Employing the phase in master slave interferometry

In this paper, we extend the master slave (MS) method, so far applied to the modulus of the spectra acquired in spectral domain interferometry, to processing complex spectra. We present the algorithm of complex master slave interferometry (CMSI) method and illustrate the importance of phase processing for signal stability and strength. We demonstrate better stability of the signal driving a direct en-face OCT image by processing both real part and imaginary part of the CMS signal. Then we show that by processing the phase, novel avenues can be opened for the master slave method. A first avenue detailed here is that of dispersion measurements.


INTRODUCTION
The key value of spectral (Fourier) domain interferometry (SDI) is its ability to encode spatial data into the spectrum at the interferometer output. There are two modalities on transducing this information from the optical domain into electrical: spectrometer-based interferometry where a broadband optical source is employed together with a spectrometer and swept source-based interferometry, where a tunable (swept) laser is used and signal is delivered by a photo-detector. Both methods are characterized by nonlinearities in transferring the modulation of the optical spectrum into an electrical signal. Such nonlinearities lead to an irregular modulation (chirp) of the electrical signal read out by the spectrometer or the photodetector while tuning the laser respectively. These nonlinearities can have two origins: the readout specificities (nonlinearities in the spectrometer or in the tuning of the swept source) and unbalanced dispersion in the interferometer and sample. Unless this chirp is compensated for, a Fourier Transform (FT), applied to the electrical signal proportional to the channeled spectra, leads to a wider and reduced amplitude of the reflectivity profile peaks.
To avoid the disadvantages mentioned above, stemming from the use of FTs, the master/slave method was proposed by Podoleanu et al. [1], to which we are referring as the Master Slave Interferometry (MSI) method. This method is based on comparing raw acquired spectra from the sample with experimentally measured spectra (real-value experimental masks Mexp) using a mirror, placed in the interferometer at equivalent OPD values where information from the sample is needed. By not performing a FT, a radical change in data processing is established. Initially, the amplitudes of the spectra were compared only via correlation methods. This meant that the phase was discarded. The initial application of the method was further improved to Complex Master Slave Interferometry (CMSI) [2] where the processing of complex signals allows phase recovery. CMSI was extended to improve the calibration stage of the method, wherefrom a reduced number of experimentally acquired channeled spectra using a mirror, any number of masks are theoretically inferred (complex-value inferred masks Mbuilt). CMSI allows novel avenues by providing more than a mask per resolution interval and calculation of any number of masks becomes possible. So far only the advantage of calculating the masks was reported in producing an OCT/SLO instrument for the eye [3], coherence revival [4], BCC of eye lids [5], Gabor filtering [6], embryology [7]. The given access to phase processing.
We demonstrate better stability of the A-scan signal, that in fact in the context of the CMSI signifies better stability of the signal that leads to a direct en-face OCT image. This was only illustrated in the initial report [2], but not used. The better stability is the consequence of processing both real part and imaginary part of the CMSI signal. Then we show that by processing the phase, novel avenues can be opened for the master slave method. A first avenue detailed here is that of dispersion measurements, as published in [8].
In the continuation of the manuscript, we will show how phase can be recovered using the CMSI, demonstrate the stability of the recovered phase with respect to the conventional MSI approach, and then demonstrate a technique to measure dispersion properties of various samples by using an "ideal" interferometer. This interferometer is perfectly compensated for unbalanced dispersion and corrected for non-linearities via the CMSI technique. By using such an ideal interferometer, any phase alteration due to insertion of an investigated sample is exclusively related to the sample dispersion.

PHASE RECOVERY IN CMSI IMPLEMENTATION
Let us consider the pixel position (x) on the spectrometer camera and the optical path difference (OPD = ) in the interferometer. CMSI consists in calculating a discrete integral between the electrical signal I(x,), converting the channeled spectrum at the output of the interferometer, and the complex-value inferred masks, Mbuilt, as follows: where  is the optical angular frequency and the symbol * signifies complex conjugate. The result is equivalent to an inverse Fourier transformation (FT -1 ) of the complex signal created by reading the channeled spectra I without_chirp expressed according to: The ultimate quantity to be obtained is the phase of the signal I(x,), (,). Using CMSI there is no need to resample the channeled spectra nor remove the uncompensated chirp because the channeled spectra are expressed as a combination of local oscillators (masks) according to , spectra that already contain the chirp measured at the Master stage. The masks represent eigenfunctions of the OCT operator that returns as eigenvalues, the reflectivity values within the A-scan at different depths. Thus, by calculating the Fourier transform and the argument of CMSI[I] (Fig. 1), it is possible to extract the phase (,) as:

STABILITY STUDY AND SIGNAL DROP-OFF IN DEPTH: COMPARISON BETWEEN MSI AND CMSI
In order to illustrate the insensitivity of CMSI to the random phase shifts, several A-scans have been recorded over time while imaging a flat mirror. The interferometer used in this experiment is that of an OCT interferometer driven by a swept source, where the digitizer is not synchronized by the source k-clock, but by its internal clock generator.
The sampling in depth is chosen equal to 0.4 m, much denser than the sampling obtained with the FT based method, estimated at 6.1 m by measuring the displacement of the peak in Fourier domain according to the displacement of the reference mirror. This massive oversampling is implemented in order to determine a well-defined reflectance profile, to accurately measure the peak width (9.0±0.2 m here).
In Fig. 2(a), MSI signal has been calculated by using the MSI approach based on cross-correlation. The cross-correlation was performed between masks and channeled spectra collected every 2 seconds. To perform the calculation, 100 channeled spectra (CSexp) have been recorded using a mirror as sample from OPD = 500 m to 540 m. These 100 CSexp are used as experimental masks. As shown in Fig. 2(a) and in Fig. 2(c) corresponding to reflectance profile for a particular time, the reflectance profile is noisy due to interferometer random phase shifts in the channeled spectra CSexp .
In Fig. 2(b), the absolute value of CMSI signal has been calculated from the same raw data previously used for the MSI.
To perform the calculation, the 100 inferred masks Mbuilt distanced at 0.4 m has been calculated using 2 CSexp measured at OPD = 500 m and 540 m. As shown in Fig. 2(b) and by the blue profile in Fig. 2(c) the CMSI reflectance profile does not present significant fluctuations.

GROUP INDEX AND DISPERSION MEASUREMENT PROCEDURE
An application of phase processing enabled by CMSI is to evaluate material dispersion. This opens two directions of applications, in metrology as well as that of compensating for dispersion variation with depth in OCT samples. We will refer to the first direction in this study.
To validate the accurate recovery of the phase through CMSI procedure, group index and dispersion measurement have been carried out. Let us consider two channeled spectra I sample (x,s) and I vacuum (x,v) recorded respectively with the sample to be investigated placed in the sample arm of the interferometer, and without. The OPD in the interferometer is adjustable by moving the sample mirror Ms using a translation stage. The corresponding OPDs s and v can be expressed as s=2(zs- The instantaneous phases  sample and  vacuum related to I sample and I vacuum can be measured according to the procedure described in Fig. 1 and the phase difference = sample - vacuum is calculated. The dependence  vs  can be fitted by a quadratic form: where c is the angular frequency at the center of the spectrum and where coefficients  and  are connected to the group index ng and group velocity dispersion GVD as follows: with e the thickness of the sample.

RESULTS & DISCUSSION
A flint glass sample SF6 is used of 20 mm thickness whose catalogue data gives a group index ng=1.8205 and group velocity dispersion GVD=173.37 fs 2 /mm at the central pixel of the camera c=2.145 rad.fs -1 , i.e. c = 0.8788 m. The channeled spectrum I vacuum is recorded without the glass being placed in the sample arm of the interferometer for the position of the mirror Ms moved along the optical axis by a linear stage, zv = 20.1 mm. When the glass is placed in the sample arm, the mirror Ms is adjusted so that the main spectral modulation of the channeled spectrum I sample is similar to that of I vacuum (zs=3.695 mm), i.e. the OPDs s and v are identical, which means coefficient 0 (Fig. 3(a)). I vacuum and I sample are recorded 10 times in order to assess the accuracy of ng and GVD measurements given in the form of (mean +/-2 standard deviation). Because fringe visibility is very weak at the edge of the spectra,  -(c) is not plotted over the all spectrum range. A fit by a quadratic polynomial centered at c, leads to ng=1.8203 +/-0.0001 and GVD=(173 +/-2) fs 2 /mm ( Fig. 3(b)). The experimental results are in good agreement with the theoretical values. The consistent deviation between the experimental and theoretical values of ng can be explained by the accuracy in measuring the thickness of the sample. If a 19.995 mm-sample was considered, then a perfect match with the theoretical value of ng would be obtained.
To assess the stability of the measurements, GVD and ng have been evaluated for different positions of the mirrors Ms and Mr, such as the OPDs with the sample (s) and without the sample (v) are always identical. In this case the variations of GVD and ng are inferior to the standard deviation, which proves the robustness of the CMSI method.

CONCLUSIONS
This paper demonstrates that the CMSI method used in SDI can efficiently be used for accurate refractive index and group velocity dispersion measurements of optically transparent samples. The method uses a continuous range of frequencies, while other methods have been reported using measurements at discrete optical frequency values [9].
Based on the results presented in this paper, other avenues are foreseeable, such as extending the CMSI method towards a matrix approach (please see the paper [10]), as well as utilization of phase recovered via the CMSI method for phase sensitive OCT and for polarization sensitive OCT. Future work should refer on how phase accuracy evolves with OPD, in the context of longer axial range proven by the CMSI [11] in comparison with conventional signal processing methods. A recent report [12] has proven the consistency of high-resolution distance measurements when using CMSI. We expect that this conclusion can be extended to phase measurements, but a demonstration remains to be reported.