## Abstract

We present a theoretical model to predict the sensitivity variation versus optical path difference (OPD) in Fourier domain spectral interferometry using configurations which produce Talbot bands. Such configurations require that the two interfering beams use different parts of the diffraction grating in the interrogating spectrometer. So far, the power distribution within the two beams in a Talbot bands experiment was considered uniform. In this report, we show that by manipulating the power distribution within the two interfering beams, the OPD value where maximum sensitivity is achieved can be conveniently tuned, as well as the sensitivity variation with OPD. Furthermore, creating a gap between the two beams leads to adjustment of the minimum detectable OPD value, while the width of the beams determine the maximum detectable OPD value. These features cannot be explained by theoretical models published so far involving spectrometer resolution elements only, while such features are correctly predicted by the model presented here.

© 2008 Optical Society of America

## 1. Introduction

Talbot Bands describe a “curious” effect, discovered by H.F. Talbot in 1837 [1], when a beam from a white light source was analyzed spectrally using a prism. By inserting a glass plate halfway into a certain side of the beam only, dark bands appeared in the spectrum. More exactly, if the plate was inserted halfway into the beam from a direction corresponding to the red end of the dispersed spectrum (the side of the beam which passes through the thinner part of the prism), dark bands were observed (Talbot Bands). If the plate was inserted from the opposite side of the beam (corresponding to the blue end of the dispersed spectrum), no bands were observed [2].

G. B. Airy dispelled any “curious” effects as mere results of interference which enhances the amplitude of some wavelengths and reduces the amplitude of others [3].

The density of bands is proportional to the thickness of glass and its index of refraction. As another property, Talbot bands exhibit a symmetric-triangular relationship between band visibility and plate thickness, a maximum visibility is observed at a thickness dependent on the beam width and the visibility goes to zero near a plate thickness of zero and also as the thickness exceeds twice the value at which the maximum occurs [4,5].

This behaviour can be explained using the configuration shown in Fig. 1. The beam from a broadband optical source is collimated by a lens CL and directed towards a spectrometer consisting of a diffraction grating in transmission, DG, and a spectrometer lens, SL, with the spectrum inspected visually. For illustration, we use normal incidence upon DG which makes the incidence angle zero.

If the diffraction maximum marked as 1 is observed, Talbot bands are obtained when a microscope slide is inserted in either of the positions A, A‵ or A‵‵. These correspond to interception of that part of the incoming beam situated towards the red side of the diffracted spectrum. If the microscope slide is inserted from the other side, no bands are obtained. If the observer looks into the order of diffraction marked as -1, then bands are obtained when the slide is placed in positions opposite to those shown at A, A‵ and A‵‵.

Let us consider a beam width which covers 2N grating lines and a temporal view of the partially coherent light as femtosecond pulses. Each diffraction grating slit generates a replica of the incoming pulse (wavelet). The incident beam at DG can be considered as an advancing front comprised of 2N simultaneously-emitted pulses, one pulse for each grating line [6,7,8]. Following diffraction, the advancing front of pulses is subject to discrete delays which are linearly dependent upon the transverse position of each slit within the beam. For a particular diffraction angle β, the spatial delay encountered by the pulse incident upon the most extreme grating line toward the blue side of the diffracted spectrum in comparison with that incident upon the most extreme grating line toward the red side of the diffracted spectrum is Ω=2Ndsinβ, as shown by the thick segment in Fig. 1. Here d is the grating period.

If we consider an imaginary line from CL separating the incoming beam into a left hand side and a right hand side, then the beam on the left of Fig. 1 will feed the part of the diffracted beam towards the blue side of the diffracted spectrum, while the beam on the right will feed the part of the diffracted beam towards the red side of the diffracted spectrum. We therefore call these two half-sides of the incoming beam and the two parts of the grating that they are incident upon the blue side and the red side respectively. Obviously, if the diffraction order denoted as -1 is used in Fig. 1, then the left side of the beam becomes the red side and the right side becomes the blue side.

Let us consider that the glass plate is introduced halfway into the beam in position A. To distinguish between the two halves of the beam created by the plate, we will call the half-beam delayed by the glass slide in Fig. 1 the “object” beam and the undelayed half-beam the “reference” beam. For the case shown, the object beam coincides with the red side and the reference beam with the blue side of the incoming beam. If the other diffraction order is observed, the spectrum is reversed and the object beam falls into the blue side and the reference beam in the red side of the incoming beam.

It follows that a compounded wavetrain can be considered for each of the diffracted beams, consisting of N pulses diffracted from N grating lines. Due to the differential delay from each individual pulse to the next, the resulting diffracted wavetrain in each half-side of the beam is much longer than the length of the incoming pulse [4]. For the first order of diffraction, the length of each diffracted wavetrain is therefore approximately Ω/2=Ndsinβ, which, using the Bragg equation, gives ~Nλ_{0}, with λ_{0} the central wavelength.

For example, let us say that the white light source emits at λ_{0}=0.7 µm, has a coherence length of 5 µm and the incoming beam covers 2N=2000 grating lines. In this case, each diffracted wavetrain consists in N=1000 individual pulses (wavelets) in each half side of the beam in Fig. 1. Each diffraction grating slit generates a replica of the incoming pulse with a spatial extention of ~5 µm. Within the diffracted beams, wavelets diffracted from successive grating lines are spatially delayed by ~0.7 µm in relation to each other and the total wavetrain length is approximately ~Nλ_{0}=0.7 mm.

From the presentation above, we distinguish two types of delay between the arrival of the two diffracted wavetrains at the observation plane Π. The delay due to the insertion of a glass plate into the beam is termed as “extrinsic” while the delay introduced upon diffraction, Ω/2, is termed as “intrinsic”. The theoretical study in [8] put forward an heuristic explanation of the band visibility as a result of the coherence length of the diffracted wavetrains. This lead to the conclusion that the Talbot Bands’ visibility depends on the amount of temporal overlap of the two diffracted wavetrains. When no glass plate is inserted into the beam, there is no extrinsic delay between the diffracted wavetrains in the two half sides of the beam in Fig. 1. However, the two diffracted wavetrains always exhibit an intrinsic delay. They are spatially displaced by their length, Nλ_{0} and therefore in the absence of an extrinsic delay there is no overlap, no interference takes place and so no bands are observed. More exactly, looking at the diffraction order 1 in Fig. 1, the diffracted wavetrain in the blue side lags that of the diffracted beam in the red side by an intrinsic delay of Nλ_{0}. Therefore, to overlap the two diffracted wavetrains [4,7,8], the diffracted wavetrain in the red side has to be extrinsically delayed in relation to the diffracted wavetrain in the blue side. This is obtained by inserting the glass plate in one of the positions A, A‵ or A‵‵ shown.

In other words, to produce Talbot Bands an extrinsic delay is introduced so as to compensate for the intrinsic delay [8]. When the extrinsic delay matches the diffracted wavetrain length, the two diffracted wavetrains are totally overlapped and Talbot Bands’ visibility is maximum.

If a glass plate which produces an extrinsic delay twice the value of the intrinsic delay is used in the same position A, the two diffracted wavetrains are again delayed in respect to each other by their length and no bands are observed. Although, this time the diffracted object wavetrain (from the red side of the grating) lags spatially behind the diffracted reference wavetrain (from the blue side of the grating) by Nλ_{0}.

If however a glass plate is inserted into the opposite half-beam in Fig. 1, this pushes the diffracted wavetrain from the blue side even more behind the diffracted wavetrain from the red side. In other words, in can be noticed that: (i) Talbot bands are not produced when the extrinsic and intrinsic delays add up leading to spatial separation of the two diffracted wavetrains and (ii) Talbot bands are produced when the intrinsic and extrinsic delays cancel each other.

For the order of diffraction 1 in Fig. 1, the reference beam is incident on the blue side of the diffraction grating and the object beam is in the red side of the diffraction grating. If the opposite order of diffraction is used, -1, then the reference beam is in the red side and the object beam in the blue side of the grating. This explains why, when the other order of diffraction is used, the glass plate has to be introduced into the opposite side of the beam to obtain Talbot Bands instead of the positions indicated by A, A‵ and A‵‵.

The properties of Talbot bands mentioned above make them ideally suited to Fourier Domain-Optical Coherence Tomography (FD-OCT), as will be explained below, a possibility which went un-noticed so far.

We have synthesized an experiment [9] to obtain Talbot bands using a modified Michelson Interferometer with opaque screens used to obscure half of each interferometer beam. A grating-CCD spectrometer was used to observe the spectrum channelling. Our experiments showed that several profiles of channelled spectrum strength versus the optical path difference (OPD) similar to those characteristic to Talbot Bands can be produced.

Talbot bands have been generally reported using white light sources. The theoretical model in [10] has shown that the visibility profile of the bands when using a narrower linewidth source deviates from the triangular shape towards a rounded profile, however with similar maximum and zeroes.

Profiles of visibility closer to Gaussian shape have been reported in recent papers on FDOCT [11,12]. FD-OCT and spectral radar are forms of spectral domain-white light interferometry (SD-WLI). Both SD-WLI and FD-OCT refer to Fourier transformation of the photo-detected signal delivered by a spectrometer at the output of a low coherence interferometer [13]. The spectrum exhibits peaks and troughs in the form of a channelled spectrum [14] and the frequency of such a modulation is proportional to the modulus of the OPD in the interferometer. SD-WLI has largely been developed for sensing applications [15,16] long before the successful demonstration of FD-OCT. FD-OCT is attractive because it eliminates the need for depth scanning which in time domain OCT is usually performed by mechanical means.

FD-OCT generally employs relatively low coherent sources with bandwidths which are a fraction of the bandwidth of white light sources; the resulting visibility exhibits a Gaussian profile the width of which is inversely dependent on the resolution of the spectrometer [11, 12].

Particular to all FD-OCT set-ups so far has been that the Gaussian visibility profile has a maximum sensitivity at an OPD of zero which gradually reduces for larger OPD values of either sign. As identical channelled spectra are generated by reflectivities of the same magnitude originating at depths of similar OPD but of a different sign, a symmetric sensitivity profile creates ambiguity when using FD-OCT to image tissue *in-vivo*. This is in general referred to as to the problem of mirror terms or mirror images [17]. In [8] we have shown that the problems of mirror terms can be addressed by using optical configurations which implement Talbot band behavior. One such set-up is that of a modified Michelson interferometer with opaque screens introduced in [9]. However, such set-up is inefficient due to power loss created by using screens. Therefore, in a letter [18] we demonstrated an optical set-up with greater power efficiency to that presented in [9], where fiber optic leads are used to send the object beam and reference beam separately to the diffraction grating. We reported again selection in the OPD sign as predicted by [8] and the theory in [10] and demonstrated that a Talbot bands arrangement of beams has potential in successfully eliminating the occurrence of mirror terms and improve the sensitivity decay with OPD. However, the letter stopped short of discussing the visibility profile dependence on the gap between the two beams sent to the diffraction grating.

As another limitation of previous reports, the theoretical models in [8, 10] considered all gratings lines equally excited, i.e. the power distribution within the beams incident on the diffraction grating was considered as described by a top hat profile. This resulted in a smoothed triangular profile with OPD in both Talbot cases and FD-OCT set-ups.

The present study addresses the limitations of our previous studies. We evaluate theoretically the consequence of power distribution within the beam profile along a direction across the grating facet, perpendicular to the grating lines. In addition to giving a more accurate account of the visibility profile, which matches the reality better, the model presented in this study predicts the interesting result of tuning the minimum of OPD range depending on the gap between the two beams, object and reference as well as suggests ways to improve the visibility profile in practice.

First, we extend the study performed in [9, 10] for cases where one or both beams in a modified Michelson configuration are obstructed using screens, by considering an arbitrary proportion of obstruction in each beam. Top hat profiles are considered for the two beams. We then extend the study to more practical cases of Talbot bands configurations designed for improved power efficiency such as the FD-OCT set-up reported in [18]. In all cases studied, we obtain the minimum and maximum OPD range, selection in OPD sign and predict the profile of visibility. We then focus our attention on improving the visibility profile and propose to redistribute the power within the section of the two beams to achieve faster rates in the visibility variation close to the minimum and maximum of OPD. If this could be made possible, then a more uniform variation of visibility within a wider OPD range results. Such study will prove valuable in improving the sensitivity of FD-OCT set-ups, intensively researched in the last three years. It is known that the decay of sensitivity with depth is an important disadvantage of FD-OCT in comparison with time domain (TD)-OCT. Addressing this issue will allow FD-OCT developers to take full advantage of the superiority of FD-OCT in comparison with TD-OCT in terms of signal to noise ratio.

## 2. Optical Configurations

The present study refers to a general configuration of SD-WLI implemented in a two-beam interferometer with a diffraction grating based interrogating spectrometer with a CCD-array detector. For the particular orientation of the grating in Fig. 2, the red part of the spectrum is diffracted up and the blue part of the spectrum down. Rays are shown for the central part of the spectrum as focused on the center of the CCD.

Essential for what follows is that the two beams coming from the interferometer may be spatially separated before entering the spectrometer. As shown in Fig. 2, the two beams excite different lines of the diffraction grating in the spectrometer. Beam 1 excites m lines counted from 1 to m and beam 2 excites M lines counted from m+h+1 to m+h+M. The un-illuminated gap between the beams cover h lines, where h may take any integer value between h=-m (a negative value signifying that the beams illuminate a number of common grating lines) and an upper limit set by the number of available grating lines. As it will be demonstrated below, the gap determines the minimum value of OPD which results in spectrum channelling. This is a feature not predicted by any previous reports on the visibility profile [11,12]. Fig. 2 also illustrates the origin of Talbot bands visibility at the core of the explanation put forward in [8], i.e. the amount of overlap of the two diffracted wavetrains.

Two cases can be defined:

#### 2.1 Beams totally superposed:

The same grating lines are excited by the two beams if h=-m and M=m. This configuration corresponds to those previously reported in SDWLI [19–23] as well as in FD-OCT imaging [11, 25–29].

#### 2.2. Beams separated on the grating:

By manipulating the value of m, h and M we can simulate all diversity of configurations, such as the classical Talbot case using a microscope slide [1–7], screens in a bulk Michelson configuration [9,10] and efficient FD-OCT [18] using the same hypothetical optical set-up based on the spectrometer configuration in Fig. 2.

### 2.2.1 Beam 2 intrinsically delayed

In this case beam 2 (profile B_{2} in Fig. 2) has an intrinsic delay with respect to beam 1. In the following we will employ the same terminology as that introduced in Fig. 1. The delay due to the OPD in the interferometer replaces that introduced using the microscope slide in the Talbot bands experiments and will be similarly termed as extrinsic. Depending on the number of lines covered by the beam incident on the diffraction grating, a cumulative intrinsic delay is built up between the most extremes rays within the transversal section of the beam. Similarly, if the two beams are spatially displaced by a number of lines, a cumulative intrinsic delay can be evaluated between the two beams. The first ray, m+h+1 in beam 2, incurs an intrinsic delay of Ω=p[d(m+h)(sinα+sinβ)] with respect to the first ray, 1, in beam 1, where p is the diffraction order, according to Eq. (2) below.

For partial superposition, -m≤h<0. For h≥0, the object and reference beams illuminate different sets of grating lines.

### 2.2.2 Beam 1 intrinsically delayed

Beam 1 is “intrinsically” delayed more than beam 2: In this case, the two beams swap positions on the grating in Fig. 2. All results can be obtained from case 2.2.1 above for the same intrinsic delay but of opposite OPD sign, according to the discussion in the introduction.

Configurations covered by cases 2.2.1 and 2.2.2 produce Talbot Bands. For instance, the classical case of Talbot bands observed using a microscope slide inserted into the beam sent to a spectrometer [1] is characterized by h=0 and the proportion of lines intercepted out of a constant m+M is altered by moving the slide laterally into the beam. In [9], using screens, m+M=constant, however a gap can be introduced by inserting one of the screens more than halfway into one of the beams.

In a more power efficient scheme for FD-OCT [18], m+M is still constant, but the grating line which generates the wavelet with the largest intrinsic delay, either m=1 or that at the other edge of the grating, m+M+h, could be conveniently changed, as well as the gap, h, without affecting the numbers m and M by simply translating fiber ends.

## 3. Theory

We follow similar steps as those in [8] and [10]. The field diffracted at an angle β is described by:

where

is the differential delay between rays diffracted by two adjacent grating lines, and d is the period of the grating. α and β are the incident and diffraction angles, both measured counter-clockwise from the grating normal (i.e. in the example presented in Fig. 1, β has a negative value).

The source optical field is described by:

where $\overline{\lambda}$
represents the wave-number and Y($\overline{\lambda}$
) the optical spectrum. Z_{1} and Z_{2} represent the effect of the diffraction grating upon each beam, as shown in [10].

In [8] we included a phase factor in one of the terms in Eq. (1) to account for the intrinsic delay introduced by the grating between the two diffracted beams. To preserve generality, the phase difference from one line to the next irrespective of the number of excited lines and of the excitation profile is now included in both Z_{1} and Z_{2} as:

where B_{1}(r) and B_{2}(r) represent the irradiance at grating line r due to beams 1 and 2 respectively, as shown in Fig. 2. Factors in (4a) and (4b) considering diffraction on the individual slits have been dropped as they are the same for both Z_{1} and Z_{2} [10].

L_{1} and L_{2} in Eq. (1) describe the effect of refractive index perturbation in the two arms of the interferometer. For the purpose of this study, we shall assume both L_{1} and L_{2} to be constant at all $\overline{\lambda}$
. L_{1} and L_{2} incorporate factors describing the extrinsic delays t_{1}, t_{2} created in the interferometer. Hence

where the differential delay,

and T_{1} and T_{2} describe the attenuation via the two interferometer arms and incorporate the reflectivity of the mirror in the reference arm and, of the sample in the object arm respectively. For multiple layer objects in the object arm, L_{2} incorporates several exponentials, each multiplied with the Fresnel reflection coefficient for that layer.

The irradiance at the face of the CCD is proportional to:

where the asterisk (*) signifies the complex conjugation operation. Eq. (7) represents a more general version of (13) in [8]. In what follows, we will show that the general expression (7) explains the Talbot bands behavior in classical set-ups (as per its original discovery [1]), the Talbot bands behavior in the Michelson interferometer with screens [9, 10] and the Talbot bands in the power efficient FD-OCT set-up as described in [18]. Eq. (7) also covers the classical case of FD-OCT set-ups where the two beams are superposed on each other in their way towards the diffraction grating [11,12,24–29]. In addition, Eq. (7) predicts some novel effects. One such effect consists in a range of OPD values where the sensitivity is zero. As another novel feature, Eq. (7) presents a route for improving the sensitivity decay with OPD as explained below.

Equation (7) describes sums of products of fields at grating lines, where:

Equation (8a) represents cross-correlation between the object and reference beam profiles. The other two coefficients in Eq. (7) represent the auto-correlation of each beam profile:

So far, in previous reports [8, 10], the excitation of grating lines was supposed uniform and a single definition for all coefficients (C_{1}, C_{2} and C_{12}) was used in Eq. (13) of reference [8]. The use of functions C_{1}, C_{2}, and C_{12} now facilitate the representation of the auto- and cross-correlation functions separately, so that each beam profile and its relative position on the grating can be represented independently. In this representation only C_{12} can be used to represent the amount of overlap of the two diffracted wavetrains.

Following the same steps as in the Appendix A in reference [10], which involve two Fourier transformations, the CCD photocurrent is written versus the angle β (or D according to Eq. (2)). These steps involve the source autocorrelation envelope:

obtained as a Fourier transformation of the power density spectrum |Y($\overline{\lambda}$
)|^{2} where σ is related to the FWHM of the source spectrum [8], [10]. Introducing the notation:

Eq. (7) becomes:

The amplitude autocorrelation function varies quickly with its argument for a low coherent source, the larger the source bandwidth, the narrower the g(x) profile. Therefore, the most relevant terms contributing to the channeled spectrum in Eq. (11), for a given OPD value, are those with zero or small argument of the correlation function g. Argument zero for g happens when:

Equation (12) illustrates the combination of extrinsic and intrinsic delays according to the comment in the introduction. When Eq. (12) is combined with the range allowed for index s in the last sum in (11), it gives the range of OPD within which a channeled spectrum is produced.

where D was approximated with the central wavelength λ_{0}, considering the first order of diffraction is used. This is a more general form for Eq. (10) in [8]. This predicts the OPD range for all Talbot bands experiments as well as for SD-WLI and FD-OCT set-ups. A novel feature is that there is a minimum OPD value, given by the gap h. No channelled spectrum results for OPD values less than (h+1) λ_{0}. The graphs presented below will illustrate this effect.

## 4. Results

The main goal of this study is to evaluate the channelled spectrum modulation amplitude versus OPD relative to the shapes of beam excitation as determined by coefficients B_{1} and B_{2} in Fig. 2. This is achieved following the same steps as in [8] and [10], by evaluating the ratio between the channelled and un-channelled intensity at the centre of the spectrum at OPD values equal to integer multiples of the central wavelength, λ_{0}.

To this goal, the summation limits in (11) are made the same by imposing that

Conditions (14a) and (14b) allow running r in all the three sums in (11) over the same range from -r_{max} to r_{max} where:

This leads to an approximation of the visibility of the channelled spectrum,

where S_{m} is the integer number which satisfies Eq. (12), according to [8], which gives the OPD value where a maximum of visibility is achieved. Equation (16) is more general than Eq. (12) in [8]. A program was written in MatLab to calculate channelled spectrum visibility using Eq. (16). We have considered T_{1}=T_{2}=1, a diffraction grating with 1200 lines/mm and a source with central wavelength of 800 nm and FWHM of 30 nm, typical for an OCT system using a superluminiscent diode at a wavelength within the therapeutic window [30]. Results are presented as pairs of graphs in Figs. 3–10 below. Each figure shows the optical arrangement under theoretical investigation, a graph displaying the power distribution of light in each beam (B_{1} and B_{2}) against grating line number (r) and a graph of the resulting visibility versus OPD, a visibility of 1 representing 100% spectrum modulation.

*Case 2.1 (Classical Michelson interferometer)*

The first case considered in Fig. 3 is that of classical set-ups of WL-SDI and FD-OCT where the two beams illuminate the same areas of the grating m=M and h=-m. An ideal shape is used for B_{1} and B_{2}, top hat, as shown in the middle. A profile of visibility, close to a triangle with the maximum sensitivity at OPD=0 is obtained. In fact, no “channelled spectrum” is obtained for OPD=0. The minimum OPD to produce at least two peaks in the channelled spectrum is the coherence length of the source, l_{c}. The profile extends up to a maximum detectable OPD value of ±mλ_{0}=±4 mm. This maximum value could be easily inferred according to the interpretation in [8] as given by the length of the diffracted wavetrain for m=5000 and λ_{0}=800 nm. This set-up presents the disadvantage of mirror terms, i.e. the same channelled spectrum results for positive and negative OPD values equal in magnitude.

*Case 2.2.1 (Michelson interferometer with screens leading to Talbot Bands)*

Now let us consider that the two beams, with top hat profile, are each obscured by screens S1 and S2 inserted halfway into the beams so that each beam illuminates different grating lines, as shown in Fig. 4.

A triangular profile of visibility is obtained, as shown in Fig. 4 right, extending over one sign of the OPD range only. This is known from the practice of Talbot bands and from our previous theoretical [10] and experimental [9] studies on the Michelson configuration with screens halfway through. The maximum OPD at which interference is observed is ~2mλ_{0} (given by double the length of the diffracted wavetrain) and the visibility peaks at OPD=mλ_{0} (when the two diffracted wavetrains are totally overlapped).

If the two screens in the modified Michelson configuration are progressively introduced into the two beams, the triangular visibility profile shifts along the OPD-axis from a peak at OPD=0 in Fig. 3 reaching the situation shown in Fig. 4 when the percentage of the beam which is blocked, η, is 50%.

Figure 5 illustrates two cases for different η. The top row refers to a case intermediate to those in Fig. 3 and Fig. 4 for which η is in the range 0%<η<50%. The bottom row refers to a case of η=75%, where the screens are introduced beyond the centers of the beams.

When η=25%, the visibility in the top right shows some mirror term due to 75% of the diffracted wavetrain length overlap before any extrinsic delay is introduced. The profiles of visibility in Figs. 3–10 in this study have been obtained using Eq. (16). However, we can equally explain these profiles using the heuristic model put forward in [8], as shown in Fig. 6. Let us explore this avenue for the case in Fig. 5 top row. With η=25% each beam illuminates 3750 lines (and generates the same number of wavelets in the diffracted wavetrains) and both beams share 2500 lines in the centre of the grating, hence m=M=3750 and h=-2500. The visibility profile (top right) shows an interference magnitude ~2/3 in OPD=0. This is consequence of only 2500 wavelets superposed out of the total of 3750 wavelets, hence a ratio of 2500/3750. The maximum interference peak is shifted to OPD=1 mm in comparison with the case of no screens in Fig. 3. Maximum interference is achieved when the two wavetrains are totally superposed, and this happens when the average intrinsic delay between the two diffracted wavetrains (h+m/2+M/2)λ_{0}=(h+m)λ_{0}=1250λ_{0} is compensated by a similar extrinsic delay value given by an OPD=1250 λ_{0}=1 mm, i.e. by making t_{1}>t_{2} to delay wavetrain 1 in Fig. 2. The visibility is brought to zero when wavetrain 1 is further delayed by the length of the wavetrain 2, i.e. by a further 3750 λ_{0}=3 mm, giving a maximum OPD=4 mm.

The two wavetrains are depicted in Fig. 6 left.

When η=75% (Fig. 6 right), the gap is h=2500 lines and each beam occupies 1250 lines (and generates the same number of wavelets in the diffracted wavetrains). Due to the gap, visibility is zero in OPD=0. Channelled spectrum is re-obtained when the two wavetrains are brought to a minimum overlap, this requires that wavetrain 1 is delayed by an OPD to compensate the intrinsic delay due to the gap. Therefore, the minimum OPD is h λ_{0}=2500λ_{0}=2 mm. The maximum visibility is obtained when the wavetrains are totally overlapped, this happens when all 1250 wavelets are superposed, achievable when the diffracted wavetrain 1 is delayed by introducing an OPD=(h+m)λ_{0}=3 mm. The width of the visibility profile is equal to double of the diffracted wavetrain lengths, i.e. 2m λ_{0}=2 mm, when the wavetrain 1 is delayed by 2 mm more from the OPD value where the gap was compensated. This gives the maximum value of OPD=hλ_{0}+2mλ_{0}=4 mm.

The graphs in Figs. 3–5 demonstrate that by manipulating the screen positions in the Michelson modified configuration, the minimum OPD value and therefore the visibility of mirror images in FD-OCT can be attenuated. Fig. 5 bottom right and Fig. 6 right exhibit a novel feature, no channelled spectrum is obtained for 0<OPD<2 mm.

Let us consider a novel configuration next, equipped with the possibility of exciting different numbers of grating lines. For instance, let us suppose that one beam excites m lines, while the other excites M (<m) lines with a gap of h lines between them. The result is shown in Fig. 7 for m=2500, M=1000 and h=500. This configuration generates a wavetrain length of 2500 λ_{0} and another wavetrain length of 1000λ_{0} according to the explanation in [8], separated for an extrinsic delay OPD=0 by an intrinsic delay 500 λ_{0}.

Here the visibility changes to a trapezoidal shape. The minimum OPD value is that for an extrinsic delay required to minimally superpose the wavetrains, equal to OPD=hλ_{0}=0.4 mm. The visibility stays at maximum from (h+M)λ_{0}=1.2 mm to (h+m)λ_{0}=2.4 mm. This is explained by a constant amount of overlap of the two wavetrains, given by the length of the shortest wavetrain, of 1000 λ_{0} in this OPD interval. The visibility maximum is evaluated simply as the ratio of the number of superposed wavelets, 2000, to the total number of wavelets, 3500 which gives 0.57. The overlap of the two diffracted wavetrains is brought to zero when the wavetrain 1 is extra delayed by the length of wavetrain 2, Mλ_{0}, i.e. for an extrinsic delay OPD=(h+m+M)λ_{0}=3.2 mm. A similar interpretation as that illustrated in Fig. 6 can be produced to explain the visibility profile in Fig. 7 right.

The modified Michelson configuration with screens is useful to illustrate transition from the Talbot bands classical case to more novel arrangements. However, such a configuration is not power efficient due to the high losses introduced by obstructing the beams. More efficient configurations can be devised in the same spirit as illustrated in [18] using fiber leads to route the two optical beams and create a gap or a controlled overlap of the two beams on the diffraction grating.

We now move to more real cases where the two beam profiles are Gaussian. Therefore, in what follows we consider the case of a more efficient power configuration using optical fiber leads and Gaussian profiles. The diagram in Fig. 8 is an example of a power-efficient optical configuration with beams that can freely illuminate any part of the grating without loss of optical power. Using different lenses at the two fiber outputs, different number of grating lines can be excited.

Light from a superluminescent source (SLD) is divided in a fiber directional coupler (DC) and launched through microscope objectives (MS1, MS2) toward a beamsplitter (BS). Light in beam 1 is transmitted through BS toward the diffraction grating (DG) while light from beam 2 is reflected toward the grating. Different configurations are achievable by moving TS1 and TS2 laterally. This lateral movement is depicted in Fig. 8 by the large unfilled arrows. A scanner and a sample can be added as in any FD-OCT set-up to any of the two arms, using a circulator or a splitter.

The graph in Fig. 8 bottom right shows the results of the theoretical model using two beams with Gaussian intensity profiles partially superimposed on the diffraction grating, this is a fiber analogue to the Talbot configuration of the modified Michelson set-up in Fig. 4. The effect of changing the beam profile from top-hat to Gaussian leads to deviation of the visibility profile from triangular. Again, good rejection of the mirror terms is obtained, but not as effective due to the overlap of the tails of the Gaussian profiles (Fig. 8 top right). If represented logarithmically, the graph in Fig. 8 bottom right exhibits -71 dB at -0.5 mm and -102 dB at -1 mm.

Case 2.2.1. (classical, no gap, total superposition of the two beams on the grating) is not shown, as this profile is similar to that shown in Fig. 3, with the only difference that the profile deviates from a triangular shape according to the same trend illustrated by comparing the visibility curves shown in Fig. 8 right bottom.

## 5. Optimization of the visibility profile

The model developed above suggests that by using Talbot bands we can control the visibility profile in FD-OCT to: (i) minimize the strength of the channelled spectrum for path differences of one sign and (ii) alter the OPD value where maximum sensitivity occurs. The main advantage of such a procedure is that the maximum visibility can be conveniently shifted inside the tissue, where it could compensate for the attenuation of signal due to tissue absorption and scattering.

The decay of sensitivity with depth is one of the main disadvantage of the classical case (beams superposed). We therefore investigated how to enhance the decay of the visibility profile. The model developed in [8] and the sketch of wavetrains in Fig. 2 suggest that by redistributing the power towards the edges of the beams and reducing the power in the center of the beams leads to redistribution of power among the wavelets in the two wavetrains. This would make the two diffracted wavetrains exhibit higher intensity at their front and back ends, in this way faster rise of visibility is achievable at the edges of the OPD range.

#### 5.1. Apodized by aperture

Due to the Gaussian profiles of the beams in Fig. 8, interference occurs for negative OPDs. This is due to the extensive tails of the profiles, B_{1} and B_{2}, which extend one into the other. This means that for OPD=0, there are parts of the two wavetrains which overlap. Therefore, mirror terms can be minimized by apodizing the beams shown in Fig. 8 to eliminate their superposition on the grating. This can be achieved using apertures placed after the microscope objectives such that beams profiles as shown in Fig. 9 left are produced. To what extent this is possible in practice remains to be seen, as preliminary experiments have shown that diffraction effects occur.

Figure 9 shows very good rejection at OPD<0, due to the absence of overlap of the profiles B_{1} and B_{2}. The maximum in visibility is at a position corresponding to the center-to-center separation of the beams (2500 λ_{0}=2 mm) and a maximum visibility of 1 is achieved as the beams are identical. However, the procedure is not power efficient.

#### 5.2. Apodized beams, shadows σ=10000, 100000, 100000

Further interest presents the introduction of shadows into the beams. The usefulness of this is that the visibility profile can be made broader and flatter around the maximum. In FD-OCT the power distribution in the reference beam is more amenable to spatial modulation procedures which are accompanied by losses, as in current practice the reference power is attenuated to ensure a good signal to noise ratio. For this reason we introduce a shadow into beam 1 only.

Presented below we have used a Gaussian shadow such that B_{1}’=B_{1}(r)[1-G(σ,r)] where B_{1}’(r) is the modified beam shape for B_{1} and G(σ,r) is a Gaussian of unit height with σ=FWHM/1√(8log2). r represents grating lines, as before. Such a shadow could be produced using a graded ND filter introduced into the middle of the beam.

Figure 10 shows that progressively flatter visibility profiles are achievable when placing broader attenuating neutral density filters into the path of one of the beams. Overall, the sensitivity of the system is reduced, and therefore a trade-off exists between the sensitivity reduction and a more uniform visibility profile.

## 6. Conclusions

So far, the optical configurations producing Talbot bands and those using a Michelson interferometer in most sensing and OCT systems have been dealt with independently. In this paper, we evaluate the visibility variation of the channelled spectrum versus OPD by taking into account the phase delay introduced by propagation to and from each grating line. The results are valid for any type of configuration, Talbot, WL-SDI or FD-OCT.

The main effect analyzed here is that of the overlap of the wavetrains after diffraction. This is the main cause of sensitivity variation with OPD and represents one of the factors often confused with the spectrometer resolution. As presented in [8], this is more fundamental than the spectrometer resolution. No report so far, based on spectrometer resolution only [11,12,31] can explain the selection in depth principle obtainable when different grating lines are used by the two beams. Report [8] can, based on the overlap of the wavetrains after diffraction. The current report represents a rigorous treatment of such overlap for different shapes of B_{1} and B_{2} as encountered in the practice of WL-SDI and in FD-OCT. The application of Talbot bands leads to optical configurations which exhibit selective sensitivity to path differences of either positive or negative sign. The potential application of this old effect for sensing and FD-OCT has so far gone unnoticed. Talbot bands offer an elegant solution to the problem of mirror terms in FD-OCT, which the OCT community attempted to address using several methods. All the methods proposed so far to reduce the mirror terms work by cancellation, where several signals are combined to null the channelled spectrum for one sign of OPD. Such techniques required either longer time (several steps for phase shifting interferometry [31, 32]) or more sophisticated hardware [17, 33, 34] and the efficiency of mirror term attenuation was affected by the sample movement. Talbot bands are simply said, not produced for one sign of OPD values. They offer a one step solution with total elimination of the mirror terms, ideally suited for moving samples.

In addition to solving the problem of mirror terms, Talbot bands may also offer a solution to improve the profile of visibility dependence with depth. This was in fact the main goal of the present study, to demonstrate that by modifying the power distribution within the two beams, the visibility profile can be altered advantageously. Shaping the power distribution within the two beams directly affect the power within the overlap of the two diffracted beams, as per the explanation put forward in [8].

We restricted the study to the main element involved in explaining the visibility profile versus depth, which is the overlap of the diffracted beams. However, other factors should also be considered in practice, such as the spectrometer detector-array resolution. At the greatest detectable frequency of spectrum modulation, at least two photodetectors per cycle of channelled spectrum should be used. According to the sampling theorem, the sensitivity of the method decays with the number of peaks in the channelled spectrum, i.e. with the OPD value. Therefore, the size of pixel in the spectrometer determines a supplementary sinc factor decay of visibility [11,12]. This presents a maximum at OPD=0 and decays for increasing OPD values either side of zero. Therefore, for accurate shapes of sensitivity versus OPD, all profiles presented in Figs. 3–10 above should be multiplied with the sinc factor in [11,12].

All conclusions drawn here are equally valid if the spectrometer uses a prism. In fact, Talbot bands were first demonstrated using a prism [1]. However, the diffraction grating allows an easier and more intuitive presentation of delays amongst the rays within the diffracted beam than the delays amongst the rays in the dispersed beams.

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