Master / slave interferometry – ideal tool for coherence revival swept source optical coherence tomography

In this paper, we demonstrate that the Master Slave (MS) interferometry method can significantly simplify the practice of coher ence revival swept source optical coherence tomography (OCT) technique. Previous implementations of the coherence revival technique required considerable resources on dispersion compensation and data resampling. The total tolerance of the MS method to nonlinear tuning, to dispersion in the interferometer and to dispersion due to the laser cavity, makes the MS ideally suited to the practice of coherence revival. In addition, enhanced versatility is allowed by the MS method in displaying shorter axial range images tha n that determined by the digital sampling of the data. This brings an immediate improvement in the speed of displaying cross-sectional images at high rates without the need of extra hardware such as graphics processing units or field programmable gate arrays. T he long axial range of the coherence revival regime is proven with images of the anterior segment of healthy human volunteers. 2016 Optical Society of America OCIS codes: (110.4500) Optical coherence tomogra phy; (170.4460) Ophthalmic optics and devices (120.3180) Interferometry; (200.4740) Optical processing. References and links 1. W. Wieser, B. R. Biedermann, T. Klein, C. M. Eigenwillig, and R. Huber, “Multi-Megahertz OCT: High quality 3D imaging at 20 million Ascans and 4.5 GVoxels per second,” Opt. Express 18(14), 14685-14704 (2010). 2. T. Klein, W. Wieser, C. M. Eigenwillig, B. R. Biedermann, and R. Huber, “Megahertz OCT for ultrawide-field retinal imaging with a 1050 nm Fourier domain modelocked laser,” Opt. Express 19(4), 3044–3062 (2011). 3. I. Grulkowski, J.J. Liu, B. Potsaid, V. Jayaraman, C.D. Lu, J. Jiang, A.E. Cable, J.S. Duker, and J.G. 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Introduction
Both implementations of spectral or Fourier domain optical coherence tomography (OCT), respectively spectrometer based (Sp) and swept source (SS), can be used to produce crosssectional (B-scan) OCT images with high speed and high sensitivity.SS-OCT has some advantages over the Sp-OCT including a higher imaging speed [1][2], longer axial range and provides a quite uniform sensitivity over the axial range, which enables deeper structures to be visualized in a single scan.Although SS-OCT technology seems to win over Sp-OCT in most of the cases, there are applications that could benefit from an even more extended imaging range, such as the imaging of the ocular anterior segment or even of the entire human eye [3].Unfortunately, due to the finite coherence length of the lasers used, the axial imaging range is still limited.An exception from this makes the new tunable vertical-cavity surface-emitting lasers (VCSEL) and akinetic light sources, which can provide a long axial range, exceeding 1 cm [4][5].However, both VCSELs and akinetic light sources are only commercially available at longer wavelengths such as 1300 and 1550 nm and are more costly than the microelectromechanical system (MEMS) SSs widely available.To our knowledge, for shorter wavelengths, around 1050 nm, there are no SSs capable of providing extended (cm) axial ranges.Recently a new VCSEL light source in the 1050 range was demonstrated but of narrow tuning bandwidth [6].
As in all implementations of OCT, SS (MEMS, VCSEL, akinetic) and Sp-OCT the generation of images is based on a Fourier transform (FT), therefore mirror terms halve the achievable axial range.Numerous methods have been demonstrated to regain the full axial imaging range [7][8][9][10][11][12][13][14][15][16][17].However, very often the improvements reported come at a price: reduced sensitivity, less axial resolution or speed, increased system complexity and associated cost, incomplete elimination of mirror terms, complex post-processing algorithms to be implemented, etc.
An elegant solution to increase the axial range in SS-OCT is the use of the coherence revival exhibited by some commercially available external cavity tunable laser (ECTL) swept sources [18].The coherence revival method employs the interference between the waves reflected from the interferometer arms that originate from waves that suffered different multiple reflections inside the swept laser cavity.For the interference to occur, the difference between the optical path lengths of the interferometer arms are mismatched by an integer multiple of the roundtrip laser cavity length.This method is easy to implement, as only the optical path difference (OPD) between the two arms of the interferometer needs to be altered.As no additional hardware is required, the axial resolution is in principle not affected while the sensitivity drop is negligible.
The coherence revival technique shifts the maximum interference obtainable around OPD = 0 to values equal to multiples of the round trip in the laser cavity.The main difference is that while conventional OCT manifests no spectrum modulation at OPD = 0, coherence revival technique presents modulation around the OPD value where maximum of sensitivity is achieved.As explained in [18], the modulation is due to phase variations created by cavity length modulation during sweeping.Under coherence revival conditions, the channeled spectrum exhibits a different modulation density for positive and negative OPD values in the interferometer while the conventional OCT delivers the same modulation for the positive and negative OPD in the interferometer, equal in modulus (mirror terms).Therefore, the coherence revival technique offers a low cost solution for removal of mirror terms.Once this is achieved, an immediate advantage is doubling the axial range.In addition, working on OPD values of different multiplicities of the laser cavity length allows simultaneous imaging of samples at different distances from the system.An important such application is simultaneous imaging of the anterior and posterior segment of the eye [19].Coherence revival is also applicable in conjunction with buffering methods to increase the sweeping of lasers with duty ratios below 50% [20,21].
However, there are some penalties in implementing the coherence revival via the conventional FT based method, as described below [18][19][20][21].
First, the k-clock equipping the ECTL swept source becomes unusable for correct data digitization, as the spectrum modulation is at least twice denser than in conventional operation.A new k-clock module needs to be re-built, or software methods of data resampling have to be employed instead [22][23][24][25][26].The larger frequency clock needed leads either to an increase in the complexity of the imaging systems, hence to increased costs, or when data is resampled, to a reduction of the rate at which images are displayed.
Second, intensive dispersion compensation is required, as twice the laser cavity length is now within the total optical path difference creating the spectral modulation to be decoded by the digitizer.While for compensating for the dispersion in the interferometer both hardware optical methods and software (numerical) methods can be implemented [27][28][29][30][31], to compensate for the dispersion due to the laser cavity, only software solutions can be used.Normally, the implementation of any solution for dispersion compensation leads to an increase in the complexity of the systems or to heavy computation.
We demonstrate here, as an alternative to the FT based method, the utilization of the recently introduced Master/Slave (MS) OCT method in combination with the coherence revival technique, which eliminates the drawbacks specified above and allows real-time production of cross-sectional images.The MS method [32] is based on comparison of shapes of the channeled spectra at the interferometer output with stored channeled spectra shapes.An immediate advantage is that the MS technique can be implemented with raw data, without the need of organizing it in linear frequency slots (as required by conventional OCT technique, based on a FT).This means that no k-clock nor any resampling procedures are needed.Moreover, the MS method is immune to any dispersion left uncompensated within the optical path difference, which here includes the fiber in the laser cavity [33].These two properties make the MS method an ideal tool for its combination with the coherence revival technique, offering the most simple and cost effective way to obtain extended axial range images without the need of data resampling or dispersion compensation.
In [34], real time production of cross-section MS-OCT images was made possible by using Graphics Processing Units (GPUs).Further progress in signal manipulation is reported here based on the simplified algorithm proposed in [35] that is incorporated here in a simple matrix multiplication.A second improvement is also based on the newest implementation of the MS method (Complex MS (CMS) [35]) that allows simplification of the preparation stage (Master).These improvements allow real-time display of processed B-scan MS-OCT images within the following frame without resorting to a GPU or a field programmable gate array, as previously reported [36].

Strategies of producing reflectivity profiles in OCT
Let us refer comparatively to the two strategies of producing reflectivity profiles in OCT: FT based (conventional) and CMS based.Let us consider a continuum of values along the wavenumber coordinate k.To obtain the reflectivity profile, A(z), at a certain axial position (z) in the A-scan of the sample investigated, the integral of the product between the channeled spectrum , () kz CS  from the sample and the kernel function jkz e has to be calculated, where , kz  is the phase of the measured channeled spectrum.
Due to nonlinearities in the decoding procedure, from spectrum to time, typical for a real spectrometer or tuning swept source, the channeled spectrum is chirped.This means that the spectrum modulation does not exhibit a regular periodicity over the wavenumber axis.A second source of nonlinearity in wavenumber is that due to dispersion in the interferometer.Therefore, the phase is not directly proportional to k, but expressed in a general way as: The parameters k g and k h take into account the non-linear dependence of the phase on the wavenumber and on dispersion left unbalanced in the interferometer respectively [35].These nonlinearities make the channeled spectrum chirped.To eliminate the effect of the chirping, two strategies can be employed: In this case, while the kernel function jkz e is left untouched, each channeled spectrum is first resampled then multiplied with a function that cancels the effect of the dispersion to obtain a linear relationship between the phase of the modified channeled spectrum (CS non-chirped ) and k.As a result, a sufficiently accurate A-scan can be produced as a Fast Fourier Transform (FFT) of the CS non-chirped : ) The advantage of this approach is the speed of the FFT.However, the preparatory steps to correct the data and produce CS non-chirped before performing FFT are computationally time expensive and prone to introducing errors.Moreover, the phase correction procedure has to be done for each acquired channeled spectrum, as illustrated in Fig. 1.Fig. 1.Procedure steps required to produce an A-scan profile using the FT based OCT (top) and procedures steps to produce a single reflectivity value using the CMS method (bottom).The CMS-OCT method requires a single processing step after data acquisition while the conventional FT-OCT method is more time demanding, as three sequential steps are required.

MS based OCT
As the MS method is immune to the amount of unbalanced dispersion in the system or to the non-linearities of the spectra, its implementation is the same in both conventional and coherence revival based SS-OCT systems.The MS strategy consists in modifying the kernel function jkz e in Eq. ( 1) rather than process the channeled spectra.In this case, to produce a single point in the A-scan, instead of a FFT, an integral of the product between the chirped channeled spectrum and a new kernel function , ) ( kz M  (denoted as complex masks) is performed [35].As reflectivity of a single point only is delivered by such an integral calculation, the MS method may not present the speed advantage of the FFT; however, the MS method eliminates several disadvantages of the FFT method.The improved version of the MS-OCT recently reported, the CMS, , with a component for each wavenumber pixel 1, 2,…Nk along the coordinate k.This procedure is documented in [35].In order to obtain the reflectivity from an axial position z, it is sufficient to calculate: As illustrated in Fig. 1, the only step to be performed after data acquisition is the dot product between the recorded CS and the complex masks.
Obviously, when there is no nonlinearity in decoding the channeled spectrum, there is a linear distribution of frequencies along time (g = k), and if in addition h = 0 (no uncompensated dispersion), Eq. ( 4) is equivalent to Eq. (3).

Procedures of producing images in OCT
In the diagram shown in Fig. 1, the complete procedures of producing A-scans, using the FT-OCT (top) and CMS-OCT (bottom) are presented.Both techniques require a "calibration" step, before data acquisition, where the FT method entails three operations, which can only be executed sequentially whilst the CMS entails a single operation as described below.

Conventional FT based OCT
Let k = 1 … Nk be the number of wavenumber sampled points used to digitize each channeled spectrum, z = 1 … Nz the axial positions where the reflectivities are evaluated from and let x = 1 … Nx be the number of A-scans in each B-scan.An A-scan is produced as non-chirped () x FFT CS .Therefore, mathematically, a B-scan image can be described by: where each of the components, non-chirped x

CS
, for x =1 .. Nx, are resampled channeled spectra of Nk components each, recorded while scanning the beam over the sample.The size in pixels of the image thus obtained is (Nx × Nk)/2 (Nx pixels laterally and Nk /2 pixels axially).To produce a cross-sectional image using conventional FTs: 1.A recalibration vector is produced (not required for swept sources equipped with k-clocks).This is a step performed prior to data acquisition.2. The recalibration vector is used to resample each channeled spectrum via a spline cubic interpolation (not required for swept sources equipped with k-clocks).Obviously, this step takes place during or after data acquisition.3. A procedure to compensate dispersion left unbalanced in the interferometer or/and dispersion due to the cavity when coherence revival is employed is needed.Again, this step can only take place during or after data acquisition.4. A B-scan is produced by performing FFTs of the channeled spectra modified according to the steps 2 and 3 above.
The time required to produce a B-scan in FT based OCT includes the timing of sequentially executing the 3 last steps presented above.For swept sources equipped with k-clocks signals, the steps 1 and 2 can be eliminated, however due to difficulties to generate a clock signal when operating in the coherence revival regime, a k-clock may not be available.

MS based OCT
Numerically, Eq. ( 4) can be rewritten to describe the reflectivity value from a scattering center at an axial position z, by: , 1 ( ) ( , ), A B-scan image can be represented in a matrix form as: where CS, is a matrix of size Nx × Nk, containing the channeled spectra CSx, acquired for all lateral pixels x =1,2,… Nx (along a laterally oriented scan), described by: Here, each row of the matrix M is a complex signal of Nk components representing a mask produced for each axial position z = 1,2,…Nz.The channeled spectra used in Eq. ( 4) to produce the B-scans, do not require any preparation, while the resulting B-scan images are completely free of eventual unbalanced dispersion or decoding nonlinearities.
A CMS based cross-sectional image is generated according to the following procedure: 1.With a high reflector as object, two or more experimental channeled spectra corresponding to different axial positions are stored.This step is performed only once for a given experimental set-up, before data acquisition.2. The experimental channeled spectra recorded at the step are then used to theoretically infer Nz complex masks (matrix M in Eq. ( 9)), as described in [35].This step, that is performed before data acquisition, needs to be repeated only if when the axial range displayed in the B-scan image is modified.3. A cross-sectional image is produced by multiplying two matrices (Eq.7).This step is performed as soon as raw data corresponding to a B-scan is acquired.
The size in pixels of the cross-sectional MS-OCT image produced in Eq. ( 7) is Nx×Nz, different from that described by Eq. ( 5) for the FT case, which is Nx×Nk/2.This axial range difference triggers a discussion on an important aspect: when using the conventional strategy, FFT based, the axial range of each A-scan scales from 0 min z  to a maximum value max z , determined by the sampling speed of the digitizer, hence by the number of sampling points Nk used to digitize the channeled spectrum.A modification of the axial range of interest (ROI), if needed, can only be achieved by effectively cropping the cross-section image, while a modification of the number of sampling points of the ROI is only possible by zero-padding the channeled spectra before FFT.In CMS, the axial length of the axial region of interest ROI is completely independent on the number of digitized points Nk.In addition, the axial range of interest as well as its coverage is adjustable by selecting the set of complex masks M in terms of their axial position and increment between their depths.
The time required to produce a CMS based B-scan image is given by the time to multiply two matrices.The FT based strategy is obviously faster in terms of producing a B-scan, when no data preparation is required before FFT.Indeed, in principle, for Nk points, a FFT based Bscan requires NxNklog2Nk operations, while a MS based B-scan obtained by matrix multiplication NxNz(2Nk-1) operations.To obtain the same axial length image in both methods, Nz=Nk/2, in which case the FFT method is a clear winner.However, in practice the channeled spectra have to be prepared before FFT, i.e. data need to be organized in equal wavenumber slots that requires extra time and resources.In addition, images in the CMS-OCT, that use Nz less than Nk/2 can be quicker produced than using the FFT based OCT.For such cases, the time to produce B-scans via Eq.( 7) is similar or shorter than the time required by FFT-OCT with resampling.

Experimental set-up
A schematic diagram of a coherence revival based SSOCT imaging system assembled for this study is depicted in Fig. 2. As optical source, a commercially available ECTL (Axsun Technologies, Billerica, MA) exhibiting coherence revival was employed.It has a central wavelength of 1060 nm, sweeping range 106 nm (quoted at 10 dB) and 100 kHz line rate.To ensure that the MS method can be employed, the stability of the source has been monitored over several months and concluded that its non-linearity function gk does not show any fluctuations.The interferometer configuration uses two single mode directional couplers, DC1 and DC2.DC1 has a ratio of 20/80 and DC2 is a balanced splitter, 50/50.DC2 feeds a balance detection receiver BPD (Thorlabs, Newton, New Jersey, model PDB481C-AC), of 1 GHz electronic bandwidth.20% from the SS power is conveyed towards the object arm, via a microscope objective MO1 (focal length 15 mm), which collimates the beam towards a galvo-scanner SX, (Cambridge Technology, Bedford, MA, model 6110) followed by a scan lens SL (Thorlabs, model LSM04-BB).The power at the object is 1.9 mW.At the other output of DC1, 80% from the SS power is directed towards the reference arm equipped with two flat mirrors, M1, M2, placed on a translation stage, TS to adjust the OPD in the interferometer.As TS, a linear actuator (Newport, Irvine, CA, model M-VP-25XA) was used, controlled by a Newport driver MM4005.
Collimating microscope objectives MO2 and MO3 are identical to MO1.The dispersion in the interferometer was left unbalanced, i.e. no glass slabs were used in the reference arm to compensate for the scan lens or other techniques were employed to compensate for the dispersion introduced by the laser cavity.
The signal from the BPD is sent to one of the two inputs of a dual input digitizer D (Alazartech, Quebec, Canada, model ATS9360).Data is digitized using the internal clock of the digitizer at 1GS/s, which leads to a maximum axial range of 12.4 mm.No data resampling procedures were implemented.Each channeled spectrum was digitized into Nk = 4096 sampling points.A number of Nx = 1000 channeled spectra were used to build each B-scan hence 10 ms acquisition per frame.The galvo-scanner is driven with a triangular waveform, where a half period of the driving signal is used to acquire data while during the second ramp, data is processed.Therefore, 20 ms taken by each frame, hence the B-scan imaging proceeds at a 50 Hz rate.Sensitivity and fall-off measurements with an optical difference of one cavity length were performed.The cavity length of the laser was measured by placing a mirror in the sample arm and adjusting the length of the reference arm.As in [17] we refer to order +1 or −1 to the situations in which the sample arm was longer (Z+1 = +11.5 cm) or shorter (Z-1= -11.5 cm) than the reference arm by a laser cavity length, respectively and order 0 (Z0 = 0) when the lengths of the two arms of the interferometer are equal.

Sensitivity drop-off and axial resolution measurements
By altering the length of the reference arm, we found out that TS has to be moved by ±11.5 cm between order 0 and +1 or -1, as illustrated in Fig. 2, which corresponds to a length of around 80 mm of fiber.A similar cavity length was reported in [18].
In order to produce the complex masks z M (z = 1 ... Nz) a number of 5 channeled spectra corresponding to 5 different axial positions of the TS separated by 0.5 mm were acquired.We used the driver of the translation stage to control exact positioning with an error below 0.1 m.However, we obtained similar results if the micrometer screw, with divisions at 5 m was used, in which case the manual positioning is performed with an error below a couple of micrometers.Then: (i) Using a simple linear regression of the first order derivative of the phase with respect to k, ( , ) k k z    , according to z, allows for removing the random noise and produces the parameters gk and hk [35].
(ii) Having the information about the non-linearity (gk) and the amount of uncompensated dispersion (hk), a digital format of Eq. ( 4) is used to produce the desired number of complex masks z M .Then, Eq. ( 6) is employed to produce axial reflectivity profiles.The number of masks to be produced Nz, has to be large enough to ensure that the distance between consecutive points in the A-scan where the reflectivities are calculated at (digital axial resolution) is smaller than the axial optical resolution.
In Fig. 3, the sensitivity drop-off profile vs. the axial position δz = z -Z+1 is shown (as the envelope of the signal for several measurements at different optical path differences, colored curves).This gives a full width at half maximum FWHM = 10 mm of the axial range.In addition, with the reference arm blocked, the axial confocal profile is shown (black curve), generated by measuring the DC signal at the photo-detector for different axial positions of a mirror as the object.The peak sensitivity is achieved at around δz = 6.0 mm, where the sensitivity reaches 101.3 dB (using the procedure detailed in [32]).The position δz = 6.0 mm determines an optical path difference between the arms of the interferometer equal to the roundtrip length of the laser cavity.
For δzmin = 0 mm, the channeled spectrum exhibits little modulation while for δzmax = 12.4 mm, a maximum modulation frequency of 500 MHz is obtained.The sensitivity drops by about 4 dB from the axial position where the peak sensitivity is measured to the edges of the axial range shown in Fig. 3 (δzmin = 0 mm and δzmax = 12.4 mm).
The long axial range of the coherence revival demands a low NA objective.It must be noticed that in practice, due to the need of good transversal resolution, the confocal gate determined by the MO1 and the SL is usually narrower than the long axial range determined by δzmax and consequently the effective imaging range is lower than this value.According to the black curve presented in Fig. 3, the width of the confocal gate measured at 1/e 2 from the maximum value is about 3.5 mm.
The procedure of producing the OCT sensitivity profiles is based on Eq. ( 6).This does not require any data resampling or numerical dispersion compensation, as the complex masks M already incorporate the information on the chirp of the channeled spectra.The FWHM corresponding to the axial resolution of the system was measured for each sensitivity profile curve leading to a value of 6.85±0.62µm (where 0.62 is the standard deviation).In a conventional SS-OCT, to avoid degradation of the axial resolution, each channeled spectrum is subject to mathematical operations before performing FFT.After data resampling and numerically compensating for dispersion, an axial resolution of 6.96±0.96µm was found, which is similar to that obtained using the CMS technique.

Time benchmarking
To evaluate the conditions in which the CMS method can provide real-time cross-sectional images, we compared the time to produce a B-scan image using the CMS method with the time required to produce the same image via FFTs, when data was resampled only but not compensated for unbalanced dispersion.Our findings are summarized in Fig. 4.
To perform the benchmarking, a LabVIEW 2015 (National Instruments, Austin Texas) project was created to run on a computer equipped with an Intel I7-5960X @ 3.0 GHz octacore processor (2 logical cores per physical core) and 16 GB of RAM.Each B-scan is built using Nx=1000 A-scans.Two situations are considered for the CMS case, Nz=Nk/2 to mimic the axial size of the B-scan obtained in FFT based OCT and also a smaller number of sampled points, Nz=Nk/4.For the conventional case of resampled FFT method (FFTR) (solid circles in Fig. 4), data were resampled via a cubic B-spline interpolation procedure before FFT.The FFT requires NxNklog2Nk operations.A fixed number of Nz = Nk /2 points was used axially.When using the CMS method, for the same number of Nz = Nk /2 axial points, a number of NxNk(2Nk -1) operations are required.Therefore, the time to produce a Bscan using the FFT approach (filled circles in Fig. 4) is shorter than the time to produce a CMS-OCT B-scan image (diamonds in Fig. 4).This curve was obtained using the standard procedure of multiplying matrices.Fortunately, high performance toolboxes that take advantage of the multicore design of the modern processors are already built within LabVIEW.By using the Multicore Analysis & Sparse Matrix toolkit (MASMT), the time performance of multiplying matrices is tremendously improved (triangles in Fig. 4).We have found that by using the same toolkit does not bring any improvement in speeding up the FFTs.
As it is demonstrated in Fig. 4, for small values of Nk, (up to around 5120 for the particular computer used in this study), the CMS technique with the adequate LabVIEW toolkit can produce B-scans faster than the procedure using the FFT after resampling (FFTR).Obviously, if data needed to be additionally compensated for dispersion, the FFT based method would have required even longer.By running our benchmarking project on computers with different, weaker CPU specifications, but with at least 4 cores, we found out that typically, CMS is faster than FFT when Nk < 3000.Typically, if the emphasis is not on a long axial range, there is no need for a high number of sampling points when digitizing data.For Nk = 4096 sampling points, with Nz = 2048 distinct points in depth, a B-scan can be produced in about 102 ms.This does not allow B-scans to be produced during the time of the next frame when using a 50-Hz galvoscanner.However, such a performance is faster than the FFTR method (120 ms) but insufficiently quick to categorize the operation as truly real-time.
An exquisite capability of the MS method is the possibility to reduce the time to display the image by reducing the number of depth points, Nz.This number is not connected to Nk and by reducing Nz, the axial resolution is not affected, the image becomes sparse only.The drawback of this operation is either a reduction of the axial range (when Nz masks are produced to encode a shorter axial range than that determined by the digitizer) or a less axial resolution (when Nz less dense masks are produced to cover the full axial range determined by the digitizer).This limitation is not essential in most applications.For example, in our case, although it is possible to achieve an axial range as long as 12.4 mm, in practice this may not be as useful, as very often the axial range is limited by the extension of the confocal gate determined by the scan lens.With squares, we show in Fig. 4 that for a reduced number Nz = Nk/4 axial points in the A-scan (which corresponds to an axial range of around 6 mm if the density of masks is kept constant), the CMS technique can provide cross-sectional images faster than its FFTR counterpart for any number of sampling points Nk used to digitize the channeled spectra.According to the benchmarking shown in Fig. 4, for the particular case when Nk = 4096, and Nz = Nk /4, a B-scan image can be produced in 52 ms.This however is still insufficient to ensure real-time operation.Further benchmarking (not presented in Fig. 5) showed that for a 50-Hz galvo-scanner, real-time operation can be achieved when Nz = Nk /10 = 409, which corresponds to an axial range of around 2.4 mm if the density of masks is kept constant.

Imaging of the eye's anterior chamber
Figure 5 shows two images of a volunteer's ocular anterior segment.Both images were produced for the revival order +1, with maximum sensitivity set at δz = 6 mm.Each image, obtained by averaging five adjacent frames was obtained by keeping the length of the reference arm constant and moving the eye axially with respect to the scan lens.In this way, the effect of confocal gate on limiting the axial range is visible.Although our full axial range is around 12.4 mm with little decay in sensitivity from δzmin = 0 to δzmax = 12.4 mm, depending on the axial position of the eye, different parts of the anterior chamber are brightened up.The coherence revival images exhibit the property that the maximum sensitivity is in the middle of the image (and of the axial range), in comparison with conventional OCT images, where the maximum sensitivity is closer to zero optical path difference.In Fig. 5(a) the middle of the image coincides with the anterior chamber.In Fig. 5(b), we bring the crystalline lens closer to the middle of the axial range.Therefore, if in Fig. 5(a) the crystalline lens is hardly visible, it becomes a lot brighter in Fig. 5(b) where the corneal stroma is dimmer.The lateral resolution of the images experimentally measured using an USAF resolution test target is 19.7 m.
To obtain Fig. 5, a number of Nz = 2048 masks (axial points) are used, hence a distance between consecutive masks (axial digital resolution) of about 5.85 µm.Laterally, each image comprises Nx = 1000 pixels, hence a lateral digital resolution of 14.5 µm.Both lateral and axial resolutions are measured in air.According to the benchmarking shown in Fig. 4, for Nk = 4096 and Nz = 2048 a B-scan can be produced in 100 ms, slightly faster than using the FT based OCT method.Consequently, the images in Fig. 5, covering an axial range of over 12 mm are produced at a frame rate of 10 Hz, however slower than the frame rate for real-time operation (50 Hz).When imaging the anterior chamber of the adult human eye, there is no need in principle for more than 6 mm axial range to image the entire segment from the corneal epithelium to the crystalline lens.Using the CMS, the axial range can be adjusted dynamically by simply restricting the axial range of the masks employed.In Fig. 6, a cross-sectional image of the anterior chamber spanning over 6 mm is shown.To produce it, the same number of Nz = Nk /2 = 2048 masks can be used, but corresponding to an axial range of 6 mm.This image was extracted from a movie (Visualization 1) produced still at 10 Hz as Nz = 2048.For an axial range of 6 mm, the axial distance between two consecutive points, when Nz = 2048 in the A-scans is 2.92 µm, which is much better than the axial optical resolution of the system.To maintain the same digital axial resolution as in Fig. 5, the number of axial points Nz can be halved to 1024, in which case, each frame can be produced at around 20 Hz.As it can be seen in Fig. 7 (and Visualization 2), although axially only Nz = 1024 are used, the axial resolution is preserved, but the speed of producing the images improved by a factor of 2 while still producing mirror free long axial range images over an axial range with little sensitivity drop-off.Fig. 7, is the first frame of a movie (Visualization 2, 20 Hz frame rate) produced while the laser beam was scanned over the whole anterior chamber.

Discussion and conclusion
We have demonstrated that CMS-OCT can be an ideal tool for coherence revival OCT imaging systems.The revival technique makes the production of long axial range, mirror term free crosssectional images in OCT possible.The CMS technique simplifies it even further, as it eliminates the need of data resampling and the need for dispersion compensation, bringing speed to the production of OCT images.
The implementation of the CMS technique itself is simple.The calibration of the system (production of the masks) involves recording a few experimental channeled spectra only, process which has to be performed only once for any given set-up.For a calibrated set-up, high resolution, high sensitivity images are produced without the need of any sophisticated mathematical procedure: the only mathematical operation involved is the multiplication of two matrices.This is made possible by the CMS method.This is based on modifying the kernel function in Eq. ( 4) (that can be done at the Master stage) instead of modifying each channeled spectrum as required by the FT based method (operation that is done during the measurement stage).
As a result, there is no need to compensate for the large dispersion introduced by the laser cavity or for the dispersion due to unmatched optics and fiber lengths between the sample and reference arm of the interferometer.Equally, there is no need to resample data or build a new costly k-clock, which may be challenging due to the pronounced chirp and high density of peaks required to produce long axial range images.The artifacts reported in [18], at large frequencies, possible due to an imperfect resampling are not visible in any of the CMS images.
When used in conjunction with a coherence revival imaging system, the CMS method can provide images over the same long axial range as its FT based counterpart.However, due to its simpler implementation, CMS has the potential to produce images less affected by artefacts and also faster without resorting to extra hardware equipment.We have considered the option of harnessing the power of the GPUs, as the speed of performing matrix multiplications is superior to that of the CPU implementation.When harnessing the power of the GPUs, the speed of performing matrix multiplications is superior to that of the CPU implementation.However, irrespective of the GPU computing power, the repetitive process of data transfer to the shared memory accessible by the GPU program does not ensure a real time operation for large number of sampling points Nk [34], which is the case when the goal is to achieve a long axial range or a fast sweeping SS.This peculiarity would recommend utilization of field programmable gate arrays, however this implementation would be complex and costly.
In MS-OCT, to speed up the production of B-scans, it is possible to act on the size of the axial region of interest to be imaged.In FT-OCT, this range is determined by the speed of the digitizer that also determines the number of sampling points Nk to digitize each channeled spectrum.
The maximum axial range is determined in both methods, conventional FFT and MS by the number of sampling points Nk used to digitize each channeled spectrum.However, after acquisition, each method offers different functionality in preparing images.In FFT based OCT, the Nk number of points dictates the axial extension of the B-scan OCT image.A smaller region of interest (ROI) is achievable only by cropping the final image.In MS-OCT, the axial ROI is determined by Nz, the number of masks to be used at the Slave stage.Consequently, it is possible to adjust the ROI easily, to match the axial size of the sample to be imaged.A useful rule of thumb for producing high axial resolution images is to assemble axial reflectivities from consecutive points in the A-scan separated by no more than half of the axial optical resolution.
In our case, as the axial resolution is around 7 µm, to produce a B-scan with a ROI of for example 2 mm, the reflectivity has to be estimated in only Nz = 570 points, instead of Nk/2=2048.The use of a large number of sampling points Nk for digitization cannot be avoided in a coherence revival system hence an increased computation time to resample data before FFT.MS-OCT solves this issue in an elegant way, by decoupling the axial ROI from Nk.In FFT based OCT, as the optical axial resolution is digitally compromised when an insufficient number of points is used to digitize the channeled spectra, very often data is zero-padded before FFT.In CMS, to tackle this inconvenient, a sufficiently large number Nz of axial reflectivities have to be estimated.For the images shown in the manuscript, we produced axial reflectivities from consecutive points (digital axial resolution) separated by 5.85 µm in Figs. 5 and 7, and 2.92 µm in Fig. 6 (to be compared with an optical axial resolution of 7 µm).Figs. 5 and 7, although showing a lower digital axial resolution, provide either long axial range or speed.As in CMS, the axial ROI can be decoupled from Nk, the number of axial reflectivities Nz can be reduced up to the level where the digital resolution equals the optical one.In FFT based method, due to zero-padding the amount of data produced cannot be limited (to truncate the image axial range).
With a speed in displaying B-scans comparable or faster to those typically reported by FT-OCT systems, with tolerance to dispersion and elimination of the need to resample the data, as detailed here, MS-OCT can become the technique of choice for producing long axial range images via coherence revival not only in medical imaging but also in measuring long distances.
In fact, the adaptation of the MS technology to the coherence revival allows using the swept source on any of the multiples of the cavity length.This simplifies especially the assembly of adaptive optics configurations, where the arms are long and where such a method would allow performing OCT at any positions within a wide layout separated by 11.5 cm apart.There is no need of any extra procedure when changing the multiple order of laser cavity used, the only specific step is that of acquiring masks for each configuration used.
The MS method is an alternative to the FT based method, but better to be combined with the coherence revival technique, as it eliminates several important drawbacks associated to the FFT technique, as no need of zero-padding, re-sampling of data or dispersion compensation is required and allows real-time production of cross-sectional images.

Fig. 4 .
Fig. 4. Time to produce a cross-sectional image (B-scan) using both methods, FFT and CMS vs the number of sampling points used to digitize each channeled spectrum.
Each column of this matrix contains the Nk components of the channeled spectrum acquired for each lateral pixel, CSx.Each of the components CSkx is obtained after digitization.M is a matrix of size Nk×Nz, described by: