Gabor fusion technique in a Talbot bands optical coherence tomography system

In this paper we show how to advantageously combine two effects to enhance the sensitivity with depth in Fourier domain (FD) optical coherence tomography (OCT): Talbot bands (TB) and Gabor-based fusion (GF) technique. TB operation is achieved by routing the two beams, from the object arm and from the reference arm in the OCT interferometer, along parallel separate paths towards the spectrometer. By adjusting the lateral gap between the two beams in their way towards the spectrometer, the position for the maximum of contrast variation of spectral modulation versus the optical path difference in the interferometer is adjusted. For five values of the focus position, the gap between the two beams is readjusted to reach maximum sensitivity. Then, similar to the procedure employed in the GF technique, a compound image is formed by stitching together the parts of the five images that exhibited maximum brightness. The smaller the diameters of the two beams, the narrower the visibility profile versus depth in Talbot bands, which brings advantages in terms of mirror terms attenuation. However, this leads to a larger spot on the linear camera, which introduces losses, therefore the combined procedure, TB/GF is investigated for four different values of the beam diameters of the two beams. Future cameras with larger pixel size may take full advantage of the TB/GF procedure proposed here. © 2012 Optical Society of America OCIS codes: (110.4500) Optical coherence tomography; (110.1085) Adaptive imaging; (120.3890) Medical optics instrumentation. References and links 1. W. Drexler and J. G. Fujimoto, Optical Coherence Tomography Technology and Applications (Springer, Berlin Heidelberg, 2008). 2. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express 11, 889–894 (2003). 3. B. Potsaid, I. Gorczynska, V. J. 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Introduction
Fourier domain optical coherence tomography (FD-OCT) is a fast developing imaging technique which has already been demonstrated in a multitude of applications [1].In comparison with time domain OCT (TD-OCT), FD-OCT provides much better sensitivity [2] and therefore higher video rate image acquisition [3].However, despite its clear advantages over TD-OCT, FD-OCT presents several drawbacks.One of the main problems in FD-OCT is mirror terms [4].Irrespective of its sign, any optical path difference (OPD) value determines the same modulation of the channeled spectrum.When zero OPD is positioned within the sample, two images are then mirrored around zero frequency.Several methods have been devised for elimination of mirror terms [5], which can be split into two categories, cancellation and non-cancellation methods [6].Cancellation methods rely on the generation of the complex signal, where its imaginary component is inferred by producing a second interferogram shifted in phase by π/2.Various techniques have been employed to generate such phase-shifted interferograms.At least two acquisition steps are required to synthesize the complex signal and then use it to cancel the term corresponding to the sign of OPD, which is to be eliminated.Such methods allow correct reconstruction of layers in depth as well as double the axial OPD range.Examples of reported methods are based on phase shifting [6][7][8][9][10], frequency shifting [11,12], using a 3×3 splitter [13], the BM-scan technique [14] or based on a numerical approach [15].A detailed presentation of different solutions to attenuate mirror terms in FD-OCT is presented in [16].
A non-cancellation method consists in laterally shifting the two interferometer beams in their way towards the spectrometer [17], to introduce a delay larger than their correlation length after dispersion/diffraction [18], method based on a principle inspired by Talbot bands (TB) [19,20].Such a solution is better suited to moving samples as it is a one step procedure, in comparison with cancellation methods that require several measurements steps.The TB method does not require either, long run stability of parameters as necessary by methods where several measurements are collected simultaneously [9,13].
Another drawback of spectral domain OCT methods, including FD-OCT, is that signals coming from all depths are captured under the same focus adjustment.This is due to acquisition of signals from all depths at once, as in fact an advantageous property of spectral domain methods.
A Gabor-based fusion (GF) technique was suggested [21] in a FD-OCT system to collect several cross section OCT images with different focus adjustments in depth, followed by selection of the highest contrast part of each image corresponding to the depth of focus range completed with stitching of all such image parts into a combined final image.The final fused image exhibits improvement of depth sensitivity with preservation of transversal resolution along depth.However, such a method is limited to improving the sensitivity up to the level allowed by the sensitivity versus OPD in the interferometer, with a decay determined by the spectrometer resolution [22].
In the present study we suggest an improved GF method, where the peak of the sensitivity profile with depth is also shifted in depth, by implementing a TB configuration.In essence, the novel method proposed here consists in moving the two profiles, confocal, C(z) and contrast of Talbot bands, C T B , together along depth, ensuring that depth position of the two peaks of the two profiles coincide.
A set-up is investigated equipped with means to perform both types of adjustments.The focus position is controlled by moving a lens in the interface optics placed between the scanning devices and the object investigated.For TB adjustment, translation stages are provided to adjust the lateral gap between the beam from the object, the object beam, and a reference beam, split from the optical source.The two beams travel parallel to each other towards the spectrometer, here implemented using a diffraction grating.If the gap between the two beams is equal or larger than the beam diameter of the two beams, then the interference takes place on the linear CCD camera used in the spectrometer, and the contrast of spectral modulation, C T B , exhibits a dependence versus OPD, characteristic to TBs [23].TBs are obtained for one sign of OPD only, i.e.TBs do not exhibit mirror terms.The sensitivity profile C T B (OPD) presents a peak, which depending on the amount of gap introduced between the two beams, can be shifted towards deeper depths in the object imaged, in opposition to standard FD-OCT where the peak of contrast modulation is at OPD = 0 [18,23].For gaps larger than the diameter of the two beams, no mirror terms exist and the maximum sensitivity is achieved in the middle of the axial range of the profile of TB contrast versus depth.
In the present study, the gap, G, between the two beams is adjusted from zero to values less than the beam diameter.Mirror terms will still exist, but what is more important in the context of this study is that maximum sensitivity is adjustably moved along the axial direction in the OCT cross section image, as demonstrated recently [16].The position in depth of the maximum of C T B (OPD) is determined by the gap G, however the width of the profile is determined by the number of grating lines illuminated, number ultimately determined by the footprints of the two beams on the diffraction grating [24].In this study, we conveniently manipulate the widths of the two beams (and so the footprints of the two beams on the grating) to control the width of the C T B profile.When reducing the size of footprints however, larger size of the spot on the linear camera results (along the coordinate axis rectangular to that of diffraction direction), which leads to losses due to the limited height of the CCD pixels.To investigate this trade-off, four different spectrometer settings are evaluated, that determine four values for the width of the TB profiles, in order to illustrate the combined Gabor/Talbot method and how to make best use of it.
A special phantom that exhibits little attenuation with depth was devised in order to evaluate the combined method, to sample the sensitivity profile of each configuration with depth in at least 8 points.A schematic diagram of the TB/GF/FD-OCT system implemented is shown in Fig. 1.As broadband source, a superluminescent diode (SLD) with a central wavelength λ = 840 nm and bandwidth Δλ = 45 nm is used.The optical signal originating from the SLD is divided into reference and object arms by a directional coupler DC (splitting ratio 90/10).In the object arm the beam is collimated by a microscope objective MO and diverted towards a microscope objective MO1 by a galvanometer scanning mirror SM.The objective MO1 is attached to a translation stage TS1.This allows control of the focus position in the object arm.Backscattered light from the sample is collected and guided to the collimator L S .The optical signal in the reference arm is diverted towards a collimator L R (of same focal length as L S ) via dispersion compensating elements placed between objectives MO2 and MO3.The distance between MO2 and MO3 is adjustable to alter the OPD in the system.A second translation stage (TS2) enables the lateral shift of launcher L R to control the value of the gap G between the interferometer beams superposed via a beam splitter BS (splitting ratio 90/10), to produce Talbot bands.Within the spectrometer, a transmission diffraction grating (TG), 1200 l/mm, Wasatch Photonics, Logan, Utah was used.After the TG, the diffracted light propagates through through a focusing element, L of focal length 20 cm and is focused on a linear camera CCD1 (Aviiva M2 CL, 12-bit, 2048-pixel (each 14 × 14 μm in size)).As focusing element L, a curved mirror or an achromatic lens are used (sketched as a lens for simplicity in Fig. 1).A 2D camera (CCD2) is also used for spectrometer alignment and monitoring of the lateral gap G between the two beams.The system is usually operated at an exposure time of 50 μs, which determines an acquisition rate of 39 Hz of B-scan OCT images when 512 lines are used per frame.

Theory
The sensitivity profile V of a conventional FD-OCT system versus OPD is determined [13,19,20] by: where and where M is the number of active pixels of the linear CCD camera while λ and Δλ denote the central wavelength and the full width at half maximum (FWHM) of the optical source spectrum respectively.The visibility range is determined by the zero points of the sinc function in Eq.
(1), denoted as full field image range FFIR in Fig. 2. The C T B factor used in Eq. ( 1) is given by the shape and the size of power distribution within the footprint of each beam incident on the diffraction grating, the gap between the two beams and the grating periodicity.More exactly, the shape of C T B is given by the correlation of the power distribution within the two beams [24].For instance if the power distribution for both reference and object beams is top-hat and G = 0, then C T B is described by a triangle shape centered at OPD = 0.By increasing the gap G, the C T B profile can be shifted axially, where the maximum of C T B is modulated by the sinc profile.The width of the C T B varies with the diameter of the beams.
In [18] the extension and position of the C T B profiles were evaluated using a wave trains representation, connected to the parameters of the experimental set-up.Assuming two beams of diameters D, which cover each N diffraction grating lines, with top-hat power distribution within each beam, the C T B base is 2Nλ .When G = 0, the C T B profile covers a range of OPD ∈ (−Nλ , +Nλ ).If P denotes the number of grating lines within the gap G, the C T B profile is shifted along the OPD range, with the position of its maximum displaced to Pλ .
The number of grating lines excited by a beam of diameter D is given by: where β is the angle between the direction of the beams and the normal to the grating and a is the grating pitch.
In conventional FD-OCT, where individual B-scan OCT images are acquired, the C T B factor is adjusted wider than the sinc range.However, when performing Talbot bands FD-OCT imaging, the narrower the C T B , the better is the attenuation of mirror terms.In a recent study [16], a TB FD-OCT system was used with extremely wide C T B , where the emphasis was not on reducing the mirror terms, but on shifting the sensitivity of the FD-OCT set-up inside the sample, by increasing the gap between the two beams.In this study, narrower profiles of C T B will be used, comparable in width with the extension of the confocal profile of the interface optics, C(z).If good transversal resolution is needed, then high NA interface optics should be used.This would shrink the confocal profile C(z) width below the axial range of the FD-OCT set-up.This is why Gabor fusion method was proposed, as a solution to perform FD-OCT with a high numerical aperture (NA) objective [21].For each focus position, a B-scan image is acquired using a narrow confocal profile translated axially within the profile given by the product of the two factors in Eq. (1).In the present study, it is proposed that Gabor fusion is applied to B-scan OCT images collected via a TB configuration with axially shifted C T B profiles, where the shift is optimally adjusted for each focus position.If a C T B profile is employed, comparable in width with the confocal profile, then there are two benefits of such a combined method: (i) enhancement of sensitivity, as C T B peak can be translated axially by increasing the gap G between the two beams and (ii) attenuation of mirror terms, as a shifted narrow C T B will extend less into the OPD range of opposite sign.Such a method requires synchronous adjustment of focus position and of the gap, G, between the two beams, as manually performed here using two translation stages, TS1 and TS2.
In GF/OCT just a fraction of each B-scan contributes to the final image.This allows here utilization of a C T B profile narrower than the sinc profile.The C T B profile is made narrow by using small beam diameters.Smaller beam diameters present other advantages, related to the possibility of employing smaller size components and a configuration that could tolerate gap G values comparable to the beam diameter, without clipping the beams.It is important for the configuration to allow achieving gap values comparable with the beam diameters, as only for gaps larger than the beam diameters the mirror terms are totally eliminated.Reduction of beam diameters is achieved by reducing the focal length of the launchers, f.This, unfortunately, leads to lower spectrometer efficiency.In a spectrometer configuration with collimators L R , L S of focal lengths f R = f S = f , the system presents a magnification m, given as m = F/ f , where F denotes the focal length of the focusing element L (Fig. 1).The optical fiber core diameter d is projected on the camera with a spot size d = md .The shorter the focal length, f, of the collimator, the bigger the magnification and the spot on the camera.When the vertical size of the spot on the camera exceeds the height of the CCD pixel, energy is lost and the efficiency of the spectrometer is lower.As it will be shown in the following, the study we performed required that the magnification m exceeded the ratio of pixel size, 14 μm to that of the fiber core, 5.6 μm.

Optimal adjustment of parameters
Before experimenting with combining GF with TB into a system, it is important to point out the possible advantages of such a system relative to its limitations.Fig. 3 and Fig. 4 describe hypothetical relative adjustments of the sinc and TB profiles.In Fig. 3 left (G = 0), the TB profile is much wider than the sinc profile.This is the case largely used in the practice of FD-OCT.In this case, the overall sensitivity profile is dominated by the sinc profile and the axial range is determined by sinc zero, z 0 , i.e. by the number of pixels in the camera.A lateral gap between the two beams may create the TB behavior, but will shift the TB profile to small sensitivity values determined by the sinc profile.The gap cannot be larger than the sinc width.Even at this extreme value, as shown in Fig. 3 right, where the sensitivity is low due to low value of the sinc factor, the attenuation of the mirror term is minimal due to the wider wings of the C T B than the wings of the sinc profile.Confocal profiles are normally much narrower, selecting a fraction from the area of the overall profile in Fig. 3.In reference [21], such a profile had a width, which allowed 5 shifted positions along depth axis before reaching the zero sinc.In Fig. 4, a different case is shown, where the C T B profile is narrower than the sinc profile.In this case, the overall sensitivity pro- (C) 2012 OSA file versus depth is determined by the C T B profile.The axial range in this case is limited by the number of grating lines used.It makes sense in this case to introduce a gap between the two beams in their way towards the grating, as shown in Fig. 4 right, where for example, the gap was adjusted to half of the sinc zero, z 0 .The sensitivity profile is moved towards deeper axial positions by the amount of the gap [24].Sensitivity improvement is obtained, when comparing the sensitivity achieved in Fig. 4 at the depth corresponding to the maximum position of the C T B , with the sensitivity for the same OPD value in Fig. 3 above.This comes down to elimination of the attenuation due to the C T B profile, within the range limited by the sinc profile.
A second improvement is in the possibility to attenuate the mirror terms.As shown in Fig. 4, right, the C T B triangle has little value for negative OPD values.In other words, attenuation of mirror terms becomes feasible.
In the experiments that follow, beam diameters will be adjusted to four values and the experimental set-up will progress from a case as that described in Fig. 3 to the case described in Fig. 4.

Experimental results
For experimental verification of the combined TB/GF method, a special phantom was built.Transparent double layers of sellotape were separated by spacers, to create air gaps where no attenuation is exhibited.This phantom is characterised by a good penetration and in combination with the objective MO1 of focal length 5 cm in the object arm, allows acquisition of images from the whole range of OPD's in Fig. 2 (OPD range: -7.5 mm, 7.5 mm), although in all presented images only half OPD range is displayed.Due to the parallelism of layers within the whole phantom, B-scan images are tolerant to lateral displacement of the sample.This is important, as the process of adjusting the collimators require often replacement of the object with a mirror and it is necessary to ensure some consistency in the aspect of B-scan images from one set-up arrangement to the next.This also allows an easy comparison of the images captured with different set-up adjustments.
The width of the C T B profile was set to four different values, by choosing four different values for the focal length of L R and L S collimators: f = 75, 40, 30 and 18.24 mm.For the first three values, achromat doublets were employed, while for the shortest focal length, a Thorlabs fiber collimator using an aspheric lens was used.When using the largest beam diameter size, in order to avoid dispersion in the lens, L when introducing a gap between the two beams, the lens was replaced with a spherical mirror of similar focal length 20 cm (and an additional cylindrical lens was used to compensate for aberrations), as described in [16].
To illustrate the improvement of the combined method in comparison to the conventional GF/OCT method, two sets of images are presented for each set of focal lengths for the two collimators.First, a B-scan OCT image is created by GF/OCT using G = 0.The focus position, z f , is adjusted in five depth positions separated by δ z f .This leads to successive B-scans with sensitivity shifted in depth by δ z f due to focus change.Then, a stripe is selected from each B-scan, of δ z f width and a compound GF/OCT image is formed by stitching the five stripes together, using a procedure similar to that described in [21].
Second, a TB/GF/OCT B-scan image is obtained using Talbot bands.To this goal, G is adjusted in steps δ G, to move the maximum of the C T B profile to coincide with the focus position determined by the confocal profile.Using a similar equation as Eq. ( 3), the step in G gap values, δ G, to shift the maximum of the C T B profile by δ z f should be: where the index of refraction of the phantom was approximated as that of the air, due to the large distance between the sellotape layers in comparison with their thickness.For δ z f = 0.6 mm, this gives a step δ G = 0.513 mm.However, for the cases presented in what follows, G was optimized experimentally to enhance the part of the B-scan corresponding to the middle of deeper stripes.The value of G is monitored using the 2D CCD camera, CCD2.In this way, five successive B-scans with sensitivity shifted in depth by the compound effect of moving both confocal, C(z) and C T B (z) profiles are obtained.Then, a compound TB/GF/OCT image is formed by stitching together the five stripes of δ z f width, each obtained for a set of pairs of G and z f values.All B-scan OCT images have a lateral size of 3.8 mm and an axial range of 3.45 mm measured in air.The process of focus adjustment introduces some deviation of the surface of equal OPD from a plan surface.For the small value of the lateral size used, the deviation of bright lines displaying the layers from vertical straight lines was insignificant.
In Fig. 5, images are shown for collimators having a focal length f = 75 mm.Figure 5(a) left shows a GF/OCT image captured for G = 0.In order to evaluate the profile of sensitivity with depth in the B-scan image, an inferred A-scan intensity profile is produced in Fig. 5(a) right, by averaging multiple adjacent A-scans (512) in the B-scans for each pixel along the lateral side (vertical coordinate).Figure 5(b) left shows the recorded B-scans for different pairs of gap G and focus, z f values.Parts which were sampled out to be used in the construction of the final TB/GF/OCT image in Fig. 5(c) are bordered by green line.Because all essential reflections in the phantom are up to 3 mm and it was decided to perform GF with 5 stripe images, each should have had a width of 0.6 mm.After the depth of 3 mm there are no more layers to show, therefore the last stripe extend from 2.4 mm to 3.45 mm, while all previous four stripes have a width of 0.6 mm.
When using collimators of focal length f = 75 mm, the beam diameter is rather large, of D = 18 mm and therefore it is not possible to achieve a sufficient gap between beams whithout clipping them by different mounts and supports in the system.Consequently the gap adjustment range was limited, some visibility improvement can be seen, but this is minimal.Initially, with the focus z advanced to the last stripe in Fig. 5(b), the gap G was experimentally optimised to achieve the best B-scan visibility.For this particular case, the values for G used to improve sensitivity at OPDs corresponding to parts 2, 3, 4 in the other B-scans in Fig. 5(b) were inferred from that at this 5th position applying a linear scale.
Using D = 18 mm, a = 0.83 μm and an angle β =30 o in Eq. ( 3), N = 25000.With such a large number of grating lines, the approximate width of the C T B factor is 2Nλ = 42 mm, much larger than the OPD width of the sinc factor, of 4z 0 = 15 mm.Even when considering the FWHM of 10.6 mm of each beam instead of their diameter D, the width in depth, Nλ /2 = 6.18 mm is still larger than the sinc width z 0 = 3.75 mm.In reality, the power distribution with the beam section is Gaussian and the C T B factors are even narrower [24].This extreme case was included here as a typical example for the adjustment illustrated in Fig. 3 where the sinc profile is narrower than the C T B profile.The decays of sensitivity with depth for such large diameter beams were presented in detail in a previous report [18].As expected from comments made on Fig. 3, minimal improvement is obtained in the TB/GF/CT image in comparison with the GF/OCT image in Fig. 5.All the other three cases illustrated below will correspond to cases where the C T B profile is similar in width (f = 40 mm), slightly narrower (f = 30 mm) and finally narrower than the sinc profile (f = 18.24 mm), with the last case typical for the case illustrated in Fig. 4, more amenable to combining the GF and TB methods.
In the following images, for other three values of f, the principle illustrated in Fig. 5(b) will be used and therefore, equivalent images will not be shown.Fig. 6 and Fig. 7 show results obtained using focal length values of f = 40 mm, and respectively f = 30 mm for the collimators L S and L R .The whole procedure was finally repeated once more for smaller beam diameters, produced by using Thorlabs fiber collimators f = 18.24 mm with results shown in Fig. 8. Fiber collimators provide narrower beams (of diameter D = 6.2 mm) that should determine narrower C T B profile.This last case ideally corresponds to that described in Fig. 4, where the axial range is determined by the C T B profile instead of the sinc profile.For all these three cases, with smaller beam diameters, no clipping took place even at large G values.We therefore show in the top right hand side of each figure, the spots on the grating, as imaged by CCD2, for the five cases of G adjustment in positions (a), (b), (c), (d) and (e).On the top of each figure in its left hand side we show for each G value the experimentally determined sensitivity profile, as described by Eq. ( 1), measured using a mirror as object, with focus z f fixed, by altering the length of the reference path only.These curves illustrate the decay of sensitivity with depth and the axial shift of sensitivity with G.While in Fig. 6 and Fig. 7 these curves are the resulting combination of both C T B and sinc factors in (1), in Fig. 8, where the sinc profile is wider than the C T B , the curve for G = 0 can be mainly assumed as due to the C T B profile.Then, by increasing G, the sinc profile intervenes more in attenuating the compound result.
Improvement in the TB/GF/OCT images in Fig. 6, Fig.This case may not be far for the GF method alone, as the C T B profile for the case in Fig. 8 was made deliberately narrower than the sinc profile, and most of the decay of sensitivity with depth in the fused B-scan in Fig. 8 middle left where only GF was applied, is due to C T B .In the practice of GF used alone, a case as that described in Fig. 3 would be used, with wider C T B than the sinc profile.The B-scan image obtained by GF alone in Fig. 5(a) would correspond to such a case.However, even when comparing the A-scan profiles from the TB/GF/OCT image in Fig. 8 bottom right with the A-scan profiles from the GF/OCT image in Fig. 5(a) right, the improvement at deep layers is larger than a factor of three.The ratio of amplitudes of layers evaluated around 2.75 mm to the amplitude of layers close to zero depth is approximately 0.3/2.5 in the B-scan GF/OCT image in Fig. 5(a) left and approximately 1.1/2.5 in the B-scan TB/GF/OCT image in Fig. 8 bottom left.This first improvement of TB/GF configuration comes together with a second improvement, that of narrower C T B profile which allows attenuation of mirror terms, as illustrated in Fig. 4.This is not quantified here, as it was presented extensively in [17] and [23] and can be estimated from the sensitivity profile with depth in the GF/OCT A-scans presented in the top left part of Figs.5-8.
By changing the focal length of the collimators L S and L R , the width of the C T B profile is altered while the axial position of its peak is dependent on G. However in practice we noticed deviations of the experimentally found G values from the values expected.For instance, irrespective of the focal length of the collimators L S and L R , any axial shift value for the C T B maximum should be obtained for the same value G.In practice however, different G values were obtained for different focal length used, f.This may be due to the misalignment of beams.The experimental results and expected values calculated using Eq. ( 4) are presented in Table 1.Superposition of beams corresponding to gap G = 0 mm on CCD2 does not necessarily ensure perfect superposition of the directions of propagation of the two beams.It was found experimentally difficult to make the two beams perfectly parallel.We have also found that small angle deviations of the two beams after BS lead to important variations of the decay of sensitivity with depth.Because both beams travel through the spectrometer together along an optical path exceeding 50 cm, just small inaccuracy in beams superposition can lead to differences between experimentally and theoretically obtained values for the peak of sensitivity position with depth for varying G values.As far as the results for the smallest focal length collimators are concerned, as they are aspheric, more aberrations are expected than with the longer focal achromats that may determine some skew in the optical power distribution of the footprint on the diffraction grating.This may explain the deviations of experimentally found G values in columns 4-7 in Table 1 from the theoretical values in the second row.
The narrower the C T B profile, the larger the attenuation of mirror terms.Narrower profiles C T B are obtained by reducing the number of grating lines, N, illuminated, achievable by reducing the beam diameter of the beams.This leads to reduction in the length of the wave trains after diffraction, approximated as 2Nλ .However, reduction in the beam diameters comes at a price, as this is achieved by reducing the focal length, f, of collimators L S and L R that translates in magnified size of the fiber core on the CCD camera.The immediate consequence is reduction in the spectrometer efficiency.To illustrate this disadvantageous effect, the last column in Table 1 shows the saturation power for each set of values for the focal length of collimators.This problem may be alleviated by using linear cameras with larger height (InGaAs cameras from Goodrich and Hamamatsu are already available with 500 μm height pixels).If larger height pixel size linear cameras are used, then the C T B profile can be made as narrow as the confocal profile without incurring power losses.
The data in columns 4-7 in Table 1 show: in the 2 nd row, theoretical values for the gap G, inferred by using Eq. ( 4); in the next rows, experimentally determined G values.
Third column presents estimation of the C T B widths theoretically estimated by considering top hat distribution of power within each section of the beam [18,24], extending up to the size as experimentally determined from the CCD2 camera, where the power in the beam profile reduced to half, values for the diameter of the two beam footprints mentioned in the second column as FWHM.The values are useful in deciding to which case, Fig. 3 or Fig. 4, each experiment for a different f value corresponds to, by comparing them with the zero of the sinc factor, z 0 = 3.75 mm.The values in this column for f = 75 mm and for f = 40 mm correspond to the case in Fig. 3, while the remaining two values for f = 30 mm and f = 18.24 mm to the case illustrated in Fig. 4.This association still holds if more realistic values are inferred for the extension of the C T B profile from experiments.For example, A-scans in the top of Figs.6-8 exhibit 20 dB attenuation for G = 0 at 2.6 mm, 2.1 mm and 1.55 mm for respectively f = 40 mm, 30 mm and 18.24 mm.
As commented above, it can be assumed that the first two values are the result of decay with depth in both factors in Eq. ( 1), while the third value is mainly determined by the decay in the C T B profile.When these values are compared with z 20dB = 2.75 mm where the sinc profile in Fig. 2 reaches 20 dB attenuation, again the same association holds, of f = 75 mm and f = 40 mm cases to the case described in Fig. 3 and of f = 30 mm and 18.24 mm to the case described in Fig. 4.These experimental values are smaller than those expected when using top hat beams and do not scale proportionally with the FWHM values of the beams in column 2. In addition to the difference in shape distribution of power, other experimental factors should be considered such as aberrations and dispersion encountered by the two beams in their way towards the grating.These factors may lead to a spread of the N diffracted wavelets within each wave-train diffracted beam with the consequence of diminished overlap of the two wave-trains [18] and quicker decay of the C T B profile than anticipated from the number of grating lines involved.
The 8 th column in Table 1 represents the height of the diffracted line on the CCD1, evaluated using the magnification of the single mode fiber core due to the ratio of focal lengths of focusing element L and that of collimators L S and L R .
The last column in Table 1 gives the experimentally measured optical powers required to saturate the camera in the spectrometer.

Conclusion
We have presented a combination of Gabor-based fusion technique with Talbot bands principle, applicable to a FD-OCT system.By using Talbot bands, the peak of visibility with depth can be moved to larger depths in synchronism with the movement of the confocal profile peak of the interface optics.In a recent report, Gabor based synthesis of B-scan OCT images was implemented for five positions of the focus [21].Here, the same number of five adjustments is made with experimental results presented for four different beam diameters.In each case, comparison is made between the contrast in the synthesized image obtained with G = 0 (corresponding to standard FD-OCT) and the contrast in the synthesized image obtained with progressive advancement of the gap G together with the focus in each of the four images with focus moved.The final fused B-scan OCT images demonstrate better contrast at larger depths when using the Talbot bands configuration.
An even more complex adjustment is possible (not demonstrated) where the width of the two beams is also adjusted together with the gap.This would require synchronous adjustment of focus position and of the gap, G, between the two beams.The two adjustments were made here manually using two translation stages, TS1 and TS2.In practice however, such adjustments can be made automatically.To do this, the average index of refraction of the object is required to estimate the movement of the confocal profile along depth.Once this is known, a suitable gap is introduced between the two beams that shifts the TB maximum position to coincide with that of the focus.Such a method if applied, can further attenuate mirror terms and reduce intensity of signal from superficial layers.The reduction of efficiency with the reduction of focal length of collimators, as shown by the power level required to saturate the camera CCD1, is correctable by using larger height pixel size in the linear camera, a trend followed by manufacturers of linear cameras for spectrometers to be used in FD-OCT.
Deliberately here, the confocal profile C(z) was devised sufficiently wide in comparison with the sinc factor to allow evaluation of axial shifts of the C T B profile in a proof of concept combination of GF and TB methods.This is because even the narrowest C T B profile here, for f = 18.24 mm, is rather wide (1.55 mm for 20 dB attenuation, i.e. the profile is larger than the width of two stripes).Obviously, the confocal profile can be made as narrow as the width of the stripe sampled out from each B-scan using higher NA MO1 objectives, to take full advantage of the GF [21] method in achieving better transversal resolution.In practice, a larger number of stripes can be used with narrower and matched profiles for both C(z) and C T B (z) in which case, better synchronization of G and z f steps will be required.

Fig. 2 .
Fig. 2. Red: Normalized sensitivity profile of sinc function (M = 1024 pixels, λ = 0.840 μm, Δλ = 0.045 μm, OPD range -7.5 mm, 7.5 mm, that determines a maximum axial range in depth, measured in air of z 0 = 3.75 mm).FFIR marks the axial range of the FD-OCT system.OPD 20dB = 2z 20dB = 5.5mm marks the OPD value where the sensitivity due to the sinc factor reduces by 10 times.All other triangular curves show the theoretical C T B factor for equal beam diameters of 10 mm and top-hat profiles for the power distribution within their section, for different gap values G.When the triangle base becomes narrower than the sinc factor, the FFIR of individual A-scans becomes smaller.

Fig. 3 .
Fig. 3. Usual adjustment of parameters in FD-OCT, where the C T B profile is wider than the sinc profile.Left: G = 0; Right: G = z 0 .

Fig. 4 .
Fig. 4. The case of C T B profile narrower than the sinc profile.

Fig. 5 .
Fig. 5. Collimators of focal length f = 75 mm.(a) GF/OCT image (left) obtained by shifting the focus into 5 equidistant depths as described in the text; (a) right: inferred A-scan intensity profile, obtained by averaging multiple adjacent A-scans in the B-scan on the left, for each pixel along the lateral side (vertical coordinate), (b) B-scan OCT images with sensitivity shifted in depth obtained by introducing Talbot bands and changing the focus in the object arm; (c) Left: TB/GF/OCT image obtained by stitching together the bordered parts within the green rectangles superposed on the images in (b); Right: corresponding inferred A-scan intensity profile.

Fig. 6 .
Fig. 6.L S and L R of focal length f = 40 mm: Top left: A-scan curves for gap G = 0, 0.94, 1.24, 1.73, 2.36 mm using a mirror as object.Top right: CCD2 images illustrating the amount of overlap of the two beams adjusted by the gap value, G. Middle left: GF/OCT image; Middle right: Average of A-scan profiles over the B-scan GF/OCT image on the left; Bottom left: TB/GF/OCT image: Bottom right: Average of A-scan profiles over the B-scan TB/GF/OCT image on the left.

Fig. 7 .Fig. 8 .
Fig. 7. L S and L R of focal length f = 30 mm: Top left: A-scan curves for gap G = 0, 0.38, 1.88, 2.84, 3.1 mm using a mirror as object.Top right: CCD2 images illustrating the amount of overlap of the two beams adjusted by the gap value, G. Middle left: GF/OCT image; Middle right: Average of A-scan profiles over the B-scan GF/OCT image on the left; Bottom left: TB/GF/OCT image: Bottom right: Average of A-scan profiles over the B-scan TB/GF/OCT image on the left.

Table 1 .
Theoretical and Experimental Parameters of the Set-up for Four Values of the Focal Length of the Collimators Launching the Two Beams Towards the Grating