This paper demonstrates that the complex master slave interferometry (CMSI) method used in spectral domain interferometry (SDI) can efficiently be used for accurate refractive index and group velocity dispersion measurements of optically transparent samples. For the first time, we demonstrate the relevance of the phase information delivered by CMSI for dispersion evaluations with no need to linearize data. The technique proposed here has been used to accurately measure the group refractive index and the group velocity dispersion of a strong dispersive sample (SF6 glass), and a weak dispersive one (distilled water). The robustness of the technique is demonstrated through the manipulation of several sets of experimental data.
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Spectral (or Fourier) domain Interferometry (SDI) is widely spread in many fields of biomedical optics and sensing. SDI is at the core of the modern Optical Coherence Tomography (OCT) [1,2] technology and of group refractive index measurement methods . The measurement of group refractive index of optically transparent samples can be carried out in various ways, very often by using a low-coherence Michelson interferometer [4–9], in which case two measurements are performed: with the sample inserted within one of the interferometer arms and without, or by a single measurement in a common-path interferometer [10,11] for thin samples.
These techniques are based on recovering the spectral phase from the electrical signal at the interferometer output. The electrical signal delivered by the spectrometer is termed in what follows as channeled spectrum, i.e. the electrical version of the optical spectrum read by a camera incorporated within a spectrometer. The channeled spectrum is usually chirped due to: (i) the nonlinear dependence of the pixel position in the camera versus the optical frequency and (ii) the unbalanced dispersion between the arms of the interferometer. Before decoding the modulations in the channeled spectrum via a Fourier transform (FT), the chirping has to be canceled out.
Faultless calibration of the spectrometer is crucial, especially when handling a large number of fringes in the spectrum, where calibration errors are amplified. Hence a calibration precision far superior to the instrument optical resolution is required . Indeed, it was shown that calibration accuracy better than one-tenth of the spacing between two consecutive pixels on the camera is often required to achieve the best possible accuracy in the measured spectral phase.
To decode the channeled spectrum and recover the spectral phase, SDI uses methods to resample and organize the data linearly along the optical frequency axis from the knowledge of the measured chirp, and apply FTs to the signal. Such decoding algorithms are based on the phase interpolation [13,14] or on more sophisticated data processing methods based on fractional Fourier transform, non-uniform fast Fourier transform and wavelet-transform analysis [15–17].
Master-Slave Interferometry (MSI)  (and its improved version, Complex Master Slave Interferometry (CMSI) [19–21]) represent novel approaches to SDI, useful in OCT signal processing. For the scope of this study, the more efficient algorithm of the CMSI will be employed. This proceeds in two stages. In a first stage (Master), several experimental channeled spectra are acquired by using a mirror as sample. These spectra are affected by the nonlinearity in transducing the optical spectrum into an electrical signal and dispersion in the interferometer, making the electrical output signal chirped. Then, digital local oscillations (named masks in this paper) are generated for as many depths (optical path differences) as required. As the algorithm employs the set of experimentally collected channeled spectra, the masks incorporate the chirp due to nonlinearity in reading the spectra and dispersion in the interferometer. In the second stage (Slave), the sample replaces the mirror. The stored masks are mixed with the channeled spectrum Isample acquired from the sample. It has been demonstrated  that, by applying CMSI, irrespective of how chirped the Isample signal is, the A-scan profile obtained is identical to that delivered by a Fourier transform of Isample for a perfectly dispersion-compensated interferometer and perfectly linearized decoder (no nonlinear dependence on the pixel position). Thus, there is no need for data resampling and the procedure is free from errors otherwise resulting from the interpolation of the experimental data. Finally, in a previous report  we demonstrated that a CMSI instrument can produce axial reflectivity profiles faster than the FFT-based conventional methods employing data resampling. This was made possible by harnessing efficient algorithms of manipulating matrices that can be implemented on computer processing units (CPU).
In this paper, we experimentally validate the relevance of phase information that can be delivered by CMSI. Thus, we demonstrate that phase information can be used to measure the group refractive index and the group velocity dispersion of optical transparent samples with high accuracy.
2. Phase reconstruction procedure by CMSI
In Fig. 1, the schematic diagram of an SDI experimental set-up is shown, containing the two main components, an in-fiber interferometer and a spectrometer. The interferometer includes a beam-splitter (shown as a single mode optical fiber directional coupler), a flat reference mirror (Mr), and a flat sample mirror (Ms). The sample to be analyzed is inserted between the sample mirror, Ms, and the achromatic lens Ls. To actuate on the optical path difference (OPD) introduced by the addition of the sample in the sample arm of the interferometer, the mirror Ms is placed on a linear translation stage.
The electrical signal at the output of the interferometer shown in Fig. 1 is typically digitized using an image acquisition board (IMAQ) connected to a linear camera via a camera link cable. This electrical signal can be expressed as a function of the pixel position (x) on the spectrometer camera and the OPD = δ in the interferometer by the following equation:
CMSI is equivalent to the steps above but carried out in a single calculation with resulting advantages in terms of procedure simplification and speed. When employing CMSI, in a first stage (Master), the sample is replaced with a mirror and several experimental channeled spectra (CSexp) are acquired for different OPD values . These experimental channeled spectra CSexp are chirped due to the non-linearity of the spectrometer, p(x), and due to the unbalanced dispersion described by ϕ0 according to Eq. (1). The purpose of the Master stage is to measure p and ϕ0 that are then used to build masks M(x,t), given by:
CMSI consists in calculating a discrete integral between the electrical signal I(x,δ), converting the channeled spectrum at the output of the interferometer, and the masks, M, as follows:17] expressed according to:Eq. (4) equal to:Equation (5) describes the phase of the signal at the output of a perfect interferometer (calibrated and perfectly balanced for dispersion), when the OPD between its arms is δ. In such a perfect interferometer, when a sample is placed in one of the arms of the interferometer, for the same δ, the alteration of the phase is exclusively due to the dispersion of that particular sample.
3. Group index and dispersion measurement procedure
Let us consider two channeled spectra Isample(x,δs) and Ivacuum(x,δv) recorded respectively with the sample to be investigated placed in the sample arm of the interferometer, and without. The corresponding OPDs δs and δv can be expressed as δs = 2(zs-zs0), where zs is the current position of the mirror Ms and zs0 is the linear stage position for OPD = 0 with the sample, and δv = 2(zv-zv0), where zv is the current position of the mirror Ms and zv0 is the linear stage position for OPD = 0 without the sample respectively. A set of masks M is built from M(x,t0) presenting several cycles over N pixels to the densest mask M(x,tN) whose number of cycles is close to N/2 (related to Nyquist frequency). As shown in Fig. 2, CMSI[Ivacuum] and CMSI[Isample] are defined with N values in time domain by an interval equal to 2π/Δω. All values of CMSI[Ivacuum] (and CMSI[Isample]) are equal to 0 except those corresponding to the product between Ivacuum (and Isample) with the set of masks M. The first mask does not start with a zero-cycle in order to remove the DC component of the channeled spectra
The instantaneous phases ϕsample and ϕvacuum related to Isample and Ivacuum can be measured according to the procedure described in Section 2 and explained in Fig. 2.
Due to the properties of CMSI, the instantaneous phases, free from the non-linearity of the spectrometer and the unbalanced dispersion of the interferometer without sample, can be written as:
The phase difference Δϕ = ϕsample - ϕvacuum can then be expressed as:
Thus, if the thickness e is known, the group index and the group velocity dispersion (GVD) can be recovered employing respectively the linear and quadratic coefficients in Eq. (9) as:
4. Experimental results
The spectrometer has been designed to measure a spectral range of 70 nm with a resolution of 0.068 nm. The camera used in the spectrometer is the line scan camera AViiVA M2 CL with 1024 pixels that operates at its maximum speed (28 kHz). As a light source, a super luminescent diode (SLD, Superlum, Cork, Ireland, model Broadlighter 890) with an 890 nm central wavelength and a 150 nm spectral bandwidth is employed. This light source uses a combination of 4 SLD modules. In this paper only the two shorter wavelength SLDs are switched on, i.e. two SLDs centered at 860 nm and 890 nm covering 70 nm of the spectrometer. The normalized spectrum of the two SLDs emitted by this particular source is shown in Fig. 3(a). The optical power is 0.4 mW in the sample arm and 0.65 mW in the reference arm. Sensitivity is 90 dB close to zero optical path difference, with a drop of 30 dB over 2 mm, which is typical for a spectrometer-based OCT system.
Because the function p measured during the Master stage is known up to an additive constant, the spectrometer is calibrated with a swept source covering a spectral range from 828 to 873 nm with a wavelength adjustment step of 0.05 nm (line-width equal to 0.045 nm). The 0.873 mm-line of the swept-source, corresponding to an angular frequency ωcalib = 2.159 rad.fs−1, has been used for the calibration. This has illuminated the 427th pixel x0 of the CCD line camera of the spectrometer (Fig. 3(b)).
In order to demonstrate that the proposed method can be used to accurately measure the group refractive index and a large range of group delay dispersion values, two types of materials were considered, with strong and weak dispersion. Two slabs from such materials were sequentially placed in one of the interferometer’s arms and their dispersion evaluated using our approach.
4.1 SF6 glass sample
The glass sample SF6 is a 20 mm-flint glass whose catalogue data gives a group index ng = 1.8205 and group velocity dispersion GVD = 173.37 fs2/mm at the central pixel of the camera ωc = 2.145 rad.fs−1, i.e. λc = 0.8788 mm .
The channeled spectrum Ivacuum (Fig. 4(a)) is recorded without the glass being placed in the sample arm of the interferometer for the position of the mirror Ms, 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 Isample (Fig. 4(c)) is similar to that of Ivacuum (zs = 3.695 mm). Ivacuum and Isample are recorded 10 times in order to assess the accuracy of ng and GVD measurements given in the form of (mean +/− 2 standard deviation). The standard deviations for ng and GVD are related to random noise coming from data-acquisition noise.
The difference between the instantaneous phases Δϕ of the two spectra is calculated by applying CMSI to Isample and Ivacuum (Fig. 4(b) and 4(d)) and using Eq. (14). Because fringe visibility is very weak at the edge of the spectra, Δϕ - Δϕ(ωc) is not plotted over the all spectrum range (Fig. 4(e)). A fit by a quadratic polynomial centered at ωc, leads to ng = 1.8203 +/− 0.0001 and GVD = (173 +/− 2) fs2/mm. The experimental results are in good agreement with the theoretical values. There is a consistent deviation between the experimental and theoretical values of ng. Its consistency and the fact that the difference is larger than the standard deviation suggest a deterministic bias and not randomness. This deviation can be explained by the accuracy in measuring the thickness of the sample. For instance, if a 19.995 mm-sample was considered, then we would obtain a perfect match with the theoretical value for ng.
These results have been obtained using channeled spectra measured for specific positions of Ms. In order to evaluate the dependence of method accuracy on positions zv and zs, GVD and ng have been computed for different sets of channeled spectra. Thus, channeled spectra were recorded, without the sample in place from zv = 20 mm to 20.4 mm in steps of 50 mm, and with the SF6 glass inserted, from zs = 3.545 mm to 3.945 mm in steps of 50 mm. Figure 5(a) shows the relative error of GVD according to the position of Ms, given by a line corresponding to β = 0. Indeed the specific set (zv = 20.1 mm, zs = 3.695 mm), described above and represented by a white circle in Fig. 5(a), has a very weak linear coefficient β (Fig. 4(e)), while the sets of positions from either side of the line (zv = 20.1 mm, zs = 3.645 mm) and (zv = 20.1 mm, zs = 3.745 mm) have a linear coefficient β different from 0 as shown in Fig. 5(b) and Fig. 5(c). The fact that β = 0 means that the OPD δs is equal to δv according to the shift theorem of Fourier transform, i.e. the main spectral modulation of Isample is identical to that obtained for Ivacuum. This condition is more general than the comparison between the position of the peaks in Fourier domain because it is related to the barycenter of the peaks under energy and it is independent from their shape.
To assess the stability of the measurement, GVD has been evaluated along the line corresponding to δs = δv in Fig. 5(a). Figure 6 shows the results, exhibiting a constant GVD value, which proves the robustness of the CMSI method. GVD is not displayed for OPDs inferior to 300 mm because the small number of modulations does not permit to completely separate the DC peak from the peak associated to the modulation, which leads to a greater inaccuracy. Moreover, the study has been carried out until an OPD equal to 1 mm. Indeed GVD should be affected by the loss of fringe visibility for high OPDs due to the finite resolution of the spectrometer. The dependence between accuracy and modulation is the same as that applicable to the conventional FT based method because the sensitivity drop off is independent from the specific method used for signal processing, FT or CMSI.
Figure 7(a) shows the relative error in determining ng according to the position of Ms. The value ng is stable irrespective of the position of the mirror Ms except for zs = 3.545 mm for which the modulation amplitude of the channeled spectrum Isample is not sufficient to perform an accurate measurement. Figure 7(b) exhibits a constant ng value, which again proves the robustness of the method.
Note that the loss of fringe visibility due to the finite resolution of the spectrometer has not been taken into account in the current study. In order to measure strong GDD, the loss of fringe visibility should be measured and compensated numerically in order to recover the spectral phase due to the sample only.
4.2 Water sample
Distilled water was used to fill in a rectangular cuvette of 10 mm in length. At the center of the spectrum, i.e. λc = 0.8788 mm, the theoretical value of the water group index is ng = 1.3412 and its group velocity dispersion is GVD = 16.98 fs2/mm at 19° C .
The Master stage is performed with the empty cuvette in the sample arm so that the dispersion due to the cuvette material itself is taken into account. Avoiding the measurement or compensation procedures for the cuvette dispersion is a practical advantage for CMSI .
The channeled spectrum Ivacuum is measured 10 times with the empty cuvette for the position of the mirror Ms, zv = 20.1 mm. Once the cuvette is filled with water, the corresponding channeled spectrum Isample is recorded 10 times by adjusting the mirror Ms so that its main spectral modulation is identical to that obtained for Ivacuum (coefficient β = 0). The position of Ms associated to Isample is then zs = 16.68 mm.
The difference between the instantaneous phases Δϕ of the two spectra is calculated from Eq. (14) and is fitted by a quadratic fit centered at ωc (Fig. 8), which leads to ng = 1.3416 +/− 0.0001 and GVD = (16.8 +/− 0.7) fs2/mm, according to Eq. (12) and (13). Here again, the experimental results are in good agreement with the theoretical values. The difference between experimental and theoretical values on ng can also be explained as being due to the limited accuracy of the sample thickness measurement. For instance, if a 10.013 mm long cuvette was considered, then we would obtain a value identical to the theoretical value of ng.
In Fig. 8 a drop in the spectral phase value can be noticed at 2.14 rad.fs−1, i.e. at the center of the spectral range seen by our spectrometer. This is a consequence of the fact that the light source employed here is actually a combination of two super luminescent emitting diodes, where the overall spectrum is not flat but drops at the center, which leads to a lower spectrometer fringe visibility. Thus, fringe visibility, as well as data acquisition noise such as camera noise and laser source noise can limit the sensitivity in measuring weak GDD values.
For β = 0, i.e. when δs = δv, GVD and ng values, (as shown in Fig. 10(a) and in Fig. 10(b) respectively), are stable and agree with the theoretical values, apart from an error in ng that is assumed to be due to an incorrect value of 10 mm of the cuvette thickness.
For the first time, processing of phase delivered by the newly introduced CMSI method is presented. Progress from the original Master Slave method to Complex Master Slave allowed conservation of phase in the signal captured, by employing complex instead of real masks. This opens several avenues for CMSI, one of them being detailed here, employed in dispersion evaluation. In summary, here we propose to use CMSI as a new approach to measure GVD and group index. Only two channeled spectra, with and without the sample placed in one of the arms of the interferometer have to be measured in order to retrieve the relative phase, followed by a quadratic polynomial fit whose first and second orders are related to group index and GVD respectively. In comparison with other methods, there is no need to linearize the data. Not needing linearization speeds up data delivery and reduces the sources of errors.
In addition, compensation of dispersion bias is automatic. The robustness of this technique has been demonstrated on several sets of channeled spectra. It is shown that the most robust results are obtained when the channeled spectrum measured with the sample is adjusted to display a similar modulation to the channeled spectrum without the sample, adjustment that makes the linear coefficient β equal to zero in the relative phase. A maximum deviation of 0.03% from the expected values was measured for the group index ng, and 1% for the GVD. These measurements could be improved by increasing the accuracy of the sample thickness measurement or by accumulating more phase within thicker samples. Another alternative could be to extend the spectral bandwidth within which the phase is measured . A 70 nm bandwidth spectrum was used in the paper but the method is applicable to any bandwidth, including ultra wide bandwidths available by using supercontinuum sources and considering higher-order terms for the fit of the spectral phase.
The measurements were made in transmission, however using a strong reflector in the sample similar measurements can be implemented in reflection. This may open the avenue of a future study on comparing the efficiency of our method with that based on the degradation of the point spread function from the speckle pattern .
Finally, our method, with more refinements and adaptations , could be used in OCT as an alternative to other methods based on the spectral phase, such as no-dye angiography based on phase methods or polarization sensitive OCT methods. These potential directions of applications should similarly, as shown in this study, take advantage of the CMS method: (i) No need for data resampling; (ii) No need for dispersion compensation. In addition, the CMS provides direct provision of information from multiple depths in the sample (when using a FT, data from outside depths of interest is discarded).
H2020 Marie Skłodowska-Curie Actions (MSCA) (625509), Engineering and Physical Sciences Research Council (EPSRC) (EP/N019229/1), Innovationsfonden (4107-00011A), H2020 European Research Council (ERC) (754695), National Institute for Health Research Biomedical Research Centre at Moorfields Eye Hospital NHS Foundation Trust (NIHR), the UCL Institute of Ophthalmology, University College London, the Royal Society Wolfson research merit award and Erasmus+.
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