High-speed high-precision and ultralong-range complex spectral domain dimensional metrology

Wen Bao; Yi Shen; Tao Chen; Peng Li; Zhihua Ding

doi:10.1364/OE.23.011013

1. Introduction

For manufacturing and packaging of multi-component optical system, precise measuring and positioning of every element are crucial to reach the ideal optical performance expected by optical design [1–3]. A precise, nondestructive tool for on-axis dimensional metrology of optical components is thus essential to the evaluation and optimization of qualities of assembled optical system. To meet this requirement, low coherence interferometric sensor (LISE) is developed [3, 4], which is based on the principle of low coherence interferometry [5]. This principle also forms the foundation of time domain optical coherence tomography (TDOCT) for medical applications [6]. Ultralong range is feasible for LISE due to a mechanical-driven delay line implemented and calibrated by an internal laser interferometer. Nevertheless, mechanical movement in LISE has inherent limitation in speed and precision. For instance, a commercial distance metrological system (LISE-LS 200) provided by FOGALE nanotech Inc. claims 2 microns accuracy and takes 10 seconds for 200 mm measurement range [7].

Spectral domain detection approach, as adopted in spectral domain optical coherence tomography (SDOCT), which is based on a spectral interferometer devoid of mechanical scanning, offers significantly improved sensitivity and speed in comparison with the time domain detection approach [8–10]. Imaging range of SDOCT is mainly limited by the spectral resolution of the spectrometer. Most of the reported spectrometers in SDOCTs use gratings as dispersive components working together with one-dimensional (1-D) line-scan cameras. Their imaging ranges are limited to several millimeters [11, 12], which are too short for measuring most optical systems. Similar to SDOCT, another well-known implementation of spectral domain detection based OCT is swept source OCT (SSOCT). Imaging range of SSOCT is mainly determined by instantaneous line-width of the swept source, and usually limited to several millimeters as well. Recently, a SSOCT system based on a vertical-cavity surface-emitting laser (VCSEL) with greatly decreased instantaneous line-width enables imaging ranges from a few centimeters up to meters [13, 14]. However, SSOCT usually has a relatively worse phase stability compared to SDOCT [11, 15]. To fulfill the aim of ultralong range and high phase stability, an orthogonal dispersive SDOCT (OD-SDOCT) system was reported in our previous works [16, 17]. The developed OD-SDOCT system was based on a VIPA-grating spectrometer, where interference spectra were dispersed into two dimensional (2-D) space by a high spectral resolution virtually-imaged phased array (VIPA) [18] and a relatively lower spectral resolution diffraction grating, and recorded by a 2-D camera [19, 20]. On the other hand, spectral domain phase microscopy, one of extensions of SDOCT, can measure large optical path difference (OPD) with high precision by phase information extracted from the complex spectra [21–24]. The averaged spectral phase method proposed by our group [24] achieved a high precision of tens of picometers with a common-path configuration under signal to noise ratio (SNR) of 61 dB. Following these ideas, a potential approach to realize high-speed high-precision and ultralong-range dimensional metrology can be feasible by use of the averaged phase measurement method in an orthogonal dispersive interferometer adapted for spectral domain detection similar to that of the OD-SDOCT system.

One problem encountered in spectral domain detection is the complex conjugate and autocorrelation, which cause artifacts and ambiguities in the reconstructed structures [11]. Various approaches based on complex signal have been proposed to remove complex conjugate and autocorrelation in SDOCT. Phase shifting [25–28] and multi-detection [29–33] were developed as the multi-frame approaches. BM-scan based on phase shifting between neighboring A-lines [34–38] was proposed as the spatial carrier approach. As to spectral domain interferometer for dimensional metrology, the phase shifting approach is readily applicable. By using a piezoelectric transducer (PZT) carrying a reflecting mirror in the reference arm, phase shift can be introduced. However, due to inaccuracy of PZT actuator, instability of interferometer, and perturbations in environment and sample, the realized phase shifts in experiments will unavoidably deviate from expectations. These deviations in phase shifts will result in degradation of suppression ratios. To reduce residual artifacts caused by phase errors, Targowski et al. [39] has ever proposed an iterative phase optimization algorithm for SDOCT. However, the iterative process is quite time consuming, and not ideal for high-speed measurement. On the other hand, since samples under measurement are usually composed of dioptric components and the measurement ranges in most cases are far longer than the penetrating depths through biological tissues realized by OCT, dimensional metrological system thus requires an even higher suppression of residual artifacts in contrast to OCT.

In this paper, we report an orthogonal dispersive spectrometer based complex spectral domain interferometric system for high-speed high-precision and ultralong-range dimensional metrology. Firstly, an orthogonal dispersive spectrometer based interferometer with ultrahigh spectral resolution is used to realize ultralong measurement range. Secondly, an improved complex method based on actual spectral phase shifts is proposed to further double the measurement range with high suppression of artifacts. Thirdly, in order to determine axial positions of interfaces and axial distances between interfaces with high precision, the averaged spectral phase measurement method is adopted. To validate the capability of our system for dimensional metrology, two typical samples are measured, and a precision of less than 60 nm within a measurement range of 200 mm is confirmed.

2. Method

The proposed complex spectral domain dimensional metrological system is schematically shown in Fig. 1. A super luminescent diode (SLD 371-HP, Superlum Diodes Ltd) emitted a beam with a center wavelength at 835 nm and a FWHM bandwidth of 45 nm. The light from the SLD was then coupled into a fiber-based Michelson interferometer with a 50/50 fiber coupler via a broadband optical circulator. The reference light was delivered to a reference mirror, which was mounted on a PZT stage (Physik Instrumente) to introduce the phase shifts. In the sample arm, the light was focused onto the sample by an objective lens with a focal length of 150 mm. The light returning from the reference and sample arms was recombining and then detected by a VIPA-grating based spectrometer, which was designed to disperse the light with a bandwidth of 30 nm and an ultrahigh spectral resolution of 2 pm, leading to a minimum measurement gap of 10 μm and a measurement range over 100 mm in half Fourier space. In the spectrometer, the collimated beam (beam-waist radius ~6.5mm) was first focused into a VIPA by a cylindrical lens (fc = 200 mm) at an angle of ~3°. The VIPA was essentially a custom-built plane-parallel solid etalon with a partially reflective film (r ≈95%) on back surface and a reflective film (R ≈100%) on front surface except for an uncoated area used as the entrance window. The FSR of the VIPA was ~0.1 nm, mainly determined by its refractive index (n = 1.51) and geometrical thickness (t = 2.40 mm). Subsequently, the light was incident onto a grating (Wasatch Photonics, d = 1/1200 mm) with an angle of ~30.1 °, and the orientation of the grating groove was normal to the focused line to achieve orthogonal dispersion. Finally, the resultant 2-D dispersed spectra were focused by an imaging lens (f = 200 mm) onto a 2-D CCD (UNIQ) with 1024 × 1024 pixels, 6.45 × 6.45 μm² pixel size and a frame rate of 30Hz. The spectral data fetched by the 2-D CCD were then cascaded to reconstruct 1-D spectra for further data processing, and the detailed method for calibration and reconstruction can be referred to [17].

Fig. 1 Schematic of the proposed complex spectral domain dimensional metrological system.

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Assume the sample under measurement is composed of $N$ discrete interfaces, and the $n$ th interface is characterized by its power reflection $R_{S n}$ and optical path length from the coupler of $z_{S n}$ . The reference mirror is assumed to have power reflectivity $R_{R}$ , and its optical path length from the coupler is $z_{R}$ . The detected spectra can be expressed as [11]

\begin{array}{l} I (k) = \frac{ρ}{4} [S (k) [R_{R} + \sum_{n = 1}^{N} R_{S n}]] + \frac{ρ}{4} [S (k) \sum_{n \neq m = 1}^{N} \sqrt{R_{S n} R_{S m}} \cos [2 k (z_{S n} - z_{S m})]] \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} + \frac{ρ}{2} [S (k) \sum_{n = 1}^{N} \sqrt{R_{R} R_{S n}} \cos [2 k (z_{R} - z_{S n})]], \end{array}

where

ρ

is the responsivity of the detector,

k

is the wavenumber,

S (k)

is the power spectral density of the light source. The first term of Eq. (1) is the non-interferometric contribution. The second term denotes the mutual interference of interfaces within the sample, and the third term represents the cross-interferometric signal. Performing inverse Fourier transform on Eq. (1) yields

\begin{array}{l} F T^{- 1} (I (k)) = \frac{ρ}{8} [Γ (z) [R_{R} + \sum_{n = 1}^{N} R_{S n}]] + \frac{ρ}{8} [Γ (z) \otimes \sum_{n \neq m = 1}^{N} \sqrt{R_{S n} R_{S m}} (δ (z \pm 2 (z_{S n} - z_{S m})))] \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} + \frac{ρ}{4} [Γ (z) \otimes \sum_{n = 1}^{N} \sqrt{R_{R} R_{S n}} (δ (z \pm 2 (z_{R} - z_{S n})))], \end{array}

where

Γ (z)

represents the envelope of the coherence function of the source. The first term of Eq. (2) is the DC term, caused by the light directly reflected or backscattered from the reference mirror and the sample. The second term represents autocorrelation term caused by mutual interference of waves scattered within the sample. The last term contains the actual axial profile of sample, which is symmetrical about zero OPD. The first two terms and complex conjugates can be regarded as artifacts that corrupt the true axial profile. To remove these artifacts is to obtain complex form of the interference spectra by phase shifting multi-frame approach. In typical three-frame technique, three consecutive frames of interference spectra

I (k, i)

are captured with phase shifts

φ_{i} (k)

introduced between interfering arms, and Eq. (1) is modified as

\begin{array}{l} I (k, i) = \frac{ρ}{4} [S (k) [R_{R} + \sum_{n = 1}^{N} R_{S n}]] + \frac{ρ}{4} [S (k) \sum_{n \neq m = 1}^{N} \sqrt{R_{S n} R_{S m}} \cos [2 k (z_{S n} - z_{S m})]] \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} + \frac{ρ}{2} [S (k) \sum_{n = 1}^{N} \sqrt{R_{R} R_{S n}} \cos [2 k (z_{R} - z_{S n}) + φ_{i} (k)]], \begin{matrix} i = 0, 1, 2. \end{matrix} \end{array}

Under assumption of

φ_{0} (k) = 0

, Eq. (3) can be written in matrix form as

[\begin{matrix} 1 & 1 & 0 \\ 1 & \cos (φ_{1} (k)) & - \sin (φ_{1} (k)) \\ 1 & \cos (φ_{2} (k)) & - \sin (φ_{2} (k)) \end{matrix}] [\begin{matrix} I_{0} (k) \\ α \\ β \end{matrix}] = [\begin{matrix} I (k, 0) \\ I (k, 1) \\ I (k, 2) \end{matrix}] \begin{matrix} . \end{matrix}

Here three unknowns

I_{0} (k) = \frac{ρ}{4} [S (k) [R_{R} + \sum_{n = 1}^{N} R_{S n}]] + \frac{ρ}{4} [S (k) \sum_{n \neq m = 1}^{N} \sqrt{R_{S n} R_{S m}} \cos [2 k (z_{S n} - z_{S m})]]

,

α = \frac{ρ}{2} [S (k) \sum_{n = 1}^{N} \sqrt{R_{R} R_{S n}} \cos [2 k (z_{R} - z_{S n})]]

and

β = \frac{ρ}{2} [S (k) \sum_{n = 1}^{N} \sqrt{R_{R} R_{S n}} \sin [2 k (z_{R} - z_{S n})]]

, can be determined from the matrix expressed by Eq. (4). Thus, the complex form of the interference spectra is obtained as defined by

\tilde{I} (k, 0) = α + j β .

Nominal phase shifts are introduced by set displacements

δ z_{i}

of the reference mirror mounted on a PZT stage, while deviations

δ {z^{'}}_{i}

from these nominal settings are unavoidable. Therefore, the actual spectral phase shifts should be expressed by

φ_{i} (k) = 2 k (δ z_{i} + δ {z^{'}}_{i}), \begin{matrix}  \end{matrix} i = 1, 2.

Obviously, these phase shifts are wavenumber dependent and are different from nominal ones in most cases. For better suppression of residual artifacts, actual spectral phase shifts should be calibrated in real-time.

To obtain actual spectral phase shifts between frames, the procedure described below is taken. Firstly, each frame of the detected spectra is Fourier transformed from spectral domain to depth domain. Secondly, the highest peak in depth domain corresponding to one well-resolved sample interface or its mirror image is selected automatically based on differentiation of two successive frames and isolated by a band-pass filter. Thirdly, inverse Fourier transform of the filtered signal is performed to get the complex spectra with spectral phase information involved, which can be expressed by

{\tilde{I}}_{p} (k, i) = \frac{ρ}{2} [S (k) \sqrt{R_{R} R_{S p}} \exp [j [2 k (z_{R} - z_{S p}) + φ_{i} (k) + θ_{0}]], \begin{matrix} i = 0, 1, 2. \end{matrix}

Where the filtered peak is supposed to be resulted from the

p

th interface of the sample, and

θ_{0}

is an arbitrary initial phase. The involved spectral phases can be written as

φ_{i}^{'} (k) = 2 k (z_{R} - z_{S p}) + φ_{i} (k) + θ_{0}, \begin{matrix} i = 0, 1, 2. \end{matrix}

Finally, these spectral phases are retrieved to determine actual spectral phase shifts between three frames as given by

φ_{i} (k) = φ_{i}^{'} (k) - φ_{0}^{'} (k), \begin{matrix} i = 1, 2. \end{matrix}

Input these calibrated spectral phases into matrix expressed by Eq. (4) to get the unknowns, accurate complex spectra is thus determined via Eq. (5). By inverse Fourier transformation of the complex spectra, axial profile with enhanced suppression of artifacts is obtained. In void of disturbing artifacts, axial positions of multiple interfaces are then determined for distance measurement by the averaged spectral phase measurement algorithm [24].

3. Results

To illustrate the procedure for spectral phase retrieval and shift determination, measurements on a sample consisted by a 2.4 mm thick BK7 plate and a mirror as depicted in Fig. 2(a) are conducted by the developed system. Three frames of interference spectra with nominal phase shift of π/2 are captured, and one of these spectra is shown in Fig. 2(b). After fast Fourier transformation applied to each of these spectra, depth profiles corresponding to interfaces as well as disturbing artifacts are obtained. One of such depth profiles is demonstrated in Fig. 2(c), where artifacts known as DC term, autocorrelation terms and mirror images are labeled.

Fig. 2 Data processing procedure for spectral phase retrieval and shift determination. (a) Sketch of the sample composition with the virtual reference mirror presented; (b) one frame of the detected interference spectra; (c) FFT of the spectra in (b) and highlighted band-pass filter for one specific peak, where “1” denotes the DC term, “2” the autocorrelation terms, “3” the mirror terms, and “4” the real interfaces; (d) wrapped spectral phase corresponding to the filtered peak in (c); (e) three linear-fittings of unwrapped phases; (f) calculated and expected spectral phase shifts. Zoom-in views of the designated parts shown in (b), (d), and (e) are also presented for clarity.

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To avoid choosing the DC and autocorrelation terms as the selected peak and enhance the sensitivity for phase retrieval, subtraction between spectral signals in two successive frames is made to help us find the highest peak in depth domain corresponding to one well-resolved sample interface or its mirror image. A band-pass filter is then adopted to isolate the peak as marked in red rectangle shown in Fig. 2(c). Inverse fast Fourier transform is performed on the filtered peak to get the complex spectra, from which the wrapped spectral phase as shown in Fig. 2(d) can be retrieved. The spectral phase is unwrapped in wavenumber domain based on the principle that we correct the phase angles by adding ± 2π when absolute jumps between consecutive phases are greater than the tolerance value (π in our method). After phase unwrapping and linear fitting according to Eq. (8), three well-separated lines of spectral phase as shown in Fig. 2(e) are obtained corresponding to three phase-shifted frames. Using these linear-fitted spectral phases, actual spectral phase shifts between successive frames are determined by Eq. (9). The calibrated spectral phase shifts are given in Fig. 2(f), where nominal phase shifts are also depicted. Deviations between actual phase shifts and nominal phase shifts are evident, which essentially lead to residual artifacts in conventional three-frame technique. However, with the calibrated spectral phase shifts, we envision accurate complex spectra obtained via Eqs. (4) and (5).

To evaluate the enhanced suppression of artifacts by the proposed method, measurements on the sample depicted in Fig. 2(a) based on different methods are performed and the results are demonstrated in Fig. 3. Direct reconstruction from real-valued spectra is given in Fig. 3(a), where disturbing DC, autocorrelation and mirror images are involved. Reconstruction from the complex method based on conventional three-frame approach is shown in Fig. 3(b), where DC and mirror images are reduced with suppression ratios of 80 dB and 30 dB, respectively. The comparison with the widely used complex method based on BM-scan approach [33] is also conducted, where 100 interference spectra with nominal phase shift of π/2 are captured. The corresponding axial profile is shown in Fig. 3(c), providing suppression ratios of DC and mirror images at 70 dB and 42 dB, respectively. Higher complex conjugate suppression is achieved, but with degraded DC suppression due to non-ignorable OPD accumulated in BM-scan. Figure 3(d) shows the reconstructed axial profile by our proposed complex method based on actual spectral phase shifts, where enhanced suppression of DC and mirror images at 80 dB and 60 dB are demonstrated, the highest achieved among existing complex methods to the best of our knowledge.

Fig. 3 Comparison of axial profiles of the sample reconstructed from different methods. (a) Direct reconstruction from real-valued spectra, (b) reconstruction from complex method based on conventional three-frame approach, (c) reconstruction from complex method based on BM-scan approach, (d) reconstruction from the proposed complex method.

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To validate the capability of our system for dimensional metrology over an ultralong measurement range, a sample consisting of a 2.4 mm thick BK7 plate, two 7.5 mm thick BK7 plates and a mirror with an overall optical path length over 200 mm is measured. The photograph of the sample is shown in Fig. 4(a) and its schematic sketch is given in Fig. 4(b). Nominal optical thicknesses of the 2.4 mm BK7 plate and the 7.5 mm BK7 plate are 3.62 mm and 11.32 mm, respectively. The reconstructed axial profile based on direct reconstruction and that based on the proposed complex method are shown in Figs. 4(c) and 4(d), respectively. Interfaces of the composed sample are correctly reconstructed without artifacts in Fig. 4(d) while hardly discernable in Fig. 4(c). To evaluate the measurement precision of the system, 200 repeated measurements are conducted, and the averaged spectral phase measurement algorithm is adopted for OPD determination. As shown in Table 1, the standard deviations of all measures are found to be within 60 nm over 200 mm depth range, and a strong correlation between standard deviation and SNR is observed, where each listed SNR is calculated from averaging of SNRs of two neighboring interfaces. It should be pointed out that 200 mm measurement range is realized by the system without movement of the sample. If axial movement of the sample and cascading of interfaces from neighboring measurements are taken, the measurement range is readily extendable. As the cascading can be realized through identical interface of the sample from neighboring measurements, no special requirement is added for the mechanics used in axial movement.

Fig. 4 Dimensional metrology over a measurement range of 200 mm. (a) A photograph of the sample, (b) schematic sketch of the sample, (c) direct reconstruction from real-valued spectra, (d) reconstruction from the proposed complex method. S1 to S7 indicate 7 interfaces of the composed sample under measurement.

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Table 1. Measured optical distances between interfaces over 200 mm range

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To confirm the feasibility of our system for dimensional metrology of dioptric components, a sample composed of two doublets (Thorlabs, AC254-150-C and Thorlabs, AC254-100-C) as schematically depicted in Fig. 5(a) is assembled for measurement. Nominal optical central distances between interfaces of two doublets are 14.77 mm, 3.21 mm, 10.78 mm and 3.21 mm, respectively. Due to dioptric properties of the sample, sharp variations in back-reflected signals are occurred within limited distance. Figure 5(b) presents the reconstructed axial profile based on the proposed complex method, where a sharp decline of neighboring interfaces from S2 to S3 is evident. Again, 200 repeated measurements are conducted on the sample for precision evaluation. The measured results for central optical distances are listed in Table 2. The standard deviations of all measures listed in Table 2 are found to be within 26 nm in this case.

Fig. 5 Dimensional metrology of the central optical distances in assembled lenses. (a) Schematic sketch of the sample, (b) reconstructed axial profile by the proposed method. S1 to S6 indicate 6 interfaces of the assembled sample.

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Table 2. Measured optical central distances of the assembled lenses

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The time taken for one measurement based on the proposed method with a computer of Inter(R) Core i7-2600K 3.40GHz CPU is 235.8 ms on average, including 120 ms for three-frame spectra capture, 3.5 ms for spectral calibration, 33.8 ms for complex spectra reconstruction, and 78.5 ms for axial positions determination. The measurement speed could be further improved with an increased frame-rate CMOS camera and a special designed software for data processing.

4. Conclusion

We have developed a high-speed high-precision and ultralong-range spectral domain dimensional metrological system based on an orthogonal interferometer. An improved complex method based on spectral phase shifts calibrated in real-time for three-step phase shifting technique is proposed. Enhanced suppression with 80 dB for DC and 60 dB for the complex conjugate is realized. To the best of our knowledge, these suppression ratios are the highest ever reported. Two typical samples are measured by the developed system and distances between interfaces are determined by the averaged spectral phase measurement algorithm. The standard deviations for the first sample are within 60 nm over 200 mm measurement range, and those for the second sample are within 26 nm over 38 mm measurement range. Ultralong measurement range of 200 mm is achieved in one measurement, and the range is readily extendable if axial movement of the sample and range cascading are further taken.

Acknowledgments

The authors would like to acknowledge the financial supports from the Chinese Natural Science Foundation (61335003, 61275196, 61327007, 11404285 and 61475143), National Hi-Tech Research and Development Program of China (2015AA020515), Zhejiang Provincial Natural Science Foundation of China (LY14F050007), the Fundamental Research Funds for the Central Universities (2014QNA5017), and Scientific Research Foundation for Returned Scholars, Ministry of Education of China.

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Gap	S1-S2	S2-S3	S3-S4	S4-S5	S5-S6	S6-S7
Optical distance (mm)	3.64	80.12	11.31	14.34	11.31	82.13
Standard deviation (nm)	51.49	32.81	16.47	23.32	41.11	51.50
SNR (dB)	37.66	46.08	56.59	55.155	51.05	42.15

Gap	S1-S2	S2-S3	S3-S4	S4-S5	S5-S6
Optical distance (mm)	14.79	3.22	5.99	10.79	3.19
Standard deviation (nm)	18.51	11.72	22.14	24.23	25.22
SNR (dB)	44.39	47.72	38.81	39.92	41.09

Gap	S1-S2	S2-S3	S3-S4	S4-S5	S5-S6	S6-S7
Optical distance (mm)	3.64	80.12	11.31	14.34	11.31	82.13
Standard deviation (nm)	51.49	32.81	16.47	23.32	41.11	51.50
SNR (dB)	37.66	46.08	56.59	55.155	51.05	42.15

Gap	S1-S2	S2-S3	S3-S4	S4-S5	S5-S6
Optical distance (mm)	14.79	3.22	5.99	10.79	3.19
Standard deviation (nm)	18.51	11.72	22.14	24.23	25.22
SNR (dB)	44.39	47.72	38.81	39.92	41.09

High-speed high-precision and ultralong-range complex spectral domain dimensional metrology

Abstract

1. Introduction

2. Method

3. Results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Tables (2)

Equations (9)

Optics Express