## Abstract

A theoretical model is developed for the Master/Slave interferometry (MSI) that is used to demonstrate its tolerance to dispersion left uncompensated in the interferometer when evaluating distances and thicknesses. In order to prove experimentally its tolerance to dispersion, different lengths of optical fiber are inserted into the interferometer to introduce dispersion. It is demonstrated that the sensitivity profile versus optical path difference is not affected by the length of fiber left uncompensated. It is also demonstrated that the axial resolution is constant within the axial range, close to the expected theoretical resolution determined by the optical source bandwidth. Then the thickness of a glass plate is measured several times in the presence of dispersion and errors in measurements are evaluated using the MSI method and the conventional Fourier transformation (FT) based method using linearized/calibrated data. The standard deviation for thickness results obtained with the MSI is more than 5 times smaller than the standard deviation for results delivered by the conventional, FT based method.

© 2015 Optical Society of America

## 1. Introduction

Spectral domain interferometry (SDI) techniques have shown beyond doubt their capacity to produce optical path difference (OPD) reflectivity profiles (A-scans) for optical coherence tomography (OCT) applications, with sufficient high accuracy. As a consequence, the SDI techniques are being used as attractive measurement tools that offer high resolution, high sensitivity and measurement speed [1]. However, the SDI methods deliver distance measurements via a conversion law between the modulation of the optical spectrum (channeled spectrum) at the interferometer output and the OPD in the interferometer. Accuracy of distance measurements therefore depends on the linearity of such conversion. A major recognized difficulty in obtaining accurate displacement measurements is the presence of dispersion in the system [2] that chirps the modulation of the channeled spectrum.

Both SDI implementations: spectrometer based (Sp)-SDI (where a broadband light source is used and the spectrum at the interferometer output is collected by a spectrometer employing a linear photo-detector array) and swept source (SS)-SDI (where the modulation of the spectrum at the interferometer output is scanned on a single point photo-detector by tuning a narrowband source through a large optical bandwidth) [3] provide axial reflectivity profiles. In both SDI implementations, the axial reflectivity profile (A-scan) is obtained by Fourier transformation (FT) of the electrical signal proportional to the shape of the channeled spectrum at the interferometer output. In a Sp-SDI system, the channeled spectrum is delivered by the linear camera in the spectrometer, while in a SS-SDI system by the photo- detector). In Sp-SDI, the geometry of dispersion or diffraction determines a nonlinear dependence of distribution of optical frequencies over the linear array of the camera employed by the spectrometer. In SS-SDI, the optical frequency tuning is nonlinear. Therefore, both SDI implementations exhibit incorrect mapping of the channeled spectra over the optical frequency coordinate. This problem is exacerbated by dispersion, especially when optical sources with large spectral bandwidth are employed. Chirping of channeled spectrum, irrespective of its origin, due to system nonlinearity as explained above or due to dispersion, affects the sensitivity and resolution of SDI. The advancement of SDI was only made possible by developing costly hardware methods [4–8] and sophisticated software algorithms [9] to ensure proper optical frequency mapping (linearization) before the FT operation.

The simplest method of dispersion compensation relies on placing the right amount of dispersion balancing material in one of the interferometer arms [10]. Compensation of the length of different materials targets at least the 2nd order dispersion (group velocity), that exercises the most significant impact on the interferometric signal [11]. The larger the bandwidth of the optical source in Sp-SDI or the tuning bandwidth in SS-SDI, the larger the deleterious effect of dispersion left uncompensated. If higher orders of dispersion have to be compensated for, choosing materials of exact length to perfectly compensate the dispersion unbalance is extremely difficult. Grating-based phase delay scanners [12] and dual optical fiber stretchers [13] were also reported for 2nd order dispersion compensation. Post processing numerical dispersion compensation techniques [14], which offer continuous adjustment capabilities and, in theory, can be optimized for any amount of dispersion are quite often preferable over the hardware solutions. Unfortunately, all the methods mentioned above require either additional expensive equipment or demand computationally expensive algorithms that limit the real time operation of the SDI systems.

In previous reports [15–17], the concept of Master/Slave Interferometry (MSI) was introduced and employed for imaging. The MSI is based on comparing electrical signals proportional to channeled spectrum shapes at the interferometer output. The higher the similarity of such shapes, the larger the MSI signal. Since no FT is required, the MSI method does not need to organize the channeled spectrum data in equal frequency slots. In all previous MSI reports [15–17], the dispersion was compensated for by matching the lengths of fiber and placing similar lenses with those used in the object arm, in the reference arm. Achieving the theoretical expected value for the axial resolution without any resampling or linearization, proved the tolerance of MSI to nonlinearities due to the tuning of the swept source [16, 18] and to the spectrometer nonlinearities [17]. This tested behavior and the fact that no FT are needed suggest that the MSI should also be tolerant to dispersion in the interferometer. Therefore, in this paper, we investigate the MSI capability to produce accurate A-scan profiles even in situations where dispersion in the interferometer is largely unbalanced. The MSI principle can be applied to any SDI technology, implemented here on a classical Sp-SDI experimental set-up (Fig. 1) [17].

To perform the comparison of channeled spectrum shapes, correlation of two electrical signals, proportional to the two channeled spectra is performed. The present report proves theoretically and experimentally that MSI can be used to generate truthful A-scans in the presence of large dispersion left uncompensated in the system, in which case more accurate displacement measurements can be performed than it is possible with the current SDI technology.

## 2. Materials and methods

#### 2.1 Experimental set-up

The Sp-SDI set-up is presented in Fig. 1. As a broadband optical source (BBS), a super- luminescent diode (Amonics, Hong Kong), central wavelength λ = 1050 nm and bandwidth Δλ = 30 nm was employed. This feeds a Michelson interferometer equipped with a single mode fiber coupler, DC. The OPD in the interferometer is defined as the difference between the object path length minus reference path length. The object path length is from DC, via a length of single mode fiber, SMF, up to the object and back to DC. The reference path length is measured from DC up to a reference flat mirror F_{R} and back to the DC. The spectrometer placed at the interferometer output incorporates a SU-LDH linear camera (Goodrich-SUI, Princeton, New Jersey) that provides 14-bit digital signal, of 1024 pixels with an acquisition speed of up to 47,000 lines per second. The amount of dispersion left unbalanced was controlled by changing the SMF length in the object arm of the interferometer.

The production of A-scan profiles using the MSI method involves two steps depending on the position of the switches K_{1} and K_{2}. In the first step (when the interferometer is used as a ‘Master interferometer’ according to [15]), both switches are placed in position 1. A flat mirror F_{O} is used as object and P channeled spectra (masks), M(OPD_{p}) = M_{p}, corresponding to a set of *P* OPD values are recorded for p = 1,2,…P. The *P* masks should be recorded at OPD values separated by half of the coherence length of the optical source or denser. All such masks are stored in the ‘Memory Block’ shown in Fig. 1. Then, in the second step, the two switches are placed in position 2, the mirror F_{O} is replaced with the object to be investigated, O, and the channeled spectrum CS is acquired and cross-correlated with all stored masks M_{p} (the interferometer is used as a ‘Slave interferometer’ according to [15]). Correlation with each mask, M_{p}, provides an output signal of amplitude A(OPD_{p}) = A_{p} in the A-scan at the OPD_{p} value used to create that mask M_{p}. The processing block shown at the bottom of Fig. 1 illustrates a diagram detailing how the amplitude of the signal originating from a certain axial position of the object is calculated.

#### 2.2 Theory

Let us consider the OPD as the difference between the object path length and the reference path length, equal fiber lengths of the directional coupler, DC, in the reference and object path and the dispersion concentrated in the SMF placed in the object arm. Under these assumptions, the OPD can be expressed as:

where *D(k)* is the optical path difference through the SMF. Its dependence on the wavenumber, k, describes the dispersion in the interferometer. $D({k}_{0})$ is the path length of the SMF for the (wavenumber k_{0}) in the center of the optical spectrum and *z* is the displacement of the reference mirror F_{R} from OPD = 0. The factor 2 takes into account the round trip of the reference beam to the reference mirror.

For any given OPD, the channeled spectrum at the output of a dispersive interferometer can be written as:

where *g(k)* is the power spectrum of the optical source employed. We demonstrate below that the MSI method can be used to recover the OPD value with no need for any linearization.

For different values 2z, the channeled spectrum exhibits different modulation. The value 2z can be considered as varying in steps *a*, so:

where *p* is an integer number and $\delta D(k)$ represents the variation in the dispersive part when the wavenumber differs from the value in the center of the spectrum. The step *a* signifies the round trip axial resolution, determined by the coherence length of the broadband source. For a Gaussian spectrum of full width half maximum (FWHM) in wavenumber, *Δk*, the coherence length is:

In MSI, in the first stage, F_{O} replaces the object (switches K_{1} and K_{2} in position 1 in Fig. 1) and the channeled spectra are measured for p = 1,2,…P OPD values in the Master interferometer:

which are stored for subsequent use. It should be noticed that the memories M_{g,p} contain the dispersion information in the modulation imprinted by D(k). The masks are divided by the spectrum for OPD = 0 to obtain the normalized masks:

In the second stage, the object to be tested replaces the mirror F_{O} and the channeled spectrum (CS) is measured. Let us consider the object made from scattering centers, P, from points r = 1 … P, separated by differential intervals *a/2*, each scattering center returning a signal of strength A_{r}. For each *OPD = ra* value, a modulation term is created in the spectrum at the interferometer output, determining a channeled spectrum:

This consists in a superposition of modulations of different periodicities with weights A_{r}. Practically, coefficients A_{r} signify the values of the A-scan at points *ra*. Let us correlate the channeled spectrum, CS(k), with the normalized mask M_{p} to obtain the correlation coefficients:

where ⊗ means correlation and:

Using the mathematical definition of the cross-correlation, ${C}_{rp}(K)$ are obtained as:

where k_{min} and k_{max} denote the minimum and maximum wavenumber respectively within the optical bandwidth. For K = 0 these become:

which, obviously reduce to:

Evaluation of the correlation coefficients C_{p}(K) for K = 0 in Eq. (8), using Eq. (12), leads to:

This shows that by correlating CS with M_{p}, with evaluation of the result in K = 0, returns C_{p}(0), proportional to the reflectivity A_{p} of the scattering center at the axial distance ap/2. Each coefficient, A_{p}, represents the strength of that component in the modulation of the channeled spectrum that pulsates with *p* cycles. The calculation above shows that the result in Eq. (13) does not depend on δD(k), so there is no dispersion dependence. An A-scan can be assembled from all amplitudes A_{p} so recovered placed at OPD values *pa*.

Additionally, by scanning the beam laterally over the object and using any of the normalized mask M_{p}, for all pixels in the transversal section of the object, an *en-face* OCT image can be produced at *OPD = pa*.

The model above represents any channeled spectrum at the interferometer output, Eq. (7), as a series development of orthogonal eigenfunctions, the masks M_{p}, given by Eq. (6). Equation (12) represents the orthogonality condition of the eigenfunctions. The modified correlation calculation presented above retrieves the eigenvalue A_{p}. In other words, in the preparation stage, switches on position 1, the eigenfunctions are measured whilst with switches in position 2, the eigenvalues are measured for the investigated object.

#### 2.3 Digital implementation

The correlation in Eq. (8) is calculated over the wavenumber axis, with the wave-number, *k* as variable.

If the channeled spectrum is read by a number U of photo-sites (pixels), then the summation of products of the two terms in Eq. (14) is performed over U points. The U pixels are covered by the channeled spectrum distributed between k_{min} and k_{max}.

Numerically, the correlation above is worked out as:

where y = 1, 2, …2U-1, while CS(n) = 0 and M_{p}(n) = 0 if n < 0 and n ≥ U. The linear camera used has U = 512 pixels, therefore for the data shown in this paper, the correlation in Eq. (15) extends over 2U-1 = 1023 wavenumber points and exhibits a maximum for y = U.

Correlation signal in Eq. (15) is high-pass filtered in HPF in Fig. 1, to remove the DC component, and rectified. The amplitude of the signal is then evaluated in *y = U*. However, as this will determine a too low strength, dependent on the phase differences between the moment the masks were acquired and the time of measurement, the amplitude of the signal is calculated as an average over a window of size *2W + 1* points around *y = U*.

The absolute values of the MSI sensitivity can be tweaked by adjusting the value of the window size *W* in Eq. (17). As previously demonstrated in [15], to achieve the best compromise between axial resolution and sensitivity, a value of *W* smaller than 20-30 has to be used. Consequently, for all data shown in this paper a value of *W = 10* was selected. When used, the axial resolution in the MSI is slightly worse, of 22-26 μm than the best achievable value, of 21 μm (please see comments below) but the sensitivity of the MSI is slightly better, by several dBs, than that of the FFT based SDI.

## 3. Results

In order to demonstrate the tolerance to dispersion of the MSI method, two cases of distance measurements are investigated: (1) data not re-sampled and (2) data re-sampled before FT, where in each case the results are compared with the MSI results obtained using the Eq. (17) above.

#### 3.1 Case 1: data not re-sampled

The sensitivity drop-off, and the variation of the axial resolution with OPD were experimentally evaluated in Fig. 2. Four cases have been considered, where increased dispersion is applied in steps of 0, 0.5, 1.0 and 1.5 m of extra single mode fiber (SMF) added to the object arm of the interferometer. For 12 axial values of the mirror position used as object (z = ± 2.4, ± 2.0, ± 1.6, ± 1.2, ± 0.8 and ± 0.4 mm), channeled spectra, CS, were acquired and 12 points A_{p} were obtained that were then used to produce A-scan profiles using the MSI method (by correlating them with masks previously recorded) and by the FT method (by Fourier transformation, with no data re-sampled, continuous curves in Fig. 2). For the same OPD values, the sensitivity values obtained with the MSI technique (filled circle shapes) are presented. The procedure of measuring the sensitivity is described in detail in [15]. According to this procedure, the reference arm signal is adjusted, such that the amplitude of the spectrum at the camera is near the saturation value. Then a neutral density filter, characterized by an optical density OD = 2 is placed into the sample arm.

For the correlation based MSI method, the sensitivity for a particular OPD_{p} is calculated as:

_{opt}. This behavior can be noticed by comparing the measurements shown in Figs. 2(b)-2(d) with that in Fig. 2(a). For a SMF length = 0.5 m in (b), z

_{opt}= 0.7 mm, for SMF length = 1 mm (c), z opt = 1.3 mm and for SMF length = 1.5m in (d), z

_{opt}= 1.9 mm. In Fig. 2(d), the sensitivity profile varies slightly non-monotonically with z as we move away from z = 0 towards positive values of OPD, it shows a minimum around z = 1.2 mm, before reaching its maximum value at z = 1.9 mm. This effect is the result of improper wavenumber mapping and unbalanced dispersion.

It is clear from Fig. 2 that the MSI method is immune to the amount of unbalanced dispersion as shown by the filled circles. The absolute values of the sensitivity do not vary with the length of the SMF, and the sensitivity profile is symmetric around OPD = 0 mm.

On the other hand, as expected, when the FT based method is employed, the maximum values of the sensitivity depend on the amount of the unbalanced dispersion and the position of maxima are placed asymmetrically around OPD = 0, unless the length of SMF is zero.

#### 3.2 Case 2: data re-sampled

In a dispersion free system, the PSF exhibits the same narrow width irrespective of the OPD value. However, in the presence of dispersion, the calibration fails, leading to deterioration of resolution and decay of sensitivity, as demonstrated below.

Let us now compare the MSI method with the FFT based method with data re-sampled. A widely used re-sampling technique is based on measuring the phase of the interferometric signal versus OPD and reorganizing the data to linearize such dependence. Results are presented in Fig. 3, where the calibration coefficients were calculated for z = 0.4 mm. Further away from z = 0.4 mm, the calibration is less efficient (cases a-c). When there is no dispersion mismatch (a), the re-sampling performs well. If little dispersion is left uncompensated, the calibration coefficients are calculated with sufficient accuracy to provide proper re-sampling, hence a constant axial resolution across the whole longitudinal range is achieved (Fig. 3(a)). When large dispersion is present, the calibration efficiency diminishes. The calibration can provide proper re-sampling only around the axial value of z = 0.4 mm used to infer the phase during the calibration process (Figs. 3(b)-3(d)).

Filled circle shapes show the values of the sensitivity obtained with the MSI technique in Fig. 3. In opposition to the FT based conventional method, the correlation based MSI method is immune to dispersion. The values of the sensitivity do not change with the length of the extra fiber SMF and are distributed symmetrically around OPD = 0 mm. For all cases presented in Fig. 2 and Fig. 3, the axial resolution (the FWHM of each individual PSF) was evaluated. The results are shown in Fig. 4. When using the correlation based MSI method, the axial resolution varies little, from around 26 μm at large OPD values, to a minimum of 22 μmaround OPD = 0. Using Δλ = 30 nm in Eq. (4), a theoretical axial resolution of δz = 16 μm is obtained. The larger values measured reflect the fact that the bandwidth of the interfering signals is less than 30 nm. It is known that the spectra from the reference and object arm via a single mode coupler, DC, are in general slightly different and the interference signal comes from the intersection of the two spectra only, narrower than the optical source spectrum. As mentioned above, we used W = 10 in all data presented here for better sensitivity, if the minimum value for W = 1 is used, then axial resolution improved to 21 μm, reaching the same best value achievable here with the FT method.

As the MSI method shows immunity to the length of the extra SMF, only a single curve is shown in Fig. 4(a) and Fig. 4(b) (line connecting filled circles).

As expected, by inspecting Figs. 2 and 3, and quantitatively demonstrated in Fig. 4, the FT based method can only provide good axial resolution (21 μm) over an extremely short axial range. When no re-sampling of data is performed, the FT method provides acceptable axial resolution around axial positions that depend on the amount of unbalanced dispersion in the system (length of the SMF in our case). If data are re-sampled, maximum axial resolution and sensitivity are obtained for that axial position where the calibration coefficients were calculated.

#### 3.3 Distance measurements

To demonstrate the capability of the MSI technique to provide displacement measurements more precisely than the conventional FT based SDI method in a system where dispersion is left unbalanced, we experimentally measured the thickness of a cover glass, from a box with a set of plates with thickness (no. 1.5) in a range (160-190 μm). The thickness of the cover slide selected was measured by using a time domain OCT, giving 170 μm. The microscope slide was placed in the focal plane of the objective L2, and its thickness was determined using both the conventional FT based SDI and the MSI methods by constructing the A-scan profiles of the slide and measuring the distance between the two peaks corresponding to the interface air/glass and glass/air. To obtain comparable results, when the FT method was employed, each channeled spectrum was zero-padded before Fourier transformation, in such a way that each point in the sensitivity profile corresponds to an axial range of 1 μm. For the MSI method, masks have been recorded every micrometer. For each of the 4 cases (no SMF and extra fiber length of 0.5, 1.0 and 1.5 m), the measurements were repeated 10 times. The microscope slide was removed and then re-positioned more or less in the same position, so the exact axial position varied from a measurement to the next.

In Fig. 5, we show results corresponding to a single set of measurements for the 4 cases. On each graph, the average value of glass thickness is shown together with its standard deviation. Values close to 170 μm were obtained when the MSI method was used, irrespective of the amount of optical fiber left unbalanced. When the FT method is used, the calibration coefficients were calculated for z = 300 μm (the plane of the first interface air/glass was approximately placed in this position). For a dispersion free system (Fig. 5(a1)), the FT based SDI method can produce a quite accurate profile for the two peaks corresponding to the discontinuities air–glass and glass-air. With dispersion (Figs. 5(a2)-5(a4)), the re-sampling of data fails outside z = 300 μm, the peak corresponding to the interface glass/air becomes wider and the precision in measuring the distance between peaks decreases.

On the other hand, MSI exhibits immunity to the amount of unbalanced dispersion, and therefore provides the same axial resolution along the entire axial range (Figs. 5(b2)-5(b4)). Hence, the distance between the two peaks determining the thickness of the slide can be better estimated in each situation. Obviously, if the plate was thicker, the second peak would have looked even broader, making the distance measurement even more inaccurate. All these results are summarized in Fig. 6, where average values of the optical thicknesses as well as their corresponding standard deviations are shown, measured using the FT based SDI and MSI methods as a function of the amount of dispersion left uncompensated (extra length of optical fiber). For illustrative purposes we show values for the thickness (dashed lines) and standard deviations (colored areas) obtained by interpolation of the values obtained for the four lengths of the dispersive material. This figure clearly demonstrates that MSI can provide displacement measurements more accurately than the currently used FT based method. The standard deviation for the MSI (Fig. 6(b)) does not exceed 0.64% of the median value obtained while the FT based method (Fig. 6(a)) produces results with standard deviations up to 3.2%.

## 4. Discussion

The long fiber used for this study is much longer than fiber mismatches in practice. This was used here to exacerbate the dispersion problem and register a significant A-scan peak enlargement in a short path, away from the calibration point by 170 μm only. If the glass plate was much thicker, shorter fiber would have been needed to obtain similar peak enlargement due to dispersion. As shown in Fig. 4, at 1.7 mm (10 times larger distance than the plate thickness measured) away from the minimum width in (a) or from the calibration point in (b) lead to hundreds of microns to mm enlargements with 50 cm of fiber. Although the A-scan peak enlargement does not vary with OPD proportionally in Fig. 4, it can be assumed that a 10 times shorter fiber, i.e. of 5 cm only, would have produced a similar enlargement for the second peak as that in Fig. 5, if it was 1.7 mm away from the first peak. An axial range of 1.5 mm is normally used in OCT imaging of the retina in the fovea region and 2.5 mm when imaging the optic nerve. In practice, lenses and beam-splitters in the object arm add to cm length mismatch. In other words, adopting the MSI method can lower the costs of assembling OCT systems, as no lenses or beam-splitters need to be replicated in the reference arm. This is in addition to the saving brought about by elimination of the clock circuit in the design of the swept source. Additionally, the tolerance to dispersion may be employed in combined OCT systems with fluorescence, where dichroic filters are used and where their replication in the reference arm becomes unnecessary.

The bandwidth of our source was quite limited. If a broader bandwidth source was available at 1050 nm, then similar deleterious effects would have been introduced by shorter length of fiber. This makes the MSI method especially useful for broadband interferometry and high axial resolution OCT instrumentation.

The only disadvantage of the MSI is the need to collect the masks. However, the cost of such a procedure is a fraction of that to equip a swept source with a clock, or of linearized spectrometers. The collection of masks needs to be done once only and the masks exhibited stability for long time. We have been using the same masks over time intervals exceeding several months and did not notice any difference in the signal obtained by redoing the mask or using a priory recorded mask.

There are no extra computing means needed, the computation effort to evaluate correlation is comparable to that of linearization in the FT based SDI. In terms of time required to evaluate an A-scan, the three prior reports [15–17] present such data, depending on three methods used to evaluate correlation, using 3 FFTs in [15], using 2 FFTs in [16] and using a fast algorithm in [17]. Here, the FFT for an A-scan of M = 512 points took 2.2 µs while MSI of the same required 256 correlation calculations demanding 1.28 ms. If distances are measured once only, the longer time of MSI in comparison with the FFT SDI is acceptable. However, if more measurements are required in a limited time, or for generation of images in OCT, advantage can be taken of the parallel nature of MSI, where all 256 correlations can be performed using the Compute Unified Device Architecture (CUDA) parallel computing platform on graphics card units (GPUs). Our benchmarking showed that using a relatively low cost GeForce GTX 780 Ti graphics card (less than £500), the time required to compute an MSI based A-scan in 256 points was reduced to the time obtainable by FFT.

A problem may be envisaged if the source bandwidth is so large that the axial resolution becomes small and a large number of masks need to be acquired. The MSI technique can be implemented irrespective of the axial resolution of the system. The only requirement for MSI is to record masks as dense as half the coherence length of the optical source. For instance, let us say that the source spectrum is as large as to determine a 1 μm axial resolution. For an axial range of 5 mm, with at least two masks per resolution interval, 5,000 masks would be needed. The recording procedure of masks would require a half-micron step translation stage, this is still of lower cost than the cost of equipping a swept source with a clock or linearizing a spectrometer. As the collection of masks is needed only once, employing a large bandwidth source amounts to lengthening the preparation stage of MSI only, from a couple of seconds to a minute. The measurement stage of applying MSI is not affected when using a large bandwidth source.

## 5. Conclusion

We demonstrated that MSI method can be used to produce accurate A-scans profiles and thickness measurements even in the presence of a large unbalanced dispersion. The MSI method does not require FTs and therefore does not need re-sampling of the recorded channeled spectra. The axial resolution provided by the MSI technique is immune to dispersion and calibration over the entire axial range. In our previous reports [15–17], the MSI was used to demonstrate its tolerance to chirped shapes of channeled spectra and to produce OCT images with no need for calibration of data to eliminate the chirp. It was assumed in all our previous reports that the chirp was mainly due to the nonlinear dependence of the channeled spectrum read versus optical frequency, as the dispersion was compensated for. Here we demonstrate that for important chirping of the channeled spectrum created by large dispersion mismatch values, the MSI still delivers accurate results. This should find applications not only in measuring distances and thicknesses of layers in OCT but also in lowering the cost of sensing solutions. As a demonstration, we measured the thickness of a microscope slide and showed that the MSI method delivers more accurate values than the conventional FT method, which fails to provide a constant axial resolution along the entire axial range.

When using the conventional method based on FT of the signal delivered by the spectrometer, the axial position of a reflector is determined by a theoretical calculation. First, the conversion coefficient between the differential displacement of the mirror reflector in the sampling arm and the density of peaks in the channeled spectrum is found experimentally. Then, any position of the reflector is inferred theoretically from the density of peaks in the channeled spectrum. This measurement assumes that there is a strict proportionality between the OPD and the frequency of the signal resulting by reading the spectrometer camera. The linearity of channeled spectrum modulation versus OPD is however as good as the calibration method. Any non-linearity in this dependence affects the accuracy of the FFT based SDI method, which depends on experimental measurements (to infer the conversion coefficient OPD to frequency) as well as on the calibration procedure. It is known from the OCT practice that linearization moves the A-scan peaks along the axial coordinate by more than a coherence length.

In opposition, the accuracy of distance measured by the MSI method is determined by the accuracy in measuring the OPD values of the method employed to acquire the masks in the 1st stage of the MSI.

## Acknowledgments

A. Bradu and A. Podoleanu acknowledge funding from the European Research Council under the Advanced Grant agreement COGATIMABIO, 249889. A. Podoleanu and M. Maria acknowledge support from the European Industrial Doctorate UBAPHODESA, 607627, supported by the European Commission. A. Podoleanu is also supported by the NIHR Biomedical Research Centre at Moorfields Eye Hospital NHS Foundation Trust and UCL Institute of Ophthalmology.

## References and links

**1. **G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photon. **4**(4), 441–471 (2012). [CrossRef]

**2. **P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Dispersion error of a beam splitter cube in white-light spectral interferometry,” Opto-Electron. Rev. **16**(4), 439–443 (2008). [CrossRef]

**3. **A. G. Podoleanu, “Optical coherence tomography,” J. Microsc. **247**(3), 209–219 (2012). [CrossRef] [PubMed]

**4. **Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. **32**(24), 3525–3527 (2007). [CrossRef] [PubMed]

**5. **K. Gaigalas, L. Wang, H.-J. He, and P. DeRose, “Procedures for wavelength calibration and spectral response correction of CCD array spectrometers,” J. Res. Natl. Inst. Stand. Technol. **114**(4), 215–228 (2009). [CrossRef]

**6. **T. J. Eom, Y. C. Ahn, C. S. Kim, and Z. Chen, “Calibration and characterization protocol for spectral-domain optical coherence tomography using fiber Bragg gratings,” J. Biomed. Opt. **16**(3), 030501 (2011). [CrossRef] [PubMed]

**7. **T.-H. Tsai, C. Zhou, D. C. Adler, and J. G. Fujimoto, “Frequency comb swept lasers,” Opt. Express **17**(23), 21257–21270 (2009). [CrossRef] [PubMed]

**8. **B. Liu, E. Azimi, and M. E. Brezinski, “True logarithmic amplification of frequency clock in SS-OCT for calibration,” Biomed. Opt. Express **2**(6), 1769–1777 (2011). [CrossRef] [PubMed]

**9. **Y. Yasuno, V. D. Madjarova, S. Makita, M. Akiba, A. Morosawa, C. Chong, T. Sakai, K.-P. Chan, M. Itoh, and T. Yatagai, “Three-dimensional and high-speed swept-source optical coherence tomography for in vivo investigation of human anterior eye segments,” Opt. Express **13**(26), 10652–10664 (2005). [CrossRef] [PubMed]

**10. **A. R. Tumlinson, B. Hofer, A. M. Winkler, B. Považay, W. Drexler, and J. K. Barton, “Inherent homogenous media dispersion compensation in frequency domain optical coherence tomography by accurate k-sampling,” Appl. Opt. **47**(5), 687–693 (2008). [CrossRef] [PubMed]

**11. **T. Hillman and D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express **13**(6), 1860–1874 (2005). [CrossRef] [PubMed]

**12. **G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. **22**(23), 1811–1813 (1997). [CrossRef] [PubMed]

**13. **S. Iyer, S. Coen, and F. Vanholsbeeck, “Dual-fiber stretcher as a tunable dispersion compensator for an all-fiber optical coherence tomography system,” Opt. Lett. **34**(19), 2903–2905 (2009). [CrossRef] [PubMed]

**14. **M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express **12**(11), 2404–2422 (2004). [CrossRef] [PubMed]

**15. **A. G. Podoleanu and A. Bradu, “Master-slave interferometry for parallel spectral domain interferometry sensing and versatile 3D optical coherence tomography,” Opt. Express **21**(16), 19324–19338 (2013). [CrossRef] [PubMed]

**16. **A. Bradu and A. G. Podoleanu, “Calibration-free B-scan images produced by master/slave optical coherence tomography,” Opt. Lett. **39**(3), 450–453 (2014). [CrossRef] [PubMed]

**17. **A. Bradu and A. G. Podoleanu, “Imaging the eye fundus with real-time en-face spectral domain optical coherence tomography,” Biomed. Opt. Express **5**(4), 1233–1249 (2014). [PubMed]

**18. **M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination,” J. Biomed. Opt. **12**(4), 041205 (2007). [CrossRef] [PubMed]