler (90/10) and then delivered to the sample and reference arms. In the reference arm, a Fourier domain optical delay line is implemented for dispersion compensation [22]. In the sample arm, collimated light of 3.27 mm 1/e^{2} diameter is focused on a test sample by an objective lens (LCPLN20XIR, Olympus, Tokyo, Japan) mounted on a precision motorized linear stage (VP-25XL, Newport, CA, USA) for axial translations of the focus. The motorized stage travels axially at a translation step of 0.1 μm over a travel range that covers focusing the objective lens through the layers of interest. 100 repeated depth profiles are captured at each translation step during a 100 ms time interval. Consequently, acquiring a total of 20,000 sets of depth profiles over a typical travel range of 2 mm takes approximately 33 minutes.

The backreflected light from the sample is collected and recombined with light from the reference arm at a second broadband fiber coupler (50/50). The time-encoded interference signal is detected by a balanced photodetector (PDB460C, Thorlabs, NJ, USA) and then digitized on one channel of a 500 MSamples/s, 12-bit analog-to-digital converter (ATS9350, AlazarTech, QC, CA). The optical path and signal power of the two fiber arms connected to the dual detection channels of the balanced photodetector were carefully matched to yield common mode noise suppression of ~22 dB.

The detected OCT interference signal requires recalibration to the linear frequency space prior to Fourier transform, which is performed in real time by using the time-frequency relation measured by an additional side MZI denoted by a dashed box in Fig. 8. Simultaneously with the detection of the main interference signal, the calibration signal is detected by a second balanced photodetector and then digitized on a second channel of the same analog-to-digital converter. By performing a Fourier transform of a single recalibrated interference spectrum, an entire depth-sectioned component reflectivity profile along the incident sample beam path is reconstructed. The maximum sensitivity of the SS-OCT system was measured to be 112 dB. The imaging depth range is about 5 mm as determined by a –10 dB sensitivity fall-off. A precision pressure and temperature data logger (DPG4000-30A, Omega Engineering, CT, USA) and a humidity monitor were used to record the environmental condition in the laboratory.

## 5. Results

#### 5.1 Monolithic layered sheet

As can be seen from the theoretical analysis, the precision and bias limit of the index measurements vary with the sample thickness. To establish an initial experimental uncertainty map for the system in measuring the indices of layers of different thicknesses, a 2.93 mm thick layered monolithic sheet manufactured with the same process as a GRIN sheet yet from stacking and thermo-compressing 108 films of a uniform polymer composition (10/90% PMMA/SAN17) was tested by the confocal-scan SS-OCT system. Measurements were carried out at a single lateral point on the sheet; however, they may be extended to profile the refractive index and the layer thicknesses of the sample in 3D when combined with x-y translation stages to laterally scan the sample.

Back-reflections occurred at the top and bottom surfaces as well as at the layer interfaces of the sheet, and each contributed to an intensity peak on an SS-OCT depth profile. The objective lens was translated by the *z*-motorized stage for a travel range of 2 mm, during which each set of 100 repeated depth profiles was captured at every 0.1 μm step of the translation and a total of 20,000 sets of depth profiles were acquired. Over the range of the translation, each surface or layer interface went in and out of focus, consecutively. Confocal intensity profiles were reconstructed from the 20,000 depth profiles to show the back-reflection intensity variations for all interfaces as a function of the axial position of the objective lens. Combining the measurements of the group optical thickness Δ*D* between two interfaces provided by the depth profiles and the translation Δ*z* needed to focus from one interface to the other, the refractive index and thickness between the two interfaces were simultaneously computed. To validate the measurement consistency and uncertainty through increasing the thickness, the first surface was fixed as the top surface of the monolithic sheet and the second surface was varied along depth from the first layer interface successively to the bottom surface of the sheet. The measured phase index and the cumulative thickness of 1 – 108 layers averaged from 100 repeated measurements with their standard deviations are plotted in Fig. 9(a) against the theoretical values. Based on the composition of the monolith, it is predicted to have a phase index of 1.5489, while the OCT-measured index (average of 100 measurements) ranges from 1.5475 to 1.5500 and deviates from the theoretical value by about 0.00003 (measuring 106 layers, 2.8378 mm thick) to 0.0014 (measuring 1 layer, 0.0502 mm thick). The standard deviation of 100 measurements varies from 0.0001 (measuring 105 layers, 2.7946 mm thick) to 0.0025 (measuring 1 layer). It is also shown in Fig. 9(a) that the measured cumulative thickness increases more rapidly near both edges of the sheet. The standard deviation of the measured cumulative thicknesses across 100 measurements varies from 0.12 μm (measuring 1 layer) to 0.28 μm (measuring 105 layers). The estimated thickness of each layer was obtained by computing the difference between the cumulative thicknesses of two consecutive numbers of layers and is plotted in Fig. 9(b), which shows a nearly parabolic profile as opposed to a theoretical uniform layer thickness of ~27 μm. The layer thickness measurements were independently validated with destructive inspection [12], i.e., cutting the sample, polishing the cut surface, and imaging the same region that was measured by OCT under a light microscope with 20 × magnification. The results of the under-compressed edge layer thicknesses measured by the destructive test agreed with the OCT measurements within 2%. The thickness discrepancy of the manufactured sheet from the design specifications indicates the thermo-compression non-uniformity of the films during thermo-compressing the film stack to a consolidated sheet. This defect in the early sample discovered by OCT prompted the manufacturer to refine the fabrication process in an autoclave guided by finite-element-analysis modeling. The new layered sheets fabricated exhibit much improved conformity to the specifications as will be shown in Section 5.2.

#### 5.2 Layered GRIN sheet

A layered polymeric GRIN sheet was tested by the confocal-scan SS-OCT system to characterize its axial refractive index distribution at a central lateral point of the sheet. The sheet was 4.8 mm thick, composed of 102 layers each with nominal thickness of 47.1 μm after thermal-compression. Based on the prescribed gradient index recipe, the PMMA/SAN17 composition and associated refractive index of the material changed every 7 or 8 layers from 100% PMMA on one end with a phase index of 1.4803 at 1318 nm to 100% SAN17 at the other end of the sample with a phase index of 1.5575 at 1318 nm. To utilize the depth range of the system with best sensitivity, the top PMMA-heavy half (layers # 1 – 51) and the bottom SAN17-heavy half (layers # 52 – 102) of the sample were measured respectively, with the sample being flipped in the latter case, such that both regions of interest were imaged using the 50 μm – 4 mm depth range (OPD) of the system. During imaging, the step size and the travel range of the objective lens were set at 0.1 μm and 1.8 mm, respectively; 100 repeated measurements were collected at every step of the objective lens.

In the analysis of the imaging data, the refractive index and thickness of each stack of layers sharing the same material composition were evaluated as shown in Fig. 10. The results were obtained by computing the group optical thickness *ΔD* between each pair of axial PSFs on depth profiles corresponding to the back-reflected signals from the boundaries of each layer stack where the composition/index changes occurred and the corresponding focal shift distance *Δz* from the confocal intensity profiles reconstructed by tracking the peak intensity of these axial PSFs. The confocal-scan SS-OCT measurements show good agreement with the phase indices measured by a Metricon refractometer [23] on the original films before thermo-compression as well as the nominal layer stack thicknesses predicted from specifications. The average discrepancy between the Metricon and OCT measured indices is 0.0005. The standard deviations of the repeated OCT index measurements range from 0.0003 – 0.0023, and are on average 0.0008, which is comparable to the precision of Metricon measurements - yet uniquely, the confocal-scan SS-OCT is capable of measuring the index profile of layers in situ. In terms of the OCT-measured stack thicknesses, they conform to the nominal values to within 1% (on average 0.3%) and the standard deviations of repeated measurements are on average 0.17 μm.

## 6. Discussion

Figures 11(a) and 11(b) summarize the results of the *n _{p}* and

*t*precisions for sample layers of thicknesses ranging from 50 – 3000 µm predicted theoretically (black line) and measured experimentally (red circles). In Fig. 11(c), the discrepancy of the measured

*n*from the specifications is plotted in red circles, which fall within the black curve denoting the predicted magnitude of

_{p}*n*bias limit. The simulated curves in Fig. 11 were estimated following the error analysis method discussed in Section 3. The experimental results were collected over the measured increasing layers of the monolithic sheet and the layer stack of the GRIN sheet as reported in Section 5. The close agreement between the experiments and the theory demonstrates the effectiveness of the analysis method presented in predicting the performance of the OCT metrology system.

_{p}On the other hand, as the precisions of the measurands Δ*D* and Δ*z* drive the sample *n _{p}* estimation precision, they define the sensitivity of the technique to index variations. Although Eq. (4) is a nonlinear quartic equation, numerical solutions show that the relationships of Δ

*D*or Δ

*z*measurement errors vs.

*n*estimation error are approximately linear in the presence of small perturbations (within ± 1%) of the measurands. The precision in measuring the optical path Δ

_{p}*D*is governed by the FD-OCT system; the uncertainty in the focal shift distance Δ

*z*is however dominated by the performance of the linear stage translating the objective lens as well as the step resolution for translation. In the current system, the stage performance drives the overall index metrology capability as shown in the error analysis that was also experimentally consolidated. For measuring the index of a thin layer (e.g., 50 μm thick), a nano-positioning stage coupled with fine

*z*sampling resolution in the nanometer level is expected to further yield even higher levels of accuracy and precision in the measurements. A trade-off for increasing the

*z*sampling number is extended data acquisition time. To speed up the acquisition, continuous

*z*-scan of the objective lens may be utilized; as such, the acquisition of the interference spectra must be synchronized with the movement of the focus going through the sample. A set of 10,000 spectra evenly acquired along the

*z*-travel of the objective (i.e., corresponding to 3 nm

*z*-sampling resolution for 30 μm

*z*-travel or 30 nm

*z*-sampling resolution for 300 μm

*z*-travel) would take as short as 0.5 s and yield the intensity variation profiles of all interfaces that are captured going in and out of focus. For the continuous

*z*-scan acquisition, the mathematical model for simultaneous refractive index and thickness metrology remains unchanged. It is important however to calibrate the linearity of the system in terms of the stage translation, data acquisition, and associated electronics. A displacement measuring interferometer may be integrated in the system to provide accurate feedback of the position of the linear translation stage and calibrate the linearity of the stage velocity to yield accurate Δ

*z*measurements. In addition, any impact of the nanometer-class

*z*-movement of the objective focus during the acquisition of an OCT interference spectrum on the final measurements needs to be carefully assessed.

Another aspect to consider is the amount of spherical aberration induced by the sample positioned in a convergent beam. Under the operating NA of 0.1782, by evaluating the aberrations caused by a BK7 plane parallel plate as an example based on the formulas given in [24], the RMS wavefront error at best focus was computed to be 0.008 waves for a 3 mm thick plate, which is considered well within the diffraction limit.

## 7. Conclusion

In this paper, we investigated the simultaneous metrology of axial index profile and thickness measurements of GRIN samples with a custom-developed hybrid confocal-scan SS-OCT system. We conducted error analysis of the system and performed simultaneous experimental measurements of the refractive indices and thicknesses of layered monolithic and GRIN sheet samples. The assessment of the metrology capability of the SS-OCT system based on the analysis methods we established yields a good prediction of the experimental results, which showed precision of 0.0008 for index measurements of a ~300 µm thick layer stack and corresponding thickness measurement precision of ~0.17 µm. An optimum working NA around 0.2 was also established in the theoretical analysis to enable achieving this precision level. The analysis methods may be applied broadly to systematically investigate the performance of various optical and photonics metrological instruments. In addition, the methodology and technique we reported is readily adaptable to measure other types of layered GRIN materials such as spherical or radial gradients. In future work, the hybrid confocal-scan SS-OCT system may be extended to profile the refractive index and the layer thicknesses of spherical GRIN samples in 3D when x-y translation and θ-φ rotation stages are incorporated in the system to scan the sample in 3D.

## Acknowledgments

This work builds on earlier support from the NYSTAR Foundation C050070 and the Manufacturable Gradient Index Optics (M-GRIN) program of the Defense Advanced Research Projects Agency (DARPA). We thank the II-VI Foundation for their support in freeform optics metrology and the NSF I/UCRC Center for Freeform Optics (CeFO) for synergic activity in the metrology of freeform optics. We thank Sydor Optics for their help with validating the thickness of a BK7 flat using a frequency scanning interferometer.

## References and links

**1. **G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. **20**(21), 2258–2260 (1995). [CrossRef] [PubMed]

**2. **S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res. **48**(27), 2732–2738 (2008). [CrossRef] [PubMed]

**3. **Y. Verma, K. Rao, M. Suresh, H. Patel, and P. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fish eye in vivo using optical coherence tomography,” Appl. Phys. B **87**(4), 607–610 (2007). [CrossRef]

**4. **A. de Castro, S. Ortiz, E. Gambra, D. Siedlecki, and S. Marcos, “Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging,” Opt. Express **18**(21), 21905–21917 (2010). [CrossRef] [PubMed]

**5. **M. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In vitro simultaneous measurement of refractive index and thickness of biological tissue by the low coherence interferometry,” IEEE Trans. Biomed. Eng. **47**(9), 1266–1270 (2000). [CrossRef] [PubMed]

**6. **S. Kim, J. Na, M. J. Kim, and B. H. Lee, “Simultaneous measurement of refractive index and thickness by combining low-coherence interferometry and confocal optics,” Opt. Express **16**(8), 5516–5526 (2008). [CrossRef] [PubMed]

**7. **Y. Zhou, K. K. H. Chan, T. Lai, and S. Tang, “Characterizing refractive index and thickness of biological tissues using combined multiphoton microscopy and optical coherence tomography,” Biomed. Opt. Express **4**(1), 38–50 (2013). [CrossRef] [PubMed]

**8. **Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. **103**(3), 1834–1841 (2007). [CrossRef]

**9. **M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure and properties,” Macromol. Symp. **294**(1), 19–32 (2010). [CrossRef]

**10. **P. Meemon, J. Yao, K. S. Lee, K. P. Thompson, M. Ponting, E. Baer, and J. P. Rolland, “Optical coherence tomography enabling non destructive metrology of layered polymeric GRIN material,” Sci. Rep. **3**, 1709 (2013). [CrossRef]

**11. **P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. **43**(10), 2063–2085 (1996). [CrossRef]

**12. **J. Yao, P. Meemon, K. S. Lee, and J. P. Rolland, “Nondestructive metrology by optical coherence tomography empowering manufacturing iterations of layered polymeric optical materials,” Opt. Eng. **52**(11), 112111 (2013). [CrossRef]

**13. **J. Yao, P. Meemon, M. Ponting, and J. P. Rolland, “Angular scan optical coherence tomography imaging and metrology of spherical gradient refractive index preforms,” Opt. Express **23**(5), 6428–6443 (2015). [CrossRef] [PubMed]

**14. **D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, and et, “Optical coherence tomography,” Science **254**(5035), 1178–1181 (1991). [CrossRef] [PubMed]

**15. **M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express **11**(18), 2183–2189 (2003). [CrossRef] [PubMed]

**16. **M. Born and E. Wolf, *Principles of Optics*, 7th ed. (Cambridge University Press, 1999).

**17. **H. H. Ku, “Notes on use of propagation of error formulas,” J. Res. Nbs. C Eng. Inst. C **70**, 263–273 (1966).

**18. **M. Gu, C. Sheppard, and X. Gan, “Image formation in a fiber-optical confocal scanning microscope,” J. Opt. Soc. Am. A **8**(11), 1755–1761 (1991). [CrossRef]

**19. **W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kärtner, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nat. Med. **7**(4), 502–507 (2001). [CrossRef] [PubMed]

**20. **J. Huang, J. Yao, N. Cirucci, T. Ivanov, and J. P. Rolland, “Performance analysis of optical coherence tomography in the context of a thickness estimation task,” J. Biomed. Opt. **20**(12), 121306 (2015). [PubMed]

**21. **K. P. Birch and M. J. Downs, “An updated Edlen equation for the refractive index of air,” Metrologia **30**(3), 155–162 (1993). [CrossRef]

**22. **K. S. Lee, A. C. Akcay, T. Delemos, E. Clarkson, and J. P. Rolland, “Dispersion control with a Fourier-domain optical delay line in a fiber-optic imaging interferometer,” Appl. Opt. **44**(19), 4009–4022 (2005). [CrossRef] [PubMed]

**23. ** Metricon Corp, http://www.metricon.com.

**24. **S. Murali, P. Meemon, K. S. Lee, W. P. Kuhn, K. P. Thompson, and J. P. Rolland, “Assessment of a liquid lens enabled in vivo optical coherence microscope,” Appl. Opt. **49**(16), D145–D156 (2010). [CrossRef] [PubMed]