## Abstract

Interferometric range measurements using a wavelength-tunable source form the basis of several measurement techniques, including optical frequency domain reflectometry (OFDR), swept-source optical coherence tomography (SS-OCT), and frequency-modulated continuous wave (FMCW) lidar. We present a phase-sensitive and self-referenced approach to swept-source interferometry that yields absolute range measurements with axial precision three orders of magnitude better than the transform-limited axial resolution of the system. As an example application, we implement the proposed method for a simultaneous measurement of group refractive index and thickness of an optical glass sample.

©2011 Optical Society of America

## 1. Introduction

Over the past two decades, optical coherence tomography (OCT) [1,2] has been widely adopted for performing noninvasive, depth-resolved imaging for a variety of biological and medical applications. With variations based on both low-coherence [3] and swept-wavelength interferometry [4, 5], OCT provides measurements of optical reflectivity as function of depth (*A*-scans) within the sample under test. By laterally scanning either the probe beam or the sample under test, depth-resolved 2- and 3-dimensional images can be produced. The high axial resolution offered by these measurements provide a unique ability to image subsurface biological structures based on the level of scattering returned by underlying tissues. For both low-coherence and swept-source (SS-) OCT, the axial resolution is inversely proportional to the frequency bandwidth of the optical source [6]. Axial resolutions on the order of 1 *μ*m have been achieved with low-coherence approaches using extremely broadband supercontinuum sources [7]. For SS-OCT, the axial resolution is typically limited to the order of 10 *μ*m due to the more limited spectral breadth available from swept-wavelength sources.

Axial displacement sensitivities greatly exceeding the axial resolution of OCT systems have been demonstrated by numerous groups using phase-sensitive frequency-domain OCT techniques [8–13]. These frequency-domain approaches include both low-coherence interferometry using spectrally dispersed detection as well as swept-wavelength interferometry. Both modes detect spectral interferograms as a function of optical frequency and produce *A*-scan data by applying a Fourier transform to the acquired fringe patterns. Small displacements of discrete reflectors can be detected by noting changes in the phase of the complex-valued *A*-scan data at the location in the data array corresponding to the reflector. These phase measurements provide a relative displacement measurement from scan to scan, and have been applied to surface profiling [9], phase imaging [11], and Doppler flow measurements [12, 13]. In all of these cases, the submicron displacements measured via phase have been relative to an arbitrary zero point within a single depth bin defined by the source-spectrum-limited axial resolution of the system. This fact has limited the application of phase-sensitive OCT to the measurement of small relative displacements only, precluding the use of phase information for increased accuracy of absolute and relative distance measurements within a single *A*-scan. In this paper, we present a novel implementation of phase-sensitive SS-OCT where phase information is used to perform measurements of lengths and thicknesses spanning multiple resolution-limited depth bins to nanometer precision. Additionally, we present a means for accurate calibration of the time (or, equivalently, spatial) domain sampling grid, leading to highly accurate optical path length measurements.

As an example application, we demonstrate the proposed phase-sensitive SS-OCT system for simultaneous measurements of group refractive index and physical thickness of optical samples. The technique employed was first introduced by Sorin and Gray [14] using low-coherence interferometry, and there has been renewed interest in the method recently using low-coherence interferometry in both the time domain [15] and spectral domain [16]. Here we present what is to the authors’ knowledge the first swept-source demonstration of simultaneous group index and thickness measurement, as well as the first implementation to utilize phase information for improved performance. The use of phase-sensitive self-referenced SS-OCT provides resolution improvements of up to two orders of magnitude for both group index and thickness over previously published works.

## 2. Principles

In SS-OCT, each *A*-scan performs an axial reflectivity measurement using optical frequency domain reflectometry (OFDR) [4]. In this technique, the output of a wavelength-tunable source is split into a fixed-length reference path and a sample path, which are then recombined allowing the light traversing each path to interfere prior to detection. As the source optical frequency is swept over a range Δ*ν* about a central frequency *ν*
_{0}, a fringe pattern is observed at the interferometer output. The frequency of the fringe pattern indicates the differential group delay between the reference path and the sample path. For a sample with *M* distinct reflectors distributed axially, the oscillating portion of the photodetector voltage at the interferometer output is

*ν*is the instantaneous frequency of the laser source,

*τ*is the group delay difference between the

_{i}*i*

^{th}reflection in the sample path and the reference path, and

*ξ*is a constant phase offset. The factor

_{i}*r*is the effective reflection coefficient of the

_{i}*i*

^{th}reflection [17]. To avoid limitations in the axial resolution of the scan due to nonlinearity of the optical frequency sweep, the fringe pattern must be sampled on a grid of equal optical frequency increments. This can be accomplished using an auxiliary interferometer to provide either instantaneous frequency data throughout the sweep in order to resample the interferogram [18], or by using the auxiliary interferometer output as a frequency clock to trigger acquisition of the fringe pattern data [19]. The equal-frequency sampling allows the sampled frequency-domain fringe pattern data to be converted to the time domain via a fast Fourier transform (FFT). In the time domain, the contribution

*Ũ*(

_{i}*τ*) due to the

*i*

^{th}reflector is

*ψ*=

_{i}*ξ*+ 2π

_{i}*ν*

_{0}

*τ*and sinc(

_{i}*x*) = sin(

*πx*)/(

*πx*). Here the sinc function arises due to the assumption of a constant amplitude over the spectral range Δ

*ν*. Other spectral shapes (or the application of a windowing function prior to the Fourier transform) will change the shape of the time domain response. The width Δ

*τ*of this response function determines the axial resolution of an

_{w}*A*-scan, and this width will generally be Δ

*τ*≈ 1/Δ

_{w}*ν*for most spectra. Note also that

*δτ*= 1/Δ

*ν*will be the sample spacing of the time domain data, so that in the best case the axial resolution will be equivalent to one temporal bin. If there are two reflectors spaced by

*δτ*or less, they will not be resolvable. For an isolated reflector, however, the location of the reflector, described by

*τ*, can be determined to within a small fraction of

_{i}*δτ*by analyzing the phase of the reflector’s contribution to the time domain reflectogram.

#### 2.1. Phase slope measurements for improved range precision

A coarse measurement of *τ _{i}* is accomplished by noting the location

*τ*

_{i}_{,}

*of the*

_{q}*i*

^{th}peak in the time domain data array. An example of such time domain data for a single

*A*-scan is plotted in Fig. 1(a) for a single isolated reflector. The precision of this coarse determination of the reflector position is

*δτ*, or one temporal bin. The true value of

*τ*is likely to lie between sampled points. This offset between the location of the peak value in the time domain data array and the true value of

_{i}*τ*can be found by applying the shift theorem of Fourier transforms [20] to a subset of time domain data surrounding the

_{i}*i*

^{th}peak. The shift theorem states that a translation in the time domain is accompanied by a corresponding linear phase factor in the frequency domain. Thus, determination of the offset between the value of

*τ*and the

_{i}*i*

^{th}peak location can be accomplished through a linear phase measurement in the frequency domain. For an isolated reflection peak, the corresponding phase contribution in the frequency domain can be found by windowing out the single peak using a digital filter and then performing an inverse FFT on the windowed data subset. Figure 1(b) is a plot of the amplitude of the windowed reflection peak selected by applying a digital filter to the positive delay peak in Fig. 1(a). The phase of the resulting frequency domain data set will wrap rapidly between 0 and 2

*π*unless the time domain subset is rotated such that the amplitude maximum occupies the first (DC) index location in the data array. Performing this rotation prior to the inverse FFT results in a slowly-varying frequency domain phase that can be straightforwardly unwrapped. Fitting a line to the unwrapped phase

*φ*(

*ν*), as shown in Fig. 1(c), gives a slope

*τ*

_{i}_{,}

*, illustrated in Fig. 1(d).*

_{q}The precision of the resulting measurement of the reflector position given by *τ _{i}* =

*τ*

_{i}_{,}

*+*

_{q}*τ*

_{i}_{,}

*as determined by the standard deviation over multiple measurements will be limited by a variety of factors, including the repeatability of the wavelength sweep as well as drift of the interferometer with changes in environmental conditions. Because measurement errors due to sweep-to-sweep variations in the optical source will be correlated for multiple reflections within a single*

_{a}*A*-scan, measurement precision can be significantly improved by performing a relative group delay measurement using one reflector within the

*A*-scan as a reference [16]. For fiber systems with a free space probe, the fiber end facet in the sample arm makes a convenient reference reflector.

#### 2.2. Sampling grid calibration for accurate absolute ranging

The accuracy with which a reflector can be located depends not only on the precision of the group delay measurement, but also the accuracy of the the time domain sampling grid that is used to perform the coarse group delay measurement, *τ _{i}*

_{,}

*. Because of the discrete Fourier transform relationship between the acquired frequency domain fringe pattern and the time domain*

_{q}*A*-scan, the range of the

*A*-scan is given by the reciprocal of the frequency domain step size,

*δν*. For an

*N*-point

*A*-scan, the time domain step size is therefore

*δτ*= (

*Nδν*)

^{−1}. As mentioned above, the fringe pattern must be sampled on a grid of equal frequency increments, either through the use of a frequency clock to trigger data acquisition or by monitoring the instantaneous frequency of source throughout a sweep and resampling the fringe data in postprocessing. Thus the uncertainty in the time domain step size depends on the accuracy with which the instantaneous optical frequency can be determined during a wavelength sweep.

The most common way to monitor the instantaneous optical frequency of a swept source is through the use of an auxiliary interferometer. Provided that the differential group delay Δ*τ* between the auxiliary interferometer paths and the mean laser sweep rate *γ* = d*ν*/d*t* are chosen such that Δ*τ*
_{2}
*γ* ≪ 1, then the output fringe pattern will be a periodic function of optical frequency with a period of 1/Δ*τ* [19]. If the fringe data is sampled or resampled at this period so that the frequency domain step size *δν* is equal to 1/Δ*τ*, then Δ*τ* will be the full range of the dual-sided time domain data set, and the Nyquist-limited measurable group delay will be Δ*τ*/2. The accuracy of the time and frequency domain sampling grids then depend on the accuracy with which Δ*τ* (or its reciprocal) can be measured. Once Δ*τ* has been determined, a range measurement with a measured delay *τ _{i}*

_{,}

*performed by locating a peak at a fractional index*

_{q}*k*in the time domain data array will have an uncertainty given by

*u*(

*x*) is used to denote the uncertainty in the quantity

*x*. Thus, the relative error in the range measurement will equal the relative error in the calibration of the auxiliary interferometer.

One way to precisely calibrate the auxiliary interferometer is by counting the number of fringe periods between well-characterized spectral features, such as molecular absorption lines. Wavelength references based on molecular absorption lines can be accurate to ±0.01 pm [21], and commercial gas cells with wavelength accuracies as good as ±0.05 pm are available in multiple spectral bands. These specifications yield a known uncertainty between two spectral features that span a frequency range Δ*ν _{c}*. For a given auxiliary interferometer with a differential group delay Δ

*τ*, the number of periods over the range Δ

*ν*will be

_{c}*m*= Δ

*ν*Δ

_{c}*τ*. Using this relationship to determine Δ

*τ*, there will be contributions to the uncertainty due to the quality of the wavelength reference

*u*(Δ

*ν*), as well an uncertainty in the determination of

_{c}*m*to a fraction of a fringe. Therefore, the total uncertainty in Δ

*τ*is

*ν*, the number of samples

*N*will be

*N*= Δ

*ν*Δ

*τ*, and the uncertainty in the time domain step size will be

*u*(

*δτ*) =

*u*(Δ

*τ*)/

*N*. Figure 2 is a plot of

*u*(

*δτ*) for some representative values over a range of interferometer path imbalances from 100

*μ*m to 100 m. This range covers typical SS-OCT systems on the short end, and typical OFDR systems designed for fiber sensing and telecommunications system testing on the upper end. The frequency sweep range is Δ

*ν*= 3.49 THz, corresponding to the spectral separation between the R20 and P20 absorption lines of H

_{c}^{13}CN at 100 Torr, a common wavelength reference material for the range between 1528 and 1562 nm. The R20 and P20 lines are located at 1530.3061 nm and 1558.0329 nm, respectively, and the uncertainty in their location is ±0.3 pm. The three curves represent the uncertainty in the time domain step size for three values of the uncertainty in the number of fringes between the absorption lines. The asymptotic value of 4.4 as is reached when the uncertainty of the wavelength reference dominates. Interestingly, this plot shows that an extremely high degree of temporal accuracy can be achieved over a range of interferometer length imbalances spanning several orders of magnitude. For short interferometers, such as those typically used for SS-OCT, more care must be taken to accurately determine the number of periods between the absorption lines to a fraction of a fringe. This is typically accomplished by fitting a Lorentzian curve to the sampled absorption line data. For longer interferometers, high temporal accuracy can be achieved without the need for fractional fringe counting.

## 3. Experimental results and discussion

The experimental phase-sensitive SS-OCT system is shown in Fig. 3. The auxiliary interferometer was calibrated using the R20 and P20 absorption lines of a 100 Torr H^{13}CN gas cell (dBm Optics model WA-1528-1562). The absorption spectrum of the gas cell was acquired using the auxiliary interferometer to trigger data acquisition. The number of samples between the R20 and P20 absorption lines was determined to a fraction of a sample by curve fitting the absorption lines to precisely locate their minima. In this process, the uncertainty of the absorption line wavelengths (known to ±0.3 pm) dominate the interferometer calibration error. The resulting measurement yielded a group delay difference between the two paths of the auxiliary interferometer of Δ*τ* = 63.9413 ± 0.0012 ns. The AC-coupled output of this interferometer was used as an analog clock to trigger data acquisition on the polarization-diverse outputs of the measurement interferometer using a National Instruments PCI-6115 data acquisition card. The tunable laser was an Agilent 81680A with a maximum sweep rate of 40 nm/s. Measurements were performed by sweeping the laser from 1500 to 1564.17 nm. This sweep range coupled with the frequency domain step size of 1/Δ*τ* = 15.6395 MHz yields *A*-scans comprising 524,288 data points.

As an application of highly precise optical range measurements, we used the experimental system to perform simultaneous measurements of group refractive index and physical thickness of an optical sample. We used a parallel fused silica plate with a nominal thickness of 3 mm as the sample under test (SUT). A reference mirror was positioned behind the SUT. To minimize scan-to-scan drift in the position of the reference mirror, the entire system was enclosed and the test plate was mounted on a motorized translation stage so it could be inserted into and removed from the measurement path without opening the enclosure. *A*-scans with the SUT present and absent are plotted in Fig. 4. The physical thickness *T* of the plate is determined from relative group delay measurements according to

*c*is the speed of light in vacuum and

*n*is the group index of the SUT, which is found using [15] Refer to Fig. 4 for the definitions of

_{g}*τ*

_{21}and

*τ*

_{43}. We employed the model described by Ciddor and Hill [22,23] to determine the group refractive index of air to be 10

^{6}(

*n*

_{g}_{,air}– 1) = 2184 ±1 at the center sweep wavelength for the atmospheric conditions present during the experiment. The value of

*τ*

_{21}is found directly in a single scan, so noise due to environmental fluctuations and scan-to-scan variations in the laser sweep cancel. Determination of

*τ*

_{43}requires two

*A*-scans, one with the SUT present and one without. By referencing the mirror range measurement to the reflection from the fiber end facet (

*τ*

_{0}in Fig. 4), scan-to-scan variations largely cancel, and the measurement noise of

*τ*

_{43}approaches that of

*τ*

_{21}. To illustrate the level of measurement noise in each group delay measurement, plots of 50 repeated measurements of referenced and unreferenced group delays defined in Fig. 4 are shown in Fig. 5. The standard deviation of unreferenced group delay measurements was 4.4 fs, whereas the standard deviations of self-referenced group delay measurements were as small as 5.2 as for

*τ*

_{21}, where the relative measurement involved two facets of a single glass plate. This value corresponds to a distance of 780 pm in air.

Using the measurement data shown in Fig. 5, we determined the group refractive index of the fused silica plate to be 1.462905 ± 0.000002. The uncertainty in this measurement includes the standard deviation of *τ*
_{21} shown in Fig. 5, as well as an increased uncertainty in *τ*
_{43} due to drift in the position of the reference mirror during the process of inserting or removing the fused silica test plate. We estimated this uncertainty to be a factor of 2 greater than the standard deviation based on the repeated measurements of *τ*
_{40} and *τ*
_{30} shown in Fig. 5. Because the group index measurement is a relative measurement (apparent from Eq. (7)), the value of the time domain step size falls out of the measurement and does not affect the final uncertainty. The small uncertainty value for this measurement is a direct result of the use of phase slope measurement and self-referencing in the determination of the group delays *τ*
_{21} and *τ*
_{43}.

To validate the group index measurement, we compared the measured value to a calculation of the group index of fused silica based on a temperature-dependent Sellmeier model. This model utilized Sellmeier coefficients provided by the manufacturer (Schott AG) along with an interpolated value for the thermo-optic coefficient based on data reported by Leviton and Frey [24]. The resulting calculated value for the group index of fused silica at the center sweep wavelength is 1.462893 ± 0.00002. The accuracy of this model is an order of magnitude less than the uncertainty of our group index measurement, and is limited by the accuracy of the Sellmeier coefficients. Our measured value for the group index of fused silica shows excellent agreement with the calculated value.

The thickness of the test plate found using Eq. (6) was 3.239584 mm ±61 nm. This is in good agreement with the value of 3.240±0.001mm found by measuring the thickness of the test plate using a mechanical micrometer. The uncertainty in the interferometric thickness measurement is dominated by the uncertainty in the calibration of the relative group delay of the auxiliary interferometer, which results in an uncertainty in the time domain step size of *u*(*δτ*) = 2.3 as. Because *τ*
_{21} is determined by the sum of an integer number of time domain samples and an adjustment of a fraction of a sample determined by the phase slope, the total uncertainty in the absolute determination of *τ*
_{21} is given by

*u*(Δ

*τ*) is independent of the magnitude of Δ

*τ*(as it is for the auxiliary interferometer calibration routine presented in the previous section), Eq. 9 reveals that the accuracy of relative distance measurements can be improved by increasing the total time domain range of the system beyond simply that which is necessary to measure the distances of interest. The overall limitation on Δ

*τ*is generally imposed by either the coherence length of the laser or the speed capability of the data acquisition system.

While the absolute accuracy of thickness measurements is determined as described in the previous paragraph, it is worth noting that the sensitivity of the measurement is significantly better than the overall uncertainty in *τ*
_{21} suggests. This is because the uncertainty in the time domain step size is constant for any given set of measurements. This can be exploited for highly precise relative measurements, such as thickness variations in a single sample. In this case *u*(*δτ*) can be ignored, and the uncertainty in the relative thickness measurement now becomes dominated by the determination of the group index. To illustrate this case, taking *u*(*δτ*) = 0 for the experimental thickness measurement of the fused silica plate, the uncertainty is reduced to ±4.5 nm. Furthermore, for relative measurements on the same sample where the group index doesn’t change, or if the group index were known exactly (for example, in a measurement of the variation in thickness of a region of vacuum between reflectors), the uncertainty is further diminished. Neglecting the group index uncertainty for the fused silica test sample results in a thickness uncertainty of ±530 pm. For monocrystalline silicon, the refractive index of 3.481 at 1550 nm would yield a thickness uncertainty of 224 pm, less than half of the crystal lattice spacing of 543 pm and comparable to the Si-Si bond length of 235 pm. Our results therefore open the door to thickness profiling of macroscopic samples with single atomic monolayer resolution.

## 4. Conclusions

We have presented a method for absolute ranging using phase sensitive swept-source interferometry yielding self-referenced group delay measurements with attosecond-level precision. This corresponds to sub-nanometer sensitivities for relative distance measurements. We have shown that the accuracy of absolute range measurements depends not only measurement noise floor, but also on accurate calibration of the time domain sampling interval. For macroscopic distance measurements, the uncertainty in this calibration dominates over the uncertainty due to measurement noise. The contribution to the uncertainty due to the sampling calibration can be reduced by increasing the time domain range of the measurement, or equivalently by sampling on a finer grid in the frequency domain. To experimentally demonstrate an application of the proposed phase-sensitive range measurements, we performed a simultaneous measurement of the group index and thickness of a fused silica plate. The uncertainty in the resulting group index measurement was ±2 × 10^{−6}, and for the thickness measurement the uncertainty was ± 61 nm.

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